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merge=lfs -text +preprints/The-Mezard-Parisi-formula-for-diluted-spin-glasses-September-23-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text +preprints/The-nonmaximal-anti-de-Sitter-Penrose-Inequality-and-original-data-rigidity-October-5-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text +preprints/The-normalized-volume-gap-in-dimension-four-September-24-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text diff --git a/preprints/The-Approximation-Threshold-for-Metric-k-Median-September-24-2026/README.md b/preprints/The-Approximation-Threshold-for-Metric-k-Median-September-24-2026/README.md new file mode 100644 index 0000000000000000000000000000000000000000..47aed40608e3c987df4858070aa4dc268b239982 --- /dev/null +++ b/preprints/The-Approximation-Threshold-for-Metric-k-Median-September-24-2026/README.md @@ -0,0 +1,16 @@ +# [The approximation threshold for metric k-median](main.pdf) + +OpenAI +September 24, 2026 + +## Citation + +```bibtex +@misc{OAI:The-Approximation-Threshold-for-Metric-k-Median-September-24-2026, + author = {{OpenAI}}, + title = {{The approximation threshold for metric $k$-median}}, + howpublished = {OpenAI Math Release preprint + \href{https://github.com/openai/math/blob/main/preprints/The-Approximation-Threshold-for-Metric-k-Median-September-24-2026/main.pdf}{OAI:The-Approximation-Threshold-for-Metric-k-Median-September-24-2026}}, + year = {2026} +} +``` diff --git a/preprints/The-Approximation-Threshold-for-Metric-k-Median-September-24-2026/build/figures/survivors.tex b/preprints/The-Approximation-Threshold-for-Metric-k-Median-September-24-2026/build/figures/survivors.tex new file mode 100644 index 0000000000000000000000000000000000000000..d6cce9ed95aed5bbb57bb08e08d8d39d9a20c381 --- /dev/null +++ b/preprints/The-Approximation-Threshold-for-Metric-k-Median-September-24-2026/build/figures/survivors.tex @@ -0,0 +1,27 @@ +\begin{figure}[tbp] +\centering +\begin{tikzpicture}[ + box/.style={draw,rounded corners=2pt,align=center,inner sep=7pt, + text width=3.95cm,font=\small}, + flow/.style={-{Stealth[length=2mm]},thick}, + note/.style={align=center,font=\small}] +\node[note] (source) {A full update at an alive source $h\in S$ opens its facility}; +\node[box,below=10mm of source,xshift=-4.8cm] (erased) + {$S\cap E_h$\\[3pt]Physically erased\\and removed from $S$}; +\node[box,right=4mm of erased] (kept) + {$i\in S\setminus E_h$\\with $ad_i0\), we give a deterministic polynomial-time +\((1+2/e+\varepsilon)\)-approximation for finite rational metric +\(k\)-median with specified candidate facilities, opening at most \(k\) +facilities. Under \(P\ne NP\), the infimum approximation factor in this +model is therefore \(1+2/e\). +\end{abstract} + +\input{sections/introduction} +\input{sections/preliminaries} +\input{sections/preparation} +\input{sections/inequality} +\input{sections/rounding} +\input{sections/derandomization} +\appendix +\input{sections/certificates} + +\bibliographystyle{plainnat} +\bibliography{references} +\end{document} diff --git a/preprints/The-Approximation-Threshold-for-Metric-k-Median-September-24-2026/build/references.bib b/preprints/The-Approximation-Threshold-for-Metric-k-Median-September-24-2026/build/references.bib new file mode 100644 index 0000000000000000000000000000000000000000..c08383fba1fb82a4054f360867c68d5c731f4eb1 --- /dev/null +++ b/preprints/The-Approximation-Threshold-for-Metric-k-Median-September-24-2026/build/references.bib @@ -0,0 +1,217 @@ +@inproceedings{AnandLee2024, + author = {Anand, Aditya and Lee, Euiwoong}, + title = {Separating {$k$}-Median from the Supplier Version}, + editor = {Vygen, Jens and Byrka, Jaros{\l}aw}, + booktitle = {Integer Programming and Combinatorial Optimization: 25th International Conference, IPCO 2024}, + series = {Lecture Notes in Computer Science}, + volume = {14679}, + pages = {14--27}, + publisher = {Springer}, + address = {Cham}, + year = {2024}, + doi = {10.1007/978-3-031-59835-7_2}, + url = {https://doi.org/10.1007/978-3-031-59835-7_2}, + note = {Full preprint: arXiv:2401.13819v1, January 24, 2024; \url{https://arxiv.org/abs/2401.13819v1}} +} + +@article{Arya2004, + author = {Arya, Vijay and Garg, Naveen and Khandekar, Rohit and Meyerson, Adam and Munagala, Kamesh and Pandit, Vinayaka}, + title = {Local Search Heuristics for {$k$}-Median and Facility Location Problems}, + journal = {SIAM Journal on Computing}, + volume = {33}, + number = {3}, + pages = {544--562}, + year = {2004}, + doi = {10.1137/S0097539702416402}, + url = {https://doi.org/10.1137/S0097539702416402} +} + +@misc{BayerTeichmann2005, + author = {Bayer, Christian and Teichmann, Josef}, + title = {The Proof of {Tchakaloff}'s Theorem}, + year = {2005}, + eprint = {math/0502473v2}, + archivePrefix = {arXiv}, + primaryClass = {math.NA}, + url = {https://arxiv.org/abs/math/0502473v2}, + note = {Version 2, 4 May 2005; published in Proceedings of the American Mathematical Society 134(10), 3035--3040 (2006), doi:10.1090/S0002-9939-06-08249-9} +} + +@article{ByrkaEtAl2017, + author = {Byrka, Jaros{\l}aw and Pensyl, Thomas and Rybicki, Bartosz and Srinivasan, Aravind and Trinh, Khoa}, + title = {An Improved Approximation for {$k$}-Median and Positive Correlation in Budgeted Optimization}, + journal = {ACM Transactions on Algorithms}, + volume = {13}, + number = {2}, + articleno = {23}, + pages = {23:1--23:31}, + year = {2017}, + doi = {10.1145/2981561}, + url = {https://doi.org/10.1145/2981561}, + note = {Corrected full manuscript: arXiv:1406.2951v4, 23 April 2016} +} + +@misc{ByrkaEtAl2026, + author = {Byrka, Jaros{\l}aw and Guo, Yuhao and Hu, Yang and Li, Shi and Wan, Chengzhang and Wang, Zaixuan}, + title = {{$k$}-Clustering via Iterative Randomized Rounding}, + year = {2026}, + eprint = {2604.06046v1}, + archivePrefix = {arXiv}, + primaryClass = {cs.DS}, + doi = {10.48550/arXiv.2604.06046}, + url = {https://arxiv.org/abs/2604.06046v1}, + note = {Version 1, April 7, 2026} +} + +@article{CGLSS2026, + author = {Cohen-Addad, Vincent and Grandoni, Fabrizio and Lee, Euiwoong and Schwiegelshohn, Chris and Svensson, Ola}, + title = {A {$(2+\varepsilon)$}-Approximation Algorithm for Metric {$k$}-Median}, + journal = {SIAM Journal on Computing}, + year = {2026}, + doi = {10.1137/25M181026X}, + url = {https://doi.org/10.1137/25M181026X}, + note = {Published online August 24, 2026. Full preprint: arXiv:2503.10972v2, May 19, 2026; \url{https://arxiv.org/abs/2503.10972v2}. First posted March 14, 2025} +} + +@inproceedings{CharikarGuha1999, + author = {Charikar, Moses and Guha, Sudipto}, + title = {Improved Combinatorial Algorithms for the Facility Location and {$k$}-Median Problems}, + booktitle = {Proceedings of the 40th Annual Symposium on Foundations of Computer Science}, + pages = {378--388}, + year = {1999}, + publisher = {IEEE Computer Society}, + doi = {10.1109/SFFCS.1999.814609}, + url = {https://doi.org/10.1109/SFFCS.1999.814609} +} + +@article{CharikarGuhaTardosShmoys2002, + author = {Charikar, Moses and Guha, Sudipto and Tardos, {\'E}va and Shmoys, David B.}, + title = {A Constant-Factor Approximation Algorithm for the {$k$}-Median Problem}, + journal = {Journal of Computer and System Sciences}, + volume = {65}, + number = {1}, + pages = {129--149}, + year = {2002}, + doi = {10.1006/jcss.2002.1882}, + url = {https://doi.org/10.1006/jcss.2002.1882}, + note = {Preliminary version in STOC 1999} +} + +@inproceedings{CharikarLi2012, + author = {Charikar, Moses and Li, Shi}, + title = {A Dependent {LP}-Rounding Approach for the {$k$}-Median Problem}, + booktitle = {Automata, Languages, and Programming}, + series = {Lecture Notes in Computer Science}, + volume = {7391}, + pages = {194--205}, + publisher = {Springer}, + year = {2012}, + doi = {10.1007/978-3-642-31594-7_17}, + url = {https://doi.org/10.1007/978-3-642-31594-7_17} +} + +@article{Feige1998, + author = {Feige, Uriel}, + title = {A Threshold of {$\ln n$} for Approximating Set Cover}, + journal = {Journal of the ACM}, + volume = {45}, + number = {4}, + pages = {634--652}, + month = jul, + year = {1998}, + doi = {10.1145/285055.285059}, + url = {https://doi.org/10.1145/285055.285059} +} + +@article{GandhiEtAl2006, + author = {Gandhi, Rajiv and Khuller, Samir and Parthasarathy, Srinivasan and Srinivasan, Aravind}, + title = {Dependent Rounding and Its Applications to Approximation Algorithms}, + journal = {Journal of the ACM}, + volume = {53}, + number = {3}, + pages = {324--360}, + year = {2006}, + doi = {10.1145/1147954.1147956}, + url = {https://doi.org/10.1145/1147954.1147956} +} + +@inproceedings{GowdaEtAl2023, + author = {Gowda, Kishen N. and Pensyl, Thomas and Srinivasan, Aravind and Trinh, Khoa}, + title = {Improved Bi-point Rounding Algorithms and a Golden Barrier for {$k$}-Median}, + booktitle = {Proceedings of the 2023 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA)}, + pages = {987--1011}, + publisher = {Society for Industrial and Applied Mathematics}, + year = {2023}, + doi = {10.1137/1.9781611977554.ch38}, + url = {https://doi.org/10.1137/1.9781611977554.ch38}, + note = {Preprint: \url{https://arxiv.org/abs/2210.13395v1}} +} + +@article{JainEtAl2003, + author = {Jain, Kamal and Mahdian, Mohammad and Markakis, Evangelos and Saberi, Amin and Vazirani, Vijay V.}, + title = {Greedy Facility Location Algorithms Analyzed Using Dual Fitting with Factor-Revealing {LP}}, + journal = {Journal of the ACM}, + volume = {50}, + number = {6}, + pages = {795--824}, + year = {2003}, + doi = {10.1145/950620.950621}, + url = {https://doi.org/10.1145/950620.950621}, + note = {Preprint: \url{https://arxiv.org/abs/cs/0207028v1}} +} + +@article{JainVazirani2001, + author = {Jain, Kamal and Vazirani, Vijay V.}, + title = {Approximation Algorithms for Metric Facility Location and {$k$}-Median Problems Using the Primal-Dual Schema and Lagrangian Relaxation}, + journal = {Journal of the ACM}, + volume = {48}, + number = {2}, + pages = {274--296}, + year = {2001}, + doi = {10.1145/375827.375845}, + url = {https://doi.org/10.1145/375827.375845} +} + +@article{LiSvensson2016, + author = {Li, Shi and Svensson, Ola}, + title = {Approximating {$k$}-Median via Pseudo-Approximation}, + journal = {SIAM Journal on Computing}, + volume = {45}, + number = {2}, + pages = {530--547}, + year = {2016}, + doi = {10.1137/130938645}, + url = {https://doi.org/10.1137/130938645}, + note = {Author manuscript dated September 11, 2014: \url{https://tcs.nju.edu.cn/shili/papers/KM-SIMPCOMP2016.pdf}; the additive-to-ordinary reduction is Theorem 4 in that version} +} + +@misc{OpenAIRecovery2026, + author = {{OpenAI}}, + title = {{Single-exponential recovery and bounded-price strictness for metric $k$-median}}, + howpublished = {OpenAI Math Release preprint + \href{https://github.com/openai/math/blob/main/preprints/Single-Exponential-Recovery-and-Bounded-Price-Strictness-for-Metric-k-Median-September-24-2026/paper.pdf}{OAI:Single-Exponential-Recovery-and-Bounded-Price-Strictness-for-Metric-k-Median-September-24-2026}}, + year = {2026} +} + +@inproceedings{Srinivasan2001, + author = {Srinivasan, Aravind}, + title = {Distributions on Level-Sets with Applications to Approximation Algorithms}, + booktitle = {Proceedings of the 42nd IEEE Symposium on Foundations of Computer Science}, + pages = {588--597}, + publisher = {IEEE Computer Society}, + year = {2001}, + doi = {10.1109/SFCS.2001.959935}, + url = {https://doi.org/10.1109/SFCS.2001.959935} +} + +@article{BBLM2005, + author = {Boucheron, St{\'e}phane and Bousquet, Olivier and Lugosi, G{\'a}bor and Massart, Pascal}, + title = {Moment inequalities for functions of independent random variables}, + journal = {The Annals of Probability}, + volume = {33}, + number = {2}, + pages = {514--560}, + year = {2005}, + doi = {10.1214/009117904000000856}, + url = {https://arxiv.org/abs/math/0503651} +} diff --git a/preprints/The-Approximation-Threshold-for-Metric-k-Median-September-24-2026/build/sections/certificates.tex b/preprints/The-Approximation-Threshold-for-Metric-k-Median-September-24-2026/build/sections/certificates.tex new file mode 100644 index 0000000000000000000000000000000000000000..4d443f498cb5ccbb5092eeed40ae3174f23ecf99 --- /dev/null +++ b/preprints/The-Approximation-Threshold-for-Metric-k-Median-September-24-2026/build/sections/certificates.tex @@ -0,0 +1,631 @@ +\section{Exact certificates for the scalar bounds} +\label{app:certificates} + +This appendix proves Lemma~\ref{lem:scalar-bounds}. All terminating +decimals in this appendix denote exact rational numbers. The argument +uses two finite tables of rational polynomial minorants. We give the +coefficient rule and the Bernstein conversion formula explicitly, so the +tables can be verified using only rational arithmetic; the accompanying +programs are supplementary implementations of these formulas. + +Throughout, $X=1-x$, $0\le x\le1$, $0\le w\le X$, $0\le r\le1$, and +$s\in\{0,1\}$. We recall the notation +\begin{align*} + P&=\int_0^1t e^{-tx}\,dt,& + A&=3-2\int_0^1e^{-tx}\,dt,& + m&=A-1-2Px,\\ + C&=1-sP,& a&=A+2Pw,& b&=(2-s)P(x+w),\qquad z=Pw^2,\\ + k_s&=(1-s)\frac{1-\sqrt{1-P}}{1+\sqrt{1-P}},& + K&=C-k_s-br+1.6z,\\ + L&=(C-m-b)r+.4z,& + M&=(1-r)a+CPr,& G&=M-L. +\end{align*} +Also set +\[ + d_0=.52,\qquad d_1=.4,\qquad + H_0=.394-.42X^2+.04X^3,\qquad H_1=.25,\qquad J_s=CH_s. +\] +The elementary integral bounds give $00. + \label{eq:cert-k-envelope} +\end{equation} +Since $00$. By~\eqref{eq:cert-integral-identity}, the right +side of~\eqref{eq:cert-first-scalar} minus its left side equals +\begin{align*} + &C(1-P)-\frac{2x(1-P)}{1+x}+sPx-(2-s)Pw+.4P^2w^2\\ + &\hspace{35mm}+P\left(\frac{C(1-P)}t+.4w^2t\right). +\end{align*} +Applying AM--GM to the last parentheses gives the lower bound +\begin{equation} + C(1-P)-\frac{2x(1-P)}{1+x}+sPx + -(2-s-2\sqrt{.4C(1-P)})Pw+.4P^2w^2. + \label{eq:cert-amgm} +\end{equation} +For $s=0$ the coefficient $2-2\sqrt{.4(1-P)}$ is at most $1.12$. +The resulting quadratic decreases for $0\le w\le X$: its derivative +is at most $-1.12P+.8P^2<0$. Substituting $w=X$ into this lower +bound gives $X T_1(P)/(1+x)$, where $T_1$ is defined above. For +$s=1$, the coefficient $1-2\sqrt{.4}(1-P)$ is at most $.38$. +Dropping the nonnegative quadratic term and using $w\le X$ gives +$T_2(P)/(1+x)$. The numerical coefficient comparisons also follow +by squaring positive quantities: $.2>.44^2$ and +$.4>.62^2$. Table~\ref{tab:cert-first}, with the substitution +budget proved below, gives $T_1(P),T_2(P)>0$. +At $X=0$, the extra factor $X$ in the $s=0$ reduction is zero, +which still proves the required non-strict inequality. + +For $r>0$, \eqref{eq:cert-first-scalar} implies +\[ + L\ge Cr\left(1-\frac{1-P}{1-Pr}\right)\ge0. +\] +For $r=0$, one has $L=.4z\ge0$ directly. + +To check $K\ge d_sG$, write $d=d_s$ and $h=1.6+.4d$. +The difference $K-dG$ is affine in $r$. With +$N_P=C-(1-s)k$, replacing $k_s$ by $(1-s)k$ only decreases this +difference. Put $\underline K=N_P-br+1.6z\le K$. The endpoint +expressions after this replacement are +\begin{align} + (\underline K-dG)|_{r=0}&=N_P-dA+Pw(hw-2d), + \label{eq:cert-r0}\\ + (\underline K-dG)|_{r=1}&=Q_1(P,A)+hP(w-u_1)^2, + \label{eq:cert-r1} +\end{align} +where +\[ + u_0=d/h,\qquad u_1=\frac{(2-s)(1+d)}{2h} +\] +and $Q_1$ is defined above. For $s=1$ we may use the unrestricted +minimum $w=u_0$ in~\eqref{eq:cert-r0}. For $s=0$, the constrained +minimum is at $w=X$ if $X\le u_0$, and at $w=u_0$ otherwise. +Equation~\eqref{eq:cert-r1} is bounded below by $Q_1(P,A)$ for +both values of $s$. These are precisely the $Q_0,Q_1$ checks in +Table~\ref{tab:cert-first}. + +We next reduce the final two scalar bounds. Recall +\[ + \kappa_s(q)=C\frac{1-q}{1-Pq}-(1-q),\qquad 0\le q\le1. +\] +For $s=1$, $k_s=0$ and $\kappa_s(q)\le0$. Thus both relevant +left sides are bounded below by $C+1.6z-b$. Its unrestricted +minimum in $w$ is the true-parameter version of $Y$ above, so the +rows $Y-H_0$ and $Y-.25$ suffice. + +For $s=0$ and $X\ge.5$, use $q\le1$ and $\kappa_0(q)\le k_0$. +Again a common lower bound is $1-k_0+1.6z-b$, whose unrestricted +minimum is the corresponding $Y$. For $s=0$ and $X\le.5$, we +have $x\ge.5$ and +\[ + P(x)\le P(.5)<.47. +\] +For example the inequality $e^{-u}\le1-u+u^2/2$ yields +$P(.5)\le1/2-1/6+1/32=35/96<.47$. +It follows that $b\ge .52P/(1-P)$, because $x+w\ge.5$ and +$1-P>.53$. Since +\[ + \kappa_0(q)=\frac{Pq(1-q)}{1-Pq} + \le\frac{P}{1-P}(1-q), +\] +we obtain $bq+.52\kappa_0(q)\le b$. The fourth scalar left side +is consequently at least +\[ + 1-.48k_0+1.6z-b. +\] +The quadratic $1.6Pw^2-2P(x+w)$ decreases on $0\le w\le X\le.5$, +so its minimum is $P(1.6X^2-2)$. Thus the fourth and fifth +inequalities follow, respectively, from $Y'+.52k-H_0\ge0$ and +$Y'-.25\ge0$ after substitution. In the former expression the +net coefficient of $k$ is $-.48$, so replacing $k_0$ by the upper +bound $k$ is conservative. + +The last scalar bound also gives positivity of $K$: since $b\ge0$, +\[ + K=C-k_s-br+1.6z\ge C-k_s-b+1.6z\ge.25>0. +\] + +\subsection{The first coefficient table and its exact verification} + +Replacing $p,v$ by $P,A$ in these expressions changes their values +by less than +\begin{equation} + \sigma=.0002. + \label{eq:cert-sigma} +\end{equation} +For completeness, the following derivative bounds hold on the line +segment between $(p,v)$ and $(P,A)$, using $0\le p,P\le.5$, +$1\le v,A\le2$, and $0\le W\le1$: +\[ +\begin{array}{c|rr|r} + T&|\partial_pT|&|\partial_vT|& + 10^6\,|T(p,v)-T(P,A)|\text{ upper bound}\\ \hline + T_1&3.64&0&27.3\\ + T_2&8.76&0&65.7\\ + Q_0(W)&3.848&.52&36.66\\ + Q_1&4.848&.52&44.16\\ + Y&2.625&0&19.6875\\ + Y'&2&0&15 +\end{array} +\] +Here $T_{1,p}=-1-1.12(1+x)+.8pX(1+x)$, with +$X(1+x)\le1$, and +$T_{2,p}=[-2(1-p)-.38X+x](1+x)+2x$. +For $Q_0$, use $h\le1.808$ and $2d\le1.04$. +For $Q_1$, $(cp-l)_p=1-2sp+sX$ has absolute value at most $2$, +$(cp-l)_v=1$, and $u_1\le1$. +The last two derivative bounds follow directly from their displayed +linear dependence on $p$. These estimates prove the claimed budget +even on the full interval, rather than only the listed subintervals. +Adding $.52k$ or subtracting $H_0,.25$ does not change this budget. +The row for $T_k$ already includes $\epsilon_P$ explicitly and does +not need this additional substitution estimate. + +Here is the finite verification mechanism used in both coefficient +tables. A row with scale $10^a$ and integers +$\lambda_0,\ldots,\lambda_n$ asserts +\begin{equation} + T(X)\ge t(X):=10^{-a}\sum_{j=0}^n\lambda_jX^j + \qquad(0\le X\le1). + \label{eq:cert-minorant} +\end{equation} +To verify it, expand $T=\sum_jc_jX^j$ and check +\begin{equation} + 10^ac_j\ge\lambda_j\quad(0\le j{\raggedright\arraybackslash}p{.39\textwidth}@{}} +\caption{First coefficient table. The scale is $10^a$, and the + listed vector is $(\lambda_0,\ldots,\lambda_n)$.} +\label{tab:cert-first}\\ +\toprule + $T$&$s$&Endpoints and splits&$a$&$\lambda$\\ +\midrule +\endfirsthead +\toprule + $T$&$s$&Endpoints and splits&$a$&$\lambda$\\ +\midrule +\endhead +\bottomrule +\endfoot + $T_k$&--&$0,.5,1$&4&$(4,73,-384,563,-184,-67)$\\ + $T_1$&--&$0,1$&2&$(14,-17,3,3,-1)$\\ + $T_2$&--&$0,1$&2&$(13,10,-22)$\\ + $Q_0(u_0)$&1&$0,1$&2&$(1,3,-1)$\\ + $Q_1$&1&$0,1$&2&$(42,-3)$\\ + $Q_1$&0&$0,1$&2&$(5,-1)$\\ + $Q_0(u_0)$&0&$u_0,1$&2&$(-2,19,-1)$\\ + $Q_0(X)$&0&$0,u_0$&2&$(2,-6)$\\ + $Y-H_0$&1&$0,1$&2&$(3,-9,45,-2)$\\ + $Y-.25$&1&$0,1$&2&$(17,-9)$\\ + $Y-H_0$&0&$.5,1$&3&$(-165,50,575,-1)$\\ + $Y-.25$&0&$.5,1$&2&$(-3,5,15,-1)$\\ + $Y'+.52k-H_0$&0&$0,.125,.25,.5$&4&$(406,-3483,7208,1786,-3)$\\ + $Y'-.25$&0&$0,.5$&2&$(14,-38,29,-1)$\\ +\end{longtable} +\endgroup + +Substitution of~\eqref{eq:cert-interpolants} into the definitions +verifies~\eqref{eq:cert-coefficient-rule} for all fourteen rows of +Table~\ref{tab:cert-first}. As additional explicit sign certificates, +the minima of the finite sets in~\eqref{eq:cert-bernstein-rule}, over +all prescribed subintervals for each row, are at least $10^{-6}$ times +the following integers, in row order: +\begin{equation} +\begin{gathered} + 114, 20000, 10000, 10000, 390000, 40000, 33818,\\ + 2743, 15000, 80000, 3625, 31250, 356, 21250. +\end{gathered} +\label{eq:cert-first-bernstein-lower} +\end{equation} +Both~\eqref{eq:cert-coefficient-rule} and these lower bounds are finite +rational comparisons specified completely by the table and +\eqref{eq:cert-bernstein-rule}. For example, the first row has +smallest Bernstein coefficient $73/640000$, and the thirteenth row +has smallest coefficient $27341/76800000$. +Consequently $T_k>0$, and every other short minorant exceeds +$\sigma=.0002$ on its indicated intervals. After the substitution +budget, this completes the first and second groups of scalar bounds +and the final two scalar bounds. + +\subsection{The third scalar bound: perturbation estimate} + +It remains to prove +\begin{equation} + ((J_s-b)r+1.6z)M+(C-k_s-J_sr)L\ge0. + \label{eq:cert-third-goal} +\end{equation} +Since $L\ge0$ is already established, replacing $k_s$ with $(1-s)k$ +only decreases the left side. We next replace $P,A$ by $p,v$. +This replacement changes the expression by at most +\begin{equation} + \sigma(r+w^2),\qquad \sigma=.0002. + \label{eq:cert-third-error} +\end{equation} +Here are full bounds for this assertion. For either endpoint choice +$(\pi,\nu)=(P,A)$ or $(p,v)$, put $c_\pi=1-s\pi$, $H=H_s$, and +write the two products as $F_1F_2+F_3F_4$, where +\begin{align*} + F_1&=[c_\pi H-(2-s)\pi(x+w)]r+1.6\pi w^2,\\ + F_2&=(1-r)(\nu+2\pi w)+c_\pi\pi r,\\ + F_3&=c_\pi(1-Hr)-(1-s)k,\\ + F_4&=[2-\nu-s\pi X-(2-s)\pi w]r+.4\pi w^2. +\end{align*} +Writing $\rho=r+w^2$, the ranges already established imply +\begin{equation} + |F_1|\le1.1\rho,\qquad |F_2|\le3,\qquad + |F_3|\le1,\qquad |F_4|\le2\rho. + \label{eq:cert-factor-bounds} +\end{equation} +For $F_1$, both nonnegative terms in its coefficient of $r$ are at +most one, while $1.6\pi\le.8$. The expression $F_2$ is a convex +combination of two quantities between zero and three. For $F_3$, +the lower bounds are $1-.394-.1719=.4341$ when $s=0$ and +$.5(1-.25)=.375$ when $s=1$, and the upper bound is one. +Finally the coefficient of $r$ in $F_4$ belongs to $[-1,1]$: +$0\le2-\nu\le1$ and $0\le s\pi X+(2-s)\pi w\le1$. + +Let $\delta F_j$ denote the change between the two choices. With +$\epsilon_P=.0000075$, direct subtraction gives +\begin{align*} + |\delta F_1|&\le2\epsilon_P\rho,& + |\delta F_2|&\le4\epsilon_P,\\ + |\delta F_3|&\le\epsilon_P,& + |\delta F_4|&\le4\epsilon_P\rho. +\end{align*} +For the first estimate, the coefficient multiplying +$\epsilon_Pr$ is $sH+(2-s)(x+w)$, at most $2$ for $s=0$ and at +most $1.25$ for $s=1$; the remaining coefficient is $1.6$. +For the second, use the convex combination in $r$, together with +$|\delta(\nu+2\pi w)|\le4\epsilon_P$ and +\[ + |\delta(c_\pi\pi)| + =|P-p|\,|1-s(P+p)|\le\epsilon_P. +\] +The third estimate is immediate from $0\le1-Hr\le1$. +For the fourth, use +$2+sX+(2-s)w\le4$ and $.4\le4$. +Combining these differences with~\eqref{eq:cert-factor-bounds} and +the identity $ab-a'b'=(a-a')b+a'(b-b')$ yields +\begin{align*} + |\delta(F_1F_2+F_3F_4)| + &\le(3\cdot15+1.1\cdot30+2\cdot7.5+30)10^{-6}\rho\\ + &=.000123\rho<\sigma\rho\qquad(\rho>0). +\end{align*} +If $\rho=0$, both expressions vanish. Thus +\eqref{eq:cert-third-error} holds throughout the domain. The factor +bounds hold for both choices, so this telescoping estimate already +accounts for products of perturbations. + +\subsection{Bernstein expansion and the second coefficient table} + +Use the polynomial quantities in~\eqref{eq:cert-polynomial-notation}, +and set +\[ + j=cH_s,\qquad b'_0=(2-s)px,\qquad b'_1=(2-s)pX, + \qquad f=2pX,\qquad Z'=pX^2. +\] +In the polynomial version of the left side of +\eqref{eq:cert-third-goal}, subtract $\sigma(r+w^2)$ and put +$w=X\eta$, $0\le\eta\le1$. Its degree-three Bernstein +coefficients in $\eta$, for $i=0,1,2,3$, are +\begin{equation} + B_0+B_1r+B_2r^2, + \label{eq:cert-eta-coefficient} +\end{equation} +where, writing $e=i/3$, $e_2=i(i-1)/6$, and +$t=\mathbf{1}_{\{i=3\}}$, the exact formulas are +\begin{equation} +\begin{aligned} + B_0={}&Z'\bigl((1.6v+.4N)e_2+1.6ft\bigr)-\sigma X^2e_2,\\ + B_1={}&(j-b'_0)v+Nl\\ + &+\bigl((j-b'_0)f-b'_1v-Nb'_1\bigr)e-b'_1fe_2\\ + &-Z'\bigl((1.6(v-cp)+.4j)e_2+1.6ft\bigr)-\sigma,\\ + B_2={}&(b'_0-j)(v-cp)-jl\\ + &+\bigl((b'_0-j)f+b'_1(v-cp)+jb'_1\bigr)e+b'_1fe_2. +\end{aligned} +\label{eq:cert-bcoeff} +\end{equation} +These formulas give all eight expansions, one for each pair $(s,i)$. +They can also be derived directly by expanding +\begin{align*} + &\bigl([j-(2-s)p(x+w)]r+1.6pw^2\bigr) + \bigl((1-r)(v+2pw)+cpr\bigr)\\ + &\quad +(N-jr)\bigl([l-(2-s)pw]r+.4pw^2\bigr) + -\sigma(r+w^2) +\end{align*} +in powers of $\eta$ after $w=X\eta$, and using the degree-three +conversion rule: if $a_h$ is the coefficient of $\eta^h$, the +$i$th Bernstein coefficient is +$\sum_{h=0}^i a_h\binom ih/\binom3h$. + +In all eight cases $B_0\ge0$. For $i=0,1$, it is zero. Otherwise +\[ + B_0=X^2\bigl(p(1.6v+.4N)-\sigma\bigr)e_2+1.6pX^2ft. +\] +Here $N\ge.5$, $v\ge1$, and $p\ge.26424112$, so the bracket is +at least $.475434016>0$ and the other term is nonnegative. + +Define +\[ + D'=4B_0B_2-B_1^2,\qquad S'=B_0+B_1+B_2. +\] +For $i\le1$, positivity of $B_1$ and $B_1+B_2$ proves +nonnegativity of~\eqref{eq:cert-eta-coefficient} on $0\le r\le1$. +For $i\ge2$, any of the following is sufficient: +\begin{enumerate} + \item $B_1,B_2\ge0$; + \item $D'>0$; + \item $S'>0$ and $B_0-B_2>0$. +\end{enumerate} +For the second criterion, $B_0\ge0$ and $D'>0$ force +$B_0,B_2>0$; completing the square proves positivity for every +real $r$. For the third, use +\[ + B_0+B_1r+B_2r^2 + =rS'+(1-r)(B_0-rB_2), +\] +where $B_0-rB_2=(1-r)B_0+r(B_0-B_2)\ge0$. + +Table~\ref{tab:cert-third} supplies these signs. It uses exactly +the coefficient rule~\eqref{eq:cert-coefficient-rule} and the +degree-six Bernstein formula~\eqref{eq:cert-bernstein-rule}. + +\begingroup +\small +\setlength{\tabcolsep}{3pt} +\renewcommand{\arraystretch}{1.18} +\begin{longtable}{@{}ccllc>{\raggedright\arraybackslash}p{.38\textwidth}@{}} +\caption{Second coefficient table. A row with $i=0,1$ supplies + the indicated certificate separately for both values of $i$.} +\label{tab:cert-third}\\ +\toprule + $s$&$i$&Endpoints and splits&$T$&$a$&$\lambda$\\ +\midrule +\endfirsthead +\toprule + $s$&$i$&Endpoints and splits&$T$&$a$&$\lambda$\\ +\midrule +\endhead +\bottomrule +\endfoot + 0&0&$0,1$&$B_1$&2&$(1,90,-39)$\\ + 0&1&$0,1$&$B_1$&2&$(1,41,-51,28,-1)$\\ + 0&$0,1$&$0,1$&$B_1+B_2$&2&$(10,-1)$\\ + 0&2&$0,.05$&$B_1$&2&$(1,-9,-98)$\\ + 0&2&$0,.05$&$B_2$&2&$(9,-23)$\\ + 0&2&$.05,1$&$D'$&4&$\begin{gathered}(-2,16,1176,-153,\\-1500,2590,-1000)\end{gathered}$\\ + 0&3&$0,.015$&$B_1$&2&$(1,-58,-262)$\\ + 0&3&$0,.015$&$B_2$&2&$(9,-5)$\\ + 0&3&$.015,.25,.5,1$&$D'$&5&$\begin{gathered}(-12,1180,2070,-32420,\\44210,33740,-14000)\end{gathered}$\\ + 1&0&$0,1$&$B_1$&2&$(5,30,-7)$\\ + 1&0&$0,1$&$B_1+B_2$&2&$(12,-4)$\\ + 1&1&$0,1$&$B_1$&2&$(5,-1)$\\ + 1&1&$0,1$&$B_1+B_2$&2&$(12,-8)$\\ + 1&2&$0,.2$&$B_1$&2&$(5,-16,-41)$\\ + 1&2&$0,.2$&$B_2$&2&$(7,-2)$\\ + 1&2&$.2,1$&$D'$&3&$(-4,17,92,87,-135)$\\ + 1&3&$0,.1$&$B_1$&3&$(54,-391,-941,-900)$\\ + 1&3&$0,.1$&$B_2$&2&$(7,-1)$\\ + 1&3&$.1,.75$&$D'$&3&$(-4,42,194,478,-995)$\\ + 1&3&$.75,1$&$S'$&2&$(12,-9)$\\ + 1&3&$.75,1$&$B_0-B_2$&2&$(-8,-32,67,-1)$\\ +\end{longtable} +\endgroup + +In the printed row order, the Bernstein coefficients of these short +minorants, over all of each row's subintervals, are at least +$10^{-6}$ times the following integers: +\begin{equation} +\begin{gathered} + 10000, 10000, 90000, 3050, 78500, 171, 710,\\ + 89250, 60, 50000, 80000, 40000, 40000, 1600,\\ + 66000, 3560, 4590, 69000, 2518, 30000, 52656. +\end{gathered} +\label{eq:cert-third-bernstein-lower} +\end{equation} +The bound $90000$ in the third position applies to both indices in +that row. These are all positive. The smallest exact Bernstein +coefficient among the table's checks is +$193875074979/3200000000000000$, in the ninth printed row. +Thus all the required strict signs follow from the finite rational +formula~\eqref{eq:cert-bernstein-rule}. + +For clarity, the negative-tail calculations needed for the four +discriminant rows are detailed next. Let +$b_{h,j}=[X^j]B_h$ and write +\[ + c_j=[X^j]D' + =\sum_u\bigl(4b_{0,u}b_{2,j-u}-b_{1,u}b_{1,j-u}\bigr), +\] +where coefficients outside the degree range are zero. For a short +minorant of degree $n$, group $\min(0,c_j)$ in consecutive blocks +of five starting at $j=n$, truncating at degree $28$. The five block +sums are bounded below by $.0001$ times the following integers: +\begin{equation} +\begin{array}{cc|c|c|rrrrr} + s&i&n&\deg D'&1&2&3&4&5\\ \hline + 0&2&6&20&-930&-11&-1&0&0\\ + 0&3&6&22&-1070&-180&-2&-1&0\\ + 1&2&4&28&-1312&-18&-1&-1&-1\\ + 1&3&4&28&-8240&-1560&-29&-1&-1 +\end{array} +\label{eq:cert-tail-blocks} +\end{equation} +These twenty rational comparisons follow by substituting +\eqref{eq:cert-bcoeff} and~\eqref{eq:cert-interpolants} into the +displayed coefficient formula. Their row sums are, respectively, +$-.0942,-.1253,-.1333,-.9831$, which are at least the final scaled +minorant coefficients $-.1,-.14,-.135,-.995$. +For all entries before the last, one checks simply +$10^a[X^j]T\ge\lambda_j$ as in +\eqref{eq:cert-coefficient-rule}. The same coefficient rule verifies +the non-discriminant rows directly. + +The tail cutoff is exact: the four discriminant degrees are +$20,22,28,28$, as recorded in~\eqref{eq:cert-tail-blocks}, so no +term above degree $28$ has been omitted. More generally, from +\eqref{eq:cert-bcoeff}, the respective triples of degrees +$(\deg B_0,\deg B_1,\deg B_2)$ are +\[ +\begin{array}{c|rrrr} + &i=0&i=1&i=2&i=3\\ \hline + s=0&(-\infty,9,9)&(-\infty,10,10)&(10,10,10)&(11,11,9)\\ + s=1&(-\infty,9,13)&(-\infty,10,13)&(10,14,13)&(11,14,12). +\end{array} +\] +Here $\deg0=-\infty$. These bounds also justify a uniform cutoff of +$28$ for every polynomial in the two coefficient tables. + +For $(s,i)=(0,2)$, the first criterion covers $X\in[0,.05]$ and +the discriminant criterion covers $X\in[.05,1]$. +For $(s,i)=(0,3)$, the corresponding split is $.015$, with the +listed additional subdivisions used only for Bernstein positivity. +For $(s,i)=(1,2)$, the split is $.2$. +For $(s,i)=(1,3)$, the first criterion covers $[0,.1]$, the +discriminant criterion covers $[.1,.75]$, and the final criterion +covers $[.75,1]$. For $i=0,1$, the full interval is covered by +$B_1$ and $B_1+B_2$. All endpoints belong to the stated intervals. +Thus every coefficient~\eqref{eq:cert-eta-coefficient} is nonnegative +for $0\le X,r\le1$. Nonnegativity and unit sum of the degree-three +Bernstein basis in $\eta$ prove the desired polynomial inequality +for $0\le w\le X$; when $X=0$ one has $w=0$ directly. +Restoring the error budget~\eqref{eq:cert-third-error} proves +\eqref{eq:cert-third-goal}. Together with the preceding reductions, +this completes the proof of Lemma~\ref{lem:scalar-bounds}. diff --git a/preprints/The-Approximation-Threshold-for-Metric-k-Median-September-24-2026/build/sections/derandomization.tex b/preprints/The-Approximation-Threshold-for-Metric-k-Median-September-24-2026/build/sections/derandomization.tex new file mode 100644 index 0000000000000000000000000000000000000000..f0c233881763e841644e695ce111780c17979ccc --- /dev/null +++ b/preprints/The-Approximation-Threshold-for-Metric-k-Median-September-24-2026/build/sections/derandomization.tex @@ -0,0 +1,585 @@ +\section{Deterministic implementation and completion} +\label{sec:derandomization} + +The preceding constructions yield an additive approximation in expectation. +We now replace their random choices by finite distributions of polynomial +support, and then choose the least expensive feasible outcome. Throughout +this section the desired approximation loss is fixed. In particular, the +grid denominator, the number of physical rounds, and all parameters chosen +in Theorems~\ref{thm:preparation} and~\ref{thm:grid-rounding} are constants +independent of the input size. All input distances are rational, and +running time includes their encoding lengths. + +\subsection{Compression of finite distributions} + +A distribution list is a finite list of states, each equipped with a +nonnegative rational weight, whose weights sum to one. The state records +the data needed to continue a construction; a feature vector records the +statistics whose expectations must be retained. + +The next lemma is the constructive finite-dimensional Carath\'eodory +argument for preserving an expectation. Its connection with moment +matching is classical~\cite{BayerTeichmann2005}; we include the exact +elimination and its bit bound because both are needed by the algorithm. + +\begin{lemma}[Affine support reduction] +\label{lem:affine-support} +Let a distribution list have states \(s_1,\ldots,s_M\), weights +\(\lambda_1,\ldots,\lambda_M\), and feature vectors +\(v(s_i)\in\mathbb Q^d\). There is an exact algorithm that replaces the +weights by nonnegative rational weights supported on at most \(d+1\) of +the same states, preserving +\[ + \sum_{i=1}^M\lambda_i v(s_i). +\] +Its running time and output encoding length are polynomial in the +encoding length of the given list and feature vectors. +\end{lemma} + +\begin{proof} +Discard zero-weight entries. If more than \(d+1\) remain, choose a +nonzero integral affine dependence \(z\) satisfying +\[ + \sum_i z_i=0, + \qquad + \sum_i z_i v(s_i)=0. +\] +Since its coefficients sum to zero, there are both positive and negative +coefficients. Set +\[ + t=\min_{i:z_i>0}\frac{\lambda_i}{z_i}, + \qquad + \lambda'_i=\lambda_i-tz_i. +\] +The new weights are nonnegative, have the same sum and feature +expectations, and include at least one additional zero. Repeating gives +the support bound. + +For completeness, the rational arithmetic has polynomial complexity. +An affine dependence can be found using at most \(d+2\) columns of the +matrix with columns \((1,v(s_i))\). Clearing denominators and using +minors gives an integral dependence of polynomial bit length. If +\(\lambda_i=b_i/D\) for a common denominator \(D\), and \(j\) attains +the minimum defining \(t\), then +\begin{equation} + \lambda'_i= + \frac{b_i z_j-b_jz_i}{Dz_j}. + \label{eq:compression-denominator} +\end{equation} +Thus one elimination adds at most the bit length of \(|z_j|\) to a +common denominator. There are at most \(M\) eliminations, and the +feature vectors do not change. The resulting bit lengths and the exact +linear-algebra computations are polynomially bounded. +\end{proof} + +Support reduction preserves every property satisfied by each state in +the list, whether or not that property appears among the features. It +does not preserve the entire joint distribution or arbitrary future +adaptive behavior. We therefore specify the retained features at each +use and verify the required estimates for the resulting process. + +\subsection{A distribution list for the preparation stage} + +Let \(M_\star\) denote the number of stars in the preparation +construction, and write \(X_e\) for the current count of star \(e\). +Each count remains within its original floor and ceiling. A weighted +pair update within a strict sign preserves the weighted sum and has +conditional mean equal to the current count vector. For every pair of +distinct coordinates \(e,f\), it also satisfies +\begin{equation} + \E[X'_e\mid X]=X_e, + \qquad + \E[X'_eX'_f\mid X]\le X_eX_f. + \label{eq:star-local-moments} +\end{equation} +Indeed, unless both coordinates are the selected pair, the product is +affine in the update parameter. If they are the selected pair, their +increments have opposite signs and the product is concave. A +one-coordinate rounding with the correct mean preserves every distinct +pair-product expectation. + +These assertions hold for every admissible state and every permissible +choice of the next pair. Thus the next pair may be selected adaptively, +for example as the first two fractional coordinates in a fixed order. +After branching one update at every state, apply +Lemma~\ref{lem:affine-support} with the features +\[ + X_e \quad\text{and}\quad X_eX_f \qquad(e0\), an interior solution is +\begin{equation} + N_i= + \frac{\sum_{h\text{ partial}}d(i,h)} + {r+\widehat P(m-\Delta)}. + \label{eq:rational-taper-radius} +\end{equation} +The breakpoints are the rational numbers \(d(i,h)\) and +\(d(i,h)/(1-\widehat P)\). Sorting these values and checking the +successive intervals finds the smallest solution, including a +breakpoint solution. If \(r=0\), the equation is constant on the +interval, and the first point of a solution interval is a breakpoint. +The separately prescribed case of sufficient +co-located mass uses \(N_i=0\). All radii, marginals, and resulting +branch probabilities have polynomial bit length. The physical states +and their feature values are determined without reference to the +list weights. + +It remains to control the list weights across repeated branching and +compression. Suppose a feature vector has \(d\) coordinates of bit +length at most \(L\). In the affine-dependence computation, clearing +denominators in each row of the matrix on at most \(d+2\) columns +produces integer entries of bit length \(O(dL)\). A nonzero kernel +vector can be chosen using minors, with bit length +\(O(d^2L+d\log(d+1))\) by the determinant bound. These bounds are +independent of the current list weights. + +By~\eqref{eq:compression-denominator}, an elimination multiplies a +common weight denominator by just one coefficient of such a +dependence. An exact branch with probability denominator \(q\) +multiplies a common denominator by at most \(q\). Branching a whole +list adds the bit lengths of only polynomially many such +denominators, and taking a product of lists adds their denominator +bit lengths. There are polynomially many list operations, including +the entire constant-depth physical history tree. Their common +denominators and numerators consequently retain polynomial bit +lengths. Exact linear algebra, comparisons, and cost evaluations +therefore give a deterministic algorithm whose bit complexity is +polynomial in the input encoding length for every fixed choice of +accuracy parameters. + +\subsection{Loss allocation and the approximation theorem} + +We first record explicitly how conditioning and deterministic +selection are combined. For a nonnegative cost \(C\) and an event +\(G\) with \(\Prob(G)\ge1-r\), +\begin{equation} + \E[C\mid G] + \le\frac{\E C}{1-r}. + \label{eq:conditioning-cost} +\end{equation} +Thus a finite distribution list contains a successful outcome whose +cost is no more than the right-hand side. No independence between +cost and budget success is required. + +\begin{theorem}[Deterministic additive approximation] +\label{thm:pseudo} +For every fixed \(\varepsilon>0\), there is an +instance-independent integer \(c\ge0\) and a deterministic +polynomial-time algorithm for metric \(k\)-median that opens at +most \(k+c\) facilities and has total connection cost at most +\[ + \left(1+\frac2e+\varepsilon\right)\operatorname{LP}, +\] +where \(\operatorname{LP}\) is the optimum value of the assignment +relaxation. In particular, its cost is at most the same factor times +\(\OPT_k\). +\end{theorem} + +\begin{proof} +Put \(\alpha_\star=1+2/e\), and choose a fixed positive rational +number +\[ + t\le\min\{1/8,\varepsilon/10\}. +\] +Choose the parameters in the constructions underlying +Theorems~\ref{thm:preparation} and~\ref{thm:grid-rounding} with +unconditioned cost factors at most \(1+t\) and +\(\alpha_\star+t\), respectively, and with their respective +budget-failure probabilities at most \(t\). The proofs of those +theorems allow the cost and failure tolerances to be made +arbitrarily small; their stated expectation guarantees absorb this +same conditioning loss. Here we keep the loss visible. + +All parameter choices can be made in a consistent order. After the +preparation parameters other than the grid are fixed, take an even +integer \(\Delta\) sufficiently large, set \(g=1/\Delta\), and +take a rational horizon of order +\(1+\log(\Delta/t)\). The requirements that the terminal error +be small and that \(\Lambda g\) be small are compatible because +\(\log\Delta/\Delta\to0\). Then choose the taper tolerance, +the selection bias, and the time step successively small. The +number of rounds is now fixed. Finally choose the heavy-source +parameter sufficiently small that the sum of the conditional +count-failure probabilities over all rounds is at most \(t\). +The resulting additive opening allowances are constants depending +only on these parameter choices. + +Let \(L=\operatorname{LP}\). Apply the preparation list to an +optimal LP solution. Its total nearest-unit cost has expectation at +most \((1+t)L\), and its budget event has probability at least +\(1-t\). Select the least expensive successful grid solution. +By~\eqref{eq:conditioning-cost}, its total nearest-unit cost +\(L_{\rm grid}\) satisfies +\[ + L_{\rm grid}\le\frac{1+t}{1-t}L. +\] +Apply the physical-rounding list to this fixed grid solution. The +summed cost bound is at most \((\alpha_\star+t)L_{\rm grid}\) +before discarding count failures, and the budget succeeds with +probability at least \(1-t\). Selecting the least expensive +successful integral output therefore gives cost at most +\[ + \frac{(1+t)(\alpha_\star+t)}{(1-t)^2}L. +\] +Since \(\alpha_\star<2\) and \(t\le1/8\), +\begin{align*} + \frac{(1+t)(\alpha_\star+t)}{(1-t)^2}-\alpha_\star + &= + \frac{(3\alpha_\star+1)t+(1-\alpha_\star)t^2} + {(1-t)^2}\\ + &\le\frac{7t}{(1-t)^2} + <10t + \le\varepsilon. +\end{align*} +The two additive allowances sum to an instance-independent +constant, which may be rounded up to an integer \(c\). + +Both minimum-cost selections are computable on polynomial-size +lists, with polynomial bit complexity as proved above. They use +total costs; the deterministic grid solution is not required to +attain every client's individual expectation bound. If \(L=0\), +the nonnegative cost has expectation zero in the preparation list +and then in the physical-rounding list, so the same argument gives +zero cost without division by \(L\). + +Every opened copy lies at an original candidate facility. Merge +copies introduced by the construction back to their original +facility. This preserves connection distances and cannot increase +the number of opened facilities. Finally, +\(\operatorname{LP}\le\OPT_k\), proving the last assertion. +\end{proof} + +\subsection{Applying the reduction on rational inputs} +\label{sec:reduction-implementation} + +Before applying Theorem~\ref{thm:li-svensson}, we record its interface +with the deterministic algorithm just proved. Every call of that reduction +uses the same clients and distances with a restricted candidate set +\(F'\subseteq F\); our pseudo-algorithm uses no additional input promise +or prescribed-open facilities. Its internal forced openings are decisions +of that call, not constraints passed from another call. Copies are merged +back to original candidates before returning its output. If +\(0<|F'|k\), and every \(i\in T'\), compute +\[ + \Delta(T',i)=\cost(T'\setminus\{i\})-\cost(T'). +\] +These numbers are nonnegative rationals of polynomial bit length. +There are \(n^{O(c)}\) such numbers. Run the reduction's deletion phase +for each distinct number as threshold, and for zero, with a fixed order +for choosing deletions; follow each run by its prescribed enumerative +completion and retain the cheapest feasible result. + +To justify this replacement, let \(B^*\ge0\) be the threshold used in +the reduction's analysis. Choose the largest enumerated value at most +\(B^*\), including equality, or zero if there is none. Every test of the form +\(\Delta(T',i)\le B^*\) has the same answer at this representative +threshold. Induction through the at most \(c\) deletions therefore +reproduces the ideal deletion history, including its fixed tie choices. +Its output is present among the enumerated outputs. The approximation +analysis still uses \(B^*\); the algorithm uses only exact rational +comparisons. This adds \(n^{O(c)}\) overhead to the stated reduction. +All candidate enumeration, distance comparisons and cost evaluations +likewise use polynomial-bit rational arithmetic. + +Finally, the approximation parameter supplied to the reduction can be a +fixed rational upper bound on our ratio, with its excess charged to the +allowed accuracy. The auxiliary positive constants in its proof may then +be chosen rational as well. These observations preserve both determinism +and polynomial binary-input time; they require no oracle for the optimum. + +\begin{proof}[Proof of Theorem~\ref{thm:main}] +Fix \(\varepsilon>0\). Apply Theorem~\ref{thm:pseudo} with +loss \(\varepsilon/4\), obtaining a deterministic +\(c\)-additive approximation, where \(c\) depends only on this fixed +loss. Choose a fixed rational number \(\alpha\) with +\[ + 1+2/e+\varepsilon/4<\alpha<1+2/e+\varepsilon/2. +\] +The pseudo-algorithm is a \(c\)-additive \(\alpha\)-approximation. +The additive-to-ordinary reduction of Theorem~\ref{thm:li-svensson}, +with a fixed positive rational loss at most \(\varepsilon/2\) and the +implementation in Section~\ref{sec:reduction-implementation}, +gives a deterministic polynomial-time +algorithm opening at most \(k\) original facilities and having +cost at most +\[ + \left(1+\frac2e+\varepsilon\right)\OPT_k. +\] +Copies can be merged back to their original facilities as above. +If exactly \(k\) facilities are required, add unused facilities +from \(F\), which is possible because \(k\le|F|\) and can only +decrease the cost. + +\end{proof} + +Combining Theorem~\ref{thm:main} with the hardness bound recalled in +Section~\ref{sec:introduction} proves Corollary~\ref{cor:threshold}: +under \(P\ne NP\), the infimum of deterministic polynomial-time +approximation factors in the specified candidate-facility model is +\(1+2/e\). The upper bound holds for every fixed positive loss and +does not require attainment of the endpoint. diff --git a/preprints/The-Approximation-Threshold-for-Metric-k-Median-September-24-2026/build/sections/inequality.tex b/preprints/The-Approximation-Threshold-for-Metric-k-Median-September-24-2026/build/sections/inequality.tex new file mode 100644 index 0000000000000000000000000000000000000000..548aa548b4bf724c834d8bfcfd96c64e63008514 --- /dev/null +++ b/preprints/The-Approximation-Threshold-for-Metric-k-Median-September-24-2026/build/sections/inequality.tex @@ -0,0 +1,365 @@ +\section{A finite inequality for the drift} +\label{sec:inequality} + +The cost analysis uses an inequality on a finite probability space. Its +comparison matrix need not be symmetric. The proof allows a positive +contribution at one index only by charging indices with smaller values; +the negative contributions at those indices pay the entire charge. + +The statement below is independent of the rounding construction. +Section~\ref{sec:rounding} will apply it to normalized client distances +and directed edge probabilities; Equation~\ref{eq:rnd-normalization} +records that substitution. We prove the finite statement first so that +the geometric analysis can use it as a single input. + +For $0\le x\le1$, define +\begin{equation} +\label{eq:analytic-coefficients} +\begin{aligned} + A(x)&=1+2\int_0^1(1-e^{-tx})\,dt,& + P(x)&=\int_0^1te^{-tx}\,dt,\\ + B(x)&=2P(x),& m(x)&=A(x)-1-2xP(x). +\end{aligned} +\end{equation} +We suppress the argument $x$ when it is fixed. Differentiation under the +integral gives $A'=B$ and +$A''(x)=-2\int_0^1t^2e^{-tx}\,dt<0$. In particular, +$00,\qquad K(r) + &\ge d_sG(r) + &&(0\le r\le1), + \label{eq:scalar-linear}\\ + \bigl((J_s-b)r+1.6z\bigr)M(r) + +(C-k_s-J_sr)L(r) + &\ge0 + &&(0\le r\le1), + \label{eq:scalar-charge}\\ + C-k_s+1.6z-bq+.52\bigl(k_s-\kappa_s(q)\bigr) + &\ge H_0 + &&(0\le q\le1), + \label{eq:scalar-type-zero}\\ + C-k_s+1.6z-b&\ge.25. + \label{eq:scalar-type-one} +\end{align} +\end{lemma} + +\begin{proof} +The scalar bounds are proved in Appendix~\ref{app:certificates} by +analytic reductions and exact polynomial certificates. +\end{proof} + +\begin{proof}[Proof of Theorem~\ref{thm:finite-inequality}] +We may discard indices of probability zero. The proof treats the +quantities with a fixed second index $i$ as a row, with $t$ indexing its +columns. + +\paragraph{Reduction of the parameters.} +For fixed $x,D,q,R,s$, the left side of +Equation~\eqref{eq:finite-inequality} is nondecreasing in $y$. +The coefficient of its linear $y$ term is nonnegative, and increasing +$y$ decreases the positive part that is subtracted. Replace $y$ there +by $\max(x,y)=x+w$, where $0\le w\le X=1-x$, and put $z=Pw^2$. +If $y0$ has $r>0$. By +Equations~\eqref{eq:scalar-linear} and +\eqref{eq:row-linear-majorant}, it also has +\begin{equation} +\label{eq:positive-row-location} + G(r)>0,\qquad D>\frac{K(r)}{G(r)}\ge d_s. +\end{equation} +In this row put +\[ + U'=\frac{Ur-(.4D+1.6)z}{Cr}. +\] +The identity +\[ + U'=\frac{D+1}{C}\left(m+b-\frac{.4z}{r}\right) + -\frac mC-\frac{1.2z}{Cr} +\] +and Equation~\eqref{eq:scalar-first} imply +\begin{equation} +\label{eq:positive-row-uprime} + U'\le\frac{(D+1)(1-P)}{1-Pr}. +\end{equation} +On the other hand, discarding the nonpositive hinge term in +Equation~\eqref{eq:row-contribution} gives +$F/C\le(1-r)R+rU'-(D+1)$. +If $R\le(D+1)/(1-Pr)$, Equation~\eqref{eq:positive-row-uprime} +makes this upper bound nonpositive. Therefore +\begin{equation} +\label{eq:positive-row-radius} + R>\frac{D+1}{1-Pr},\qquad U'\le R(1-P). +\end{equation} +In particular $r=1$ is impossible for a positive row, since +Equation~\eqref{eq:positive-row-uprime} would then give $U'\le D+1$. + +For every column with $q<1$, Equations~\eqref{eq:finite-comparison} +and \eqref{eq:positive-row-radius} now give +\[ + (1-q)R+qU'\le R(1-Pq)\le D+Z. +\] +For $q=1$, the excess of the left side over $D+Z$ is exactly +$U'-D-Z$. Using $\E_tZ=1$ and again discarding the nonpositive +hinge term, we obtain +\begin{equation} +\label{eq:positive-row-excess} + F\le C\E_t\big[\ind_{\{q=1\}}(U'-D-Z)_+\big] + \le C\E_t(U'-D-Z)_+. +\end{equation} +This treats the unconstrained columns with $q=1$ explicitly. + +It remains to bound the largest value to which this row can charge. +An exact rearrangement gives +\[ + U'-D=\frac{br-1.6z-L(r)D}{Cr}. +\] +Set $T=(J_s-b)r+1.6z$. The expression in +Equation~\eqref{eq:scalar-charge} is +\begin{equation} +\label{eq:charge-scalar-rearrangement} + TM(r)+(C-k_s-J_sr)L(r)=TG(r)+K(r)L(r)\ge0. +\end{equation} +Using $L(r)\ge0$ and Equation~\eqref{eq:positive-row-location}, +we conclude that +\[ + br-1.6z-L(r)D + \le br-1.6z-\frac{L(r)K(r)}{G(r)}\le J_sr. +\] +Thus $U'-D\le H_s$. Since $C\le1$, +Equation~\eqref{eq:positive-row-excess} proves +\begin{equation} +\label{eq:outgoing-charge} + F_i\le\E_t(H_{s_i}-D_t)_+\qquad(F_i>0). +\end{equation} +The function $H_0$ decreases from $.394$ to $.014$ on $[0,1]$, +and $H_1=.25$. Consequently every nonzero charge in +Equation~\eqref{eq:outgoing-charge} goes to an index with $D_t<.394$. +This separates every possible receiving index from all positive rows, +which have $D_i>d_{s_i}\ge.4$. It remains to prove that the negative +contribution of each receiving row covers the total charge it receives. + +\paragraph{Negative rows cover the incoming charges.} +Put $c_*=.394$ and define +\[ + \mathcal P=\{i:F_i>0\},\qquad + \mathcal S=\{i:D_i\le c_*\},\qquad + \Gamma_{it}=\ind_{\{i\in\mathcal P\}}(H_{s_i}-D_t)_+. +\] +Equation~\eqref{eq:positive-row-location} shows that +$\mathcal P\cap\mathcal S=\varnothing$, because $d_s\ge.4>c_*$. +We claim that every $i\in\mathcal S$ satisfies +\begin{equation} +\label{eq:incoming-charge} + F_i+\E_t\Gamma_{ti}\le0. +\end{equation} + +Fix such an $i$ and a column $t$. Hold $q=q_{ti}$, $Z=D_t$, +the two types $s_i,s_t$, and $\chi=\ind_{\{F_t>0\}}$ fixed. +As a function of a variable $D\in[0,c_*]$, +\[ + I_{s_i}(D;q,Z)+\chi(H_{s_t}-D)_+ +\] +is convex: the first term is affine and the second is a positive part +multiplied by a nonnegative constant. It suffices to bound this +expression at $D=0,c_*$. This check concerns the explicit majorant; +it does not require the comparison hypothesis to remain valid when +the variable $D$ changes. + +At $D=c_*$, the incoming charge is zero. By +Equation~\eqref{eq:pointwise-linear-majorant}, the remaining term is +at most $c_*G(q)-K(q)\le0$. Indeed, if $G(q)<0$, use $K(q)>0$; +otherwise use $K(q)\ge d_{s_i}G(q)$ and $c_*d_0=.52$. Since $k_s-\kappa_s(q)\ge0$, +Equation~\eqref{eq:scalar-type-zero} gives +$I_{s_i}(0;q,Z)\le-H_0$, which covers its incoming charge. +In the other cases the charge is either zero or $H_1=.25$. +Equation~\eqref{eq:scalar-type-one}, together with $q\le1$, +gives +\[ + I_{s_i}(0;q,Z)\le b-1.6z-C+k_s\le-.25, +\] +which covers these cases as well. Convexity proves the pointwise +bound throughout $[0,c_*]$. Apply it at $D=D_i$ and average over +$t$ in Equation~\eqref{eq:centered-row-bound} to obtain +Equation~\eqref{eq:incoming-charge}. + +Finally, the incoming charge is $\Gamma_{ti}$ by definition, and +exchanging the two finite sums requires no symmetry of $q$. +Equation~\eqref{eq:outgoing-charge} bounds all +positive rows by their outgoing charges; +Equation~\eqref{eq:incoming-charge} makes all receiving rows pay +their incoming charges; all other rows have nonpositive contribution. +Every nonzero $\Gamma_{it}$ has $i\in\mathcal P$ and +$t\in\mathcal S$. Exchanging the two indices in the finite sum with +weights $\pi_i\pi_t$ therefore gives +\[ + \E_iF_i\le\E_{i,t}\Gamma_{it}-\E_{i,t}\Gamma_{ti}=0. +\] +This proves the stronger bound $2z$ after the parameter reduction and, +since $z\le\Theta$, proves Equation~\eqref{eq:finite-inequality}. +\end{proof} diff --git a/preprints/The-Approximation-Threshold-for-Metric-k-Median-September-24-2026/build/sections/introduction.tex b/preprints/The-Approximation-Threshold-for-Metric-k-Median-September-24-2026/build/sections/introduction.tex new file mode 100644 index 0000000000000000000000000000000000000000..2689fb78ae0dd651dd17193a3bb75ca84fb9634a --- /dev/null +++ b/preprints/The-Approximation-Threshold-for-Metric-k-Median-September-24-2026/build/sections/introduction.tex @@ -0,0 +1,196 @@ +\section{Introduction}\label{sec:introduction} + +In metric \(k\)-median, a finite set \(J\) of clients and a finite set \(F\) +of candidate facilities lie in a common metric space. Given an integer \(k\), +one chooses at most \(k\) facilities to minimize the sum of the distances +from clients to their nearest chosen facility. For a nonempty set +\(S\subseteq F\), write +\[ + d(j,S)=\min_{i\in S}d(j,i),\qquad + \cost(S)=\sum_{j\in J}d(j,S),\qquad + \OPT_k=\min_{\substack{S\subseteq F\\1\le|S|\le k}}\cost(S). +\] +The input distances are rational and encoded in binary, and +\(1\le k\le |F|\). Both \(k\) and the metric are unrestricted. +Candidate facilities are specified as part of the input; we do not +require \(F=J\). The empty-client instance has cost zero and can be +handled immediately. + +The facility budget is a central difficulty: a solution with small +connection cost can be much easier to obtain when extra facilities are +allowed. Our approach first permits a constant additive surplus, makes +that approximation deterministic, and then removes the surplus using an +established reduction. + +\begin{theorem}\label{thm:main} +For every fixed \(\varepsilon>0\), there is a deterministic algorithm which, +given a finite rational metric \(k\)-median instance with \(1\le k\le |F|\), +returns \(S\subseteq F\), \(|S|\le k\), such that +\[ + \sum_{j\in J}d(j,S)\le + \left(1+\frac2e+\varepsilon\right)\OPT_k. +\] +Its running time is polynomial in the binary encoding length of the instance. +The polynomial may depend on \(\varepsilon\). +\end{theorem} + +Hardness below \(1+2/e\) in this facility model follows from the +Max-\(k\)-Coverage gap: covered clients have distance one and uncovered +clients have distance at least three. We use the formulation under \(P\ne NP\) +recorded by Anand and Lee~\cite[Sections~1 and~1.1]{AnandLee2024}, based on +Feige's perfect-completeness coverage gap~\cite[Theorem~5.3]{Feige1998}. +For an explicit metric completion, represent sets by facilities and +universe elements by a disjoint client set. Give distinct points within +either part distance two, and give a facility--client pair distance one +for incidence and three otherwise. This is a metric: a same-part edge +of length two is bounded by \(1+1\), and a cross-part edge of length at +most three is bounded by \(2+1\). For a universe of size \(m\), the +connection cost is exactly \(3m-2c\) when the chosen sets cover \(c\) +elements; a perfect cover has cost \(m\). Thus a gap between +covering every element and covering at most a +\(1-1/e+\delta\) fraction gives the lower bound +\(1+2/e-2\delta\), for every fixed positive \(\delta\). +Together with Theorem~\ref{thm:main}, +this proves the following consequence. + +\begin{corollary}\label{cor:threshold} +If \(P\ne NP\), the infimum of deterministic polynomial-time approximation +factors for metric \(k\)-median with specified candidate facilities is \(1+2/e\). +\end{corollary} + +The infimum in Corollary~\ref{cor:threshold} does not assert attainment of the +endpoint factor. The hardness assumption and candidate-facility model are part +of the statement. In particular, the corresponding lower-bound question with +the restriction \(F=J\) is different~\cite{AnandLee2024}. + +\subsection{Historical context and method} + +Metric \(k\)-median has been a testing ground for LP rounding, +primal-dual algorithms, and local search. Charikar, Guha, Tardos and +Shmoys gave the first constant-factor approximation through LP +rounding~\cite{CharikarGuhaTardosShmoys2002}. Jain and +Vazirani connected the cardinality constraint with metric facility +location through Lagrangian relaxation and the primal-dual +method~\cite{JainVazirani2001}. The combinatorial improvements of +Charikar and Guha~\cite{CharikarGuha1999} and the greedy dual-fitting +approach of Jain et al.~\cite{JainEtAl2003} developed this connection +further. A different route, the bounded-swap local search of Arya et +al.~\cite{Arya2004}, achieves factor \(3+\varepsilon\). + +Li and Svensson~\cite{LiSvensson2016} made constant-additive +pseudo-approximations a route past the factor-three barrier: a constant +number of extra facilities can be removed at an arbitrarily small +additional approximation loss. Subsequent bi-point rounding, including +the corrected \(2.675+\varepsilon\) guarantee of Byrka et +al.~\cite{ByrkaEtAl2017}, refined this approach. Gowda et +al.~\cite{GowdaEtAl2023} obtained \(2.613\) in their SODA 2023 version. +Cohen-Addad et al.~\cite{CGLSS2026} reached +\(2+\varepsilon\), first announced in 2025. Byrka et +al.~\cite{ByrkaEtAl2026} subsequently developed a graph-based iterative +rounding framework that also gives \(2+\varepsilon\) for metric +\(k\)-median and extends to other clustering objectives. +Theorem~\ref{thm:main} reaches every fixed factor above the +\(1+2/e\) hardness boundary in the general candidate-facility model. + +Our construction follows the iterative graph framework, with the +cost analysis replaced by a directed comparison inequality and a +potential that retains some surviving assignments after an opening. +The earlier framework's general transfer theorem uses a parameter at +least \(2\) in the metric \(k\)-median case +\cite[Theorem~6.1]{ByrkaEtAl2026}; it does not supply the smaller factor +proved here. We establish the grid preparation, physical updates, +comparison, and deterministic implementation together. + +A separate companion combines anchor recovery with bounded-price payment +accounting to give an independent randomized approximation below factor +\(2\) while respecting the facility budget~\cite[Theorem~1.2]{OpenAIRecovery2026}. + +Two other methodological antecedents are useful to distinguish. +Splitting fractional facilities and organizing assignments around +separated representatives and bundles are standard LP-rounding +techniques; Charikar and Li~\cite{CharikarLi2012} provide a close +predecessor. The pair updates used in our preparation belong to the +dependent-rounding approach of Srinivasan~\cite{Srinivasan2001} and +Gandhi et al.~\cite{GandhiEtAl2006}, with weighted variants studied by +Byrka et al.~\cite{ByrkaEtAl2017}. We prove the particular moment bounds +and coupled supplier construction needed here. Likewise, reducing the +support of a distribution while matching moments is a standard +Carath\'eodory argument; see Bayer and Teichmann~\cite{BayerTeichmann2005} +for the finite-cubature perspective. Our implementation specifies which +moments suffice throughout the physical process and proves the required +rational encoding bounds. + +\subsection{Proof overview} + +The proof separates the preparation of fractional openings, their +iterative rounding, and the removal of randomness and surplus. The +intermediate guarantee is measured against the standard assignment LP: +its variable \(y_i\) is the amount of opening at facility \(i\), and each +client receives a unit of this opening mass. The LP has total opening +mass at most \(k\) and cost at most \(\OPT_k\). + +\paragraph{Preparing a constant grid.} +Section~\ref{sec:preparation} replaces the LP openings by multiples of +\(g=1/\Delta\), where the even integer \(\Delta\) depends only on the +accuracy. The total mass increases by at most \(3g\), and each client's +expected assignment cost increases by an arbitrarily small factor. +Separated representative clients define disjoint bundles of nearby +opening mass. A bundle's deficit is the part missing from a unit; its +suppliers are the facilities or other bundles holding that part of the +representative's assignment. Rounding deficits together with their +suppliers controls both the opening budget and the cost of restoring a +full unit. This coupling must allow one supplier to serve many clients, +because assignment mass is not a shared capacity. + +\paragraph{Rounding the grid.} +View every positive grid opening as copies of mass \(g\). In each round, +a directed graph specifies which copies a source erases when that source's +facility opens. Each target has exactly \(\Delta\) incoming sources, +chosen with distance-dependent probabilities. Opening and erasing are +balanced so that only a constant additional number of facilities is +needed. This source-opening and neighbor-erasure framework, including +balanced batches and forced large updates, builds on Byrka et +al.~\cite[Sections~3--6]{ByrkaEtAl2026}. + +The connection-cost analysis needs a different comparison below factor +two. For each client, we track surviving copies from its initial assigned +unit and a persistent bound on its eventual connection distance. A +potential combines their remaining assignment cost with that bound. Its +coefficient depends on elapsed time as well as surviving mass. When one +of the client's own copies opens, we continue tracking the cheaper +survivors; this produces a subtraction that pays for other possible +erasures. The finite inequality in Section~\ref{sec:inequality} captures +that payment for arbitrary directed comparison coefficients. Its proof +transfers contributions from positive rows to lower-distance rows, whose +negative contributions cover them. Appendix~\ref{app:certificates} +proves every scalar estimate by explicit rational polynomial certificates. +The geometric substitution and the potential analysis are given in +Section~\ref{sec:rounding}. + +A source erasing many copies is handled by forcing its facility open. +Because this choice depends on the graph, its potential change is bounded +for each realized graph first. Only then do we average over graphs. This +order lets the finite directed inequality control the averaged drift +without conditioning on a graph-dependent forcing event. + +\paragraph{A deterministic solution and the exact budget.} +The preceding analysis uses randomness, but only specified low-order +moments of each local choice enter its bounds. Section~\ref{sec:derandomization} +constructs polynomial-size rational distributions preserving these +moments. All physical histories are expanded for a constant number of +rounds, and the cheapest output satisfying the additive budget is chosen. +The separate rational-arithmetic analysis proves polynomial time in the +binary input length. The result is a deterministic approximation opening +at most \(k+c\) facilities for an accuracy-dependent constant \(c\). +Li and Svensson's reduction then removes this surplus with an arbitrarily +small additional loss. + +\paragraph{Organization.} +Section~\ref{sec:preliminaries} fixes the model and states the external +additive-to-ordinary reduction. Section~\ref{sec:preparation} prepares the +openings. Section~\ref{sec:inequality} proves the finite drift inequality, +with its scalar certificates in Appendix~\ref{app:certificates}. +Section~\ref{sec:rounding} gives and analyzes the iterative procedure. +Section~\ref{sec:derandomization} supplies the deterministic implementation +and completes the proof of Theorem~\ref{thm:main}. + diff --git a/preprints/The-Approximation-Threshold-for-Metric-k-Median-September-24-2026/build/sections/preliminaries.tex b/preprints/The-Approximation-Threshold-for-Metric-k-Median-September-24-2026/build/sections/preliminaries.tex new file mode 100644 index 0000000000000000000000000000000000000000..599c5e00d1fdd9459d275709f0c40f795de32a29 --- /dev/null +++ b/preprints/The-Approximation-Threshold-for-Metric-k-Median-September-24-2026/build/sections/preliminaries.tex @@ -0,0 +1,104 @@ +\section{Preliminaries and the reduction}\label{sec:preliminaries} + +We use the client set \(J\), candidate set \(F\), connection cost +\(\cost\), and optimum \(\OPT_k\) defined in the introduction. +Polynomial time includes the binary encodings of all distances. +We henceforth assume \(J\ne\varnothing\) and \(1\le k\le |F|\). +This section fixes the fractional representation and states the external +reduction that will remove the final constant additive surplus. + +\subsection{The fractional solution} + +We use the standard relaxation +\begin{equation}\label{eq:lp} +\begin{aligned} + \text{minimize}\quad &\sum_{j\in J}\sum_{i\in F}d(j,i)x_{ij}\\ + \text{subject to}\quad& + \sum_{i\in F}x_{ij}=1 &&(j\in J),\\ + &0\le x_{ij}\le y_i\le1 &&(i\in F,\ j\in J),\\ + &\sum_{i\in F}y_i\le k. +\end{aligned} +\end{equation} +Its rational optimum can be found in polynomial time. For fixed openings +\(y_i\), assign each client to its nearest unit of opening mass, using a fixed +order to break ties. If \(h_j(t)\) is the distance needed to accumulate mass +\(t\), put +\[ + V_j=\int_0^1h_j(t)\,dt. +\] +Then \(L=\sum_jV_j\le\OPT_k\) for an optimal solution of~\eqref{eq:lp}. + +As in LP-rounding constructions such as Charikar--Li~\cite[Section~2]{CharikarLi2012}, +it is convenient to split an opening into co-located pieces so that each +client's nearest-unit assignment uses whole pieces. There are only +polynomially many needed split points: each client uses at most one partial +piece in a fixed distance order. Refine each facility simultaneously at all +the resulting cut points. Denote the whole-piece assignment of client \(j\) +by \(F_j\). For a set \(U\) of pieces, write \(y(U)=\sum_{i\in U}y_i\); +thus \(y(F_j)=1\) and \(V_j=\sum_{i\in F_j}y_id(j,i)\). + +Internal pieces at one location are allowed to be distinct, so their distance +can be zero. All geometric arguments use only nonnegativity, symmetry, and +the triangle inequality. Whenever a procedure returns open pieces, merge +them by their original facility identity. Merging preserves every client's +connection cost and cannot increase the number of facilities. + +\subsection{Constant-additive pseudo-approximation} + +\begin{definition} +A \(c\)-additive \(\alpha\)-approximation for metric \(k\)-median returns +at most \(k+c\) facilities of cost at most \(\alpha\OPT_k\). +The additive allowance \(c\) is independent of the input instance. +\end{definition} + +We use the following established result. + +\begin{theorem}[Li--Svensson~\cite{LiSvensson2016}, Theorem~4 of the 2014 manuscript] +\label{thm:li-svensson} +Let \(\alpha>1\), and suppose \(\mathcal A\) is a \(c\)-additive +\(\alpha\)-approximation algorithm for metric \(k\)-median. +For every \(\eta>0\), there is an ordinary +\((\alpha+\eta)\)-approximation algorithm with running time +\(n^{O(c/\eta)}\) times the running time of \(\mathcal A\). +For fixed \(\alpha,c,\eta\), this is polynomial overhead. +\end{theorem} + +The theorem permits separate client and candidate sets and applies the +pseudo-algorithm to restricted candidate instances. Our construction works +on every such instance. We will obtain a deterministic pseudo-algorithm +before applying Theorem~\ref{thm:li-svensson}; the external reduction is not +being used to convert an expected-cost statement into determinism. + +For each fixed accuracy we permit all grid denominators, numbers of rounds, +and additive allowances to be constants of any size. They may depend on +that accuracy, but not on \(k\), the number of input points, their distances, +or their bit lengths. Thus it suffices to prove, for each fixed +\(\tau>0\), a deterministic \(c_\tau\)-additive +\((1+2/e+\tau)\)-approximation. Our analysis in fact bounds the +pseudo-solution against \(L\), the initial fractional cost. This does not +assert the same factor for rounding the original relaxation to exactly +\(k\) facilities: the additive allowance and the subsequent reduction +are essential. + +\subsection{Probability and bookkeeping} + +We use \(\E\), \(\Prob\), and \(\Var\) for expectation, probability, and variance. +Every distribution used below has finite support. Randomized procedures +first provide the estimates; Section~\ref{sec:derandomization} replaces their +choices by explicit polynomial-size lists. + +Two elementary facts will be used repeatedly. First, for a nonnegative random +cost \(C\) and an event \(E\) of probability at least \(1-\rho>0\), +\begin{equation}\label{eq:conditioning} + \E[C\mid E]\le\frac{\E C}{1-\rho}. +\end{equation} +No cost--event independence is required. Second, a finite distribution +contains an outcome whose cost is at most its expected cost. These facts +allow us to discard budget failures and select a single output by total +cost; we never require all individual client expectation bounds to hold +simultaneously at that output. + +The notation \(O_\Delta(\cdot)\) means that its implicit constant may depend +on the fixed grid denominator \(\Delta\), and on no input parameter. +All unqualified constants in the geometric and analytic estimates are +absolute, unless an explicit previously fixed parameter is indicated. diff --git a/preprints/The-Approximation-Threshold-for-Metric-k-Median-September-24-2026/build/sections/preparation.tex b/preprints/The-Approximation-Threshold-for-Metric-k-Median-September-24-2026/build/sections/preparation.tex new file mode 100644 index 0000000000000000000000000000000000000000..81532d1a0b01184c32a8fb65a23b2fc8b18a8618 --- /dev/null +++ b/preprints/The-Approximation-Threshold-for-Metric-k-Median-September-24-2026/build/sections/preparation.tex @@ -0,0 +1,526 @@ +\section{Preparing the openings}\label{sec:preparation} + +We first put the fractional openings on a sufficiently fine constant grid. +The use of separated representatives and bundles is motivated by the +discretization approach in~\cite{ByrkaEtAl2026}. The construction below +rounds a bundle's missing assignment mass together with the facilities +that can supply it. This coupling provides both an additive opening +bound and a multiplicative connection-cost bound. + +Throughout this Section, the client set is nonempty; an instance without +clients requires no cost analysis. We start with feasible fractional +openings $0\le y_i\le1$. For each client $j$, choose a nearest-unit +assignment $x_{ij}$, resolving ties in a fixed order, and write +\[ + V_j=\sum_i d(i,j)x_{ij}. +\] +The feasibility constraints are $\sum_i x_{ij}=1$ and $x_{ij}\le y_i$; +there is no supply constraint shared by different clients. + +\begin{theorem}\label{thm:preparation} +For every $\eta>0$, there is a constant $\Delta_0$ with the following +property. Given a finite rational metric instance and feasible rational +openings $0\le y_i\le1$ with a nearest-unit assignment of cost $V_j$ +for each client $j$, fix any constant even integer $\Delta\ge\Delta_0$ and let +$g=1/\Delta$. There is a randomized procedure, running in expected +polynomial time on rational inputs, that produces openings +$\widetilde y_i\in[0,1]\cap g\mathbb Z$, allowing polynomially many +co-located labeled facility pieces, such that +\[ + 1\le\sum_i\widetilde y_i\le\sum_i y_i+3g +\] +for every output, and the expected nearest-unit assignment cost of every +client $j$ is at most $(1+\eta)V_j$. +In particular the grid can be required to satisfy any additional +positive instance-independent upper bound on $g$. +\end{theorem} + +The expectation in Theorem~\ref{thm:preparation} is taken after rejecting +outcomes that exceed the opening budget. Section~\ref{sec:derandomization} +gives a deterministic implementation that preserves the corresponding +bound on the sum of the client costs. + +We use the following elementary dependent-rounding observation, in both +weighted and unweighted form. The mean-preserving pair update and its +product-supermartingale argument follow the standard +approach~\cite{Srinivasan2001,GandhiEtAl2006}; weighted rounding with one +residual fraction is also studied in~\cite[Section~2]{ByrkaEtAl2017}. +We give the proof to specify exactly the moment guarantees used here. + +\begin{lemma}\label{lem:prep-weighted-rounding} +Let $q_e\ge0$ be rational counts and let $\beta_e>0$ be rational weights. +One can randomly update the counts within the bounds +$[\lfloor q_e\rfloor,\lceil q_e\rceil]$, preserving +$\sum_e\beta_e q_e$ on every update, until at most one count is +fractional. Every first moment is preserved, and the expectation of +every product of two distinct counts is at most the product of their +initial values. Rounding the possible remaining fractional count with +its correct mean preserves these moment assertions. +For the resulting integer counts $Q_e$, +\[ + \Var(Q_e)\le q_e,\qquad + \E[Q_eQ_f]\le q_eq_f\quad(e\ne f). +\] +The procedure uses linearly many two-coordinate updates. +\end{lemma} + +\begin{proof} +When two coordinates are fractional, move them along the line +\[ + (q_e,q_f)\longmapsto + (q_e+t,q_f-(\beta_e/\beta_f)t) +\] +to one of the two endpoints allowed by their fixed floor and ceiling +bounds. Choose the endpoint probabilities so that the conditional +mean of $t$ is zero. At least one coordinate becomes integral, and +integral coordinates are not updated subsequently. Each coordinate +is a martingale. The product of the two updated coordinates is a +concave quadratic in $t$; a product involving exactly one updated +coordinate has unchanged conditional mean. Thus every distinct-pair +product is a supermartingale. Rounding the last fractional coordinate +with the correct conditional mean leaves all these products unchanged +in expectation. Finally, an integer random variable supported on +$\{m,m+1\}$, with mean $m+\theta$, has variance +$\theta(1-\theta)\le m+\theta$. This proves the variance assertion. +\end{proof} + +\begin{proof}[Proof of Theorem~\ref{thm:preparation}] +We first form disjoint bundles containing most of each representative's +assigned mass. We then round their deficits jointly with the openings +that supply them, reserving a small amount of budget for clipping large +deficits. After rounding inside bundles, every representative can again +receive one unit. The final part of the proof compares connection costs +at two representative-distance scales and conditions on the budget event. + +We will choose positive rational constants in the order +\[ + \delta,\quad M,\quad T_1,\quad\gamma,\quad g, +\] +with $\delta$ small, $M$ and then $T_1$ large, and $\gamma$ and then $g$ +small. All choices depend only on $\eta$; the last choice means that +every sufficiently fine even grid is permitted. During the proof we +require +\begin{equation}\label{eq:prep-parameter-basics} + 0<\delta\le\tfrac14,\qquad 0<\gamma\le1,\qquad + \theta:=\frac1{M\delta}\le\tfrac18,\qquad + \alpha:=\frac M{\delta T_1}\le\tfrac14. +\end{equation} +Set $p_0=1-\theta$ and $\beta=M/T_1=\delta\alpha$. + +\paragraph{Splitting and representatives.} +For each original facility, view its opening $y_i$ as an interval of +length $y_i$ and its assignment to client $j$ as the initial subinterval +of length $x_{ij}$. Split at the endpoints of these subintervals for all +clients. There are at most $|J|+1$ positive-mass pieces per original +facility, and every assignment now uses whole pieces. Retain the +notation $y_i$ for the mass of a piece, and let $F_j$ be the set of pieces +used by client $j$. Thus +\[ + y(F_j)=1,\qquad + V_j=\sum_{i\in F_j}y_i d(i,j),\qquad + y(W):=\sum_{i\in W}y_i. +\] +Pieces retain the distances of their original facility. They may be +treated as distinct candidates throughout the rounding; merging +co-located integral openings at the end preserves all connection +distances and can only decrease the number of opened facilities. + +Process the clients in nondecreasing order of $V_j$. Select each +remaining client $s$ as a representative and absorb every remaining +client $j$ for which $d(j,s)\le MV_j$. Write $\mathcal S$ for the set of +representatives. A client $j$ assigned to $s$ satisfies +\begin{equation}\label{eq:prep-representative-properties} + V_s\le V_j,\qquad d(j,s)\le MV_j. +\end{equation} +Distinct representatives satisfy +\[ + d(s,t)>M\max(V_s,V_t): +\] +indeed, the later one was not absorbed by the earlier one. Let $L_s$ be +the distance to a nearest other representative, taking $L_s=\infty$ +when $s$ is the only representative. Define +\[ + P_s=\{i\in F_s:d(s,i)\le\delta L_s\},\quad + p_s=y(P_s),\quad c_s=1-p_s,\quad A_s=F_s\setminus P_s, +\] +and let $U$ consist of all pieces outside $\bigcup_s P_s$. +For $L_s=\infty$, interpret $P_s=F_s$ and $c_s=0$. + +The bundles $P_s$ are disjoint. Otherwise a piece in $P_s\cap P_t$ +would imply +\[ + d(s,t)\le\delta(L_s+L_t)\le2\delta d(s,t), +\] +which is impossible. For finite $L_s$, the mass in $A_s$ has distance +at least $\delta L_s$ from $s$, so +\begin{equation}\label{eq:prep-deficit-small} + \delta L_s c_s\le V_s,\qquad + 0\le c_s\le\theta,\qquad p_s\ge p_0. +\end{equation} +These bounds also cover $V_s=0$: in that case $c_s=0$. + +\paragraph{Supplier stars and signed counts.} +For the raw opening count, give every bundle a base level of one. A +supplier star couples planned deficits in its target bundles to the +mass that will supply them. One planned count unit subtracts mass $g$ +from each target bundle and adds mass $g$ at the supplier if it lies in $U$. +A bundle supplier incurs no additional opening in this accounting. +Thus a star with $d$ targets changes the opening mass by $(1-d)g$ for +an outside supplier and by $-dg$ for a bundle supplier. + +Make a list $\mathcal E$ of stars; a star $e$ has weight $w_e$, a flag +$a_e\in\{0,1\}$ indicating an outside supplier, a target set +$D_e\subseteq\mathcal S$, and a specified supplier: +\begin{enumerate} + \item For each $i\in U$, take $w_e=y_i$, $a_e=1$, + $D_e=\{s:i\in A_s\}$, and supplier $i$. + \item For each pair $s,t$ with $y(A_s\cap P_t)>0$, take + $w_e=y(A_s\cap P_t)$, $a_e=0$, $D_e=\{s\}$, and supplier $P_t$. + Necessarily $t\ne s$. + \item For each $s$ with $c_s>0$, take $w_e=\gamma c_s$, $a_e=0$, + $D_e=\{s\}$, and use as supplier the bundle of a nearest other + representative. Such a representative exists because $c_s>0$. +\end{enumerate} +The last stars are called extra stars. The planned count of star $e$ +is $q_e=w_e/g$. With $u_e=a_e-|D_e|$, the planned raw opening mass is +\begin{equation}\label{eq:prep-raw-identity} + |\mathcal S|+\sum_e u_e w_e + =\sum_i y_i-\gamma\sum_s c_s. +\end{equation} +To see this, the flag-one weights sum to $y(U)$, while the total weight +targeting $s$ is $(1+\gamma)c_s$. + +Apply Lemma~\ref{lem:prep-weighted-rounding} independently to the +positive and negative signs of $u_e$, using weights $|u_e|$. Complete +the rounding of the positive class by rounding its possible last +fraction with the correct mean. Round the zero-coefficient counts +independently between their floor and ceiling. Leave the last +fractional count of the negative class, if any, temporarily unrounded. +All counts are then integral except possibly this one. + +If there is no remaining fraction, the resulting counts are both the +nominal counts and the actual activation counts. Otherwise, write the +remaining count as $m+\xi$, where $0<\xi<1$, and let its star have +$d=|D_e|$ targets. Its nominal count is $m+\operatorname{Bernoulli}(\xi)$. +If its flag is one, open $(m+1)g$ at its supplier, irrespective of the +nominal coin. Activate $m$ units at each target, and activate one +additional unit at every target in a random subset of size +\[ + H\in\{\lfloor d\xi\rfloor,\lceil d\xi\rceil\}, + \qquad \E H=d\xi, +\] +whose membership marginal is $\xi$ at every target. Such a subset is +obtained by choosing the indicated size with the correct mean and +then taking a uniformly shifted interval of that size in a fixed +cyclic ordering of the targets. The nominal coin can be independent +of this choice. All other stars use their integral counts for both +openings and activations. + +This modification bounds the raw opening error independently of the +number of targets. For the exceptional star its actual signed +contribution, in count units, minus its planned contribution is +\[ + a_e(m+1)-dm-H-(a_e-d)(m+\xi) + =a_e(1-\xi)+d\xi-H<2. +\] +A positive coefficient $u_e$ necessarily equals one, so the last +rounding in the positive class adds less than one further count unit. +The zero class has no signed contribution. If $\widetilde y_i$ denotes +the actual opening on $U$, and $Z_s$ denotes the total activated deficit +mass at $s$, we therefore have on every outcome +\begin{equation}\label{eq:prep-raw-bound} + \sum_{i\in U}\widetilde y_i+\sum_s(1-Z_s) + \le\sum_i y_i-\gamma\sum_s c_s+3g. +\end{equation} +Moreover, each actual outside opening dominates both its nominal +opening and every activation demand placed on it by a single target. + +\paragraph{Deficit moments and clipping.} +The signed count already controls raw opening mass. The remaining +obstacle is that an activated deficit might leave too little mass in its +own bundle. We cap that deficit and pay for the added bundle mass with +the reserve in Equation~\ref{eq:prep-raw-bound}. +For every $s$, let $A_{es}$ be the integer activation count from a star +targeting $s$. Then +\begin{equation}\label{eq:prep-deficit-moments} + \E Z_s=(1+\gamma)c_s, + \qquad \Var(Z_s)\le g(1+\gamma)c_s. +\end{equation} +Here it matters that the last negative star may depend on the preceding +random choices. Condition on the complete count vector immediately +before its exceptional rounding. At most one coordinate is +fractional, so for distinct $e,f$ targeting $s$, at least one of +$A_{es},A_{fs}$ is deterministic under this conditioning. Each has +conditional mean equal to its current count. Their conditional product +expectation is therefore the product of those two current counts. +The distinct-pair product bound in +Lemma~\ref{lem:prep-weighted-rounding} consequently still holds for +these activations. Their individual means are $q_e$, and each is +supported on $\{\lfloor q_e\rfloor,\lceil q_e\rceil\}$. Thus +\[ + \Var(Z_s) + \le g^2\sum_{e:s\in D_e}\Var(A_{es}) + \le g^2\sum_{e:s\in D_e}q_e + =g(1+\gamma)c_s, +\] +as asserted. The nominal counts also retain the same first and +distinct-pair moment bounds. + +Clip the deficit at one half: +\[ + Z_s^*=\min(Z_s,\tfrac12),\qquad p'_s=1-Z_s^*. +\] +Discard any excess activation units. This can be done in whole units +because $\Delta$ is even. By Equation~\ref{eq:prep-parameter-basics}, +$\mu_s:=\E Z_s\le1/4$. For any random variable $X$ with mean $\mu0,\\R_s,&c_s=0.\end{cases} +\] +If $V_j=0$, Equation~\ref{eq:prep-representative-properties} gives +$D=V_s=0$. All pieces of $F_s$ then have distance zero from $j$, +$c_s=0$, and $Z_s=0$ surely. The rounded bundle $P_s=F_s$ still has +one unit at distance zero. Hence the new cost is identically zero. +We may therefore assume $V_j>0$ in the remaining cost analysis. + +Suppose first that $K_s>T_1V_j$. We have $R_s\ge\delta K_s$: this is +immediate if $c_s=0$, and otherwise some assigned mass lies beyond +$\delta L_s$. Consequently $D/R_s\le\alpha<1$. This controls the +cost of exchanging the portions of $F_j$ and $F_s$ that are not shared. +Leave the shared pieces matched identically. The masses of +$F_s\setminus F_j$ and $F_j\setminus F_s$ are equal. Each piece in the +former set is at distance at most $R_s+D$ from $j$. Each piece in the +latter set is at distance at least $R_s-D$ from $j$, because the +nearest-unit assignment at $s$ uses all closer mass before stopping at +radius $R_s$. This argument remains valid at ties. Comparing the two +residual costs shows that using the old assignment $F_s$ for client $j$ +has cost at most +\begin{equation}\label{eq:prep-far-exchange} + C_j(F_s):=\sum_{i\in F_s}y_i d(j,i) + \le\frac{1+\alpha}{1-\alpha}V_j. +\end{equation} + +Serve $j$ with the feasible assignment just constructed for $s$. +For each $i\in P_s$, +\[ + \E\widetilde y_i + =y_i\E[p'_s]/p_s\le y_i/p_s. +\] +For an ordinary star, the expected retained activation mass is at most +its original weight. Thus outside suppliers do not increase their +expected cost charge. It remains to compare arbitrary new supply +inside a bundle $P_t$ with the old pieces in $A_s\cap P_t$. +This case only arises when $c_s>0$. Then $D\le\beta L_s$, and for any +old supplier piece $i\in P_t$, +\[ + d(j,i)\ge d(s,t)-D-\delta L_t + \ge(1-\beta-\delta)d(s,t). +\] +Any two pieces of $P_t$ are at distance at most $2\delta L_t$. +Because $L_t\le d(s,t)$ and $1-\beta-\delta>0$, every possible new +supplier piece has distance to $j$ at most +\[ + \kappa\,d(j,i),\qquad + \kappa:=1+\frac{2\delta}{1-\beta-\delta}. +\] +The comparison holds for every old piece, so it also holds relative to +the weighted mean distance of the old pieces aggregated by that star. +Multiplying by its expected retained activation mass bounds its +expected cost by $\kappa$ times its old cost contribution. + +Finally, for the extra star at $s$, its supplier $P_t$ satisfies +$d(s,t)=L_s$ and $L_t\le L_s$. Every supply position is therefore at +distance at most $(\beta+1+\delta)L_s$ from $j$. By +Equation~\ref{eq:prep-deficit-small}, its expected cost is at most +\[ + (\beta+1+\delta)\gamma c_s L_s + \le\frac{(\beta+1+\delta)\gamma}{\delta}V_j. +\] +There is no extra star when $c_s=0$. +Combining these bounds with Equation~\ref{eq:prep-far-exchange}, the +unconditional expected nearest-unit cost in this case is at most +$B_{\mathrm{far}}V_j$, where +\begin{equation}\label{eq:prep-far-factor} + B_{\mathrm{far}} + :=\max\{p_0^{-1},\kappa\}\frac{1+\alpha}{1-\alpha} + +\frac{(\beta+1+\delta)\gamma}{\delta}. +\end{equation} + +\paragraph{Cost comparison at a nearby representative scale.} +Now suppose $K_s\le T_1V_j$. Consider the mass on the old set $F_j$, +using nominal rounded masses on $U$ and actual rounded masses in the +bundles. This mass is available in the actual opening vector. Its +unconditional expected cost is at most $V_j/p_0$: nominal outside +openings have mean $y_i$, and a bundle piece has expected mass at most +$y_i/p_s\le y_i/p_0$. + +We bound the expected shortfall of this available mass from one. +Write $M_U$ for its nominal mass on $U\cap F_j$. The first and pair +moments of the nominal counts imply +\begin{equation}\label{eq:prep-outside-noise} + \E M_U=y(U\cap F_j),\qquad + \Var(M_U)\le g\,y(U\cap F_j)\le g. +\end{equation} +For a fixed complete star outcome, put +\[ + a_t=y(F_j\cap P_t),\qquad + \overline M_P=\sum_t a_t p'_t/p_t, +\] +and write $M_P$ for the actual rounded mass on these bundle pieces. +Conditional on the star outcome, bundle rounding has nonpositive pair +covariances within each bundle and is independent between bundles. +The variance of a single grid-rounded mass of conditional mean +$\overline y_i$ is at most $g\overline y_i$. Therefore +\begin{equation}\label{eq:prep-conditional-noise} + \E[M_P\mid\text{star outcome}]=\overline M_P,\qquad + \Var(M_P\mid\text{star outcome}) + \le g\overline M_P\le g/p_0. +\end{equation} + +Clipping at $1/2$ cannot increase distance from $c_t\le1/2$, so +Equation~\ref{eq:prep-deficit-moments} gives +\begin{align} + \E|p'_t-p_t| + &=\E|c_t-Z_t^*|\notag\\ + &\le\E|c_t-Z_t| + \le\gamma c_t+\sqrt{g(1+\gamma)c_t}. + \label{eq:prep-rescaling-noise} +\end{align} +Since $\sum_t a_t\le1$, it follows that +\[ + \E\left|\overline M_P-\sum_t a_t\right| + \le\sum_t\frac{a_t}{p_t}\E|p'_t-p_t| + \le\frac{\gamma+\sqrt{g(1+\gamma)}}{p_0}. +\] +Use the identity $y(U\cap F_j)+\sum_t a_t=1$, the triangle inequality, +and the second-moment bounds in +Equations~\ref{eq:prep-outside-noise}--\ref{eq:prep-conditional-noise}. +The expected shortfall is at most +\begin{equation}\label{eq:prep-shortfall} + \E(1-M_U-M_P)_+\le H, + \qquad + H:=\sqrt g+\sqrt{g/p_0} + +\frac{\gamma+\sqrt{g(1+\gamma)}}{p_0}. +\end{equation} +In particular no independence between outside counts and rescaled +bundle masses is being assumed. + +Trim any surplus mass on $F_j$, which cannot increase its cost. To +fill a remaining shortfall, there is always one unit of available +opening within distance +\[ + QV_j,\qquad Q:=M+(1+\delta)T_1, +\] +from $j$. Indeed, if $c_s=0$, there are no activations at $s$ and $P_s$ +has exactly one unit, all at distance at most +$D+R_s\le(M+T_1)V_j$. If $c_s>0$, use $P_s$ and the bundle $P_t$ of a +nearest other representative. They are disjoint and each has at +least half a unit. Their supply lies within distance +$D+(1+\delta)L_s\le QV_j$. +This supply can fill the shortfall even if some of it is already used: +if the current assignment has mass $1-q$, a ball containing at least +one opening unit contains at least $q$ unused units. Allocate those +units to finish the assignment. By Equation~\ref{eq:prep-shortfall}, +the unconditional expected nearest-unit cost is at most +$B_{\mathrm{near}}V_j$, where +\begin{equation}\label{eq:prep-near-factor} + B_{\mathrm{near}}:=p_0^{-1}+QH. +\end{equation} + +\paragraph{Choosing parameters and enforcing the budget.} +Equations~\ref{eq:prep-far-factor} and~\ref{eq:prep-near-factor} give +explicit bounds that approach one in the stated order of parameter +choices. More precisely, first choose $\delta$ small to make the +limiting value of $\kappa$ close to one. Next choose $M$ large so that +$p_0^{-1}$ is close to one, and then choose $T_1$ large so that $\alpha$ +and $\beta$ are small. With these constants fixed, choose $\gamma$ +small enough to control both the last term of $B_{\mathrm{far}}$ and +the term $Q\gamma/p_0$ in $B_{\mathrm{near}}$. Finally choose $g$ small +enough to control all square-root terms and $\rho$ in +Equation~\ref{eq:prep-budget-probability}. All constants may be +rational, and all sufficiently small $g=1/\Delta$ with even $\Delta$ +satisfy the resulting strict bounds. In particular, arrange +\[ + \rho<\tfrac12,\qquad + \frac{\max\{B_{\mathrm{far}},B_{\mathrm{near}}\}}{1-\rho} + \le1+\eta. +\] + +Reject a star-rounding outcome outside $\mathcal G$ and repeat. +For the purpose of the preceding unconditional analysis, the subsequent +bundle rounding can be imagined on every outcome. Let +$\widetilde V_j$ denote the resulting nearest-unit cost. Its +nonnegativity, rather than any independence from $\mathcal G$, gives +\begin{equation}\label{eq:prep-cost-final} + \E[\widetilde V_j\mid\mathcal G] + \le\frac{\E\widetilde V_j}{\Prob(\mathcal G)} + \le(1+\eta)V_j. +\end{equation} +The already established supply and opening bounds hold on every +accepted outcome. + +The split representation, star list, and number of pair updates are +polynomial in the input size. Pair updates and subset choices use +rational probabilities and rational linear operations. Their +encodings have polynomial bit length: each weighted update introduces +only ratios of integer weights bounded by $|\mathcal S|$, and there +are polynomially many updates; the subsequent rescalings divide by +the rational masses $p_s$. Exact rational sampling has expected +polynomial running time. The acceptance probability is at least +one half, so rejection multiplies the expected running time by at +most two. This proves the theorem. +\end{proof} diff --git a/preprints/The-Approximation-Threshold-for-Metric-k-Median-September-24-2026/build/sections/rounding.tex b/preprints/The-Approximation-Threshold-for-Metric-k-Median-September-24-2026/build/sections/rounding.tex new file mode 100644 index 0000000000000000000000000000000000000000..6aff02d7014d946ba60f98988b063e1e27d5b3b3 --- /dev/null +++ b/preprints/The-Approximation-Threshold-for-Metric-k-Median-September-24-2026/build/sections/rounding.tex @@ -0,0 +1,871 @@ +\section{Tapered iterative rounding}\label{sec:rounding} + +This section rounds a sufficiently fine grid solution. The graph used for +rounding gives every active copy one unit of possible incoming erasers. +Its distance-dependent taper makes the opening that accompanies an erasure +comparable to the client's surviving candidates. A potential that retains +some cheap candidates after an opening converts this comparison into an +application of Theorem~\ref{thm:finite-inequality}. + +\begin{theorem}[Rounding a grid solution]\label{thm:grid-rounding} +For every fixed $\tau>0$ and all sufficiently large constant integers +$\Delta$, put $g=1/\Delta$. Suppose a finite rational metric instance has openings +$\widetilde y_i\in[0,1]\cap g\mathbb Z$ of total mass $M\ge1$. +For a client $j$, let $V_{j,0}$ be the cost of its nearest unit of this mass. +There is a constant $C=C(\tau,\Delta)$ and a randomized algorithm of +expected polynomial running time that outputs a set $O$ of facilities with +\[ + |O|\le M+C, + \qquad + \E d(j,O)\le(1+2/e+\tau)V_{j,0}\quad\text{for every client }j. +\] +All parameters and the additive constant are independent of the instance. +In particular, an initial mass at most $k+c_0$, for a fixed constant +$c_0$, gives at most $k+c_0+C$ facilities. +The deterministic implementation of the resulting total-cost guarantee +is given in Section~\ref{sec:derandomization}. +\end{theorem} + +We prove the theorem using the functions $A,P,B,m$ defined in the preceding +section. Thus $B(x)=2P(x)=A'(x)$ and +$m(x)=A(x)-1-xB(x)$. We use, in particular, +\begin{equation}\label{eq:rnd-function-identities} + \begin{gathered} + 00,\qquad m(x)\ge0,\\ + (1+x)A(x)+xB(x)-1=3x\qquad(0\le x\le1). + \end{gathered} +\end{equation} +The identities and derivative bounds used below follow directly from +the defining integrals. Constants implicit in $O_\Delta(\cdot)$ may +depend on $\Delta$, but not on the instance, the round, or the subsequent +heavy-source threshold. + +\subsection{The graph and the physical process} + +The source-opening and neighbor-erasure operation follows the framework +of Byrka et al.~\cite[Definitions~3.2 and~5.1]{ByrkaEtAl2026}. +We specify the graph, taper, and batches here because their precise +properties will be used in the sub-two drift analysis. + +Represent each opening by active copies of mass $g$, grouped by facility. +Each group has at most $\Delta$ active copies. Co-located facilities +introduced by earlier splitting may remain distinct groups throughout +the process; they can be merged in the output. A group always means one +such labeled facility group, rather than an entire zero-distance class. +An integral group has exactly $\Delta$ copies. + +Fix a small time step $\ell>0$ and a large time horizon $\Lambda$. +There are $T=\lceil\Lambda/\ell\rceil$ rounds. At the start of round +$r\in\{0,\ldots,T-1\}$ let $\xi=r\ell$ and $x=e^{-\xi}$. +For that round choose a fixed rational number $\widehat P$ satisfying +\begin{equation}\label{eq:rnd-taper-approximation} + 0<\widehat P\le\tfrac12, + \qquad |\widehat P-P(x)|\le\zeta, + \qquad 0<\zeta<\tfrac14. +\end{equation} +The finitely many rational approximations may be fixed in advance. +The analytic value $x$ is used only in the analysis. + +For each active target copy $i$, independently of the other targets +conditional on the current state, choose a set $\mathcal H_i$ of exactly +$\Delta$ active source copies. If the colocated active mass is below one, +choose $N_i>0$ minimally so that +\begin{equation}\label{eq:rnd-edge-marginals} + q_{hi}=\min\left\{1,\frac{(1-d(i,h)/N_i)_+}{\widehat P}\right\}, + \qquad \sum_h gq_{hi}=1, +\end{equation} +and give $\mathcal H_i$ these marginals. Such a radius exists: the sum +is continuous and nondecreasing in the positive radius, starts below +one, and reaches the total active mass at a finite radius. This last +observation also handles total active mass exactly one. The equation +is piecewise linear in $1/N_i$. Fixed-sum rounding realizes the +marginals since their unscaled sum is the integer $\Delta$. + +If the colocated active mass is already at least one, set $N_i=0$ and +choose $\Delta$ colocated copies, including every active copy in the +target's own group. Any fixed such choice suffices in this case; denote +its marginals by $q_{hi}$ as well. In either case every own-group copy +belongs to $\mathcal H_i$ with certainty, and +\begin{equation}\label{eq:rnd-taper-comparison} + \begin{aligned} + N_i(1-\widehat Pq_{hi})&\le d(i,h)&&\text{if }q_{hi}<1,\\ + d(i,h)&\le N_i(1-\widehat Pq_{hi})&&\text{if }q_{hi}>0. + \end{aligned} +\end{equation} +For $N_i>0$ these statements follow from the three cases of the clipped +linear formula; for $N_i=0$ all sources of positive marginal are +colocated. Define the out-set of source $h$ by +\[ + E_h=\{i:h\in\mathcal H_i\}. +\] +A full update at $h$ erases $E_h$ and opens the facility of $h$. + +We use balanced and biased unbalanced batches, with forced openings for +large updates, as in the approach of Byrka et +al.~\cite[Algorithms~4--5 and Section~5.4]{ByrkaEtAl2026}. +The exact batch and chunk rules below are part of our construction. +To implement many updates while controlling the total opening mass, put +\[ + u_h=1-g|E_h|, + \qquad + \mathcal Z=\{h:u_h=0\},\quad + \mathcal P=\{h:u_h>0\},\quad + \mathcal N=\{h:u_h<0\}. +\] +Every target has $\Delta$ incoming sources, so $\sum_hu_h=0$. Write +\begin{equation}\label{eq:rnd-balance-size} + A_* =\sum_{h\in\mathcal P}u_h + =\sum_{h\in\mathcal N}|u_h|, + \qquad + |\mathcal P\cup\mathcal N|\le 2A_*/g. +\end{equation} +The last inequality uses that each nonzero $u_h$ is a nonzero multiple +of $g$. Choose a further small constant $b_*>0$, to be fixed last, and put +\[ + B_* =\max\{1,b_*A_*\}. +\] +A source is \emph{heavy} if $g|E_h|>B_*$. Force open the facilities of +all heavy sources, whether or not any random update is triggered. + +All random choices and erasure sets in a round refer to its pre-round +graph and active copies. Set $p_*=2\ell g$, and fix a small bias +$0<\phi\le1$. Flip an independent fair coin to choose one of the following +two types of round. +\begin{enumerate} +\item In a balanced round, propose each $h\in\mathcal Z$ independently +with probability $p_*$. Select a proposed source only if no other +proposed source has an out-set intersecting its out-set. +\item In an unbalanced round, select sources of $\mathcal P$ +independently with probability $p_*$, and nonheavy sources of +$\mathcal N$ independently with probability $p_*(1+\phi)$. +For each heavy source, partition its out-set into chunks of mass at +most $B_*+1$ and trigger each chunk independently with probability +$p_*(1+\phi)$. A triggered chunk erases just that chunk; its source +facility is already included among the forced openings. +\end{enumerate} +All probabilities are at most one for the parameter choices below. +Take the union of the selected full out-sets and triggered chunks and +erase it. Then reset each forced or selected facility to $\Delta$ +active copies. Copies removed by either erasure or reset are distinct +from the regenerated copies. + +Two invariants are useful. First, an integral target group's own +$\Delta$ copies exhaust each of its incoming sets. Therefore an +erasure of an integral group is caused by one of its own sources; that +source either is selected or is heavy. The group is reset in either +case. Thus integral facilities persist, including when other facilities +are colocated with them. Second, every erasure has an associated +selected or forced opening. Hence total active mass always stays at +least one. The construction of the next round's graph is consequently +well defined. + +\subsection{The opening-count bound} + +We first bound the number of facilities independently of connection cost. +The next lemma controls the total active mass with high probability; +completion will later turn that mass bound into a facility-count bound. + +\begin{lemma}[Constant additive count]\label{lem:rnd-count} +Fix $g,\phi,\ell,T$ as above, with $\ell$ sufficiently small as a +function of $\Delta$ and $\phi$. For every fixed $\eta>0$, one can +choose $b_*>0$ and a constant $C$ such that the final active mass is at +most $M+C$ with probability at least $1-\eta$. The bound holds +uniformly over all starting states of total mass $M$. +\end{lemma} + +\begin{proof} +Condition on the entire pre-round state and its realized graph. +Write $A=A_*$, $B=B_*$, $p=p_*$, and $m_h=g|E_h|$ for this proof. +If $n$ is the number of active copies, then $\sum_hm_h=n$. +Each source in the zero class has $m_h=1$, so removing that class gives +\[ + \sum_{h\in\mathcal P\cup\mathcal N}m_h + =|\mathcal P\cup\mathcal N|. +\] +Every heavy source lies in $\mathcal N$ because $B\ge1$. If $H$ is +their number, then +\begin{equation}\label{eq:rnd-heavy-count} + H\le\frac{|\mathcal P\cup\mathcal N|}{B} + \le\frac{2A}{g\max\{1,b_*A\}} + \le\frac{2}{gb_*}. +\end{equation} +Thus forced resets add at most a constant mass. If $A=0$, there are +no heavy sources and the same conclusions hold directly. + +In a balanced round, the selected erasure sets are disjoint and each +has mass one. Opening one facility for each selected source adds at +most one unit per set, so there is no mass increase except the forced +allowance. + +For an unbalanced round index the ordinary selections and heavy +chunks by actions $e$. Let $I_e$ be the trigger indicator, $S_e$ its +erased set, $m_e=g|S_e|$, and $a_e=1$ for an ordinary selection or $0$ +for a chunk. Resetting facilities adds at most one unit per ordinary +selection in addition to the forced allowance. More explicitly, the +change in active mass is at most +\[ + H+\sum_e a_e I_e + -g\left|\bigcup_{e:I_e=1}S_e\right|. +\] +The union is a set of pre-round copy identities. A reset may remove +additional old copies outside this union, which only decreases the +change. Counting a selected or forced group more than once likewise +only enlarges the upper bound. Truncated inclusion--exclusion bounds +the erased union from below. Consequently the mass increase, apart +from forcing, is at most +\begin{equation}\label{eq:rnd-count-statistic} + Z_* =\sum_e I_e(a_e-m_e) + +\sum_{e1/b_*$, then $B=b_*A>1$. Combining +\eqref{eq:rnd-count-mean} and \eqref{eq:rnd-count-variance} yields +\begin{equation}\label{eq:rnd-count-failure} + \Prob(Z_*>0) + \le O\!\left(\frac{\Delta^2b_*}{p\phi^2g}\right). +\end{equation} +If instead $A\le1/b_*$, there are at most $2/(gb_*)$ ordinary +sources in the nonzero classes. The actual positive mass change is +then bounded deterministically by this constant plus the forced +allowance; a deterministic bound on the quadratic statistic is +unnecessary. + +Choose $b_*$ sufficiently small that the union bound for +\eqref{eq:rnd-count-failure} over the $T$ rounds is at most $\eta$. +On the complementary event the total increase is at most, for example, +$4T/(gb_*)$. This proves the lemma with a constant $C$. +The proof was conditional on every state and graph, so the same union +bound applies to the adaptive sequence of rounds. +\end{proof} + +For later use, the mean and variance of $Z_*$ depend only on indicator +moments through degree four, since $Z_*$ is a quadratic polynomial. +The probability bounds for balanced selections below need only +first and second moments of their proposal indicators. These are +the moment requirements used in Section~\ref{sec:derandomization}. + +\subsection{A persistent backup and final completion} + +We next establish a persistent bound on each client's eventual +connection distance, including trajectories on which none of its assigned +copies opens. This is the backup-and-completion role in iterative +rounding~\cite[Section~3.3 and Lemmas~6.3 and~6.5]{ByrkaEtAl2026}; +the following argument proves the bound for our tapered graph and reset +rule. + +Fix a client for the remainder of the analysis. Let $S_0$ be the +$\Delta$ copies in its initial nearest unit, chosen using any fixed +tie-breaking rule. Write $d_i$ for its distance to copy $i$, +\[ + V_0=\sum_{i\in S_0}gd_i, + \qquad R=\max_{i\in S_0}d_i. +\] +In particular $R\le\Delta V_0$. + +\begin{lemma}[Backup and completion]\label{lem:rnd-completion} +Once a copy of $S_0$ is physically removed, an integral facility within +distance $5R$ of the client exists and persists to the end. +At the end of the process one can retain all integral facilities and +open further facilities, using no more than the total final active +mass, so that every client is connected within its own distance $5R$. +\end{lemma} + +\begin{proof} +Consider the round of the first physical removal from $S_0$. Just +before that round all of $S_0$ is still active. For $i,t\in S_0$, +$d(i,t)\le d_i+d_t\le2R$. At radius $2R/(1-\widehat P)$, the +$\Delta$ copies of $S_0$ all have incoming marginal one for target +$i$. Minimality of the taper radius gives +\[ + N_i\le2R/(1-\widehat P)\le4R. +\] +If $R=0$, the zero-radius rule gives the same conclusion directly. +An erasing source $h$ has $d(i,h)\le N_i$ by +\eqref{eq:rnd-taper-comparison}, and hence $d_h\le5R$. +Its facility is either selected or forced open. A removal by reset +of $i$'s own group instead opens a facility within $R$. +The persistence invariant proves the first assertion. + +At the end, consider the clients not already served within $5R$ by +an integral facility. The first assertion implies that every such +client still has all its original $S_0$ copies. Its radius-$R$ +neighborhood of active copies therefore has mass at least one. +That neighborhood contains no integral group, since such a group +would itself provide the required connection. + +Process these neighborhoods in nondecreasing order of their radii. +Accept a neighborhood if it is disjoint from all previously accepted +ones and open any facility in each accepted neighborhood. These +disjoint neighborhoods each contain at least one unit of mass and +contain no integral group. Their openings, together with the retained +integral groups, therefore fit within the final active mass. +If a radius-$R$ neighborhood was skipped, it meets an accepted +radius-$R'$ neighborhood with $R'\le R$. Through an intersection +point, the distance to the accepted neighborhood's chosen facility +is at most $R+2R'\le3R$. This proves the asserted $5R$ bound for +every client, also when some radii are zero. +\end{proof} + +At any intermediate time define $b$ to be the distance from the fixed +client to its nearest integral facility, capped at $5R$; when no +integral facility exists use $b=5R$. By persistence and +Lemma~\ref{lem:rnd-completion}, the eventual completed solution has +connection distance at most $b$ at every time on the trajectory. +The cap need not already be attained by an open facility. It is a +valid bound on the eventual connection because completion supplies +$5R$ on every trajectory. The quantity $b$ never increases. + +\subsection{The potential and its single-action bounds} + +For analysis only, track an alive set $S\subseteq S_0$ of original +copies. A physically removed copy leaves $S$ permanently, even when +its facility is reset with regenerated copies. We also permit the +additional discards specified below. Put +\[ + y=g|S|,\qquad V=\sum_{i\in S}gd_i. +\] +At elapsed time $\xi$, with $x=e^{-\xi}$, define +\begin{equation}\label{eq:rnd-potential} + f=aV+(1-y)b, + \qquad + a=A(x)+B(x)(y-x)=1+m(x)+B(x)y. +\end{equation} +The coefficient $a$ is the value at $y$ of the tangent line to the +concave function $A$ at $x$; $x$ is a deterministic clock, while $y$ +is the realized alive mass. Its affine dependence on $y$ makes the +erasure correction quadratic in removed mass and cost, so the graph +average of the ideal bound below requires only first and second edge +moments. +In particular $a\ge1$. The initial potential is +\begin{equation}\label{eq:rnd-initial-potential} + f_0=A(1)V_0=(1+2/e)V_0. +\end{equation} +If $C_j$ denotes the actual connection distance in the completed +output, the preceding cap property gives +\begin{equation}\label{eq:rnd-terminal-correction} + C_j-f_T\le y_Tb_T-a_TV_T\le5Ry_T. +\end{equation} +The potential and every single-action bound below are $O(R)$ uniformly +over time: $V\le R$, $0\le y\le1$, $b\le5R$, and $a$ is bounded. + +We first freeze time and the graph. Write $B=B(x)$ and +$b_h=\min\{b,d_h\}$. For a source $h\notin S$, let +\[ + \delta_h=g|E_h\cap S|, + \qquad Y_h=\sum_{i\in E_h\cap S}gd_i. +\] +If only this full source update affects $S$, its removal changes the +coefficient to $a-B\delta_h$ and makes the new cap at most $b_h$. +Its potential change is consequently at most +\begin{align*} + &(a-B\delta_h)(V-Y_h)+(1-y+\delta_h)b_h-f\\ + &\quad=\delta_hb_h-aY_h-B\delta_h(V-Y_h) + +(1-y)(b_h-b)\\ + &\quad\le\delta_hb_h-aY_h-B\delta_h(V-Y_h). +\end{align*} +The same statement holds when $h$ is not alive but its group contains +alive copies: all those copies belong to $E_h$ by the own-group edges, +so its reset creates no unaccounted removal. + +If an alive source $h\in S$ opens, retain among the physical survivors +only copies satisfying $ad_i0. + \end{gathered} +\] +Thus forcing supplies a fixed fraction of $Q$ as free savings. + +After force accounting, the remaining alive cost is at most $V$ and +the cap is at most $V/g$. Any further frozen-time increase has +magnitude $O_\Delta(V)$, and a change to the alive set still has +probability $O_\Delta(\ell)$. If the alive set does not change, the +cap only improves. Subsequent actions and time advance therefore +have expected increase at most $O_\Delta(\ell Q)$. + +For comparison with the pre-force ideal expression, we also have +\begin{equation}\label{eq:rnd-force-generator-size} + |\dot f|+|\mathcal B|\le O_\Delta(Q). +\end{equation} +For inside sources, $d_h\le V/g$. If $y<1$, the grid gives +$1-y\ge g$, so an external cap charge is bounded using $b\le Q/g$. +If $y=1$, the alive set itself is a full unit; its maximum distance +is at most $V/g$. The radius argument from +Lemma~\ref{lem:rnd-completion} then bounds any source erasing an alive +copy within $5V/g$ of the client. Only $O_\Delta(1)$ sources have +nonzero contributions. These observations prove +\eqref{eq:rnd-force-generator-size}. + +Choose $\ell$ sufficiently small in terms of these constants and +$c_\Delta$. The savings in \eqref{eq:rnd-force-savings} absorb both +the subsequent expected increase and the possible negative quantity +$\ell(\dot f+\mathcal B)$. This proves +\eqref{eq:rnd-batch-drift} also in the forced case, without an +additional error term. If $Q=0$, all the quantities in this +comparison vanish, so no division by $Q$ is needed. More explicitly, +if $V=0$ then $b'=0$, and the force savings remove $(1-y)b$; when +also $y=1$, relevant source distances and all generator terms are +zero. The argument is uniform in the later choice of $b_*$. +\end{proof} + +The order of conditioning in Lemma~\ref{lem:rnd-batch} is essential. +Whether a group is forced depends on the graph. We have proved the +drift comparison for \emph{every} realized graph, in both force cases; +we will now average that comparison over the original graph law, +without conditioning the law on a forcing event. + +\subsection{Averaging the graph and applying the finite inequality} + +The physical batch has now been compared with the ideal generator for +every realized graph. We average that generator using edge marginals and +cross-target second moments, then verify the hypotheses of the abstract +finite comparison inequality. The resulting bound is the cost estimate +that will be summed over rounds. + +\begin{lemma}[Averaged ideal drift]\label{lem:rnd-average} +Conditional on the pre-round history, the ideal expression satisfies +\begin{equation}\label{eq:rnd-average-drift} + \dot f+\E_{\mathrm{graph}}\mathcal B + \le BgV+12\zeta V. +\end{equation} +\end{lemma} + +\begin{proof} +The pre-round quantities $S,y,V,b,x$ are fixed throughout this proof. +Define +\begin{equation}\label{eq:rnd-graph-statistics} + \begin{aligned} + w_i&=\sum_{t\in S}gq_{ti},\\ + K_{it}&=\sum_{h\notin S}gq_{hi}q_{ht},\\ + s_i&=\frac{\sum_{h\notin S}gq_{hi}^2}{1-w_i} + \quad\text{if }w_i<1, + \qquad s_i=0\quad\text{if }w_i=1. + \end{aligned} +\end{equation} +We have $0\le w_i\le y\le1$ and $0\le s_i\le1$. +The inequality $2uv\le u^2+v^2$ gives +\begin{equation}\label{eq:rnd-kernel-bound} + K_{it}\le\tfrac12\bigl[s_i(1-w_i)+s_t(1-w_t)\bigr]. +\end{equation} +No symmetry of the directed marginals $q_{hi}$ is assumed. + +To average the positive quadratic erasure term, put +$X_{hi}=\ind_{\{h\in\mathcal H_i\}}$. Choices for distinct targets +are independent. Expanding the two removed quantities consequently +gives the exact formula +\begin{align} + \E_{\mathrm{graph}}\sum_{h\notin S}g\delta_hY_h + &=\sum_{i,t\in S}g^2K_{it}d_t + +\sum_{i\in S}g^2d_i\sum_{h\notin S}gq_{hi}(1-q_{hi})\notag\\ + &\le\sum_{i,t\in S}g^2K_{it}d_t+gV. + \label{eq:rnd-diagonal-correction} +\end{align} +Here $\sum_hgq_{hi}=1$ bounds the diagonal correction. + +For the cap contribution, the external incoming mass at $i$ is +$1-w_i=(1-y)+(y-w_i)$. Split it in these proportions. On its first +part use $b_h\le b$, and on its second part use, whenever $q_{hi}>0$, +\[ + b_h\le d_h\le d_i+d(i,h) + \le d_i+(1-\widehat Pq_{hi})N_i. +\] +This yields +\begin{equation}\label{eq:rnd-backup-average} + \sum_{h\notin S}gq_{hi}b_h + \le(1-y)b+(y-w_i)\bigl[d_i+(1-\widehat Ps_i)N_i\bigr]. +\end{equation} +If $w_i=1$, then $y=1$ and both sides are zero, so the convention +for $s_i$ requires no division in that case. + +For completeness, averaging \eqref{eq:rnd-ideal-generator} before +using the cap bound gives, apart from the additive term $BgV$, +\begin{align*} + &[1-(1+y)a-By]V-y(1-y)b\\ + &\quad+\sum_{i\in S}g\sum_{h\notin S}gq_{hi}b_h + +a\sum_{i\in S}gw_id_i+BV\sum_{i\in S}gw_i + +B\sum_{i,t\in S}g^2K_{it}d_t\\ + &\quad-\sum_{i,t\in S}g^2(1-q_{ti})(d_t-ad_i)_+. +\end{align*} +The first term of \eqref{eq:rnd-backup-average}, summed over $i$, +cancels $-y(1-y)b$. The coefficient of $V$ after adding +\eqref{eq:rnd-time-derivative} simplifies using +\begin{equation}\label{eq:rnd-coefficient-identity} + (1+y)a+yB+x(y-x)B'-1=3y+B(y-x)^2. +\end{equation} +Indeed \eqref{eq:rnd-function-identities} and its derivative give +\[ + (1+x)A+xB-1=3x, + \qquad A+(2+x)B+xB'=3. +\] +Expanding the left side of \eqref{eq:rnd-coefficient-identity} in +$y-x$ has these as its constant and linear coefficients, and has +quadratic coefficient $B$. + +It follows that $\dot f+\E_{\mathrm{graph}}\mathcal B-BgV$ is at most +\begin{equation}\label{eq:rnd-expanded-drift} +\begin{aligned} + &-[3y+B(y-x)^2]V\\ + &\quad+\sum_{i\in S}g\Bigg\{ + (y-w_i)\bigl[d_i+(1-\widehat Ps_i)N_i\bigr]+aw_id_i + +BVw_i+B\sum_{t\in S}gK_{it}d_t\\ + &\hspace{43mm} + -\sum_{t\in S}g(1-q_{ti})(d_t-ad_i)_+\Bigg\}. +\end{aligned} +\end{equation} + +If $y=0$, the ideal generator and time derivative are zero. If +$y>0$ but $V=0$, then every $d_i$ with $i\in S$ is zero. For any +$i\in S$ with $N_i>0$, all alive sources are colocated with $i$, so +$q_{ti}=1$ for every $t\in S$ and $w_i=y$. If $N_i=0$, its radius +term already vanishes. Every term of +\eqref{eq:rnd-expanded-drift} is then zero. Thus the lemma holds +in these degenerate cases without normalization. + +Assume now $V>0$, and set +\begin{equation}\label{eq:rnd-normalization} + \mu=V/y,\qquad + D_i=d_i/\mu,\qquad + r_i=w_i/y,\qquad + R_i=(1-2\zeta)N_i/\mu,\qquad C_i=1-Ps_i. +\end{equation} +Give each index $i\in S$ probability $g/y$. Expectations over these +indices, written $\E_i$ and $\E_t$, satisfy +$\E_iD_i=1$ and $r_i=\E_tq_{ti}$. +For $q=q_{ti}<1$, the taper approximation implies +\[ + (1-2\zeta)(1-Pq)\le1-\widehat Pq, +\] +because the difference is at least +$2\zeta(1-Pq)-\zeta q\ge2\zeta(1-q)\ge0$. +The triangle inequality and \eqref{eq:rnd-taper-comparison} give +\begin{equation}\label{eq:rnd-normalized-comparison} + R_i(1-Pq_{ti})\le D_i+D_t\qquad(q_{ti}<1). +\end{equation} +These are precisely the radius hypotheses of +Theorem~\ref{thm:finite-inequality}. + +The error in changing the taper coefficient is controlled without +any bound on individual normalized radii. Multiply the first +inequality in \eqref{eq:rnd-taper-comparison} by +$(g/y)(1-q_{ti})$, sum over $t\in S$, and use +$1-\widehat Pq_{ti}\ge1/2$ and $d(i,t)\le d_i+d_t$. This proves +\begin{equation}\label{eq:rnd-radius-error-control} + (1-r_i)N_i/\mu\le2(D_i+1). +\end{equation} +Since +\[ + 0\le(1-\widehat Ps_i)-(1-2\zeta)(1-Ps_i)\le3\zeta, +\] +the replacement of +$(1-r_i)(1-\widehat Ps_i)N_i/\mu$ by +$C_i(1-r_i)R_i$ costs at most $6\zeta(D_i+1)$, whose expectation +is at most $12\zeta$. + +Divide \eqref{eq:rnd-expanded-drift} by $yV=y^2\mu$ and put +$\Theta=P(y-x)^2/y$. By \eqref{eq:rnd-kernel-bound}, the normalized +kernel term is at most +\[ + B\E_{i,t}K_{it}D_t + \le P\E_i\bigl[s_i(1-yr_i)(1+D_i)\bigr]. +\] +After the radius replacement, the entire normalized expression is +therefore at most +\begin{align*} + -3-2\Theta+\E_i\Big[& + (1-r_i)(D_i+C_iR_i)+ar_iD_i+2Pyr_i\\ + &+Ps_i(1-yr_i)(1+D_i) + -\E_t(1-q_{ti})(D_t-aD_i)_+\Big]+12\zeta. +\end{align*} +Using $a=1+m+2Py$ and $\E_iD_i=1$, this is exactly +\begin{align*} + \E_i\Big[&C_i(1-r_i)R_i + +r_i\{mD_i+(2-s_i)Py(D_i+1)\}-C_i(D_i+1)\\ + &-\E_t(1-q_{ti})(D_t-(1+m+2Py)D_i)_+\Big] + -2\Theta+12\zeta. +\end{align*} +Theorem~\ref{thm:finite-inequality} bounds the expectation by +$2\Theta$. Multiplication by $yV$, with $y\le1$, proves +\[ + \dot f+\E_{\mathrm{graph}}\mathcal B + \le BgV+12\zeta yV\le BgV+12\zeta V, +\] +as claimed. +\end{proof} + +\subsection{Survival, accumulated error, and parameter choices} + +\begin{lemma}[Decay of the alive mass]\label{lem:rnd-survival} +For sufficiently small $\ell$ depending only on $\Delta$, every +client's analysis process satisfies +\begin{equation}\label{eq:rnd-survival} + \E y_T\le(1-\ell/2)^T\le e^{-\Lambda/2}. +\end{equation} +\end{lemma} + +\begin{proof} +Condition on a history, a graph, and a currently alive target. +If its group is forced, the original target identity is removed +certainly. Otherwise its $\Delta$ incoming source copies correspond +to $\Delta$ distinct possible actions: each ordinary source gives +its selection action, and each heavy source gives the unique chunk +containing this target. The marginal and pair estimates in the +proof of Lemma~\ref{lem:rnd-batch} give an erasure probability at least +\[ + \Delta\ell g\bigl(1-O_\Delta(\ell)\bigr) + -O(\Delta^2\ell^2g^2)\ge\ell/2. +\] +Here the lower bound is the union bound with pair intersections +subtracted. A simultaneous reset does not restore the original +copy identity, and additional analysis discards only reduce survival. +The conditional expected surviving mass is thus at most +$(1-\ell/2)y$. Iteration, starting from $y_0=1$, and $T\ell\ge\Lambda$ +prove the lemma. +\end{proof} + +We finish the proof of Theorem~\ref{thm:grid-rounding}. +First run the process without rejecting count failures, and complete +every output by Lemma~\ref{lem:rnd-completion}. Average +Lemma~\ref{lem:rnd-batch} over the graph and use +Lemma~\ref{lem:rnd-average}. Since $V\le V_0$, $B\le1$, and +$R\le\Delta V_0$, summing over $T$ rounds gives +\[ + \E f_T\le + \left[1+2/e+(\Lambda+1) + \{g+12\zeta+C_\Delta\Delta(\phi+\ell)\}\right]V_0 +\] +for a constant $C_\Delta$, provided $\ell\le1$. +Equations~\eqref{eq:rnd-terminal-correction} and +\eqref{eq:rnd-survival} consequently imply the unconditional estimate +\begin{equation}\label{eq:rnd-total-error} + \E C_j\le + \left[1+2/e+5\Delta e^{-\Lambda/2} + +(\Lambda+1)\{g+12\zeta+C_\Delta\Delta(\phi+\ell)\}\right]V_0. +\end{equation} +All these estimates also apply when $V_0=0$: then $R=0$, the completed +connection cost is zero, and no normalization by $V_0$ is used. + +Here is an explicit order of choices that avoids circular dependence. +It suffices to consider $0<\tau\le1$. +\begin{enumerate} +\item Choose a sufficiently large constant integer $\Delta$ and put +$g=1/\Delta$. For each such choice take, for example, +\[ + \Lambda=\left\lceil2\log(40\Delta/\tau)\right\rceil+1. +\] +Then $5\Delta e^{-\Lambda/2}\le\tau/8$. Taking $\Delta$ sufficiently +large also makes $(\Lambda+1)g\le\tau/8$, since +$g\log(1/g)\longrightarrow0$. Any additional upper bound on $g$ +required by the preparation stage may be imposed at the same time. +\item With $\Delta,\Lambda$ fixed, choose a small positive rational +$\zeta$ so that $12(\Lambda+1)\zeta\le\tau/8$. +Next choose a small positive rational $\phi$, and then a small positive +rational $\ell$, so that +\[ + C_\Delta\Delta(\Lambda+1)(\phi+\ell)\le\tau/8. +\] +Require in addition every upper bound on $\ell$ used in the count, +batch, and survival lemmas, and $p_*(1+\phi)\le1$. +The batch constants are uniform in the remaining threshold $b_*$, +so these requirements can be imposed before choosing that threshold. +This fixes the constant number of rounds $T=\lceil\Lambda/\ell\rceil$. +\item Put $\alpha=1+2/e$ and choose a fixed +$0<\eta\le\tau/(4(\alpha+\tau))$. +Finally choose rational $b_*>0$ as in Lemma~\ref{lem:rnd-count} so +that the final mass is at most $M+C$ with probability at least +$1-\eta$. +\end{enumerate} +The unconditional cost in \eqref{eq:rnd-total-error} is then at most +$(\alpha+\tau/2)V_0$ for every client. Repeat the entire process on +failure of the final mass test. The accepted output is the +unconditional output conditioned on that test, whose success +probability is at least $1-\eta$. Since connection costs are +nonnegative, +\[ + \E[C_j\mid\text{mass test succeeds}] + \le\frac{\alpha+\tau/2}{1-\eta}V_0 + \le(\alpha+\tau)V_0. +\] +Completion uses at most the final active mass, so an accepted output +has at most $M+C$ facilities. This proves the count and cost assertions +of Theorem~\ref{thm:grid-rounding}. + +For fixed accuracy all parameters are instance-independent constants. +The number of current facilities never exceeds the number in the +split input, and every group has at most $\Delta$ active copies. +The graph, its marginals, a fixed-sum rounding, the action sets, +their intersections, and the greedy completion can therefore be +computed in polynomial time. The radius equations and all physical +probabilities are rational; the approximations in +\eqref{eq:rnd-taper-approximation} are fixed constants. Only the +analysis uses the transcendental functions or alive-set discards. +The expected number of repetitions is at most $1/(1-\eta)$, giving +expected polynomial running time. Section~\ref{sec:derandomization} +replaces the local random choices by polynomial-size lists and obtains +a deterministic output with the corresponding total-cost bound. diff --git a/preprints/The-Approximation-Threshold-for-Metric-k-Median-September-24-2026/main.pdf b/preprints/The-Approximation-Threshold-for-Metric-k-Median-September-24-2026/main.pdf new file mode 100644 index 0000000000000000000000000000000000000000..7392c1b45000da40b8d063aac6b8fd85a78369ff --- /dev/null +++ b/preprints/The-Approximation-Threshold-for-Metric-k-Median-September-24-2026/main.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:c8176d1671beb83d3a459ff65e5c632d94364ec7e542930bf19fad0b75524d50 +size 562378 diff --git a/preprints/The-Artinian-Lex-Plus-Powers-Betti-Theorem-September-23-2026/README.md b/preprints/The-Artinian-Lex-Plus-Powers-Betti-Theorem-September-23-2026/README.md new file mode 100644 index 0000000000000000000000000000000000000000..2ff95b7e4ae058d4829cc535d7e4190993e36492 --- /dev/null +++ b/preprints/The-Artinian-Lex-Plus-Powers-Betti-Theorem-September-23-2026/README.md @@ -0,0 +1,18 @@ +# [The Artinian Lex-Plus-Powers Betti Theorem](paper.pdf) + +**Author:** OpenAI + +**Date:** September 23, 2026 + +## Citation + +```bibtex +@misc{OAI:The-Artinian-Lex-Plus-Powers-Betti-Theorem-September-23-2026, + author = {{OpenAI}}, + title = {{The Artinian Lex-Plus-Powers Betti Theorem}}, + howpublished = {OpenAI Math Release preprint + \href{https://github.com/openai/math/blob/main/preprints/The-Artinian-Lex-Plus-Powers-Betti-Theorem-September-23-2026/paper.pdf}{OAI:The-Artinian-Lex-Plus-Powers-Betti-Theorem-September-23-2026}}, + year = {2026} +} +``` + diff --git a/preprints/The-Artinian-Lex-Plus-Powers-Betti-Theorem-September-23-2026/build/bibliography.tex b/preprints/The-Artinian-Lex-Plus-Powers-Betti-Theorem-September-23-2026/build/bibliography.tex new file mode 100644 index 0000000000000000000000000000000000000000..c48e7a106101de0f5c69e65daf0f0d9646ddecdf --- /dev/null +++ b/preprints/The-Artinian-Lex-Plus-Powers-Betti-Theorem-September-23-2026/build/bibliography.tex @@ -0,0 +1,130 @@ +\begin{thebibliography}{99} +\bibitem{Abedelfatah15} +A.~Abedelfatah, +\emph{On the Eisenbud--Green--Harris conjecture}, +Proc. Amer. Math. Soc. \textbf{143} (2015), no.~1, 105--115. +\href{https://doi.org/10.1090/S0002-9939-2014-12216-7}{doi:10.1090/S0002-9939-2014-12216-7}. +Numbered statements refer to +\href{https://arxiv.org/abs/1212.2653v1}{arXiv:1212.2653v1}. + +\bibitem{Bigatti} +A.~M.~Bigatti, +\emph{Upper bounds for the Betti numbers of a given Hilbert function}, +Comm.\ Algebra \textbf{21} (1993), no.~7, 2317--2334. +\href{https://doi.org/10.1080/00927879308824679}{doi:10.1080/00927879308824679}. + +\bibitem{BCR} +A.~M.~Bigatti, A.~Conca, and L.~Robbiano, +\emph{Generic initial ideals and distractions}, +Comm.\ Algebra \textbf{33} (2005), no.~6, 1709--1732. +\href{https://doi.org/10.1081/AGB-200058217}{doi:10.1081/AGB-200058217}. +Author version: +\href{https://arxiv.org/abs/math/0308004}{arXiv:math/0308004}. + +\bibitem{CDeS20} +G.~Caviglia and A.~De Stefani, +\emph{The Eisenbud--Green--Harris conjecture for fast-growing degree sequences}, +Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) \textbf{25} (2024), no.~3, +1753--1762. +\href{https://doi.org/10.2422/2036-2145.202203_025}{doi:10.2422/2036-2145.202203\_025}. +Numbered statements refer to +\href{https://arxiv.org/abs/2007.15467v2}{arXiv:2007.15467v2}. + +\bibitem{CK14} +G.~Caviglia and M.~Kummini, +\emph{Poset embeddings of Hilbert functions and Betti numbers}, +J. Algebra \textbf{410} (2014), 244--257. +\href{https://doi.org/10.1016/j.jalgebra.2014.04.007}{doi:10.1016/j.jalgebra.2014.04.007}. +Numbered statements refer to +\href{https://arxiv.org/abs/1210.5562v2}{arXiv:1210.5562v2}. + +\bibitem{CM08} +G.~Caviglia and D.~Maclagan, +\emph{Some cases of the Eisenbud--Green--Harris conjecture}, +Math. Res. Lett. \textbf{15} (2008), no.~3, 427--433. +\href{https://www.math.purdue.edu/~gcavigli/Articles-PDF/%5B4%5D.pdf}{Author manuscript}. + +\bibitem{CS} +G.~Caviglia and A.~Sammartano, +\emph{On the Lex-plus-powers Conjecture}, +Adv.\ Math.\ \textbf{340} (2018), 284--299. +\href{https://doi.org/10.1016/j.aim.2018.10.005}{doi:10.1016/j.aim.2018.10.005}. +Numbered statements refer to +\href{https://arxiv.org/abs/1802.03035v2}{arXiv:1802.03035v2}. + +\bibitem{CL} +G.~F.~Clements and B.~Lindstr\"om, +\emph{A generalization of a combinatorial theorem of Macaulay}, +J. Combin. Theory \textbf{7} (1969), no.~3, 230--238. +\href{https://doi.org/10.1016/S0021-9800(69)80016-5}{doi:10.1016/S0021-9800(69)80016-5}. + +\bibitem{EGH} +D.~Eisenbud, M.~Green, and J.~Harris, +\emph{Higher Castelnuovo theory}, +Ast\'erisque \textbf{218} (1993), 187--202. +\url{https://www.numdam.org/item/AST_1993__218__187_0/}. + +\bibitem{FR} +C.~A.~Francisco and B.~P.~Richert, +\emph{Lex-plus-powers ideals}, +in I.~Peeva (ed.), \emph{Syzygies and Hilbert Functions}, +Lecture Notes in Pure and Applied Mathematics \textbf{254}, +Chapman \& Hall/CRC, Boca Raton, FL, 2007, 113--144. +Author version: +\url{https://math.okstate.edu/people/chris/lpp-updated-final.pdf}. + +\bibitem{Hulett} +H.~A.~Hulett, +\emph{Maximum Betti numbers of homogeneous ideals with a given Hilbert function}, +Comm.\ Algebra \textbf{21} (1993), no.~7, 2335--2350. +\href{https://doi.org/10.1080/00927879308824680}{doi:10.1080/00927879308824680}. + +\bibitem{Kuzmanovski} +N.~Kuzmanovski, +\emph{A Structural Property of Generic Initial Ideals}, +preprint, 18 August 2026. +\href{https://arxiv.org/abs/2608.17281v1}{arXiv:2608.17281v1}. + +\bibitem{MM} +J.~Mermin and S.~Murai, +\emph{The Lex-Plus-Powers Conjecture holds for pure powers}, +Adv.\ Math.\ \textbf{226} (2011), no.~4, 3511--3539. +\href{https://doi.org/10.1016/j.aim.2010.08.022}{doi:10.1016/j.aim.2010.08.022}. +Author version: \url{https://s-murai.w.waseda.jp/papers/LPP.pdf}. + +\bibitem{MPS} +J.~Mermin, I.~Peeva, and M.~Stillman, +\emph{Ideals containing the squares of the variables}, +Adv.\ Math.\ \textbf{217} (2008), no.~5, 2206--2230. +\href{https://doi.org/10.1016/j.aim.2007.11.014}{doi:10.1016/j.aim.2007.11.014}. +Author version: +\url{https://pi.math.cornell.edu/~mermin/papers/ideals_containing_the_squares_of_the_variables.pdf}. + +\bibitem{Murai} +S.~Murai, +\emph{Borel-plus-powers monomial ideals}, +J. Pure Appl. Algebra \textbf{212} (2008), no.~6, 1321--1336. +\href{https://doi.org/10.1016/j.jpaa.2007.09.010}{doi:10.1016/j.jpaa.2007.09.010}. +Author version: \url{https://s-murai.w.waseda.jp/papers/BPP.pdf}. + +\bibitem{ArtinianEGH} +OpenAI, +\emph{Commuting Division-Coefficient Forms and the Artinian +Eisenbud--Green--Harris Conjecture}, +OpenAI Math Release preprint +\href{https://github.com/openai/math/blob/main/preprints/Commuting-Division-Coefficient-Forms-and-the-Artinian-Eisenbud-Green-Harris-Conjecture-September-23-2026/paper.pdf}{OAI:Commuting-Division-Coefficient-Forms-and-the-Artinian-Eisenbud-Green-Harris-Conjecture-September-23-2026}, +2026. + +\enlargethispage{2\baselineskip} +\bibitem{Pardue} +K.~Pardue, +\emph{Deformation classes of graded modules and maximal Betti numbers}, +Illinois J. Math.\ \textbf{40} (1996), no.~4, 564--585. +\href{https://doi.org/10.1215/ijm/1255985937}{doi:10.1215/ijm/1255985937}. + +\bibitem{Richert} +B.~P.~Richert, +\emph{A study of the lex plus powers conjecture}, +J. Pure Appl. Algebra \textbf{186} (2004), no.~2, 169--183. +\href{https://doi.org/10.1016/S0022-4049(03)00130-0}{doi:10.1016/S0022-4049(03)00130-0}. +\end{thebibliography} diff --git a/preprints/The-Artinian-Lex-Plus-Powers-Betti-Theorem-September-23-2026/build/figures/endpoint-slices.tex b/preprints/The-Artinian-Lex-Plus-Powers-Betti-Theorem-September-23-2026/build/figures/endpoint-slices.tex new file mode 100644 index 0000000000000000000000000000000000000000..9f83d80b896292fab72a9604245c1f700a454025 --- /dev/null +++ b/preprints/The-Artinian-Lex-Plus-Powers-Betti-Theorem-September-23-2026/build/figures/endpoint-slices.tex @@ -0,0 +1,22 @@ +\begin{figure}[htbp] +\centering +\begin{tikzpicture}[>=Stealth,font=\small] +\node (u) at (-3.3,0) {$uz^s\in H$}; +\node (xu) at (3.3,0) {$x_iuz^s\in zH$}; +\node (transfer) at (3.3,-1.5) {$x_iuz^{s-1}\in H$}; +\draw[->] (u) -- node[above] {$\times x_i$} (xu); +\draw[->] (transfer) -- node[right] {$\times z$} (xu); +\draw[->,dashed] (u) -- + node[pos=.47,below,sloped] {transfer $z$ to $x_i$} (transfer); +\node[align=center] at (-3.2,-1.55) + {$x_i[uz^s]=0$ in $H/zH$\\[3pt] + $1\le sh}C_{h\ell}t_\ell + \qquad(1\le h\le n). +\end{equation} +Here \(c_h\in k^\times\), \(B_{h\ell},C_{h\ell}\in A\), and every summand +has degree \(a_h\). Moreover, +\[ + \{t^\alpha=t_1^{\alpha_1}\cdots t_n^{\alpha_n}: + \alpha\in\N^n\} +\] +is a graded left-\(\Delta\) basis of the \emph{full} algebra \(N\). +\end{theorem} + +The basis assertion in Theorem~\ref{thm:forms} is stronger than a basis of the finite +complete-intersection quotient: every monomial \(t^\alpha\), without +an upper bound on its exponents, occurs in the decomposition of \(N\). +This is what will make \(N\) a free module over a polynomial ring and +allow us to transfer an entire resolution. The Hilbert-function +consequence of the foundation theorem alone would not suffice. + +\begin{remark}\label{rem:coefficients} +Commutativity of the forms does not mean that their individual +\(\Delta\)-coefficients commute. The proof below always places those +coefficients on the left and multiplies by the \(t_i\) on the right. +The construction itself has no hypothesis on the additional generators +of \(I\). +\end{remark} diff --git a/preprints/The-Artinian-Lex-Plus-Powers-Betti-Theorem-September-23-2026/build/sections/01-introduction.tex b/preprints/The-Artinian-Lex-Plus-Powers-Betti-Theorem-September-23-2026/build/sections/01-introduction.tex new file mode 100644 index 0000000000000000000000000000000000000000..c542492b19215d0b3c0c4c6e4ad9ec3cb3fe1eed --- /dev/null +++ b/preprints/The-Artinian-Lex-Plus-Powers-Betti-Theorem-September-23-2026/build/sections/01-introduction.tex @@ -0,0 +1,227 @@ +\section{Introduction}\label{sec:introduction} + +A Hilbert function records the dimensions of a graded quotient; its graded +Betti numbers also record the relations among its generators and all higher +syzygies. The lex-plus-powers conjecture predicts one extremal ideal for +both kinds of data when the original ideal contains a regular sequence. +We prove its full Artinian form over \(\C\), and deduce the full +characteristic-zero statement for regular sequences of arbitrary positive +length by the established Artinian reduction and coefficient-field descent. + +Let +\[ + S=\C[x_1,\ldots,x_n],\qquad + 2\le a_1\le\cdots\le a_n,\qquad + P=(x_1^{a_1},\ldots,x_n^{a_n}), +\] +where each variable has degree one. Lexicographic order always has +\(x_1>\cdots>x_n\), and a lex segment starts with the \emph{greatest} +monomials. Suppose that a homogeneous ideal \(I\subseteq S\) contains a +homogeneous regular sequence \(f_1,\ldots,f_n\) with \(\deg f_i=a_i\). +Here regularity means that multiplication by \(f_i\) is injective on +\(S/(f_1,\ldots,f_{i-1})\) for each \(i\). Its full length makes +\(S/I\) Artinian: the complete-intersection Hilbert series is +\(\prod_i(1+z+\cdots+z^{a_i-1})\). +For each \(d\ge0\), prescribe +\begin{equation} + J_d=L_d+P_d,\qquad \dim_\C J_d=\dim_\C I_d, + \label{eq:prescription} +\end{equation} +where \(L_d\) is the largest lex-segment subspace with this property. +Part of the assertion below is that these dimensions can be attained +and that the prescribed spaces form an ideal. + +For a standard graded polynomial ring \(A\) over a field \(F\), write +\[ + \beta^A_{p,j}(B)=\dim_F\Tor^A_p(B,F)_j, + \qquad F=A/A_{>0}. +\] +The index \(p\) is the homological degree and \(j\) is the internal degree. + +\begin{theorem}\label{thm:main} +For every \(n\ge1\), every ordered degree sequence +\(2\le a_1\le\cdots\le a_n\), and every homogeneous ideal +\(I\subseteq\C[x_1,\ldots,x_n]\) containing a regular sequence of these +degrees, the spaces in \eqref{eq:prescription} exist and +\(J=\bigoplus_{d\ge0}J_d\) is a homogeneous ideal. For this single ideal, +\[ + \beta^S_{p,j}(S/I)\le\beta^S_{p,j}(S/J) + \qquad\text{for every \(p\ge0\) and \(j\in\Z\)}. +\] +There is no restriction on the degrees or number of the additional +generators of \(I\). +\end{theorem} + +Thus Theorem~\ref{thm:main} resolves the Artinian lex-plus-powers +conjecture positively over \(\C\). +Its Hilbert-function assertion is the Artinian Eisenbud--Green--Harris +statement; the Betti inequalities are stronger. + +The Artinian restriction can be removed from these two conclusions. +Caviglia and Maclagan proved the Artinian reduction for the +Eisenbud--Green--Harris conjecture \cite[Proposition~10]{CM08}; +Caviglia and Kummini proved the corresponding reduction for +lex-plus-powers Betti numbers \cite[Section~4, Theorem~4.1]{CK14}. +Together with a finite coefficient-field descent, these reductions give +the following form over every characteristic-zero field. + +\begin{corollary}[Eisenbud--Green--Harris and lex-plus-powers in characteristic zero] +\label{cor:characteristic-zero} +Let \(F\) be a field of characteristic zero, let +\[ + S_F=F[x_1,\ldots,x_n],\qquad + 1\le c\le n,\qquad 2\le a_1\le\cdots\le a_c, + \qquad P_F=(x_1^{a_1},\ldots,x_c^{a_c}), +\] +with the standard grading and lexicographic order \(x_1>\cdots>x_n\). +Suppose that a homogeneous ideal \(I\subseteq S_F\) contains a homogeneous +regular sequence \(f_1,\ldots,f_c\) with \(\deg f_i=a_i\). +For \(d\ge0\), put +\[ + q_d=\dim_F I_d-\dim_F(P_F)_d. +\] +This integer lies between zero and the number of degree-\(d\) monomials +outside \(P_F\). Let \(J_d\) be the sum of \((P_F)_d\) and the span of the +\(q_d\) lexicographically greatest such monomials. Then +\(J=\bigoplus_{d\ge0}J_d\) is a homogeneous ideal, and for this single ideal +\[ + \dim_F(S_F/I)_d=\dim_F(S_F/J)_d\quad(d\ge0),\qquad + \beta^{S_F}_{p,j}(S_F/I)\le\beta^{S_F}_{p,j}(S_F/J) + \quad(p\ge0,\ j\in\Z). +\] +There is no restriction on the degrees or number of the additional +generators of \(I\). +\end{corollary} + +Corollary~\ref{cor:characteristic-zero} includes non-Artinian quotients +and fields that are not algebraically closed. The case with no prescribed +regular sequence is the classical lex-ideal theorem cited below. +The division-coefficient construction itself remains the full-length +construction over \(\C\); Section~\ref{sec:characteristic-zero} proves +the corollary from Theorem~\ref{thm:main}. + +\subsection{Context and previous results} + +The Eisenbud--Green--Harris (EGH) Hilbert-function conjecture arose in +their work on higher Castelnuovo theory \cite[Section~4]{EGH}. +The lex-plus-powers conjecture, generally attributed to Charalambous +and Evans, strengthens that prediction to graded Betti numbers. +Francisco and Richert discuss its origins and identify its first +printed appearance in the work of Evans and Richert +\cite[Section~5]{FR}; see also the public formulations in +\cite[Section~1]{Richert} and +\cite[Conjectures~1.1 and 1.2]{CS}. +Without a prescribed regular sequence, the extremality of lex ideals +at fixed Hilbert function is the theorem of Bigatti, Hulett, and +Pardue \cite{Bigatti,Hulett,Pardue}. +The additional powers encode the degrees of the regular sequence and +give the sharper comparison studied here. +For ideals already containing these powers, Clements--Lindstr\"om +compression supplies the Hilbert-function comparison \cite{CL}. + +The prescribed-powers Betti comparison developed through several stages. +Mermin, Peeva, and Stillman proved the comparison for ideals containing +the squares of the variables in characteristic zero +\cite[Theorem~1.1]{MPS}; Murai established it for strongly stable ideals +plus the prescribed powers over any field \cite[Theorem~1.1]{Murai}. +Mermin and Murai then proved the lex-plus-powers comparison for ideals +containing a monomial regular sequence, again over every field +\cite[Theorems~3.1 and 8.1]{MM}. They also isolated the reduction to a +Hilbert-matching monomial ideal with no smaller graded Betti numbers +\cite[Conjecture~10.2]{MM}. It is this reduction that we obtain from +division coefficients. Our proof of the subsequent monomial comparison +is included for completeness; its conclusion is not new. + +For nonmonomial regular sequences, the Hilbert-function and Betti problems +have had different ranges of known cases. Abedelfatah proved EGH for full +regular sequences whose forms split into linear factors +\cite[Corollary~4.3]{Abedelfatah15}. Caviglia and Maclagan established +EGH for fast-growing degrees \cite{CM08}, and Caviglia and De Stefani +weakened the required growth to +\(a_i\ge\sum_{j0}$. In such a resolution, the +number of free generators of degree $j$ in homological position $p$ is +$\beta^A_{p,j}(B)$. If $H\subsetneq A$ is a homogeneous ideal, then +\begin{equation} + \beta^A_{p+1,j}(A/H)=\beta^A_{p,j}(H) + \qquad(p\geq 0,\ j\in\Z). + \label{filt:ideal-quotient} +\end{equation} +If $H$ is monomial, a minimal resolution of $A/H$ can be chosen +homogeneous for the exponent multigrading, with all generator +multidegrees in $\Z_{\geq0}^r$. +\end{lemma} + +\begin{proof} +Lift a homogeneous $F$-basis of $B/A_{>0}B$ to $B$ and map the free module +on these lifts to $B$. This map is surjective: in each degree, an element +not yet expressed in the lifts differs from such an expression by a sum +of positive-degree multiples of elements of smaller degree, to which +induction applies. Its kernel is finitely generated because $A$ is +Noetherian. Repeating the construction for successive kernels gives a +free resolution. At each stage, reduction modulo $A_{>0}$ identifies the +chosen free generators with a basis of the corresponding module modulo +$A_{>0}$, so the next kernel lies in $A_{>0}$ times that free module. +Thus every differential between free modules has positive-degree +entries. Tensoring with $F$ makes these differentials zero and proves +the assertion about $\Tor$. + +A minimal resolution of a proper homogeneous ideal $H$, followed by +the inclusion $H\hookrightarrow A$ and the quotient $A\twoheadrightarrow +A/H$, is a minimal resolution of $A/H$: the additional entries also +have positive degree since $H\subseteq A_{>0}$. This proves +\eqref{filt:ideal-quotient}, including $H=0$. + +For a monomial quotient, carry out the same construction with +multihomogeneous lifts. The initial free module is $A$ in multidegree +zero. A free module whose generator multidegrees are nonnegative has +no component in a multidegree outside $\Z_{\geq0}^r$; neither does its +kernel or a quotient of that kernel. Induction therefore gives the +last assertion. +\end{proof} + +\begin{lemma}[Koszul calculation] +\label{filt:koszul} +Let $K_\bullet(B)$ be the Koszul complex on $y_1,\ldots,y_r$ with +coefficients in $B$. Its term of internal degree $j$ in position $p$ is +\[ + K_p(B)_j=\bigoplus_{\substack{\sigma\subseteq\{1,\ldots,r\}\\ + |\sigma|=p}} B_{j-p}e_\sigma, +\] +where $e_\sigma$ has degree $p$. Its homology satisfies +\[ + \dim_F H_p(K_\bullet(B)_j)=\beta^A_{p,j}(B). +\] +\end{lemma} + +\begin{proof} +For $\sigma=\{i_1<\cdots0}=NT_{>0}. + \label{mono:positive-degree} +\end{equation} + +The left $S$-action commutes with the right $T$-action by associativity +of multiplication in $N$. Moreover, +$N=\Delta\otimes_\C S$ as a left $S$-module, so $N$ is left $S$-free +and therefore flat. Each element of $I$ is central in $N$, whence +$IN$ is a two-sided ideal. In particular it is a right $T$-submodule, +and +\[ + Q\simeq\Delta\otimes_\C(S/I). +\] +Taking a complex basis of $(S/I)_d$ proves +\eqref{mono:dimension-transfer}. + +Let $F_\bullet\to S/I$ be a minimal graded $S$-free resolution. +Since $S$ is commutative, regard its terms as right $S$-modules and +tensor with the $(S,T)$-bimodule $N$. Flatness gives a resolution +\[ + F_\bullet\otimes_SN\longrightarrow Q +\] +by graded free right $T$-modules. An entry $f$ of an original +differential acts on $N$ by left multiplication. This map is right +$T$-linear, again by associativity. + +Choose a $k$-basis $b_1,\ldots,b_m$ of $\Delta$, which is a right +$T$-basis of $N$ under \eqref{mono:free-map}. If $f$ is nonzero and +has degree $e>0$, then each $fb_u$ belongs to $N_e$. By the graded +decomposition \eqref{mono:basis}, its coefficients in the basis +$b_1,\ldots,b_m$ are homogeneous polynomials in $T$ of degree $e$. +They therefore lie in $T_{>0}$. Every nonzero original differential +entry has positive degree, so the transferred resolution is minimal. +Each summand $S(-s)$ of $F_p$ becomes $m$ summands $T(-s)$. +Lemma~\ref{filt:minimal} proves \eqref{mono:betti-transfer} for every +$p,j$. If $I=S$, both quotients and their resolutions may be taken +to be zero, and the same formulas hold. +\end{proof} + +The next proposition supplies the monomial ideal used by the rest of +the proof. Its ideal is allowed to depend on $I$, but it is fixed +simultaneously for all homological and internal degrees. + +\begin{proposition}[Monomial reduction] +\label{mono:bound} +Let $S=\C[x_1,\ldots,x_n]$, let +$2\leq a_1\leq\cdots\leq a_n$, and let $I\subseteq S$ be a +homogeneous ideal containing a homogeneous regular sequence +$f_1,\ldots,f_n$ with $\deg f_h=a_h$. Put +$P=(x_1^{a_1},\ldots,x_n^{a_n})$. +There is a monomial ideal $M\subseteq S$ containing $P$ such that +\begin{align} + \dim_\C(S/M)_d&=\dim_\C(S/I)_d + &&(d\in\Z),\label{mono:hilbert-bound}\\ + \beta^S_{p,j}(S/I)&\leq\beta^S_{p,j}(S/M) + &&(p\geq0,\ j\in\Z). + \label{mono:betti-bound} +\end{align} +\end{proposition} + +\begin{proof} +If $I=S$, take $M=S$. Assume henceforth that $I$ is proper. +Apply Theorem~\ref{thm:forms} to the given regular sequence. It +supplies $k$, $\Delta$, and commuting degree-one forms $t_h$ satisfying +the basis property \eqref{mono:basis} and homogeneous identities +\begin{equation} + t_h^{a_h}=c_hf_h+\sum_{\ellh}C_{h\ell}t_\ell, + \qquad c_h\in k^\times, + \label{mono:triangular} +\end{equation} +where the multipliers belong to $N=\Delta[x_1,\ldots,x_n]$. +In particular $C_{h\ell}$ is homogeneous of degree $a_h-1$ when +nonzero. Define $T=k[y_1,\ldots,y_n]$, $m=\dim_k\Delta$, +$W=IN$, and $Q=N/W$ as in Lemma~\ref{mono:transfer}. + +Under \eqref{mono:free-map}, view $N$ as the free $T$-module with +coefficient space $V=\Delta$. In each total degree, order exponents +by placing the lexicographically smallest tuple +\[ + (\alpha_n,\ldots,\alpha_1) +\] +first. This order is compatible with addition. Apply +Proposition~\ref{filt:comparison} to the homogeneous $T$-submodule +$W$ with this order. + +For an exponent $\alpha$, let $V_\alpha\subseteq\Delta$ be its +leading coefficient space. It is closed under left multiplication +by $\Delta$: if $w\in W$ witnesses a coefficient $b$ at $\alpha$ +with no earlier coefficients, then $\lambda w\in W$ witnesses +$\lambda b$. Here $W$ is left $\Delta$-linear because it is an ideal +of $N$. Hence $V_\alpha$ is a left ideal of $\Delta$. Since +$\Delta$ is a division algebra, +\[ + V_\alpha=0\quad\hbox{or}\quad\Delta. +\] +Explicitly, if $0\ne b\in V_\alpha$, every $c\in\Delta$ is +$(cb^{-1})b$ and therefore also belongs to $V_\alpha$. + +The set +\[ + \mathcal U=\{\alpha\in\Z_{\geq0}^n:V_\alpha=\Delta\} +\] +is upward closed under addition of exponent vectors, since the leading +object is a $T$-submodule. Define $M\subseteq S$ to be the monomial +ideal with exponent set $\mathcal U$, and define +$M_k\subseteq T$ by the same exponent set. Then +\begin{equation} + N/\operatorname{in}_{\prec}W + \simeq\Delta\otimes_k(T/M_k) + \simeq(T/M_k)^{\oplus m} + \label{mono:initial-quotient} +\end{equation} +as graded right $T$-modules. + +We next verify the power containment. Since every $f_\ell$ lies +in $I$, Equation~\eqref{mono:triangular} gives +\[ + t_h^{a_h}-\sum_{\ell>h}C_{h\ell}t_\ell\in W. +\] +Expand each $C_{h\ell}$ in the left $\Delta$-basis $t^\alpha$. +Right multiplication by $t_\ell$ turns a term $bt^\alpha$ into +$bt^{\alpha+e_\ell}$, using only commutativity of the forms. +Every resulting exponent has a positive coordinate of index greater +than $h$. It consequently lies strictly after $a_he_h$ in the +chosen order, because $a_he_h$ has zero coordinates at all indices +greater than $h$. The displayed element therefore has leading +position $a_he_h$ with coefficient $1$. Thus $a_he_h\in\mathcal U$ +for every $h$, proving $P\subseteq M$. + +It remains to compare the numerical invariants. The scalar extension +of $S/M$ from $\C$ to $k$, with $x_i$ renamed $y_i$, is $T/M_k$. +Its Hilbert function is unchanged. Tensoring the Koszul complex over +$\C$ with the field $k$ also preserves exactness and commutes with +homology, so +\begin{align*} + \dim_k(T/M_k)_d&=\dim_\C(S/M)_d,\\ + \beta^T_{p,j}(T/M_k)&=\beta^S_{p,j}(S/M). +\end{align*} +Equations \eqref{mono:dimension-transfer} and +\eqref{mono:initial-quotient}, together with +\eqref{filt:hilbert}, now give +\[ + m\dim_\C(S/I)_d + =\dim_k Q_d + =\dim_k(N/\operatorname{in}_{\prec}W)_d + =m\dim_\C(S/M)_d. +\] +Likewise, \eqref{mono:betti-transfer}, +\eqref{filt:betti}, and \eqref{mono:initial-quotient} yield, for +every $p\geq0$ and every $j\in\Z$, +\[ + m\beta^S_{p,j}(S/I) + =\beta^T_{p,j}(Q) + \leq\beta^T_{p,j}(N/\operatorname{in}_{\prec}W) + =m\beta^S_{p,j}(S/M). +\] +Cancelling the positive integer $m$ proves both assertions. +\end{proof} + +The right action in Lemma~\ref{mono:transfer} and the placement of the +$\Delta$-coefficients on the left are deliberate. The argument uses +commutativity among the forms $t_i$, but never requires an individual +coefficient in $\Delta$ to commute with a form. Proposition~\ref{mono:bound} +completes the reduction to an ideal containing $P$. The remaining +comparison is the prescribed-powers theorem of Mermin and Murai +\cite[Theorem~3.1]{MM}. Sections~\ref{sec:stabilization}--\ref{sec:tor} +give a direct proof by stabilization, box compression, and a Koszul-rank +induction. diff --git a/preprints/The-Artinian-Lex-Plus-Powers-Betti-Theorem-September-23-2026/build/sections/04-stabilization.tex b/preprints/The-Artinian-Lex-Plus-Powers-Betti-Theorem-September-23-2026/build/sections/04-stabilization.tex new file mode 100644 index 0000000000000000000000000000000000000000..8b1ab768f310d8165cfd17e8893146006f0eeed5 --- /dev/null +++ b/preprints/The-Artinian-Lex-Plus-Powers-Betti-Theorem-September-23-2026/build/sections/04-stabilization.tex @@ -0,0 +1,216 @@ +\section{Stabilization inside the exponent box} +\label{sec:stabilization} + +We now start with a monomial ideal containing the prescribed powers. +The aim of this section is to impose stability inside the finite exponent +box while preserving the Hilbert function and weakly increasing every +graded Betti number of the quotient. This is the pure-power +deformation strategy of Mermin and Murai \cite[Section~3]{MM}. + +Let +\[ + S=\C[x_1,\ldots,x_n],\qquad + 2\le a_1\le\cdots\le a_n,\qquad + P=(x_1^{a_1},\ldots,x_n^{a_n}),\qquad b_i=a_i-1. +\] +For $1\le r\le n$ and $d\in\Z$, put +\[ + \mathcal B_r(d) + =\{u\in\Z_{\ge0}^r: |u|=d,\ 0\le u_i\le b_i + \text{ for }1\le i\le r\}. +\] +In particular, this set is empty when $d<0$. + +\begin{definition} +\label{stab:definition} +A subset $X\subseteq\mathcal B_r(d)$ is \emph{stable} if +\[ + u\in X,\quad h<\ell,\quad + u+e_h-e_\ell\in\mathcal B_r(d) + \quad\Longrightarrow\quad u+e_h-e_\ell\in X. +\] +A monomial ideal containing $(x_1^{a_1},\ldots,x_r^{a_r})$ is +\emph{stable in the box} if its included exponent vectors in +$\mathcal B_r(d)$ form a stable set for every $d$. +This definition imposes no transfer condition on monomials outside the box. +\end{definition} + +The modification used below acts on one pair of variables. Fix +$h<\ell$, write $\xi=x_h$ and $q=x_\ell$, and choose a primitive +$a_\ell$-th root of unity $\zeta\in\C$. Define a graded +$\C$-linear map $D:S\to S$ on monomials by +\begin{equation} +\label{stab:distraction-formula} + D(x^u)=\left(\prod_{i\ne\ell}x_i^{u_i}\right) + \prod_{s=0}^{u_\ell-1}(q-\zeta^s\xi), + \qquad u\in\Z_{\ge0}^n, +\end{equation} +with the empty product equal to $1$. +The resolution argument below is the nested-factor mechanism used for +distractions in~\cite[Theorem~2.19 and Corollary~2.20]{BCR}. +We give the argument directly because the +root-of-unity factors here are periodic, rather than eventually constant. + +\begin{lemma} +\label{stab:distraction} +For every monomial ideal $K\subseteq S$, its vector-space image $D(K)$ +is a homogeneous ideal. Moreover, +\begin{gather*} + \dim_\C(S/D(K))_d=\dim_\C(S/K)_d + \quad(d\in\Z),\\ + \beta^S_{p,j}(S/D(K))=\beta^S_{p,j}(S/K) + \quad(p\ge0,\ j\in\Z). +\end{gather*} +The map also satisfies $D(P)=P$. +\end{lemma} + +\begin{proof} +The expansion of $D(x^u)$ has coefficient $1$ at $x^u$, and every +other monomial is lexicographically larger than $x^u$. Thus $D$ is +unitriangular on each finite-dimensional graded piece and is invertible. +Multiplication by a variable other than $q$ commutes with $D$, while +\begin{equation} +\label{stab:multiplication} + qD(x^u)=D(qx^u)+\zeta^{u_\ell}\xi D(x^u). +\end{equation} +Since $K$ is spanned by monomials and is an ideal, these identities show +that $D(K)$ is an ideal. The invertibility and degree preservation of +$D$ give the Hilbert-function equality. + +To check preservation of $P$, first observe that +\[ + \prod_{s=0}^{a_\ell-1}(q-\zeta^s\xi) + =q^{a_\ell}-\xi^{a_\ell}\in P. +\] +Here the second term belongs to $P$ because $a_h\le a_\ell$. +If a monomial is divisible by $q^{a_\ell}$, its image under $D$ is +divisible by this displayed polynomial. If it is divisible by another +generator of $P$, that generator remains a factor of its image. +Consequently $D(P)\subseteq P$. In each degree this inclusion is an +inclusion between vector spaces of the same finite dimension, so it is +an equality. + +It remains to compare Betti numbers. If $K=S$, both quotients are zero. +Otherwise, choose a minimal exponent-multigraded free resolution +$F_\bullet\to S/K$ with $F_0=S$, as in +Lemma~\ref{filt:minimal}. A nonzero differential entry from a generator +of multidegree $u$ to one of multidegree $v$ has the form +$c x^{u-v}$, where $u\ge v$ coordinatewise. Keeping the ordinary +graded free modules, replace this entry by +\begin{equation} +\label{stab:ratio-entry} + c\frac{D(x^u)}{D(x^v)}. +\end{equation} +The ratio is a polynomial: all unchanged monomial factors are nested, +as are the products in~\eqref{stab:distraction-formula}. + +For each free summand whose generator has multidegree $v$, define a +linear change on its coefficient polynomials by +\begin{equation} +\label{stab:coefficient-change} + C_v(x^{w-v})=\frac{D(x^w)}{D(x^v)},\qquad w\ge v. +\end{equation} +More explicitly, the right-hand side is +\[ + \left(\prod_{i\ne\ell}x_i^{w_i-v_i}\right) + \prod_{s=v_\ell}^{w_\ell-1}(q-\zeta^s\xi). +\] +It has the original monomial $x^{w-v}$ with coefficient $1$ and only +lexicographically larger additional terms. Hence $C_v$ is an +invertible graded vector-space map. These maps need not be +$S$-linear; the new differential defined by~\eqref{stab:ratio-entry} +is $S$-linear. + +The changes on the free summands intertwine the old and new +differentials. Indeed, for a source coefficient $x^{w-u}$ and a +nonzero entry $c x^{u-v}$, the required equality is precisely +\[ + \frac{D(x^w)}{D(x^u)} + \left(c\frac{D(x^u)}{D(x^v)}\right) + =c\frac{D(x^w)}{D(x^v)}. +\] +Thus the new maps form a complex that is exact in every positive +homological degree. At $F_0=S$ the change is $C_0=D$, so its image +from $F_1$ is $D(K)$. We have obtained a free resolution of $S/D(K)$ +with the same ordinary graded free modules as $F_\bullet$. +Every replaced nonzero entry still has its original positive degree, +so this resolution is minimal. Its free ranks in each degree give +the asserted equality of all graded Betti numbers. +\end{proof} + +\begin{proposition} +\label{stab:main} +Let $S=\C[x_1,\ldots,x_n]$, let $2\le a_1\le\cdots\le a_n$, +and put $P=(x_1^{a_1},\ldots,x_n^{a_n})$. +For every monomial ideal $K\supseteq P$, there is a monomial ideal +$H\supseteq P$ stable in the box such that +\[ + \dim_\C(S/H)_d=\dim_\C(S/K)_d\quad(d\in\Z) +\] +and +\[ + \beta^S_{p,j}(S/K)\le\beta^S_{p,j}(S/H) + \qquad(p\ge0,\ j\in\Z). +\] +\end{proposition} + +\begin{proof} +If $K$ is stable in the box, take $H=K$. Otherwise choose a failed +transfer from $q=x_\ell$ to $\xi=x_h$, with $h<\ell$, and form +the map $D$ above. Replace $K$ by +\[ + K'=\operatorname{in}_{\mathrm{lex}}D(K), +\] +where the leading monomial is the lexicographically largest one. +By Lemma~\ref{stab:distraction}, $P\subseteq D(K)$, and hence +$P\subseteq K'$. That lemma and Proposition~\ref{filt:comparison} +show that the Hilbert function is preserved and +\[ + \beta^S_{p,j}(S/K)\le\beta^S_{p,j}(S/K') + \qquad(p\ge0,\ j\in\Z). +\] +We prove that repeated replacements terminate by measuring progress +in every lexicographic prefix. + +Fix a degree $d$ and a prefix $E$ of the degree-$d$ monomials, listed +from greatest to least. Let $\pi_E$ be the projection onto their span. +Gaussian elimination with this ordered list of columns gives +\[ + \#\{m\in E:m\in K'\} + =\dim_\C\pi_E(D(K)_d). +\] +For the monomials $m\in K\cap E$, the vectors $\pi_E(D(m))$ +are linearly independent: their coordinates in their original +monomial positions form a triangular matrix with diagonal entries $1$. +Consequently +\begin{equation} +\label{stab:prefix-growth} + \#(K'\cap E)\ge\#(K\cap E) +\end{equation} +for every such prefix in every degree. + +There is a strict inequality for at least one prefix. Choose a +monomial $x^u\in K$ inside the box witnessing the failed transfer, so +\[ + m'=\frac{\xi}{q}x^u + \quad\text{lies inside the box but does not belong to }K. +\] +Its existence implies $1\le u_\ell\sum_h b_h\). We also set +\(\mathcal B_0(0)=\{()\}\) and \(\mathcal B_0(d)=\varnothing\) for +\(d\ne0\). No ordering of the bounds is required in this section. + +Vectors are ordered lexicographically, with the first coordinate +taking precedence, and lex segments start with the largest vectors. +For \(X\subseteq\mathcal B_r(d)\), let +\(\operatorname{Lex}(X)\) be the lex segment of cardinality \(|X|\). +Recall that \(X\) is \emph{stable} if +\[ + u\in X,\quad h<\ell,\quad + u+e_h-e_\ell\in\mathcal B_r(d) + \quad\Longrightarrow\quad u+e_h-e_\ell\in X. +\] +Every lex segment is stable, since each allowed transfer increases +the vector in lex order. Define the bounded shadow and the slices by +\[ + \begin{split} + \partial X + &=\{u+e_h\in\mathcal B_r(d+1):u\in X,\ 1\le h\le r\},\\ + X(s) + &=\{u\in\mathcal B_{r-1}(d-s):(u,s)\in X\} + \qquad(0\le s\le b_r). + \end{split} +\] +The bounds in \(\mathcal B_{r-1}\) are the first \(r-1\) bounds. +Empty sets have empty shadows; thus these definitions also cover +negative degrees and empty slices. + +\begin{lemma}[Shadows and end slices] +\label{box:inequalities} +Let \(r\ge1\), let \(b_1,\ldots,b_r\) be positive integers, and let +\(d\in\Z\). If \(X\subseteq\mathcal B_r(d)\) is stable and +\(Y=\operatorname{Lex}(X)\), then \(\partial Y\) is a lex segment and +\begin{equation} + \label{box:three-inequalities} + |\partial X|\ge|\partial Y|, + \qquad |X(0)|\ge|Y(0)|, + \qquad |X(b_r)|\le|Y(b_r)|. +\end{equation} +\end{lemma} + +\begin{proof} +We first prove the assertion about lex shadows, without assuming +stability of \(X\). Empty shadows need no argument. For +\(v\in\mathcal B_r(d+1)\), with \(d\ge0\), its lex largest +degree-\(d\) predecessor is +\[ + p(v)=v-e_{\max\{h:v_h>0\}}. +\] +Every predecessor belongs to the bounded box. Moreover, the map +\(p\) preserves weak lex order. Indeed, suppose \(v>w\), with first +difference at coordinate \(k\). Subtraction at the last nonzero +coordinate cannot change an earlier coordinate of \(v\). If the +operation changes a coordinate of \(w\) before \(k\), then +\(p(v)>p(w)\) already at that coordinate. Otherwise the strict +inequality at \(k\) can disappear only if \(v_k=w_k+1\) and the tail +of \(v\) after \(k\) is zero. Equality of total degrees then says +that the tail of \(w\) contains exactly one unit. Subtracting its +last nonzero coordinate gives \(p(v)=p(w)\). In all other cases the +first strict inequality remains. Membership of \(v\) in the shadow +of a lex segment is equivalent to membership of \(p(v)\) in that +segment, proving that its shadow is lex. + +We prove the three inequalities simultaneously by induction on +\(r\). If \(\mathcal B_r(d)\) is empty, they are immediate. For +\(r=1\), each degree box has at most one vector, so \(X=Y\). +Assume henceforth that \(r>1\). A prime will indicate lex +replacement or shadow in the first \(r-1\) coordinates. + +Each slice \(X(s)\) is stable, and stability between slices gives +\begin{equation} + \label{box:original-slices} + \partial'X(s)\subseteq X(s-1) + \qquad(1\le s\le b_r). +\end{equation} +Replace each slice by \(Z(s)=\operatorname{Lex}'(X(s))\), and let +\(Z\subseteq\mathcal B_r(d)\) be the set with these slices. +By the induction hypothesis and \eqref{box:original-slices}, +\[ + |\partial'Z(s)|\le|\partial'X(s)|\le|X(s-1)|=|Z(s-1)|. +\] +The shadow on the left and the slice on the right are lex segments +in the same degree box. Consequently +\begin{equation} + \label{box:compressed-slices} + \partial'Z(s)\subseteq Z(s-1) + \qquad(1\le s\le b_r). +\end{equation} +Thus a unit in the last coordinate of a member of \(Z\) may be moved +to any earlier coordinate with available capacity, while remaining +in \(Z\). + +\smallskip\noindent +\emph{The zero slice.} +If \(Z(0)=\mathcal B_{r-1}(d)\), then +\(|Y(0)|\le|Z(0)|=|X(0)|\). Otherwise let \(v\) be the largest +vector of \(\mathcal B_{r-1}(d)\) omitted from \(Z(0)\); all members +of \(Z(0)\) are strictly larger than \(v\). + +We claim that every \((u,s)\in Z\) is strictly larger than +\((v,0)\). Equality is excluded by the choice of \(v\). If instead +\((u,s)<(v,0)\), let \(k0\), +since every member of the lex slice \(Z(0)\) is larger than \(v\). +We will +move the \(s\) units of the last coordinate into the first \(r-1\) +coordinates to obtain a vector \(w\le v\). + +Let +\[ + c=\sum_{kc\), fill all those +positions and put the remaining \(s-c\) units at coordinate \(k\). +Since \(|u|+s=|v|=d\) and the prefixes before \(k\) agree, +\begin{equation} + \label{box:tail-filling} + s-c + =v_k-u_k-\sum_{k0}B=0$. +\end{lemma} + +\begin{proof} +Write $d_p(B)_j:K^R_p(B)_j\to K^R_{p-1}(B)_j$ for the Koszul +differential. Since its homology computes $\Tor^R(B,\C)$, +\[ + \beta^R_{p,j}(B) + =\dim_\C K^R_p(B)_j + -\operatorname{rank}d_p(B)_j + -\operatorname{rank}d_{p+1}(B)_j. +\] +This proves the rank formula. The Koszul complex of $U$ embeds +termwise into that of $B$, and the image of each of its differentials +embeds into the corresponding differential image for $B$. +For the quotient $V$, the termwise surjections give +\[ + \operatorname{im}d_p(V)_j + =\text{the image of }\operatorname{im}d_p(B)_j + \text{ in }K^R_{p-1}(V)_j. +\] +Thus each of the two ranks can only decrease under either operation. +Direct sums and grading shifts commute with the Koszul complex. +Finally, if all the variables annihilate $B$, every Koszul differential +is zero. +\end{proof} + +\begin{theorem} +\label{tor:stable} +Let $r\ge1$, let $2\le a_1\le\cdots\le a_r$, and let +$A=\C[x_1,\ldots,x_r]$. Suppose that $H\subseteq A$ is a +monomial ideal containing $(x_1^{a_1},\ldots,x_r^{a_r})$ and is +stable in the box. Let $G=\Lambda H$ be its degreewise lex +replacement inside the box, together with the prescribed pure powers. +Then, for every $p\ge0$ and $j\in\Z$, +\[ + \beta^A_{p,j}(H)\le\beta^A_{p,j}(G), + \qquad + \beta^A_{p,j}(A/H)\le\beta^A_{p,j}(A/G). +\] +\end{theorem} + +\begin{proof} +By Proposition~\ref{box:ideal}, $G$ is an ideal stable in the box +and has the same Hilbert function as $H$. If $H=A$, then $G=A$ +and both comparisons are equalities. For proper $H$, the ideal +comparison implies the quotient comparison in positive homological +degrees by Lemma~\ref{filt:minimal}: +\begin{equation} +\label{tor:ideal-quotient} + \beta^A_{p+1,j}(A/H)=\beta^A_{p,j}(H), + \qquad + \beta^A_{p+1,j}(A/G)=\beta^A_{p,j}(G). +\end{equation} +The two quotient Betti numbers in homological degree zero are both +$1$ when $j=0$ and $0$ otherwise. We prove the two comparisons +simultaneously by induction on $r$, establishing the ideal comparison +and then using~\eqref{tor:ideal-quotient}. When $r=1$, a monomial +ideal containing $x_1^{a_1}$ is a power ideal or the full ring, and +$H=G$. + +Assume $r>1$ and $H$ proper, and put +\[ + R=\C[x_1,\ldots,x_{r-1}],\qquad z=x_r,\qquad a=a_r. +\] +For $s\ge0$, define coefficient ideals +\[ + H^{(s)}=\{u\in R:uz^s\in H\},\qquad + G^{(s)}=\{u\in R:uz^s\in G\}. +\] +These are ascending monomial ideals containing the earlier prescribed +powers, and they equal $R$ for $s\ge a$. For $s0}H^{(s)}\subseteq H^{(s-1)},\qquad + R_{>0}G^{(s)}\subseteq G^{(s-1)}. +\end{equation} +To see the first inclusion, it is enough to multiply a monomial of +$H^{(s)}$ by an earlier variable $x_i$. If its earlier-variable +exponent, or that exponent after multiplication by $x_i$, is outside +the box, the earlier powers already imply the required membership. +Otherwise stability moves one factor of $z$ to $x_i$. The proof +for $G$ is identical. + +We next pass from ideals over $A$ to modules over $R$. Multiplication +by $z$ is injective on $H$, since $H$ is an ideal in the domain $A$. +Set +\[ + B_H=H/zH,\qquad B_G=G/zG. +\] +Reducing a minimal $A$-free resolution of $H$ modulo $z$ gives a +minimal $R$-free resolution of $B_H$. Here is a direct exactness +check. Lift a cycle in positive homological degree modulo $z$ to +an element $y$ of the free resolution, and write $dy=zv$. +Applying the next differential shows that $v$ is a cycle, using +injectivity of $z$ on the free modules; in degree zero, apply the +augmentation and use injectivity of $z$ on $H$ instead. +Exactness gives $v=du$, so $y-zu$ is a cycle and hence a boundary. +Its reduction modulo $z$ is the original cycle. Right exactness +identifies the cokernel with $B_H$, and all differential entries +remain of positive degree or become zero, so minimality is preserved. +The same argument applies to $G$. Therefore +\begin{equation} +\label{tor:reduction} + \beta^A_{p,j}(H)=\beta^R_{p,j}(B_H),\qquad + \beta^A_{p,j}(G)=\beta^R_{p,j}(B_G). +\end{equation} +The injective multiplication maps also give +\[ + \dim_\C(B_H)_d=\dim_\C H_d-\dim_\C H_{d-1},\qquad + \dim_\C(B_G)_d=\dim_\C G_d-\dim_\C G_{d-1}. +\] +Consequently $B_H$ and $B_G$ have equal Hilbert functions. + +The monomial decomposition by powers of $z$ describes these modules +more precisely; this is the decomposition appearing in +\cite[Lemma~3.7]{CS}. The coefficient of $z^0$ in $H/zH$ is $H^{(0)}$, +and its coefficient of $z^s$ for $s\ge1$ is +$H^{(s)}/H^{(s-1)}$. Since $H^{(s)}=R$ for $s\ge a$, this gives +an isomorphism of graded $R$-modules +\begin{equation} +\label{tor:slice-decomposition} + B_H\cong H^{(0)} + \oplus\bigoplus_{s=1}^{a-1} + (H^{(s)}/H^{(s-1)})(-s) + \oplus(R/H^{(a-1)})(-a). +\end{equation} +There are no terms for $s>a$. The shifts record that multiplication +by $z^s$ raises ordinary degree by $s$. By +\eqref{tor:slice-annihilation}, all middle summands are annihilated +by the variables of $R$, as illustrated in +Figure~\ref{fig:endpoint-slices}. The decomposition is $R$-linear +because multiplying by an earlier variable preserves the exponent of +$z$. The middle summands still contribute to $\Tor$, but their +Koszul differentials have rank zero. Thus they contribute nothing to +the deficits. Lemma~\ref{tor:deficit} yields +\begin{equation} +\label{tor:endpoint-deficits} + D^R_{p,j}(B_H) + =D^R_{p,j}(H^{(0)}) + +D^R_{p,j-a}(R/H^{(a-1)}), +\end{equation} +and the identical formula holds for $G$. + +\input{figures/endpoint-slices} + +It remains to compare the two endpoint contributions. Write +$\Lambda'$ for lex replacement with the earlier prescribed powers +in $R$. The induction hypothesis applies to both $H^{(0)}$ and +$H^{(a-1)}$. It gives an ideal comparison for the first and a +quotient comparison for the second. Their respective lex +replacements have the same Hilbert functions, and hence the same +Koszul term dimensions. Subtracting the Betti inequalities from +these dimensions gives +\begin{align} + D^R_{p,j}(H^{(0)}) + &\ge D^R_{p,j}(\Lambda'H^{(0)}), + \label{tor:induction-zero}\\ + D^R_{p,j-a}(R/H^{(a-1)}) + &\ge D^R_{p,j-a}(R/\Lambda'H^{(a-1)}). + \label{tor:induction-top} +\end{align} +This also covers any coefficient ideal equal to $R$: its lex +replacement is $R$, the corresponding ideal comparison is equality, +and its quotient is zero. + +The slice inequalities of Lemma~\ref{box:inequalities} imply +\begin{equation} +\label{tor:endpoint-inclusions} + G^{(0)}\subseteq\Lambda'H^{(0)},\qquad + \Lambda'H^{(a-1)}\subseteq G^{(a-1)}. +\end{equation} +Indeed, fix an earlier-variable degree $e\ge0$. For the first +inclusion, apply the zero-slice inequality to the included exponent +sets of $H$ and $G$ in total degree $e$. For the second, apply +the top-slice inequality in total degree $e+a-1$. Each slice of +$G$ is already a lex segment in the earlier-variable box, so its +cardinality comparison with the corresponding slice of $H$ becomes +the stated containment in the appropriate lex replacement. +All monomials outside the earlier-variable box belong to both ideals. +This proves the inclusions in every degree. + +Apply Lemma~\ref{tor:deficit} to the first inclusion in +\eqref{tor:endpoint-inclusions} and to the quotient map furnished +by the second: +\[ + R/\Lambda'H^{(a-1)}\longrightarrow R/G^{(a-1)}. +\] +Together with~\eqref{tor:induction-zero} and +\eqref{tor:induction-top}, this gives +\begin{align*} + D^R_{p,j}(H^{(0)})&\ge D^R_{p,j}(G^{(0)}),\\ + D^R_{p,j-a}(R/H^{(a-1)}) + &\ge D^R_{p,j-a}(R/G^{(a-1)}). +\end{align*} +Adding and using~\eqref{tor:endpoint-deficits}, we obtain +\[ + D^R_{p,j}(B_H)\ge D^R_{p,j}(B_G). +\] +The equal Hilbert functions of $B_H$ and $B_G$ imply +\[ + \dim_\C K^R_p(B_H)_j + =\binom{r-1}{p}\dim_\C(B_H)_{j-p} + =\dim_\C K^R_p(B_G)_j, +\] +with the binomial coefficient interpreted as zero outside its range. +Subtracting the deficit inequality from these equal dimensions gives +$\beta^R_{p,j}(B_H)\le\beta^R_{p,j}(B_G)$. +Equation~\eqref{tor:reduction} proves the ideal comparison over $A$, +and~\eqref{tor:ideal-quotient} completes the quotient comparison. +Every step applies to every $p\ge0$ and $j\in\Z$; in particular, +negative internal degrees and Koszul terms outside their homological +range introduce no additional cases. +\end{proof} diff --git a/preprints/The-Artinian-Lex-Plus-Powers-Betti-Theorem-September-23-2026/build/sections/07-assembly.tex b/preprints/The-Artinian-Lex-Plus-Powers-Betti-Theorem-September-23-2026/build/sections/07-assembly.tex new file mode 100644 index 0000000000000000000000000000000000000000..104ea56060381765ff84c2a9b923c65f5d0cd4c7 --- /dev/null +++ b/preprints/The-Artinian-Lex-Plus-Powers-Betti-Theorem-September-23-2026/build/sections/07-assembly.tex @@ -0,0 +1,53 @@ +\section{The prescribed lex-plus-powers ideal}\label{sec:assembly} + +We now combine the comparisons, keeping the same ideal at every +homological and internal degree. + +\begin{proof}[Proof of Theorem~\ref{thm:main}] +If \(I=S\), then \(L_d=S_d\) is allowed in every degree and \(J=S\); +all quotient Betti numbers are zero. Assume \(I\ne S\). +Proposition~\ref{mono:bound} gives a monomial ideal \(M\supseteq P\) +with the Hilbert function of \(I\) and +\[ + \beta^S_{p,j}(S/I)\le\beta^S_{p,j}(S/M) + \qquad(p\ge0,\ j\in\Z). +\] +Proposition~\ref{stab:main} gives a box-stable monomial ideal +\(H\supseteq P\), with the same Hilbert function, and +\[ + \beta^S_{p,j}(S/M)\le\beta^S_{p,j}(S/H). +\] +By Proposition~\ref{box:ideal}, the degreewise box-lex replacement +\(G=\Lambda H\) is an ideal, with the same Hilbert function. +Theorem~\ref{tor:stable} gives +\[ + \beta^S_{p,j}(S/H)\le\beta^S_{p,j}(S/G). +\] + +It remains to check that \(G\) is the ideal prescribed in +\eqref{eq:prescription}. +In degree \(d\), list all monomials in decreasing lex order and remove +those belonging to \(P_d\). The remaining list is exactly +\(\mathcal B_n(d)\) in decreasing lex order. +The monomials of \(G_d\) outside \(P_d\) are the first +\[ + q_d=\dim_\C I_d-\dim_\C P_d +\] +positions of this remaining list. This integer lies between zero and +\(|\mathcal B_n(d)|\), because \(H\supseteq P\) has the same Hilbert +function as \(I\). + +Every box prefix is obtained by intersecting some full-list prefix with +the box. Thus a lex-segment subspace \(L_d\) with +\(\dim_\C(L_d+P_d)=\dim_\C I_d\) exists. Enlarging that full-list prefix +to the largest one of the same resulting dimension adds only monomials +already in \(P_d\), and does not change the sum. Consequently +\(L_d+P_d=G_d\) for every \(d\), including the empty and full box cases. +Hence \(J=G\). The displayed comparisons prove all the asserted Betti +bounds for this single \(J\). +\end{proof} + +The argument proves idealness and all syzygy bounds together. +The use of \(\C\) enters the commuting-form construction and supplies the +roots of unity for stabilization; no conclusion in positive +characteristic is asserted for an arbitrary regular sequence. diff --git a/preprints/The-Artinian-Lex-Plus-Powers-Betti-Theorem-September-23-2026/build/sections/08-characteristic-zero.tex b/preprints/The-Artinian-Lex-Plus-Powers-Betti-Theorem-September-23-2026/build/sections/08-characteristic-zero.tex new file mode 100644 index 0000000000000000000000000000000000000000..2d58540b11cf9b8de5ba77741e190c2bdce5c3ca --- /dev/null +++ b/preprints/The-Artinian-Lex-Plus-Powers-Betti-Theorem-September-23-2026/build/sections/08-characteristic-zero.tex @@ -0,0 +1,85 @@ +\section{Arbitrary regular-sequence length and characteristic-zero fields} +\label{sec:characteristic-zero} + +The new input of this paper is the Artinian theorem over \(\C\). We now +apply the published Artinian reduction and record the coefficient-field +argument needed for Corollary~\ref{cor:characteristic-zero}. + +\begin{proof}[Proof of Corollary~\ref{cor:characteristic-zero}] +If \(I=S_F\), the prescription gives \(J=S_F\), and all quotient Betti +numbers vanish. We may therefore assume that \(I\) is proper. + +First let \(F=\C\). After a linear change of coordinates, one may arrange +that +\[ + f_1,\ldots,f_c,x_{c+1},\ldots,x_n +\] +is a regular sequence; the list of variables is empty when \(c=n\). +Write \(\overline f_i\) for the images of the \(f_i\) in +\(\overline S=\C[x_1,\ldots,x_c]\). They form a full regular sequence +of degrees \(a_1,\ldots,a_c\). Theorem~\ref{thm:main}, applied in +\(\overline S\) to every homogeneous ideal containing this sequence, +gives both the Eisenbud--Green--Harris and lex-plus-powers assertions for +\(\overline f_1,\ldots,\overline f_c\). + +The Artinian reduction of Caviglia--Maclagan +\cite[Proposition~10]{CM08} transfers the Hilbert-function assertion to +\(f_1,\ldots,f_c\). Theorem~4.1 of Caviglia--Kummini \cite{CK14} +transfers the Betti assertion as well. Their formulation uses Betti +numbers of ideals. For any proper homogeneous ideal \(K\) in a standard +graded polynomial ring \(A\), the exact sequence +\(0\to K\to A\to A/K\to0\) gives +\[ + \beta^A_{p+1,j}(A/K)=\beta^A_{p,j}(K) + \qquad(p\ge0,\ j\in\Z). +\] +The remaining quotient Betti number is \(\beta^A_{0,0}(A/K)=1\). +Thus their convention is equivalent to the quotient convention used +here. The lexicographic embedding modulo \(P_F\) in that reduction +selects exactly the prescribed \(q_d\) greatest monomials outside \(P_F\) +in each degree. A linear change of coordinates preserves Hilbert functions +and graded Betti numbers, so this proves the corollary over \(\C\). + +Now let \(F\) be any characteristic-zero field. Choose homogeneous +generators \(g_1,\ldots,g_r\) of \(I\), and let \(F_0\subseteq F\) be +the field generated over \(\Q\) by all coefficients of the \(g_i\) and +the \(f_i\). Set +\[ + S_0=F_0[x_1,\ldots,x_n],\qquad + I_0=(g_1,\ldots,g_r,f_1,\ldots,f_c)\subseteq S_0. +\] +Then \(I_0S_F=I\). The sequence \(f_1,\ldots,f_c\) is regular in +\(S_0\): after tensoring each multiplication map with the faithfully +flat extension \(F/F_0\), it is the corresponding injective map over +\(S_F\). The finitely generated characteristic-zero field \(F_0\) +admits an embedding into \(\C\). Extending by this embedding preserves +regularity, so the case just proved applies to +\[ + S_{\C}=\C\otimes_{F_0}S_0,\qquad I_{\C}=I_0S_{\C}. +\] +No embedding of \(F\) into \(\C\) is needed. + +Let \(J_{\C}\) be the resulting lex-plus-powers ideal. It is a monomial +ideal, so let \(J_0\subseteq S_0\) be generated by the same monomials and +put \(J_F=J_0S_F\). For \(E=F\) or \(E=\C\), let +\(S_E=E\otimes_{F_0}S_0\). Graded field base change preserves the +dimensions of homogeneous pieces. It also gives, for \(M_0=S_0/I_0\) +and for \(M_0=S_0/J_0\), +\[ + E\otimes_{F_0}\Tor^{S_0}_p(M_0,F_0)_j + \;\cong\; + \Tor^{S_E}_p(E\otimes_{F_0}M_0,E)_j. +\] +For example, this follows by tensoring the Koszul complex of the +variables with \(E\); its graded homology commutes with this flat +extension. Consequently the Hilbert equalities and Betti inequalities +over \(\C\) give the same equalities and inequalities over \(F\). +Finally, the monomials of \(P_F\) and the lexicographic order are +determined by their exponents, while the dimensions defining \(q_d\) +are unchanged by these field extensions. Hence \(J_F\) is precisely +the degreewise ideal \(J\) prescribed in the corollary. +\end{proof} + +The argument changes the coefficient field only through extensions of +the common characteristic-zero field \(F_0\). It makes no assertion for +arbitrary regular sequences in positive characteristic. diff --git a/preprints/The-Artinian-Lex-Plus-Powers-Betti-Theorem-September-23-2026/paper.pdf b/preprints/The-Artinian-Lex-Plus-Powers-Betti-Theorem-September-23-2026/paper.pdf new file mode 100644 index 0000000000000000000000000000000000000000..cc9bd52fba7e7da97d2537cc37f5b2daefbfab88 --- /dev/null +++ b/preprints/The-Artinian-Lex-Plus-Powers-Betti-Theorem-September-23-2026/paper.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:81874e1994ff2773543f714989feac78650ddc902f7f36047c6ce8d3288a9c05 +size 447798 diff --git a/preprints/The-Bass-trace-conjecture-for-complex-group-rings-September-24-2026/README.md b/preprints/The-Bass-trace-conjecture-for-complex-group-rings-September-24-2026/README.md new file mode 100644 index 0000000000000000000000000000000000000000..71eafda7e1ddf3bbf45d1ce098137ecf124a714e --- /dev/null +++ b/preprints/The-Bass-trace-conjecture-for-complex-group-rings-September-24-2026/README.md @@ -0,0 +1,18 @@ +# [The Bass trace conjecture and the characteristic-zero Kaplansky idempotent conjecture](The-Bass-trace-conjecture-for-complex-group-rings-September-24-2026.pdf) + +**Author:** OpenAI + +**Date:** September 24, 2026 + +## Citation + +```bibtex +@misc{OAI:The-Bass-trace-conjecture-for-complex-group-rings-September-24-2026, + author = {{OpenAI}}, + title = {{The Bass trace conjecture and the characteristic-zero Kaplansky idempotent conjecture}}, + howpublished = {OpenAI Math Release preprint + \href{https://github.com/openai/math/blob/main/preprints/The-Bass-trace-conjecture-for-complex-group-rings-September-24-2026/The-Bass-trace-conjecture-for-complex-group-rings-September-24-2026.pdf}{OAI:The-Bass-trace-conjecture-for-complex-group-rings-September-24-2026}}, + year = {2026} +} +``` + diff --git a/preprints/The-Bass-trace-conjecture-for-complex-group-rings-September-24-2026/The-Bass-trace-conjecture-for-complex-group-rings-September-24-2026.pdf b/preprints/The-Bass-trace-conjecture-for-complex-group-rings-September-24-2026/The-Bass-trace-conjecture-for-complex-group-rings-September-24-2026.pdf new file mode 100644 index 0000000000000000000000000000000000000000..36f654acd01c7cf81539a6d450a0c44e5201e3bc --- /dev/null +++ b/preprints/The-Bass-trace-conjecture-for-complex-group-rings-September-24-2026/The-Bass-trace-conjecture-for-complex-group-rings-September-24-2026.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:1c4d90b5ed4de2ef66b4b07b3d5c3e2bfe8822a538177c64ee1d3c3e091120bf +size 457936 diff --git a/preprints/The-Bass-trace-conjecture-for-complex-group-rings-September-24-2026/build/main.tex b/preprints/The-Bass-trace-conjecture-for-complex-group-rings-September-24-2026/build/main.tex new file mode 100644 index 0000000000000000000000000000000000000000..8df4bcc4ee73329437cd4d551bdda445f6462fcd --- /dev/null +++ b/preprints/The-Bass-trace-conjecture-for-complex-group-rings-September-24-2026/build/main.tex @@ -0,0 +1,63 @@ +\documentclass[11pt]{article} +\usepackage[T1]{fontenc} +\usepackage{lmodern} +\usepackage[margin=1.05in]{geometry} +\usepackage{amsmath,amssymb,amsthm,mathtools} +\usepackage{microtype,enumitem,booktabs} +\usepackage{tikz} +\usetikzlibrary{arrows.meta,positioning} +\usepackage[colorlinks=true,linkcolor=blue!45!black,citecolor=blue!45!black,urlcolor=blue!45!black]{hyperref} +\usepackage[nameinlink,noabbrev]{cleveref} +\newtheorem{theorem}{Theorem}[section] +\newtheorem{lemma}[theorem]{Lemma} +\newtheorem{proposition}[theorem]{Proposition} +\newtheorem{corollary}[theorem]{Corollary} +\theoremstyle{definition} +\newtheorem{definition}[theorem]{Definition} +\theoremstyle{remark} +\newtheorem{remark}[theorem]{Remark} +\newcommand{\C}{\mathbb C} +\newcommand{\R}{\mathbb R} +\newcommand{\Z}{\mathbb Z} +\newcommand{\N}{\mathbb N} +\DeclareMathOperator{\HS}{HS} +\DeclareMathOperator{\HH}{HH} +\DeclareMathOperator{\tr}{tr} +\DeclareMathOperator{\supp}{supp} +\DeclareMathOperator{\con}{con} +\DeclareMathOperator{\id}{id} +\numberwithin{equation}{section} +\setlist{itemsep=3pt,topsep=5pt} +\setlength{\emergencystretch}{1.5em} +\pdfinfoomitdate=1 +\pdftrailerid{} +\pdfsuppressptexinfo=15 +\hypersetup{pdftitle={The Bass trace conjecture and the characteristic-zero Kaplansky idempotent conjecture},pdfauthor={OpenAI}} +\title{The Bass trace conjecture and the characteristic-zero Kaplansky idempotent conjecture} +\author{OpenAI} +\date{September 24, 2026} +\begin{document} +\maketitle +\begin{abstract} +We prove the complex group-ring Bass trace conjecture for every discrete +group: the Hattori--Stallings trace of a finitely generated projective module +over its complex group ring is supported on conjugacy classes of finite-order +elements. As a consequence, for every torsion-free group $G$ and every +commutative unital domain $R$ of characteristic zero, the only idempotents +in $RG$ are $0$ and $1$. This proves Kaplansky's idempotent conjecture in +characteristic zero. +\end{abstract} +\tableofcontents +\input{sections/01-introduction} +\input{sections/02-framework} +\input{sections/03-metric} +\input{sections/04-filtrations} +\input{sections/05-surgery} +\input{sections/06-chains} +\input{sections/07-idempotents} +\input{sections/08-detector} +\input{sections/09-completion} +\input{sections/10-torsion-free} +\bibliographystyle{plain} +\bibliography{references} +\end{document} diff --git a/preprints/The-Bass-trace-conjecture-for-complex-group-rings-September-24-2026/build/references.bib b/preprints/The-Bass-trace-conjecture-for-complex-group-rings-September-24-2026/build/references.bib new file mode 100644 index 0000000000000000000000000000000000000000..188eb206693f202b9579d10381757fa086194665 --- /dev/null +++ b/preprints/The-Bass-trace-conjecture-for-complex-group-rings-September-24-2026/build/references.bib @@ -0,0 +1,267 @@ +@article{Bass1976, + author = {Bass, Hyman}, + title = {{Euler} characteristics and characters of discrete groups}, + journal = {Inventiones Mathematicae}, + volume = {35}, + year = {1976}, + pages = {155--196}, + doi = {10.1007/BF01390137} +} + +@incollection{Bass1979, + author = {Bass, Hyman}, + title = {Traces and {Euler} characteristics}, + editor = {Wall, C. T. C.}, + booktitle = {Homological Group Theory}, + series = {London Mathematical Society Lecture Note Series}, + volume = {36}, + publisher = {Cambridge University Press}, + address = {Cambridge}, + year = {1979}, + pages = {1--26}, + doi = {10.1017/CBO9781107325449.003} +} + +@article{Hattori1965, + author = {Hattori, Akira}, + title = {Rank element of a projective module}, + journal = {Nagoya Mathematical Journal}, + volume = {25}, + year = {1965}, + pages = {113--120}, + doi = {10.1017/S002776300001148X} +} + +@article{Stallings1965, + author = {Stallings, John}, + title = {Centerless groups---an algebraic formulation of {Gottlieb}'s theorem}, + journal = {Topology}, + volume = {4}, + number = {2}, + year = {1965}, + pages = {129--134}, + doi = {10.1016/0040-9383(65)90060-1} +} + +@article{BCM2004, + author = {Berrick, A. J. and Chatterji, Indira and Mislin, Guido}, + title = {From acyclic groups to the {Bass} conjecture for amenable groups}, + journal = {Mathematische Annalen}, + volume = {329}, + number = {4}, + pages = {597--621}, + year = {2004}, + doi = {10.1007/s00208-004-0521-6}, + url = {https://arxiv.org/abs/1004.1941v1} +} + +@article{BCM2006, + author = {Berrick, A. J. and Chatterji, Indira and Mislin, Guido}, + title = {Homotopy idempotents on manifolds and {Bass}' conjectures}, + journal = {Geometry \& Topology Monographs}, + volume = {10}, + year = {2007}, + pages = {41--62}, + doi = {10.2140/gtm.2007.10.41}, + url = {https://arxiv.org/abs/0903.4341v1} +} + +@incollection{CM2011, + author = {Chatterji, Indira and Mislin, Guido}, + title = {{Hattori--Stallings} trace and {Euler} characteristics for groups}, + booktitle = {Geometric and Cohomological Methods in Group Theory}, + editor = {Bridson, Martin R. and Kropholler, Peter H. and Leary, Ian J.}, + series = {London Mathematical Society Lecture Note Series}, + volume = {358}, + pages = {256--271}, + publisher = {Cambridge University Press}, + year = {2009}, + doi = {10.1017/CBO9781139107099.007}, + url = {https://arxiv.org/abs/0910.4419v1} +} + +@article{Linnell1983, + author = {Linnell, P. A.}, + title = {Decomposition of augmentation ideals and relation modules}, + journal = {Proceedings of the London Mathematical Society}, + series = {3}, + volume = {47}, + number = {1}, + year = {1983}, + pages = {83--127}, + doi = {10.1112/plms/s3-47.1.83} +} + +@article{BH2015, + author = {Berrick, A. J. and Hesselholt, Lars}, + title = {Topological {Hochschild} homology and the {Bass} trace conjecture}, + journal = {J. reine angew. Math.}, + volume = {704}, + year = {2015}, + pages = {169--185}, + doi = {10.1515/crelle-2013-0051}, + url = {https://arxiv.org/abs/1305.6438v2} +} + +@article{BV1998, + author = {Burger, Marc and Valette, Alain}, + title = {Idempotents in complex group rings: theorems of {Zalesskii} and {Bass} revisited}, + journal = {Journal of Lie Theory}, + volume = {8}, + number = {2}, + year = {1998}, + pages = {219--228}, + doi = {10.5802/jolt.142}, + url = {https://jolt.centre-mersenne.org/articles/10.5802/jolt.142/} +} + +@article{OW2023, + author = {{\"O}inert, Johan and Wagner, Stefan}, + title = {Complex group rings and group {$C^*$}-algebras of group extensions}, + journal = {Journal of Algebraic Combinatorics}, + volume = {58}, + year = {2023}, + pages = {387--397}, + doi = {10.1007/s10801-022-01183-6} +} + +@article{Connes1985, + author = {Connes, Alain}, + title = {Non-commutative differential geometry}, + journal = {Publications Math\'ematiques de l'IH\'ES}, + volume = {62}, + year = {1985}, + pages = {41--144}, + doi = {10.1007/BF02698807} +} + +@article{Burghelea1985, + author = {Burghelea, Dan}, + title = {The cyclic homology of the group rings}, + journal = {Commentarii Mathematici Helvetici}, + volume = {60}, + year = {1985}, + pages = {354--365}, + doi = {10.1007/BF02567420} +} + +@article{Eckmann1986, + author = {Eckmann, Beno}, + title = {Cyclic homology of groups and the {Bass} conjecture}, + journal = {Commentarii Mathematici Helvetici}, + volume = {61}, + year = {1986}, + pages = {193--202}, + doi = {10.1007/BF02621911} +} + +@misc{Alpert2025, + author = {Alpert, Hannah}, + title = {Simplicial volume and 0-strata of separating filtrations}, + year = {2024}, + eprint = {2211.06362}, + archivePrefix = {arXiv}, + primaryClass = {math.DG}, + note = {arXiv:2211.06362v3}, + url = {https://arxiv.org/abs/2211.06362v3} +} + +@article{Papasoglu2020, + author = {Papasoglu, Panos}, + title = {Uryson width and volume}, + journal = {Geometric and Functional Analysis}, + volume = {30}, + number = {2}, + pages = {574--587}, + year = {2020}, + doi = {10.1007/s00039-020-00533-5} +} + +@article{BraunSauer2021, + author = {Braun, Sabine and Sauer, Roman}, + title = {Volume and macroscopic scalar curvature}, + journal = {Geometric and Functional Analysis}, + volume = {31}, + number = {6}, + pages = {1321--1376}, + year = {2021}, + doi = {10.1007/s00039-021-00588-y} +} + + +@article{Formanek1973, + author = {Formanek, Edward}, + title = {Idempotents in {Noetherian} group rings}, + journal = {Canadian Journal of Mathematics}, + volume = {25}, + number = {2}, + year = {1973}, + pages = {366--369}, + doi = {10.4153/CJM-1973-037-6} +} + +@article{FarrellLinnell2003, + author = {Farrell, F. Thomas and Linnell, Peter A.}, + title = {Whitehead groups and the {Bass} conjecture}, + journal = {Mathematische Annalen}, + volume = {326}, + year = {2003}, + pages = {723--757}, + doi = {10.1007/s00208-003-0424-y} +} + +@article{BartelsLuckReich2008, + author = {Bartels, Arthur and L{\"u}ck, Wolfgang and Reich, Holger}, + title = {On the {Farrell--Jones} Conjecture and its applications}, + journal = {Journal of Topology}, + volume = {1}, + number = {1}, + year = {2008}, + pages = {57--86}, + doi = {10.1112/jtopol/jtm008} +} + +@article{MineyevYu2002, + author = {Mineyev, Igor and Yu, Guoliang}, + title = {The {Baum--Connes} conjecture for hyperbolic groups}, + journal = {Inventiones Mathematicae}, + volume = {149}, + number = {1}, + year = {2002}, + pages = {97--122}, + doi = {10.1007/s002220200214} +} + +@article{Puschnigg2002, + author = {Puschnigg, Michael}, + title = {The {Kadison--Kaplansky} conjecture for word-hyperbolic groups}, + journal = {Inventiones Mathematicae}, + volume = {149}, + number = {1}, + year = {2002}, + pages = {153--194}, + doi = {10.1007/s002220200216} +} + +@article{Kantorovich2006, + author = {Kantorovich, L. V.}, + title = {On the translocation of masses}, + journal = {Journal of Mathematical Sciences}, + volume = {133}, + number = {4}, + year = {2006}, + pages = {1381--1382}, + doi = {10.1007/s10958-006-0049-2}, + note = {Translation of the 1942 original} +} + +@article{HjorthMolberg2006, + author = {Hjorth, Greg and Molberg, Mats}, + title = {Free continuous actions on zero-dimensional spaces}, + journal = {Topology and its Applications}, + volume = {153}, + number = {7}, + year = {2006}, + pages = {1116--1131}, + doi = {10.1016/j.topol.2005.03.003} +} diff --git a/preprints/The-Bass-trace-conjecture-for-complex-group-rings-September-24-2026/build/sections/01-introduction.tex b/preprints/The-Bass-trace-conjecture-for-complex-group-rings-September-24-2026/build/sections/01-introduction.tex new file mode 100644 index 0000000000000000000000000000000000000000..d2517f194d204700451c132684e158548e4c9412 --- /dev/null +++ b/preprints/The-Bass-trace-conjecture-for-complex-group-rings-September-24-2026/build/sections/01-introduction.tex @@ -0,0 +1,286 @@ +\section{Introduction} +\label{sec:introduction} + +The rank of a projective module over a group ring contains information +about individual conjugacy classes. This refinement of ordinary rank is +a basic ingredient of group-ring Euler characteristics: alternating sums +of projective-module ranks retain information that a numerical Euler +characteristic forgets \cite{Bass1976,Bass1979}. The Bass trace conjecture +asks how much of that information can occur at elements of infinite order. +We first specify the complex version of this question. + +For a unital complex algebra $A$, let $K_0(A)$ be the Grothendieck group of +finitely generated projective right $A$-modules, and set +\[ + \HH_0(A)=A/[A,A], +\] +where $[A,A]$ is the complex linear span of commutators. An idempotent +$e=(e_{ij})\in M_q(A)$ represents the projective right module +$eA^q$, with matrices acting on columns by left multiplication. Its +Hattori--Stallings trace is +\[ + \HS_A([eA^q])=\left[\sum_{i=1}^q e_{ii}\right]\in\HH_0(A), +\] +extended additively to virtual projective classes. + +Let $G$ be a discrete group and write $\con(G)$ for its set of conjugacy +classes. The quotient $\HH_0(\C G)$ identifies with +$\bigoplus_{C\in\con(G)}\C C$ by +\begin{equation}\label{eq:class-trace} + \left[\sum_{h\in G}a_hh\right] + \longmapsto + \sum_{C\in\con(G)}\left(\sum_{h\in C}a_h\right)C. +\end{equation} +Indeed, a commutator has zero sum on each conjugacy class, and the +differences of conjugate group elements span the kernel of the displayed +map. Such differences are commutators: $tst^{-1}-s=[t,st^{-1}]$. +All sums arising from an element of $\C G$ have finite support. +We write $\HS_G(x)(C)$ for the coefficient at $C$. + +The complex group-ring Bass conjecture asks whether these coefficients +vanish on every infinite-order conjugacy class, for every group and every +class in $K_0(\C G)$; see +\cite[Conjecture 2.1]{BCM2004} and \cite[Conjecture 2]{CM2011}. + +\begin{theorem}\label{thm:bass} +Let $G$ be any discrete group. For every $x\in K_0(\C G)$ and every +infinite-order element $g\in G$, +\[ + \HS_G(x)([g]_G)=0. +\] +Equivalently, if $\con_{\mathrm{fin}}(G)$ denotes the conjugacy classes of +finite-order elements, then +\[ + \HS_G\bigl(K_0(\C G)\bigr) + \subseteq\bigoplus_{C\in\con_{\mathrm{fin}}(G)}\C C. +\] +\end{theorem} + +This resolves the complex group-ring formulation positively. Finite-order +coefficients need not vanish away from the identity: for a finite group +$F$, the idempotent $|F|^{-1}\sum_{h\in F}h$ has coefficient +$|C|/|F|$ at every conjugacy class $C$. The conclusion therefore differs +from the identity-only support assertion for integral group rings. +That integral assertion follows in Section~\ref{sec:completion} by combining +Theorem~\ref{thm:bass} with the known restriction on its finite-order +coefficients. + +\subsection{Trace rank and the idempotent conjecture} + +For a torsion-free group the identity is the only finite-order element, +so Theorem~\ref{thm:bass} also has numerical and ring-theoretic +consequences. Define the canonical trace and the augmentation by +\[ + \tau_G\!\left(\sum_{g\in G}a_g g\right)=a_1,\qquad + \epsilon_G\!\left(\sum_{g\in G}a_g g\right)=\sum_{g\in G}a_g. +\] +For a matrix $b=(b_{ij})\in M_q(\C G)$, put +$\tau_{G,q}(b)=\sum_{i=1}^q\tau_G(b_{ii})$, and let +$\epsilon_{G,q}(b)\in M_q(\C)$ be the entrywise augmentation. +The coefficient at $[1]_G$ of the Hattori--Stallings trace defines +\[ + \tau_{G,*}:K_0(\C G)\longrightarrow\C,\qquad + \tau_{G,*}([e(\C G)^q])=\tau_{G,q}(e). +\] +The augmentation also defines the integer-valued augmentation rank +\[ + \operatorname{rk}_{\epsilon}:K_0(\C G) + \xrightarrow{(\epsilon_G)_*}K_0(\C) + \xrightarrow{\dim_{\C}}\Z. +\] + +The idempotent conjecture commonly attributed to Kaplansky asks whether, +for every torsion-free group $G$ and every field $k$, an element +$e\in kG$ satisfying $e^2=e$ must be $0$ or $1$; see +\cite[Problem 3]{OW2023}. It concerns elements of the group ring itself, +not arbitrary idempotent matrices over that ring. + +\begin{corollary}\label{cor:torsion-free} +Let $G$ be any torsion-free discrete group. +\begin{enumerate}[label=\textup{(\roman*)}] + \item For every $q\geq1$ and every idempotent $e\in M_q(\C G)$, + \[ + \tau_{G,q}(e)=\operatorname{rank}_{\C}\bigl(\epsilon_{G,q}(e)\bigr). + \] + Consequently $\tau_{G,*}=\operatorname{rk}_{\epsilon}$, with + $\Z$ viewed as a subgroup of $\C$, and + \[ + \tau_{G,*}\bigl(K_0(\C G)\bigr)=\Z. + \] + \item For every commutative unital domain $R$ of characteristic zero, + every $e\in RG$ satisfying $e^2=e$ is either $0$ or $1$. +\end{enumerate} +\end{corollary} + +Part \textup{(ii)} resolves the characteristic-zero Kaplansky idempotent +conjecture positively, and includes coefficient domains as well as fields. +Section~\ref{sec:torsion-free} derives both parts from the trace theorem. +The integer range in part \textup{(i)} concerns algebraic $K_0(\C G)$; +its relation to the reduced $C^*$-completion is made explicit there. + +The integral trace consequence also links the theorem to fixed-point +theory. A map $f$ is a \emph{homotopy idempotent} if $f\circ f\simeq f$. +Berrick, Chatterji and Mislin's equivalences \cite[Theorems 1--2]{BCM2006} +then imply that every homotopy idempotent on a connected closed smooth +oriented manifold of dimension greater than two is freely homotopic to +a map with exactly one fixed point. They also identify ordinary and +$L^2$-Lefschetz numbers for pointed homotopy idempotents of connected +closed smooth oriented manifolds that induce the identity on the +fundamental group. +Corollary~\ref{cor:homotopy-idempotents} states these consequences precisely; +Section~\ref{sec:completion} explains their connection to projective +Euler characteristics. + +\subsection{The two parts of the proof} + +We first describe the algebraic construction to explain the geometric +statement it needs. The body of the paper begins with that geometric +statement, formulated independently of group rings, and completes its +proof before constructing the cycles and cocycles to which it applies. + +For a finitely generated group $G$, the algebraic construction starts +with an idempotent $e\in M_q(\C G)$ and one infinite-order element +$g\in G$. The finitely many nonzero group coefficients of $e$ determine +a finite set of steps. Products of matrix +coefficients along paths from $h$ to $gh$, followed by the ordinary +matrix trace, give complex weights on finite lists of group elements. +Idempotency says that summing over any intermediate vertex removes that +vertex. Cyclicity of matrix trace gives a second identity, a cyclic shift +whose last vertex is multiplied by $g$. + +Project the vertices to the coset set $Y=\langle g\rangle\backslash G$. +The group $H=C_G(g)/\langle g\rangle$, where $C_G(g)$ is the centralizer +of $g$, acts freely on $Y$. For every positive even dimension $2m$, the weighted +lists define a simplicial cycle modulo this action. Its simplices use +connected sets of vertices in one graph of bounded degree, fixed as $m$ +increases. A Pfaffian cocycle evaluates on this cycle as +$2^{-m}\HS_G([e(\C G)^q])([g]_G)$. Here a cocycle means an alternating +function on vertex lists that vanishes on simplicial boundaries. The +explicit construction and the evaluation occupy +Sections~\ref{sec:idempotents} and~\ref{sec:detector}. + +The geometric part, proved first in Sections~\ref{sec:framework}--\ref{sec:chains}, +shows that every invariant cocycle evaluates to zero on cycles of this +kind in sufficiently high dimension. We cut the supporting simplicial +complex successively into pieces of bounded diameter, choosing the +separating sets nearly to minimize their areas. A volume estimate and +repeated slicing make the expected number of points in the final +zero-dimensional set arbitrarily small. A subdivision construction turns +this point count into a bound for cocycle evaluations. Equivariant chain +homotopy keeps the evaluation unchanged throughout the construction. + +The fixed graph degree makes increasing the dimension useful. Through +each vertex there are only exponentially many connected vertex sets of a +given size. On an $n$-simplex, Euclidean structures chosen from spanning +trees of its vertex set give volume $1/n!$; repeated slicing produces the +same factorial in the comparison with a point of the final separating +set. After these factors cancel, radii chosen in terms of the fixed +degree bound make the remaining volume ratio tend to zero with $n$. +The local replacement argument turns this strict volume margin into the +small expected point count. No homological-dimension bound on $H$ is +assumed. + +Two features permit this argument for arbitrary groups. A compact +probability space with a free action separates overlapping translates of +local modifications, without any amenability assumption. Also, each top +simplex has its own averaged Euclidean area functional. The areas need +not match along common faces; the successive cuts keep the relevant +dimensional measures away from those faces. +The cocycles need not be globally bounded: near each original simplex, +the construction uses only a fixed finite set of possible vertices. +Bounding the cocycle on those lists suffices for the evaluation estimate. + +The final assembly in Section~\ref{sec:completion} chooses a large even +dimension, forcing the individual trace coefficient to vanish. Passing +from finitely generated groups to arbitrary groups uses the exact sum +over subgroup conjugacy classes contained in an ambient conjugacy class. +Section~\ref{sec:torsion-free} then extracts the augmentation rank from +the identity coefficient. In the scalar case this rank is $0$ or $1$, +and the standard faithful-trace argument forces the idempotent itself +to be $0$ or $1$. A finite-coefficient reduction extends that conclusion +from complex coefficients to every commutative characteristic-zero domain. + +\subsection{Relation to previous work} + +The rank construction comes from Hattori and Stallings +\cite{Hattori1965,Stallings1965}; Bass developed its relation to Euler +characteristics and formulated the integral conjecture in +\cite{Bass1976}; see also \cite[Conjecture (4.5)]{Bass1979}. +Bass proved complex trace vanishing at infinite-order elements for linear +groups \cite[Theorem (6.1)(f)]{Bass1979}. For the integral trace, +Linnell established a restriction on the possible nonzero coefficients +\cite[Lemma 4.1]{Linnell1983}, later recovered through topological +Hochschild homology by Berrick and Hesselholt \cite[Theorem A]{BH2015}. +Its finite-order consequence is the input used in +Section~\ref{sec:completion}. + +The scalar idempotent problem also has a long history of trace methods. +Formanek proved the characteristic-zero conclusion for torsion-free +groups satisfying the ascending chain condition on subgroups +\cite[Theorem 1]{Formanek1973}. Burger and Valette revisit the theorems of +Kaplansky and Zalesskii: the canonical trace of a complex scalar +idempotent lies in $[0,1]$, its endpoints force the idempotent to be +trivial, and its value is rational +\cite[Theorems 1.3 and 2.1]{BV1998}. They also obtain +$\tau_{G,*}(K_0(\C G))\subseteq\mathbb Q$ +\cite[Corollary 1.2]{BV1998}. Rationality still permits nonintegral +values strictly between zero and one. For torsion-free groups, +Theorem~\ref{thm:bass} identifies the trace with augmentation rank, +which excludes those values. The subsequent faithful-trace argument +is classical and is included in Section~\ref{sec:torsion-free}. + +Broader group classes have been treated by algebraic and analytic +methods. Farrell and Linnell proved the complex Bass statement for +elementary amenable groups as part of a result over integral domains +\cite[Theorem 1.6]{FarrellLinnell2003}. Berrick, Chatterji and Mislin +obtained it for amenable groups through rational surjectivity of the +degree-zero Bost assembly map for countable groups +\cite[Theorems 1.3--1.4 and Lemma 1.5]{BCM2004}. Their result concerns +the stronger trace assertion on $\ell^1(G)$, with finite support on +finite-order classes. An algebraic route uses induction from finite +subgroups: the Farrell--Jones conjecture with complex coefficients +implies the complex Bass statement +\cite[Theorems 1.5 and 1.9]{BartelsLuckReich2008}. +The present proof uses the finite support of group-ring idempotents +throughout; it requires neither an assembly hypothesis nor a statement +about a completed group algebra. + +For some groups, analytic methods give a stronger idempotent conclusion +in the reduced group $C^*$-algebra $C_r^*(G)$, the operator-norm closure +of $\C G$ acting on $\ell^2(G)$. Using Lafforgue's Banach $KK$-theory, +Mineyev and Yu proved this for torsion-free subgroups of hyperbolic groups +\cite[Theorems 19--21]{MineyevYu2002}. Puschnigg independently obtained +the torsion-free hyperbolic case through local cyclic homology +\cite{Puschnigg2002}; see also \cite[Section 6]{MineyevYu2002}. +Our conclusion for arbitrary torsion-free groups concerns the group ring +and the trace on its algebraic $K_0$, as distinguished in +Section~\ref{sec:torsion-free}. + +The higher-dimensional algebraic construction has antecedents in the +cyclic-homological approach to the conjecture. Connes constructed a pairing +of even cyclic cocycles with projective classes +\cite[Part II, \S2, Proposition 14]{Connes1985}. Burghelea's decomposition +of cyclic homology by conjugacy classes identifies the infinite-order +components through the reduced centralizers $C_G(g)/\langle g\rangle$ +\cite[Proposition 1.8 and Theorem I]{Burghelea1985}. Eckmann used +higher-degree vanishing to obtain trace vanishing over rational group +rings for specified classes of groups of finite rational homological +dimension \cite[\S3, Theorems 3.1 and 3.3]{Eckmann1986}. +Sections~\ref{sec:idempotents}--\ref{sec:detector} give direct finite-chain +constructions in this tradition, including the normalization of the +Pfaffian detector; no cyclic-homology theorem is needed as an input. +The sparse-chain theorem replaces a homological-dimension hypothesis +with connected-support control on the particular cycles. + +The geometric proof develops the separating-set method of Papasoglu +\cite[\S2, especially Lemma 2.5]{Papasoglu2020}, the separating filtrations and simplex-counting +argument of Alpert \cite[Lemmas 4, 6--7]{Alpert2025}, and the +Cantor-parameter approach of Braun and Sauer +\cite[\S\S2, 4--5]{BraunSauer2021}. Alpert's equivariant formulations +\cite[\S4, especially Lemmas 11--12]{Alpert2025} provide the closest model +for the local comparisons. Alpert's filtration results concern manifold +geometry and compatible metrics. We give the full piecewise-linear argument needed +for simplex-dependent areas and arbitrary cycles in the coinvariant +chain complex. The separate Pfaffian calculation converts this +geometric vanishing into a statement about each conjugacy-class +coefficient. diff --git a/preprints/The-Bass-trace-conjecture-for-complex-group-rings-September-24-2026/build/sections/02-framework.tex b/preprints/The-Bass-trace-conjecture-for-complex-group-rings-September-24-2026/build/sections/02-framework.tex new file mode 100644 index 0000000000000000000000000000000000000000..9d631e6a5334ac0e6d03ddf2ec3167db78382a73 --- /dev/null +++ b/preprints/The-Bass-trace-conjecture-for-complex-group-rings-September-24-2026/build/sections/02-framework.tex @@ -0,0 +1,75 @@ +\section{A vanishing theorem for sparse simplicial cycles} +\label{sec:framework} + +We formulate the geometric statement independently of group rings. +Its input is a graph with a free group action and a simplicial complex +whose top-dimensional vertex sets are connected in that graph. Its output +is the vanishing of every invariant cocycle evaluation in sufficiently +high dimension. + +For a set $Y$, let $\C^{(Y)}$ be the vector space with basis +$\{\delta_y:y\in Y\}$, consisting of finite linear combinations. The +augmented chains of the full simplex on $Y$ are +\[ + E_j=\bigwedge^{j+1}\C^{(Y)}\quad(j\geq0),\qquad E_{-1}=\C. +\] +The differential is contraction with the functional +$\alpha(\sum_y c_y\delta_y)=\sum_y c_y$: +\begin{equation}\label{eq:boundary} + \partial(\delta_{y_0}\wedge\cdots\wedge\delta_{y_j}) + =\sum_{i=0}^{j}(-1)^i + \delta_{y_0}\wedge\cdots\wedge\widehat{\delta_{y_i}} + \wedge\cdots\wedge\delta_{y_j}. +\end{equation} +In degree zero this is the augmentation to $\C$; the empty wedge is $1$. +For a simplicial subcomplex $K$, its augmented oriented chains +$C_*(K)$ form a subcomplex of $E_*$. + +If a group $H$ acts on $Y$, it acts on both chain complexes by permuting +vertices, with the signs prescribed by the exterior product. For an +$H$-module $V$, its coinvariants are +\[ + V_H=V/\operatorname{span}_{\C}\{hv-v:h\in H,\ v\in V\}. +\] +The differentials induce differentials on coinvariants. A cycle in +$C_n(K)_H$ is a class whose boundary is zero there; an arbitrary finite +chain representing it need not have zero boundary before passage to +coinvariants. An $H$-invariant linear functional $\psi:E_n\to\C$ +descends to $(E_n)_H$. We call it a cocycle when +$\psi\partial=0$ on $E_{n+1}$. + +\begin{theorem}[Sparse-chain vanishing]\label{thm:sparse} +For each integer $D\geq2$ there is an integer $n_0(D)$ with the following +property. Let $Y$ be the vertex set of a countable connected simple graph +of degree at most $D$, and let a countable group $H$ act by graph +automorphisms, freely on vertices. Suppose $n\geq n_0(D)$ and $K$ is an +$H$-invariant subcomplex of the full simplex on $Y$ such that: +\begin{enumerate}[label=\textup{(\roman*)}] + \item $K$ is generated by its $n$-simplices and has finitely many + $H$-orbits of simplices; + \item the vertices of each $n$-simplex span a connected subgraph of $Y$. +\end{enumerate} +Then every $H$-invariant cocycle $\psi:E_n\to\C$ evaluates to zero on +the image in $(E_n)_H$ of every cycle in $C_n(K)_H$. +\end{theorem} + +The bound $n_0(D)$ depends only on the graph degree, not on $H$, $K$ or +$\psi$. There is no boundedness hypothesis on $\psi$. We will bound it +only on a finite set of basis wedges associated with each fixed original +simplex. This distinction is necessary in the application. + +For the proof assume $K\ne\varnothing$; the empty case has no positive +degree chains. Its realization $|K|$ consists of finitely supported +probability vectors $p:Y\to[0,1]$ whose supports are simplices of $K$. +A simplex will also denote its closed affine realization; $\sigma^\circ$ +denotes its relative interior. We call simplices of $K$ \emph{original} +to distinguish them from the later subdivisions, and call its +$n$-simplices \emph{top simplices}. + +The proof has three steps. First, we choose a metric and simplexwise +areas for which balls have small volume in high dimension. +Second, we construct separating filtrations with very few points in +their final stratum, on average over a free probability parameter. +Third, we use their subdivisions to construct equivariant chain maps +homotopic to inclusion whose cocycle evaluations approach zero. Each +approximation uses a finite filtration; no limiting filtration is needed. diff --git a/preprints/The-Bass-trace-conjecture-for-complex-group-rings-September-24-2026/build/sections/03-metric.tex b/preprints/The-Bass-trace-conjecture-for-complex-group-rings-September-24-2026/build/sections/03-metric.tex new file mode 100644 index 0000000000000000000000000000000000000000..3199f4544d69d395b51e9a569734d8127e737660 --- /dev/null +++ b/preprints/The-Bass-trace-conjecture-for-complex-group-rings-September-24-2026/build/sections/03-metric.tex @@ -0,0 +1,262 @@ +\section{A metric and volume bound for connected supports} +\label{sec:metric} + +We work with the graph \(Y\), group \(H\), and nonempty +\(n\)-dimensional complex \(K\) of Theorem~\ref{thm:sparse}, where +\(n\geq1\). We need two kinds of geometric control: a metric with +finite neighborhoods in which to make cuts, and Euclidean area +functionals for which those cuts satisfy a slicing inequality. +We will use a scaled transportation metric on \(|K|\) and, +separately inside each top simplex, Euclidean structures determined +by its spanning trees. Each top simplex will have volume \(1/n!\), +while every affine branch of a distance function will have gradient +norm at most one in each of these structures. Counting the possible +connected vertex supports will then give the ball-volume estimate +used in the separating argument. + +\subsection{The metric and its finite local geometry} + +Represent a point of \(|K|\) by a probability vector \(p\) on \(Y\) whose +support lies in a simplex of \(K\). Let \(d_Y\) denote graph distance and +define +\begin{equation} + \rho(p,q)=\frac1{\sqrt n} + \sup_{\operatorname{Lip}_{d_Y}(\ell)\leq1} + \sum_{v\in Y}\ell(v)\bigl(p(v)-q(v)\bigr). + \label{eq:transport-metric} +\end{equation} +This is the scaled Kantorovich--Rubinstein metric on finitely +supported probability measures, in its Lipschitz-potential form +\cite[pp.~1381--1382]{Kantorovich2006}. We establish its needed +properties directly. +Only finitely many terms occur in the pairing. To see that the +supremum is finite and describe the distance function on each simplex, +let \(V\subset Y\) be a nonempty finite set and put +\[ + \mathcal P_V= + \left\{\ell\in\R^V: + \sum_{v\in V}\ell(v)=0,\quad + |\ell(v)-\ell(w)|\leq d_Y(v,w)\quad(v,w\in V)\right\}. +\] +This is a nonempty compact polytope. Indeed, its difference constraints +and zero mean bound every coordinate by +\(\max_{v,w\in V}d_Y(v,w)\). Each \(\ell\in\mathcal P_V\) extends to +a \(1\)-Lipschitz function on \(Y\) by +\[ + \widetilde\ell(y)=\min_{v\in V}\bigl(\ell(v)+d_Y(v,y)\bigr). +\] +The difference constraints show that this extension agrees with +\(\ell\) on \(V\), and the triangle inequality proves that it is +\(1\)-Lipschitz. Conversely, a global \(1\)-Lipschitz function restricts, +after subtraction of its mean, to an element of \(\mathcal P_V\). +Constants pair to zero with \(p-q\). Consequently, when \(V\) contains +both supports, the supremum in \eqref{eq:transport-metric} is the maximum +over \(\mathcal P_V\). + +Symmetry follows by replacing \(\ell\) with \(-\ell\), and the triangle +inequality follows by applying each potential to +\(p-q=(p-u)+(u-q)\). The indicator of a single vertex is +\(1\)-Lipschitz for \(d_Y\), so +\begin{equation} + |p(v)-q(v)|\leq\sqrt n\,\rho(p,q) + \qquad(v\in Y). + \label{eq:coordinate-control} +\end{equation} +Thus \(\rho\) separates points and is a metric. It is \(H\)-invariant +because graph automorphisms permute the admissible potentials. On +probability vectors supported in a fixed finite set, it is the +restriction of a norm on the finite-dimensional space of vectors with +coordinate sum zero. It therefore gives the usual topology on every +finite subcomplex. + +For \(c\in|K|\), write +\[ + \begin{aligned} + f_c(p)&=\rho(c,p),& U_s(c)&=\{p:f_c(p)0\). By +\eqref{eq:coordinate-control}, the ball of radius +\(\eta/(2\sqrt n)\) about \(p\) lies in the finite star of +\(\supp p\). Within this star the metric and realization topologies +agree. The open star of \(\supp p\) is also a neighborhood of \(p\) +in the locally finite realization, so the agreement is local in +both topologies. A closed metric ball is a closed subset of the +finite union of closed simplices it meets, and is therefore compact. + +If \(L_s(hc)\) meets \(\kappa\), \eqref{eq:support-distance} supplies +\(a\in A\) and \(b\in V(\kappa)\) with +\(d_Y(ha,b)\leq s\sqrt n\). For each \(a\), the map \(h\mapsto ha\) +is injective because \(H\) acts freely on vertices. The finitely +many pairs \(a,b\) and finite graph balls therefore allow only +finitely many \(h\). Finally, for \(v\in V(\kappa)\), the same +injection maps \(H_\kappa\) into \(V(\kappa)\). In particular, +\(|H_\kappa|\leq|V(\kappa)|\). +\end{proof} + +We will also use the following consequence. For fixed \(c\) and \(s\), +only finitely many \(h\in H\) satisfy +\begin{equation} + L_s(hc)\cap L_s(c)\ne\varnothing. + \label{eq:finite-ball-overlaps} +\end{equation} +Indeed, an intersection gives \(\rho(hc,c)\leq2s\), and then +\eqref{eq:support-distance} puts some vertex of \(h\supp c\) +within distance \(2s\sqrt n\) of \(\supp c\). The same +injectivity argument applies. + +\subsection{Simplex areas and distance gradients} + +The metric now provides finite neighborhoods in which we can make +cuts. For area comparisons, we use Euclidean structures separately +inside the top simplices. They need not agree on shared faces. + +Fix a top simplex \(\sigma\). For each spanning tree \(T\) of the +connected graph on \(V(\sigma)\), equip its affine span with the +Euclidean structure in which the tree-edge vectors +\(\delta_v-\delta_w\), with either orientation for each edge, form +an orthonormal basis of the translation space. For \(0\leq j\leq n\), +let \(a_j^\sigma\) be the average, over these finitely many trees, +of Euclidean \(j\)-area on polyhedral subsets of \(\sigma^\circ\) +of dimension at most \(j\). We allow relatively open faces in +these sets; pieces of dimension less than \(j\) have zero +\(j\)-area, and \(a_0^\sigma\) is counting measure. These area +functionals are covariant under \(H\), since the action permutes +the spanning trees. + +\begin{lemma}\label{lem:tree} +In every tree Euclidean structure on a top simplex \(\sigma\), its +\(n\)-volume is \(1/n!\). Every affine branch of \(f_c|_\sigma\) +has gradient of norm at most one, for every \(c\in|K|\). +The same gradient bound holds after restriction to any affine +subspace of the span of \(\sigma\). +\end{lemma} + +\begin{proof} +Root the tree at \(v_0\), orient its edges away from the root, and +order the remaining vertices so that each parent precedes its +children. Put +\[ + t_i=\delta_{v_i}-\delta_{\operatorname{parent}(v_i)}, + \qquad b_i=\delta_{v_i}-\delta_{v_0} + \quad(1\leq i\leq n). +\] +Each \(b_i\) is the sum of the \(t_j\) along its root path. +Thus the change-of-basis matrix is triangular with diagonal +entries one. The \(t_i\) are an orthonormal basis in the chosen +metric, so the parallelepiped spanned by the \(b_i\) has volume +one. The simplex has volume \(1/n!\). + +An affine branch has derivative along \(t_i\) equal, up to sign, to +\[ + \frac{\ell(v_i)-\ell(\operatorname{parent}(v_i))}{\sqrt n} +\] +for a \(1\)-Lipschitz potential \(\ell\). Since the two vertices +are adjacent, this derivative has absolute value at most +\(1/\sqrt n\). The squared gradient norm is the sum of the +squares of these \(n\) derivatives and is at most one. +Restriction to an affine subspace replaces the gradient by its +orthogonal projection, which cannot increase its norm. +\end{proof} + +\subsection{The volume margin in high dimension} + +We next bound the total top-dimensional area of a metric ball by +counting its possible connected vertex supports. Fix +\[ + R=4D^2,\qquad r=R/2. +\] +For a specified vertex \(v\), there are at most \(D^{2n}\) +connected sets of \(n+1\) vertices containing \(v\). To prove +this, choose a spanning tree of each such set rooted at \(v\). +A traversal using each tree edge once in each direction is a +walk of length \(2n\), starting at \(v\), whose visited set is +exactly the prescribed set. Choosing one such walk per set +gives an injection into the length-\(2n\) walks from \(v\), +of which there are at most \(D^{2n}\). + +Set \(k=\lfloor R\sqrt n\rfloor\). A graph ball of radius \(k\) +has at most +\[ + \sum_{j=0}^kD^j\leq D^{k+1} +\] +vertices, since \(D\geq2\). As \(|\supp c|\leq n+1\), there are +at most \((n+1)D^{k+1}\) vertices within distance \(k\) of +\(\supp c\). By \eqref{eq:support-distance}, each top simplex +meeting \(L_R(c)\) contains one of them. The connected-set +count and Lemma~\ref{lem:tree} therefore give, for every \(c\in|K|\), +\begin{equation} + \sum_{\sigma\ \mathrm{top}} + a_n^\sigma\bigl(L_R(c)\cap\sigma^\circ\bigr) + \leq V_R:= + \frac{(n+1)D^{\lfloor R\sqrt n\rfloor+1}D^{2n}}{n!}. + \label{eq:volume-bound} +\end{equation} +Counting a simplex more than once only enlarges this upper bound. + +With \(D\) fixed, the ratio of this estimate to +\((R-r)^n/n!\) is +\[ + \frac{V_R}{(R-r)^n/n!} + =\frac{(n+1)D^{\lfloor4D^2\sqrt n\rfloor+1}}{2^n} + \longrightarrow0. +\] +Indeed, its logarithm is +\(-n\log2+4D^2(\log D)\sqrt n+O_D(\log n)\). +We may therefore choose \(n_0(D)\geq1\) so that, for every +\(n\geq n_0(D)\), +\begin{equation} + V_R<\frac{(R-r)^n}{n!}. + \label{eq:strict-volume} +\end{equation} +For the remainder of the geometric proof, fix such an \(n\). +The graph degree bound \(D\), and hence \(R\) and \(r\), remain +fixed as dimensions increase. The strict margin +\eqref{eq:strict-volume} will convert successive area comparisons +into a bound on the number of points at the bottom of a +separating filtration. diff --git a/preprints/The-Bass-trace-conjecture-for-complex-group-rings-September-24-2026/build/sections/04-filtrations.tex b/preprints/The-Bass-trace-conjecture-for-complex-group-rings-September-24-2026/build/sections/04-filtrations.tex new file mode 100644 index 0000000000000000000000000000000000000000..a76b4de188a4c9c88dceec2947503d50b037c175 --- /dev/null +++ b/preprints/The-Bass-trace-conjecture-for-complex-group-rings-September-24-2026/build/sections/04-filtrations.tex @@ -0,0 +1,276 @@ +\section{Equivariant separating filtrations}\label{sec:filtrations} + +We retain the complex $K$, the metric $\rho$, and the radii $r0$ there is an admissible equivariant locally constant +filtration $(Z_j,T_j)_{j=0}^n$ such that, for every original top simplex +$\sigma$ of $K$, +\begin{equation}\label{eq:zero-strata} + \int_X\#\bigl(Z_0(x)\cap\sigma\bigr)\,d\mu(x) + \le C_\sigma\epsilon. +\end{equation} +Here $C_\sigma$ depends on the fixed geometric data but is independent of +$\epsilon$ and of the complexity of the chosen triangulations. +\end{proposition} + +\subsection{Successive near minimizers} + +The comparison with a cut through a ball follows the separating-set +method of Papasoglu \cite[Section 2, especially Lemma 2.5]{Papasoglu2020} +and its iterated and equivariant forms in Alpert +\cite[Lemmas 6--7 and 11--12]{Alpert2025}. We prove the comparisons for our +simplex-dependent areas, retaining the error from each near minimizer. + +\begin{proof}[Proof of Proposition~\ref{prop:zero-strata}] +For an equivariant family $Z$ of dimension at most $j$, define +\begin{equation}\label{eq:normalized-area} + \mathcal I_j(Z)= + \sum_{\sigma\ \mathrm{top}\bmod H}\frac{1}{|H_\sigma|} + \int_X a_j^\sigma\bigl(Z(x)\cap\sigma^\circ\bigr)\,d\mu(x). +\end{equation} +The sum runs over representatives of the original top simplices, and +$H_\sigma$ is the finite setwise stabilizer of $\sigma$. This quantity is +finite for an admissible family. There are finitely many representatives, +and, on each, local constancy and compactness of $X$ give finitely many +finite polyhedral configurations, each of finite area. + +Start with $(Z_n,T_n)=(|K|,K)$. At step $d=n,\ldots,1$, keep all previously +chosen data fixed and choose the admissible pair $(Z_{d-1},T_{d-1})$ so that +$\mathcal I_{d-1}(Z_{d-1})$ is within $\epsilon$ of the infimum over the +admissible choices at this step. Lemma~\ref{lem:filtration-exists} ensures +that this class is nonempty. Its infimum is a finite nonnegative number, +so such a choice exists. Consequently, for every admissible replacement +$Z'$ at this step, +\begin{equation}\label{eq:near-minimality} + \mathcal I_{d-1}(Z_{d-1}) + \le \mathcal I_{d-1}(Z')+\epsilon. +\end{equation} +Neither an attained minimum nor a limit of filtrations is used. + +\subsection{Disjoint replacements in the parameter and spatial variables} + +Fix a center $c\in|K|$. The set +\[ + \mathcal F_c= + \{h\in H\setminus\{1\}:L_R(hc)\cap L_R(c)\ne\varnothing\} +\] +is finite. Indeed such an intersection implies $\rho(hc,c)\le2R$; +the support-distance bound, the finite graph balls and freeness on vertices +then leave only finitely many possibilities for $h$. + +There is a finite clopen partition of $X$ into sets $A$ satisfying +$A\cap hA=\varnothing$ for every $h\in\mathcal F_c$. To construct one, +use the free action and the zero-dimensional Hausdorff topology to choose, +around each $x\in X$, a clopen neighborhood disjoint from its translates +by this finite set. A finite subcover exists by compactness; the atoms of +its common refinement have the required property. Thus +\begin{equation}\label{eq:product-disjointness} + \{hA\times L_R(hc):h\in H\} + \quad\text{is a pairwise disjoint family.} +\end{equation} +This partition is fixed independently of $\epsilon$. The assertion +concerns the product of parameter and spatial variables; the sets $hA$ +alone need not be disjoint. + +Fix one atom $A$ and any clopen subset $B\subseteq A$. For $r\le s\le R$ +put +\begin{equation}\label{eq:local-areas} + M_j(s)=\int_B\sum_{\sigma\ \mathrm{top}} + a_j^\sigma\bigl(Z_j(x)\cap\sigma^\circ\cap L_s(c)\bigr)\,d\mu(x), + \qquad 0\le j\le n. +\end{equation} +For a generic $t\in(r,R)$, also put +\[ + N_{d-1}(t)=\int_B\sum_{\sigma\ \mathrm{top}} + a_{d-1}^\sigma + \bigl(Z_d(x)\cap\sigma^\circ\cap S_t(c)\bigr)\,d\mu(x). +\] +These sums involve only finitely many original simplices. Unlike +\eqref{eq:normalized-area}, they run over all such simplices and carry no +stabilizer weights. + +At stage $d$, write $P=Z_d$ and $Z=Z_{d-1}$. For each $x$, make the +simultaneous replacement +\begin{equation}\label{eq:local-replacement} + Z'(x)= + \left(Z(x)\setminus\bigcup_{h:x\in hB}U_t(hc)\right) + \cup + \left(P(x)\cap\bigcup_{h:x\in hB}S_t(hc)\right). +\end{equation} +The active closed balls are pairwise disjoint by +\eqref{eq:product-disjointness}. The families of balls and levels are +locally finite, and their use is conditioned on clopen sets, so the +construction is equivariant and locally constant on every original +simplex. Apply Lemma~\ref{lem:refinement} with $T=T_d$, +$S=T_{d-1}$, $P=Z_d$, $Z=Z_{d-1}$, and the centers and activation +sets $(hc,hB)$. Outside a countable set of levels, it represents +$Z'$ by an admissible common refinement: the inserted levels have +the required dimension drop on faces of $T_d$ and of the old finer +triangulation $T_{d-1}$, and the lemma preserves the diameter bound +on components of $P(x)\setminus Z'(x)$. + +We next compute the change in~\eqref{eq:normalized-area}. Genericity +against the faces of $T_{d-1}$ implies that the old set $Z$ meets any +inserted level in dimension at most $d-2$. When $d=1$, that intersection +is empty. Hence the removed $(d-1)$-area is unchanged if the open ball is +replaced by its closure, and the inserted cut has zero $(d-1)$-area +overlap with the retained old set. + +The normalization in~\eqref{eq:normalized-area} makes the removed area +exactly $M_{d-1}(t)$ and the inserted area exactly $N_{d-1}(t)$. Here is +the orbit-counting identity behind this assertion. Let $Q$ denote the +relevant equivariant set, and let $W(c)$ denote either the ball or the +level. Covariance of area and invariance of $\mu$ give +\begin{align*} + &\sum_{\sigma\ \mathrm{top}\bmod H}\frac1{|H_\sigma|} + \sum_{h\in H}\int_{hB} + a_j^\sigma\bigl(Q(x)\cap\sigma^\circ\cap W(hc)\bigr)\,d\mu(x) + \\ + &\quad=\sum_{\sigma\ \mathrm{top}\bmod H}\frac1{|H_\sigma|} + \sum_{h\in H}\int_B + a_j^{h^{-1}\sigma} + \bigl(Q(x)\cap(h^{-1}\sigma)^\circ\cap W(c)\bigr)\,d\mu(x) + \\ + &\quad=\int_B\sum_{\tau\ \mathrm{top}} + a_j^\tau\bigl(Q(x)\cap\tau^\circ\cap W(c)\bigr)\,d\mu(x). +\end{align*} +The map $h\mapsto h^{-1}\sigma$ takes each value in the simplex orbit +exactly $|H_\sigma|$ times. All local contributions are nonnegative, +and only finitely many translates meet any fixed representative simplex. +The disjointness and zero-overlap statements above therefore yield +\[ + \mathcal I_{d-1}(Z')-\mathcal I_{d-1}(Z) + =N_{d-1}(t)-M_{d-1}(t). +\] +Combining this identity with~\eqref{eq:near-minimality} gives +\begin{equation}\label{eq:local-comparison} + M_{d-1}(t)\le N_{d-1}(t)+\epsilon + \quad\text{for almost every }t\in(r,R). +\end{equation} + +\subsection{Slicing and the strict volume margin} + +We have compared the area of a separating set inside a ball with that +of a cut through the preceding set. Slicing now connects these +comparisons across the successive dimensions. + +Work first in one original top simplex $\sigma$ and one of its tree +metrics. Decompose $Z_d(x)\cap\sigma^\circ$ into affine polyhedral +pieces on which $f_c=\rho(c,\cdot)$ is affine. On a $d$-dimensional +piece $Q$, its tangential gradient has norm $\alpha\le1$ by +Lemma~\ref{lem:tree}. Euclidean slicing perpendicular to that gradient +gives +\[ + \int_r^b\operatorname{vol}_{d-1}(Q\cap\{f_c=t\})\,dt + =\alpha\operatorname{vol}_d(Q\cap\{r0$, use the filtration supplied by +Proposition~\ref{prop:zero-strata}. We will turn it into an +$H$-equivariant chain map +\[ + F^\epsilon:C_*(K)\longrightarrow E_* +\] +that is equivariantly chain homotopic to inclusion. Its evaluation by +a fixed invariant cocycle will tend to zero. These two properties +prove Theorem~\ref{thm:sparse}. Throughout this section $n$ remains +fixed while $\epsilon$ tends to zero. + +\subsection{Component averages and their finite support} + +For a fixed parameter $x$, take the barycentric subdivision of the +final triangulation $T_0(x)$. Its vertices are the barycenters $b_F$ +of nonempty simplices $F$ of $T_0(x)$. Define +\[ + \ell(F)=\min\{i:b_F\in Z_i(x)\}, + \qquad Z_{-1}(x)=\varnothing. +\] +All $Z_i(x)$ are subcomplexes of $T_0(x)$, so $b_F\in Z_i(x)$ +is equivalent to $F\subseteq Z_i(x)$. Let $U_F$ be the connected +component of $Z_{\ell(F)}(x)\setminus Z_{\ell(F)-1}(x)$ containing +$b_F$. For any such component $U$, set +\begin{equation}\label{eq:component-average} + J_U=\bigcup_{p\in U}\supp p, + \qquad + u_U=\frac{1}{|J_U|}\sum_{v\in J_U}\delta_v. +\end{equation} +The next lemma proves that $J_U$ is finite and controls these +labels as $x$ varies. The intended image of an oriented barycentric +simplex is +\begin{equation}\label{eq:exterior-replacement} + [b_{F_0},\ldots,b_{F_j}] + \longmapsto u_{U_{F_0}}\wedge\cdots\wedge u_{U_{F_j}}\in E_j. +\end{equation} + +This choice uses the separating filtration in a specific way. +If nested faces $F\subset F'$ have the same level $i$, the segment +joining their barycenters lies in $Z_i\setminus Z_{i-1}$. +Except at its endpoint $b_F$, it lies in the relative interior +of $F'$, which the subcomplex $Z_{i-1}$ misses; the endpoint also +avoids $Z_{i-1}$ by its level. The two barycenters therefore receive +the same component vector, making any exterior term containing +both zero. Along a complete flag $F_0\subset\cdots\subset F_n$, +with $\dim F_i=i$, the levels are nondecreasing: if +$F\subset F'\subseteq Z_i$, then $F\subseteq Z_i$. +A nonzero image must consequently use all levels $0,1,\ldots,n$, +beginning at a point of $Z_0$. We will bound the number of these +flags after establishing that the labels define finite chains. + +\begin{lemma}\label{lem:finite-component-labels} +For every original simplex $\kappa$ of $K$, there is a finite set +$W_\kappa\subseteq Y$, independent of $\epsilon$ and $x$, such that +$J_{U_F}\subseteq W_\kappa$ whenever $b_F\in\kappa$. The triangulation +over $\kappa$, together with all its assigned vectors $u_{U_F}$, +is locally constant in $x$. These assignments are equivariant. +\end{lemma} + +\begin{proof} +Each component of $Z_i(x)\setminus Z_{i-1}(x)$ has +$\rho$-diameter at most $2R$. For $i=0$ the components are +singletons, since $Z_0(x)$ is a locally finite zero-dimensional +subcomplex. If $b_F\in\kappa$ and $p\in U_F$, the support-distance +bound from Lemma~\ref{lem:proper} gives +\[ + \min d_Y(\supp b_F,\supp p) + \leq\sqrt n\,\rho(b_F,p)\leq 2R\sqrt n. +\] +Choose an original top simplex containing $p$. Its $n+1$ vertices +span a connected graph, so every vertex in $\supp p$ is within +graph distance $n$ of the vertex furnished by this inequality. +Consequently we may take +\[ + W_\kappa= + \left\{v\in Y: + d_Y\bigl(v,V(\kappa)\bigr) + \leq\left\lceil 2R\sqrt n+n\right\rceil\right\}. +\] +The bounded degree of $Y$ makes this set finite. Notice that the +support of $p$ need not itself span a connected graph; we used +the containing top simplex. + +Let $K[W_\kappa]$ be the finite subcomplex consisting of the simplices +whose vertices belong to $W_\kappa$. The whole component $U_F$ +lies in its realization. It follows that $U_F$ is exactly the +component containing $b_F$ in +\[ + \bigl(Z_{\ell(F)}(x)\setminus Z_{\ell(F)-1}(x)\bigr) + \cap |K[W_\kappa]|. +\] +Indeed $U_F$ is a connected subset of this restriction, and every +connected subset of the restriction containing $b_F$ also lies in +the original component. Thus restricting to this finite subcomplex +does not cut off a part of the component needed to compute $J_{U_F}$. + +There are finitely many original simplices in $K[W_\kappa]$. +A common refinement of their finite clopen parameter partitions +makes all their triangulation and filtration data constant. +On each resulting piece the components and their sets of support +vertices are constant as well. This proves local constancy of +the assignments over $\kappa$. Finally, +$J_{hU}=hJ_U$, and hence $u_{hU}=h u_U$, proving equivariance. +\end{proof} + +Subdivide each oriented original simplex into its fundamental +subdivision chain and apply the exterior assignment +\eqref{eq:exterior-replacement} to every term. This defines a map +$f_x:C_*(K)\to E_*$. Both operations commute +with the augmented boundary. For subdivision, interior facets +cancel in pairs and the boundary facets give the compatible +subdivision on each original face. For the second operation, +each assigned vector has coefficient sum one, so contraction +with the coefficient-sum functional gives +\[ + \partial(u_0\wedge\cdots\wedge u_j) + =\sum_{i=0}^j(-1)^i + u_0\wedge\cdots\wedge\widehat{u_i}\wedge\cdots\wedge u_j. +\] +Thus $f_x$ is an augmented chain map. + +By Lemma~\ref{lem:finite-component-labels}, its value on any +original simplex is constant on a finite clopen partition and +is supported in a fixed finite exterior space. We may therefore +define +\begin{equation}\label{eq:averaged-chain-map} + F^\epsilon(c)=\int_X f_x(c)\,d\mu(x). +\end{equation} +This integral is an actual finite chain: for every $c$ it is a +finite linear combination of finitely many chain values. +It commutes with the augmented boundary. Moreover, +$f_{hx}(hc)=h f_x(c)$, so invariance of $\mu$ shows that +$F^\epsilon$ is $H$-equivariant. + +\subsection{Counting the surviving flags} + +We now bound the evaluation of $f_x$ on a top simplex. The number +of subdivision simplices may depend on $\epsilon$; the following +bound depends instead on the number of zero-stratum points. +The component-labeling and flag-counting method is modeled on +Alpert \cite[Lemmas 4 and 10]{Alpert2025}; the carrier argument below +supplies the bound without a manifold hypothesis. + +\begin{lemma}\label{lem:surviving-flags} +For every original top simplex $\sigma$ and every $x$, at most +\[ + 2^n\#\bigl(Z_0(x)\cap\sigma\bigr) +\] +terms in its fundamental barycentric subdivision chain have +nonzero images under the exterior replacement map. +\end{lemma} + +\begin{proof} +Suppress $x$. Such a term corresponds to a complete flag +\[ + F_0\subset F_1\subset\cdots\subset F_n, + \qquad \dim F_i=i, +\] +in the restriction of $T_0$ to $\sigma$. As shown when defining +\eqref{eq:exterior-replacement}, a nonzero term has levels exactly +$0,1,\ldots,n$. In particular $F_0=\{z\}$ for a point +$z\in Z_0\cap\sigma$ and $F_i\subseteq Z_i$ for every $i$. + +By Lemma~\ref{lem:carrier}, $z$ lies in the relative interior of +a unique $i$-simplex $S_i$ of $T_i$ contained in $Z_i$. To see why +this carrier controls the final refinement, let $F_i$ occur in a +surviving flag through $z$. The $T_i$-carrier of a relative-interior +point of $F_i$ contains $F_i$ by refinement and belongs to $Z_i$, +because $Z_i$ is a subcomplex. That carrier has dimension at least +$i$, since it contains $F_i$, and at most $i$, since $\dim Z_i\leq i$. +It contains $z$, so it equals $S_i$. Hence $F_i\subseteq S_i$. + +For a fixed preceding face $F_{i-1}$, the possible $F_i$ are thus +full-dimensional simplices in a triangulation of the single affine +$i$-simplex $S_i$, having $F_{i-1}$ as a facet. There is at most one +on either side of the affine hyperplane spanned by that facet. +Otherwise two simplices on the same side would have intersecting +relative interiors near an interior point of the facet. There +are therefore at most two choices at each stage and at most $2^n$ +surviving flags through $z$. Summing over $z$ proves the assertion. +This argument concerns one carrier simplex at a time and uses no +manifold assumption on $K$ or on the strata. +\end{proof} + +Figure~\ref{fig:flags} illustrates the two choices at each of the two +stages when $n=2$. + +\begin{figure}[htbp] +\centering +\begin{tikzpicture}[scale=1.05] + \coordinate (A) at (-2,0); + \coordinate (B) at (2,0); + \coordinate (C) at (0,3); + \coordinate (L) at (-1.3333,1); + \coordinate (R) at (1.3333,1); + \coordinate (z) at (0,1); + \fill[blue!9] (L)--(z)--(C)--cycle; + \fill[blue!18] (z)--(R)--(C)--cycle; + \fill[orange!13] (L)--(z)--(A)--cycle; + \fill[orange!24] (z)--(R)--(A)--cycle; + \draw[thin,gray] (z)--(A) (A)--(R) (z)--(C); + \draw[thick] (A)--(B)--(C)--cycle; + \draw[very thick,blue!65!black] (L)--(R); + \fill (z) circle (2pt) node[below right=2pt,fill=white,inner sep=1pt] {$z$}; + \node at (-0.44,1.68) {$F_2$}; + \node at (0.44,1.68) {$F_2$}; + \node at (-1.08,0.62) {$F_2$}; + \node at (-0.25,0.67) {$F_2$}; + \node[above=2pt] at (-0.66,1) {$F_1$}; + \node[above=2pt] at (0.66,1) {$F_1$}; +\end{tikzpicture} +\caption{The flag count in dimension two. The point $z\in Z_0$ has +two possible incident edges $F_1$ along its one-dimensional carrier. +Each edge has at most two incident triangles $F_2$ inside the original +two-dimensional carrier. The four shaded triangles give the four possible +flags through $z$ in this example. The unshaded triangle has no +one-dimensional-stratum edge through $z$.} +\label{fig:flags} +\end{figure} + +Let $\psi:E_n\to\C$ be the invariant cocycle in +Theorem~\ref{thm:sparse}. For an original top simplex $\sigma$, put +\[ + M_\sigma= + \max_{v_0,\ldots,v_n\in W_\sigma} + \left|\psi(\delta_{v_0}\wedge\cdots\wedge\delta_{v_n})\right|. +\] +This is finite, even though $\psi$ need not be bounded on all +basis simplices of the infinite vertex set. Every vector in +\eqref{eq:component-average} is a nonnegative convex combination +of vertices in $W_\sigma$. Multilinearity therefore bounds the +evaluation of each surviving term by $M_\sigma$. The preceding +lemma and Proposition~\ref{prop:zero-strata} give +\begin{align} + \left|\psi\bigl(F^\epsilon(\sigma)\bigr)\right| + &\leq 2^n M_\sigma + \int_X\#\bigl(Z_0(x)\cap\sigma\bigr)\,d\mu(x) + \notag\\ + &\leq 2^n M_\sigma C_\sigma\epsilon + \longrightarrow 0.\label{eq:small-chain-evaluation} +\end{align} +All constants here are independent of $\epsilon$. Since chains +have finite support, the same convergence holds on every fixed +$c\in C_n(K)$. + +\subsection{Equivariant comparison and passage to coinvariants} + +It remains to show that the small evaluations just obtained equal +the original cocycle evaluation on every coinvariant cycle. +We do this by constructing a homotopy before taking coinvariants. + +\begin{lemma}\label{lem:equivariant-homotopy} +Let $K$ be a simplicial complex with an $H$-action and finite +setwise simplex stabilizers. Let $E_*$ be the augmented exterior +chain complex on a nonempty $H$-set. Any two $H$-equivariant +augmented chain maps $A,B:C_*(K)\to E_*$ that agree in degree $-1$ +are $H$-equivariantly chain homotopic. +\end{lemma} + +\begin{proof} +The target is acyclic as an augmented complex. Indeed, for any +fixed vertex $v$, the operator $a\mapsto\delta_v\wedge a$ satisfies +\[ + \partial(\delta_v\wedge a)+\delta_v\wedge\partial a=a. +\] +This contraction need not be equivariant. We use it only to choose +finite fillers and then impose equivariance by finite averaging. + +Set $h_{-1}=0$. Suppose the homotopy has been constructed below +degree $j$. For an oriented $j$-simplex $e$, the residual +\[ + r(e)=A(e)-B(e)-h_{j-1}(\partial e) +\] +is a cycle, by the chain-map identities and the previously +constructed homotopy. Choose a finite chain $t$ with +$\partial t=r(e)$. Let $S$ be the setwise stabilizer of $e$ and +let $\chi:S\to\{1,-1\}$ be its orientation character, so that +$s e=\chi(s)e$. The residual is equivariant, and hence +$s r(e)=\chi(s)r(e)$. Define +\[ + \overline t=\frac1{|S|}\sum_{s\in S}\chi(s)\,s t. +\] +Then $\partial\overline t=r(e)$ and +$s\overline t=\chi(s)\overline t$. Choose such a filler for one +oriented representative of every simplex orbit, put +$h_j(e)=\overline t$, and extend equivariantly and linearly. +The orientation-character identity makes the extension +well-defined, including when the stabilizer reverses orientation. +This gives $A-B=\partial h+h\partial$ in degree $j$ and completes +the induction. +\end{proof} + +\begin{proof}[Proof of Theorem~\ref{thm:sparse}] +The geometric construction applies for every sufficiently large +$n$ depending only on $D$, as fixed by \eqref{eq:strict-volume}. +For nonempty $K$, construct $F^\epsilon$ as above. The stabilizers +are finite by Lemma~\ref{lem:proper}, and $F^\epsilon$ and inclusion +$\iota$ both induce the identity in augmented degree $-1$. +Lemma~\ref{lem:equivariant-homotopy} supplies an equivariant +homotopy +\[ + \iota-F^\epsilon=\partial h+h\partial. +\] + +Let $\overline c$ be a cycle of $C_n(K)_H$ and choose a finite +chain lift $c\in C_n(K)$. We do not assume that $c$ is a cycle +before taking coinvariants. Its boundary is a finite sum of +differences $ha-a$. Equivariance of the homotopy makes +$h_{n-1}(\partial c)$ another such sum, which the invariant +functional $\psi$ annihilates. The cocycle identity annihilates +$\partial h_n(c)$. Thus, for every $\epsilon>0$, +\[ + \psi\bigl(\iota(c)\bigr) + =\psi\bigl(F^\epsilon(c)\bigr). +\] +The right-hand side tends to zero by +\eqref{eq:small-chain-evaluation}, whereas the left-hand side +does not depend on $\epsilon$. Hence it is zero. The evaluation +depends only on the coinvariant class because $\psi$ is invariant. +For empty $K$ there is no nonzero top chain to consider. +\end{proof} diff --git a/preprints/The-Bass-trace-conjecture-for-complex-group-rings-September-24-2026/build/sections/07-idempotents.tex b/preprints/The-Bass-trace-conjecture-for-complex-group-rings-September-24-2026/build/sections/07-idempotents.tex new file mode 100644 index 0000000000000000000000000000000000000000..e050bcb2d87ecc376d11bf30560eebb4d5a53ef3 --- /dev/null +++ b/preprints/The-Bass-trace-conjecture-for-complex-group-rings-September-24-2026/build/sections/07-idempotents.tex @@ -0,0 +1,258 @@ +\section{Cycles from an idempotent} +\label{sec:idempotents} + +We now return to group rings. For an idempotent and one infinite-order +conjugacy class, we will construct cycles in every positive even degree. +Their vertices follow steps in one fixed graph of bounded degree. In +sufficiently high dimension, the supporting complexes will therefore +satisfy the hypotheses of Theorem~\ref{thm:sparse}. We first record why +it suffices to perform this construction for finitely generated groups. + +The use of cyclic idempotent expressions and reduced centralizers has +its conceptual background in Connes's pairing with projective classes +\cite[Part II, Section 2, Proposition 14]{Connes1985} and Burghelea's +conjugacy-class decomposition \cite[Theorem I]{Burghelea1985}. Our finite +path identities and the detector's normalization are proved directly; +no cyclic-homology identification is assumed. + +\begin{lemma}[Subgroup locality]\label{lem:locality} +Let $i:L\hookrightarrow G$ be an inclusion of discrete groups. Extension +of scalars defines $i_*:K_0(\C L)\to K_0(\C G)$, and for every +$y\in K_0(\C L)$ and $C\in\con(G)$, +\begin{equation}\label{eq:class-aggregation} + \HS_G(i_*y)(C) + =\sum_{\substack{B\in\con(L)\\ B\subseteq L\cap C}} + \HS_L(y)(B). +\end{equation} +Only finitely many terms are nonzero. Moreover, every +$x\in K_0(\C G)$ is induced from a finitely generated subgroup, which may +be required to contain any prescribed finite subset of $G$. +Consequently, the finite-order-support assertion of +Theorem~\ref{thm:bass} for all finitely generated groups implies that +assertion for all discrete groups. +\end{lemma} + +\begin{proof} +We regard free right modules as columns: an idempotent +$e\in M_r(\C L)$ acts on the left, with image $e(\C L)^r$. +Tensoring the split decomposition defined by $e$ with $\C G$ identifies +the extended image with $e(\C G)^r$. The coefficient sum in an ambient +class $C$ is the sum of its coefficient sums over all $L$-classes contained +in $L\cap C$. This proves \eqref{eq:class-aggregation} for the image of an +idempotent and then, by subtraction, for every virtual class. Finiteness +follows from the finite coefficient support of the matrices representing +that class. + +Write an arbitrary $x\in K_0(\C G)$ as the difference of the classes of +two finite idempotent matrices. The group generated by their coefficient +supports and the prescribed finite subset is finitely generated. The +same matrices are idempotent over its group ring, since that ring embeds +in $\C G$, and their difference induces $x$. Finally, if an ambient class +consists of infinite-order elements, so does every subgroup class occurring +in \eqref{eq:class-aggregation}: subgroup inclusion preserves the order of +an element. Vanishing of each subgroup coefficient therefore gives +vanishing of the ambient coefficient. This uses all the subgroup classes +in the sum, since an ambient conjugacy class can split into several of +them. +\end{proof} + +\subsection{A fixed graph and finite path sums} + +For the rest of this section, fix a finitely generated group $G$, an +idempotent $e\in M_r(\C G)$, and an infinite-order element $g\in G$. +Write +\[ + e=\sum_{s\in G}A_s s,\qquad A_s\in M_r(\C),\qquad + S_0=\{s\in G:A_s\ne0\}. +\] +The set $S_0$ is finite. Idempotence says that +$\sum_{uv=t}A_uA_v=A_t$ for every $t\in G$. Thus the matrix-valued kernel +\[ + a(h,h')=A_{h^{-1}h'} +\] +is invariant under simultaneous left translation and satisfies +\begin{equation}\label{eq:kernel-idempotence} + \sum_{k\in G}a(h,k)a(k,h') + =\sum_{uv=h^{-1}h'}A_uA_v + =a(h,h'). +\end{equation} +All these sums are finite. In particular, the order of the matrix +factors agrees with the column convention for right modules. + +Put +\[ + J=C_G(g),\qquad H=J/\langle g\rangle,\qquad + Y=\langle g\rangle\backslash G, + \qquad \bar h=\langle g\rangle h. +\] +Choose a finite symmetric generating set $S\subset G\setminus\{1\}$ +containing $S_0\setminus\{1\}$ and its inverses. Give $Y$ the simple +undirected graph whose edges join $\bar h$ to $\overline{hs}$ for $s\in S$; +loops are omitted and repeated edges identified. The graph is connected +and has degree at most the fixed integer +\[ + D=\max\{2,|S|\}. +\] +The group $H$ acts by left multiplication. This is well-defined because +$J$ centralizes $g$, and it preserves right-step edges. It is free on +vertices: $\overline{ch}=\bar h$ with $c\in J$ implies $ch=g^q h$ for an +integer $q$, and hence $c=g^q$. Both $H$ and $Y$ are countable. We keep +this graph and this value of $D$ throughout all subsequent choices of +dimension. + +For $k\ge1$, let $Q_k=J\backslash G^k$, with simultaneous left +translation on tuples. Define the weight of a tuple by +\begin{equation}\label{eq:path-weight} + w_k(h_0,\ldots,h_{k-1}) + =\tr\bigl(a(h_0,h_1)\cdots + a(h_{k-2},h_{k-1})a(h_{k-1},gh_0)\bigr), +\end{equation} +where $w_1(h_0)=\tr(a(h_0,gh_0))$. These weights are $J$-invariant. +They define a finite complex linear combination $v_k$ of elements of +$Q_k$. Indeed, a nonzero weight has increments +\[ + s_i=h_i^{-1}h_{i+1}\quad(0\le i1$, take $s=t$. +Then $h^{t^a}=1$, so the restriction would make $h$ conjugate to $1$, +a contradiction. Thus every nonidentity finite-order coefficient also +vanishes. Taking differences of projective classes finishes the proof. +\end{proof} + +\subsection{Homotopy idempotents and Lefschetz numbers} + +The integral support statement also has fixed-point consequences. +A self-map $f$ is a \emph{homotopy idempotent} if $f\circ f\simeq f$. +Such a map on a connected finite CW complex $X$ admits maps +$u:X\to Y$ and $v:Y\to X$ with +\[ + v\circ u\simeq f,\qquad u\circ v\simeq\id_Y +\] +\cite[Section 3]{BCM2006}. The second identity says that $Y$ is +\emph{finitely dominated} by $X$. For a pointed homotopy idempotent +with $f_\#=\id_G$, where $G=\pi_1(X)$, the induced maps $u_\#$ and +$v_\#$ identify $\pi_1(Y)$ with $G$. The universal-cover chains of $Y$ +are chain homotopy equivalent to a finite complex $P_*$ of finitely +generated projective $\Z G$-modules. The sum +$\sum_i(-1)^i[P_i]\in K_0(\Z G)$ is its projective Euler class. The +Hattori--Stallings trace of this class records the fixed-point class indices of $f$: +it is the Reidemeister trace, with conjugacy classes inverted when +passing from the left-module convention of \cite[Lemma 4.1]{BCM2006} +to ours. The augmentation trace and the identity coefficient of this projective +class give the ordinary and $L^2$-Lefschetz numbers, respectively +\cite[Lemmas 8.1--8.2]{BCM2006}. Berrick, Chatterji and Mislin use these +relations to deduce the following consequences of the integral support +statement and its numerical trace consequence. + +\begin{corollary}[Berrick--Chatterji--Mislin consequences] +\label{cor:homotopy-idempotents} +Let $M$ be a connected closed smooth oriented manifold. +\begin{enumerate}[label=\textup{(\roman*)}] + \item If $\dim M>2$ and $f:M\to M$ satisfies $f\circ f\simeq f$ + by free homotopy, then some self-map freely homotopic to $f$ has + exactly one fixed point. + \item Choose $x_0\in M$ and put $G=\pi_1(M,x_0)$. If + $f:(M,x_0)\to(M,x_0)$ satisfies $f\circ f\simeq_* f$ by based + homotopy and $f_\#=\id_G$, then + \[ + L(f)=L^{(2)}(\widetilde f). + \] + Here $\widetilde f$ is the lift to the universal cover fixing a chosen + point above $x_0$; it is $G$-equivariant. The left side is the ordinary + Lefschetz number, and the right side is the alternating sum of von + Neumann traces on the induced $L^2$-homology maps, as in + \cite[Definition 7.1]{BCM2006}. +\end{enumerate} +\end{corollary} + +\begin{proof} +For $x\in K_0(\Z G)$ write $h_C(x)=\HS_{\Z G}(x)(C)$. The +augmentation trace $\epsilon$ is the sum of these coefficients, while +the Kaplansky trace $\kappa$ is the identity coefficient. The sum is +finite, and Corollary~\ref{cor:integral} gives +\[ + \epsilon(x)=\sum_{C\in\con(G)}h_C(x)=h_{[1]_G}(x)=\kappa(x). +\] +This is the weak Bass conjecture for every group. The cited paper uses +projective left modules, whereas we use right modules. The anti-involution +$g\mapsto g^{-1}$, together with transpose on idempotent matrices, +transports between these conventions and relabels the coefficient at +$[g]_G$ by $[g^{-1}]_G$. It preserves identity support and both +$\epsilon$ and $\kappa$. Thus Corollary~\ref{cor:integral} gives the +classical Bass conjecture in the cited convention, and the displayed +equality gives weak Bass there. The forward implications of +\cite[Theorem 1]{BCM2006} and \cite[Theorem 2]{BCM2006} now give +\textup{(i)} and \textup{(ii)}, respectively. +\end{proof} diff --git a/preprints/The-Bass-trace-conjecture-for-complex-group-rings-September-24-2026/build/sections/10-torsion-free.tex b/preprints/The-Bass-trace-conjecture-for-complex-group-rings-September-24-2026/build/sections/10-torsion-free.tex new file mode 100644 index 0000000000000000000000000000000000000000..de4b6d7b5edcd7a706ff03854395c74b12d13f51 --- /dev/null +++ b/preprints/The-Bass-trace-conjecture-for-complex-group-rings-September-24-2026/build/sections/10-torsion-free.tex @@ -0,0 +1,121 @@ +\section{Torsion-free groups: trace rank and idempotents} +\label{sec:torsion-free} + +For a torsion-free group, the support theorem leaves only the identity +conjugacy class. We first identify its coefficient with augmentation +rank, and then use the two possible scalar ranks to determine scalar +idempotents. + +\begin{proof}[Proof of Corollary~\ref{cor:torsion-free}\textup{(i)}] +Let $e\in M_q(\C G)$ be idempotent and put +$x=[e(\C G)^q]\in K_0(\C G)$. Since $G$ is torsion-free, +Theorem~\ref{thm:bass} says that $\HS_G(x)$ is supported on $[1]_G$. +Therefore +\begin{align} + \tau_{G,q}(e) + &=\HS_G(x)([1]_G)\notag\\ + &=\sum_{C\in\con(G)}\HS_G(x)(C)\notag\\ + &=\epsilon_G\!\left(\sum_{i=1}^q e_{ii}\right) + =\tr\bigl(\epsilon_{G,q}(e)\bigr) + =\operatorname{rank}_{\C}\bigl(\epsilon_{G,q}(e)\bigr). + \label{eq:torsion-free-rank} +\end{align} +The class sum is finite because the entries of $e$ have finite group +support. The last +equality holds because $\epsilon_{G,q}(e)$ is an idempotent complex +matrix: its image-kernel decomposition makes its trace equal to its +rank. Taking differences of projective classes gives +$\tau_{G,*}=\operatorname{rk}_{\epsilon}$. This map has range exactly +$\Z$, since its values are integers and $[\C G]$ has value $1$. +\end{proof} + +For matrices the conclusion is numerical. For example, +$\operatorname{diag}(1,0)\in M_2(\C G)$ is an idempotent other than zero +and the identity, with augmentation rank one. The scalar case is special +because its augmentation rank can only be zero or one. + +\begin{proof}[Proof of Corollary~\ref{cor:torsion-free}\textup{(ii)}] +First take $R=\C$ and let $e\in\C G$ be idempotent. Equation +\eqref{eq:torsion-free-rank} with $q=1$ gives +\[ + \tau_G(e)=\epsilon_G(e)\in\{0,1\}. +\] +We use the standard faithful-trace argument, as in +\cite[Theorem 2.1 and Remark 2.2]{BV1998}, spelling out a similarity +to a projection. Embed $\C G$ by left convolution in the group von +Neumann algebra $\mathcal N(G)$, the commutant of the right regular +representation on $\ell^2(G)$. Its canonical normalized trace is +\[ + \tau_G(T)=\langle T\delta_1,\delta_1\rangle . +\] +Its faithfulness can be seen directly: if +$T\geq0$ and $\tau_G(T)=0$, then $T^{1/2}\delta_1=0$. Since $T^{1/2}$ +commutes with right translations and their translates of $\delta_1$ +span a dense subspace, $T^{1/2}=0$, hence $T=0$. + +Set +\[ + a=e^*e+(1-e)^*(1-e). +\] +For every $\xi\in\ell^2(G)$, +\[ + \langle a\xi,\xi\rangle + =\|e\xi\|^2+\|(1-e)\xi\|^2 + \geq \tfrac12\|\xi\|^2. +\] +Thus $a$ is positive and invertible in $\mathcal N(G)$, and a direct +multiplication gives $ae=e^*a$. With $s=a^{1/2}$, the element +$p=ses^{-1}$ satisfies $p^2=p$ and +$p^*=s^{-1}e^*s=ses^{-1}=p$. Hence $p$ is a projection, and +traciality gives $\tau_G(p)=\tau_G(e)$. If this value is zero, +faithfulness applied to $p=p^*p$ gives $p=0$. If it is one, +faithfulness applied to $1-p$ gives $p=1$. Similarity then gives +$e=0$ or $e=1$. + +Now let $R$ be any commutative unital domain of characteristic zero +and write an idempotent $f\in RG$ as +$f=\sum_{g\in S}r_g g$ with $S$ finite. Put +\[ + R_0=\Z[r_g:g\in S]\subseteq R,\qquad F=\operatorname{Frac}(R_0). +\] +The field $F$ is finitely generated over $\mathbb Q$, and therefore +embeds in $\C$. Indeed, send a finite transcendence basis of +$F/\mathbb Q$ to an algebraically independent tuple of complex +numbers; the resulting embedding of the rational function field +extends across the finite algebraic extension $F$ because $\C$ is +algebraically closed. The coefficientwise maps +$R_0G\hookrightarrow RG$ and $R_0G\hookrightarrow\C G$ are injective. +Thus $f^2=f$ already holds in $R_0G$, and its image in $\C G$ is an +idempotent. The complex case makes that image $0$ or $1$, and +injectivity gives the same conclusion for $f$. Only the finitely +generated coefficient field $F$ was embedded; no embedding of all +of $R$ into $\C$ is required. +\end{proof} + +\begin{remark}[Algebraic and completed trace ranges] +Continue to assume that $G$ is torsion-free. Let +$j:\C G\hookrightarrow C_r^*(G)$ be the inclusion into the reduced +group $C^*$-algebra, and let +\[ + j_*:K_0(\C G)\longrightarrow K_0^{\mathrm{top}}(C_r^*(G)) +\] +be the comparison map to topological $K$-theory. Write +$\tau_*^{\mathrm{top}}:K_0^{\mathrm{top}}(C_r^*(G))\to\R$ for the +canonical trace pairing. It agrees with $\tau_{G,*}$ on this image: +the same positive similarity used above, applied in a matrix algebra +over $C_r^*(G)$, produces a projection representing the image of a +matrix idempotent, and traciality preserves its trace. Consequently +\[ + \tau_*^{\mathrm{top}}\!\left(j_*K_0(\C G)\right)=\Z. +\] +This is the integer range on the algebraic image, not a statement +about every class of $K_0^{\mathrm{top}}(C_r^*(G))$. In particular, +the argument does not presume that the algebraic augmentation extends +to the reduced completion. The stronger completed idempotent question +is the Kadison--Kaplansky conjecture; see \cite[Section 1]{OW2023}. + +The coefficient reduction above uses characteristic zero. An idempotent +$e\notin\{0,1\}$ would give nonzero zero divisors through $e(1-e)=0$, +so the zero-divisor conjecture would imply the idempotent conjecture. +The reverse implication is not established here. +\end{remark} diff --git a/preprints/The-Blockwise-Alperin-Weight-Conjecture-September-23-2026/README.md b/preprints/The-Blockwise-Alperin-Weight-Conjecture-September-23-2026/README.md new file mode 100644 index 0000000000000000000000000000000000000000..a8fdf46e5439c9ad44b399fdf221878f63cac3dc --- /dev/null +++ b/preprints/The-Blockwise-Alperin-Weight-Conjecture-September-23-2026/README.md @@ -0,0 +1,18 @@ +# [The Blockwise Alperin Weight Conjecture](paper.pdf) + +**Author:** OpenAI + +**Date:** September 23, 2026 + +## Citation + +```bibtex +@misc{OAI:The-Blockwise-Alperin-Weight-Conjecture-September-23-2026, + author = {{OpenAI}}, + title = {{The Blockwise Alperin Weight Conjecture}}, + howpublished = {OpenAI Math Release preprint + \href{https://github.com/openai/math/blob/main/preprints/The-Blockwise-Alperin-Weight-Conjecture-September-23-2026/paper.pdf}{OAI:The-Blockwise-Alperin-Weight-Conjecture-September-23-2026}}, + year = {2026} +} +``` + diff --git a/preprints/The-Blockwise-Alperin-Weight-Conjecture-September-23-2026/build/figures/field-tower.tex b/preprints/The-Blockwise-Alperin-Weight-Conjecture-September-23-2026/build/figures/field-tower.tex new file mode 100644 index 0000000000000000000000000000000000000000..3d09d055b1ebb532e17e91a204776b3f2d36cf72 --- /dev/null +++ b/preprints/The-Blockwise-Alperin-Weight-Conjecture-September-23-2026/build/figures/field-tower.tex @@ -0,0 +1,40 @@ +\begin{figure}[!ht] +\centering +\begin{tikzpicture}[ + every node/.style={font=\small}, + field/.style={inner sep=4pt}, + inclusion/.style={-{Stealth[length=4pt]},semithick} +] +\begin{scope}[xshift=-3.45cm] + \node[font=\small\bfseries] at (0,4.65) {Characteristic zero}; + \node[field] (kb) at (0,0) {$K_b$}; + \node[field] (ks) at (-1.45,1.35) {$K_s$}; + \node[field] (kbp) at (1.45,1.35) {$K_b'$}; + \node[field] (l) at (-1.45,2.8) {$L$}; + \node[field] (lp) at (0,4) {$L'=LK_b'$}; + \draw[inclusion] (kb)--(ks); + \draw[inclusion] (kb)--(kbp); + \node[font=\footnotesize,align=left] at (1.85,.35) + {unramified\\degree $n$}; + \draw[inclusion] (ks)--(l); + \draw[inclusion] (l)--(lp); + \draw[inclusion] (kbp)--(lp); +\end{scope} +\begin{scope}[xshift=3.45cm] + \node[font=\small\bfseries] at (0,4.65) {Residue fields}; + \node[field] (f0) at (0,0) {$F_0$}; + \node[field] (fp) at (1.45,1.35) {$F'$}; + \node[field] (f) at (-1.45,2.8) {$F$}; + \node[field] (s) at (0,4) {$S=FF'$}; + \draw[inclusion] (f0)--(fp); + \draw[inclusion] (f0)--(f); + \draw[inclusion] (f)--(s); + \draw[inclusion] (fp)--(s); +\end{scope} +\node[font=\footnotesize,align=center] at (0,-.85) + {$\operatorname{res}(K_b)=\operatorname{res}(K_s)=F_0, + \qquad [F:F_0]=[L:K_s],\qquad [L:K_s]_pm\bigl(h+\log_p m\bigr). + \label{eq:genus-rounding-threshold} +\end{equation} +Let $F/F_0$ be finite with $[F:F_0]_p\le p^h$, and let $Z$ be the +normalization of $E_F$ in a finite purely inseparable extension of its +function field of degree $d\le m$. Choose a compositum $S=FF'$. +The curve $Z_S$ is integral and regular. Let $r_i$ be the ramification +index of $Z_S\to E_S$ at the unique point over $t_i$. +With $\nu(Z)$ computed over $F$, one has +\begin{equation} + \frac{\nu(Z)}d\ge\sum_{i=1}^n\left(1-\frac1{r_i}\right). + \label{eq:inseparable-genus-bound} +\end{equation} +The bound is uniform in $F$, $Z$, and the tuple-class multiset used +later in the counting argument. +\end{theorem} + +The proof has two parts. Absolute differentials give the same +inequality with an error of at most $h+\log_p d$. After multiplication +by $d$, the difference between the two sides is a difference of +divisor degrees over $F$, hence a multiple of $p^{s-h}$. The strict +threshold then forces this difference to be nonnegative. We develop +the differential identities first, together with the separable +ramification bound needed later in \Cref{sec:specialization}. + +\subsection{Divisor degrees, duality, and separable ramification} + +The required identities hold over more general fields than the +twisted-curve construction. Throughout this and the next subsection, +let $\kappa$ be any perfect field of characteristic $p$, let +$S/\kappa$ be finitely generated, and put +$\rho=\operatorname{trdeg}_\kappa S$. Curves and degrees in these two +subsections are over $S$. + +The classical theory of inseparable genus change begins with Tate +\cite{Tate1952}; Schr\"oer \cite{Schroer2009} gives a modern account +for geometrically integral curves. Here absolute differentials retain +arithmetic genus without a geometric-reducedness hypothesis and also +measure ramification at the marked points. + +\begin{lemma}[Degrees over the stated ground field] +\label{lem:curve-degrees} +For an invertible sheaf $\mathcal L$ on $C$, +\begin{equation} + \chi(C,\mathcal L)=\deg_S\mathcal L+\chi(C). + \label{eq:curve-euler-degree} +\end{equation} +A finite dominant morphism $f:C\to C'$ of such curves is finite flat. If +its function-field degree is $\delta$, then +\[ + \deg_S f^*D=\delta\deg_S D + \quad\text{and}\quad + \deg_S f_*D'=\deg_S D' +\] +for divisors $D$ on $C'$ and $D'$ on $C$. These assertions, and Euler +characteristics, add over disjoint unions of regular integral curves. +After a finite separable extension $S_1/S$, divisor and line-bundle degrees +and coherent Euler characteristics have the same numerical values over +$S_1$ as before extension over $S$. +\end{lemma} + +\begin{proof} +For a closed point $x$ put $\deg_S(x)=[\kappa(x):S]$. For any divisor $D$ +the quotient in +\[ + 0\longrightarrow\mathcal O_C(D) + \longrightarrow\mathcal O_C(D+x) + \longrightarrow\mathcal Q_x\longrightarrow0 +\] +has length one over the DVR $\mathcal O_{C,x}$, so its dimension over $S$ +is $[\kappa(x):S]$. Finite-support sheaves have no higher cohomology. +Adding and subtracting closed points proves +$\chi(\mathcal O_C(D))=\chi(C)+\deg_S D$. A rational trivialization of +$\mathcal L$ identifies it with $\mathcal O_C(D)$ for a divisor $D$, giving +\eqref{eq:curve-euler-degree}. Applied to a principal divisor, the same +formula gives degree zero, so this degree depends only on $\mathcal L$. + +Locally on $C'$, the finite module $f_*\mathcal O_C$ is torsion-free over +a DVR and therefore free, of generic rank $\delta$. +For a closed point $x\in C'$ and a uniformizer $t$ there, reduction of +this free module modulo $t$ gives +\begin{equation} + \sum_{y\mapsto x}e_y[\kappa(y):\kappa(x)]=\delta, + \label{eq:local-degree-formula} +\end{equation} +where $e_y$ is the valuation of $t$ in $\mathcal O_{C,y}$. +Indeed the summand at $y$ has a filtration of length $e_y$ with +successive quotients $\kappa(y)$. Multiplication by +$[\kappa(x):S]$ proves the pullback degree formula. The pushforward +formula follows from +$f_*y=[\kappa(y):\kappa(x)]x$ and the tower formula for residue degrees. + +Finite separable scalar extension preserves regularity, and a regular +curve after such extension is a disjoint union of its integral +components. Flat base change for coherent cohomology +in \Cref{foundation:cohomology} gives +\[ + H^i(C_{S_1},\mathcal G_{S_1}) + =H^i(C,\mathcal G)\otimes_S S_1. +\] +Thus the dimensions over the respective fields, and hence Euler +characteristics, agree. Applying +\eqref{eq:curve-euler-degree} proves the assertion for line bundles and +their Cartier divisors. In particular it also applies to a reduced +finite divisor: separable scalar extension preserves its reducedness. +\end{proof} + +\begin{proposition}[The absolute determinant computes arithmetic genus] +\label{prop:absolute-genus} +The absolute cotangent sheaf $\Omega_{C/\kappa}$ is locally free of rank +$\rho+1$, and +\begin{equation} + \deg_S\det\Omega_{C/\kappa}=\nu(C). + \label{eq:absolute-determinant-genus} +\end{equation} +\end{proposition} + +\begin{proof} +An affine coordinate ring of $C$ is a localization of a finitely +generated integral $\kappa$-algebra: choose a finitely generated domain +with fraction field $S$, clear the finitely many coefficients in a +presentation over $S$, and take the contraction of the defining prime +ideal before localizing. Its local rings are consequently essentially +of finite type over $\kappa$. The regularity criterion over a perfect +field in \Cref{foundation:regularity} says that these regular local rings +are local rings at smooth points of the corresponding +$\kappa$-variety. Their absolute differentials are free of rank +\[ + \operatorname{trdeg}_\kappa\kappa(C)=\rho+1. +\] +The sheaf is coherent: the relative differentials over $S$ are coherent, +and $\Omega_{S/\kappa}$ is finite-dimensional, so the cotangent sequence +gives a finite presentation locally. This proves the asserted local +freeness without making a smoothness assumption over $S$. + +Choose a projective embedding $i:C\hookrightarrow\mathbb P^b_S$ and let +$\mathcal I$ be its ideal sheaf. This is a regular immersion of +codimension $b-1$. Here is a local verification. Let $R$ be an ambient +regular local ring, $I$ the local ideal, and $R/I$ the regular local ring +of $C$. Their dimensions differ by $b-1$, by the dimension formula at +the common residue field. Since both are regular, the kernel of +\[ + \mathfrak m_R/\mathfrak m_R^2 + \longrightarrow + \mathfrak m_{R/I}/\mathfrak m_{R/I}^2 +\] +has dimension $b-1$. Choose $b-1$ elements of $I$ lifting a basis of this +kernel. They form part of a regular system of parameters of $R$ and +cut out a regular local quotient of dimension $\dim(R/I)$. +The remaining ideal defining $R/I$ in that quotient is prime and must +be zero: a nonzero prime in a finite-dimensional local domain strictly +lowers the quotient dimension. Thus the chosen elements generate $I$. +It follows that $\mathcal I/\mathcal I^2$ is locally free of rank $b-1$. + +The absolute conormal sequence is in fact short exact: +\begin{equation} + 0\longrightarrow\mathcal I/\mathcal I^2 + \longrightarrow i^*\Omega_{\mathbb P^b_S/\kappa} + \longrightarrow\Omega_{C/\kappa}\longrightarrow0. + \label{eq:absolute-conormal} +\end{equation} +To check the left end, the kernel of the last arrow is locally free, +since the last module is locally free. Its rank is +$(b+\rho)-(1+\rho)=b-1$. The conormal sequence surjects +$\mathcal I/\mathcal I^2$ onto this kernel. A surjection between locally +free modules of the same rank is an isomorphism, proving +\eqref{eq:absolute-conormal}. + +The projective space is obtained from $\mathbb P^b_\kappa$ by extending +constants. Its absolute cotangents therefore split into relative +cotangents over $S$ and the constant bundle of $\Omega_{S/\kappa}$. +Taking determinants in \eqref{eq:absolute-conormal} gives +\[ + \det\Omega_{C/\kappa} + \simeq\mathcal L_c\otimes_S\det\Omega_{S/\kappa}, + \qquad + \mathcal L_c= + \mathcal O_C(-b-1)\otimes + \det(\mathcal I/\mathcal I^2)^\vee. +\] +The last tensor factor is a constant line and has degree zero. + +We compute the degree of $\mathcal L_c$ using the +projective-space duality in \Cref{foundation:duality}. +A Koszul resolution for the local regular sequence defining $C$ shows +that +\[ + \mathcal E xt^v_{\mathbb P^b_S} + (i_*\mathcal O_C,\mathcal O(-b-1)) + =\begin{cases} + i_*\mathcal L_c,&v=b-1,\\ + 0,&v\ne b-1. + \end{cases} +\] +After dualizing the Koszul resolution, its top cohomology +is the dual of the top exterior power of the free module of equations, +restricted to $C$. A change of those equations acts on this exterior +power by the determinant of their change on $\mathcal I/\mathcal I^2$. +Thus the local top cohomology modules glue to +$\det(\mathcal I/\mathcal I^2)^\vee\otimes\mathcal O_C(-b-1)$. + +The local-to-global Ext spectral sequence has only this one nonzero +row. Projective-space duality consequently gives, for $r=0,1$, +\[ + H^r(C,\mathcal O_C)^\vee + \simeq + \operatorname{Ext}^{b-r}_{\mathbb P^b_S} + (i_*\mathcal O_C,\mathcal O(-b-1)) + \simeq H^{1-r}(C,\mathcal L_c). +\] +Hence $\chi(C,\mathcal L_c)=-\chi(C)$. Applying +\eqref{eq:curve-euler-degree} yields +$\deg_S\mathcal L_c=-2\chi(C)$, which proves +\eqref{eq:absolute-determinant-genus}. +\end{proof} + +\begin{lemma}[A separable determinant bound, including the wild term] +\label{lem:separable-different} +Let $f:C\to C'$ be a finite dominant, generically separable morphism of +regular integral projective curves over $S$, of degree $\delta$. +There is an effective divisor $R_f$ on $C$ such that +\[ + \nu(C)=\delta\nu(C')+\deg_S R_f. +\] +If $y\in C$ has ramification index $a_y$ over $f(y)$, then +\begin{equation} + \operatorname{ord}_y R_f\ge a_y-1+\mathbf 1_{p\mid a_y}. + \label{eq:separable-different-bound} +\end{equation} +No separability of $\kappa(y)/\kappa(f(y))$ is required. +The formula and the inequalities can be added componentwise after a +finite separable extension of $S$. +\end{lemma} + +\begin{proof} +The morphism +$f^*\Omega_{C'/\kappa}\to\Omega_{C/\kappa}$ is generically an isomorphism: +at the function fields a finite separable extension has no relative +differentials, and both absolute differential spaces have dimension +$\rho+1$. Its determinant is therefore a nonzero regular morphism of +line bundles. The divisor of this morphism is effective; call it $R_f$. +Its degree is the difference of the two determinant degrees, which is +$\nu(C)-\delta\nu(C')$ by +\Cref{lem:curve-degrees,prop:absolute-genus}. + +Write $x=f(y)$ and choose a uniformizer $t$ of the DVR +$B=\mathcal O_{C',x}$. Its absolute differential is primitive in +$\Omega_{B/\kappa}$, meaning that it is part of a $B$-basis. Indeed the +conormal sequence for the residue field is +\[ + (t)/(t^2)\longrightarrow + \Omega_{B/\kappa}\otimes_B\kappa(x) + \longrightarrow\Omega_{\kappa(x)/\kappa}\longrightarrow0. +\] +The middle term has dimension $\rho+1$ and the right term dimension +$\rho$, since $\kappa(x)$ is finite over $S$ and +\Cref{foundation:fields} computes absolute differential dimension over +the perfect field $\kappa$. Thus the first map has nonzero +one-dimensional image, generated by $dt$. Consequently $dt$ is +nonzero modulo the maximal ideal, and it extends to a basis of the +free module $\Omega_{B/\kappa}$. + +In $A=\mathcal O_{C,y}$ write $t=u\tau^{a_y}$, where $u$ is a unit and +$\tau$ a uniformizer. Absolute differentiation gives +\[ + dt=\tau^{a_y}\,du+ + a_yu\tau^{a_y-1}\,d\tau. +\] +All coordinates of this column of the differential matrix are +divisible by $\tau^{a_y-1}$; when $p\mid a_y$, the second term is zero, +so they are divisible by $\tau^{a_y}$. Expanding the determinant in +this column proves \eqref{eq:separable-different-bound}. Every step +uses absolute differentials over $\kappa$, so an inseparable extension +of residue fields causes no change to the argument. +\end{proof} + +\subsection{The differential calculation for a height-one map} + +We have established the determinant formula for arithmetic genus and +the separable ramification bound. The remaining ingredient is an +exact genus identity for a purely inseparable map of exponent one; +successive such maps will handle arbitrary purely inseparable covers. + +Write $T_C=(\Omega_{C/\kappa})^\vee$ for the absolute tangent bundle. +It has rank $\rho+1$, even when the relative tangent sheaf over $S$ has +different behavior. + +The calculation below is a curve-level form of the canonical-bundle +formula for a $p$-closed foliation; compare +\cite[Section~I.1, Proposition~1.1 and Equations~(1.2)--(1.3)]{Ekedahl1988}. +We prove the version needed here directly from local presentations, +using absolute differentials over the perfect constants. In this way +the kernel, the Frobenius-power cokernel, and all inseparable residue +extensions remain explicit. + +\begin{lemma}[Height-one kernels and cokernels] +\label{lem:height-one} +Let $\pi:C\to C'$ be a finite dominant morphism of regular integral +projective curves over $S$ such that +$\kappa(C)^p\subseteq\kappa(C')$. Put +$q=[\kappa(C):\kappa(C')]=p^a$ and +\[ + \mathcal F=\ker(T_C\longrightarrow\pi^*T_{C'}), + \qquad + \mathcal Q=\operatorname{coker}(T_C\longrightarrow\pi^*T_{C'}). +\] +Both $\mathcal F$ and $\mathcal Q$ are locally free of rank $a$, and +\begin{equation} + \det\mathcal Q\simeq(\det\mathcal F)^{\otimes p}. + \label{eq:height-one-cokernel} +\end{equation} +Consequently +\begin{equation} + \nu(C)=q\nu(C')+(p-1)\deg_S\mathcal F. + \label{eq:height-one-genus} +\end{equation} +\end{lemma} + +\begin{proof} +\textit{The integral presentation.} +On an affine open of $C'$, every element of the finite upstairs +coordinate ring has its $p$th power in the downstairs ring: the power +lies in the downstairs fraction field and is integral, so normality +puts it in that ring. There is therefore a unique prime upstairs over +each downstairs prime, since membership of an element is determined +by membership of its $p$th power. At a closed point we may consequently +write $B\subset A$ for the downstairs and upstairs DVRs, with $A$ still +finite over $B$. It is finite free of rank $q$ and satisfies +$A^p\subset B$. + +Let $\tau$ be an upstairs uniformizer, let $\epsilon$ be the +ramification index, and put $f=[\kappa(A):\kappa(B)]$. +The element $\tau^p\in B$ has upstairs valuation $p$; all nonzero +elements of $B$ have upstairs valuation in $\epsilon\mathbb Z$. +Thus $\epsilon\mid p$, so $\epsilon$ is $1$ or $p$. +Reduction of the rank-$q$ free module modulo a downstairs uniformizer +gives $q=\epsilon f$ by \eqref{eq:local-degree-formula}. +The residue extension has exponent at most one. Successively +adjoining a generator outside the preceding residue field gives a +degree-$p$ extension each time. Choose residue generators +$\bar b_1,\ldots,\bar b_r$ in this manner, so their monomials with +exponents in $\{0,\ldots,p-1\}$ form a $\kappa(B)$-basis of $\kappa(A)$ +and $f=p^r$. Lift them to $b_1,\ldots,b_r\in A$. If $\epsilon=p$, +append $b_{r+1}=\tau$; if $\epsilon=1$, append nothing. In both cases +there are exactly $a$ generators, and $c_\ell=b_\ell^p$ belongs to $B$. + +For a downstairs uniformizer $t$, the ideal $tA$ is +$\tau^\epsilon A$. Filter $A/tA$ by the powers of $\tau$ from $0$ to +$\epsilon$. Each successive quotient is $\kappa(A)$, and the lifted +residue monomials give its basis after multiplication by the +corresponding power of $\tau$. It follows that the $q$ monomials +\[ + b_1^{i_1}\cdots b_a^{i_a},\qquad 0\le i_\ell1$. + +\textit{The Frobenius tower and its recurrence.} +Let $M_e=S(E)$ and $M_z=S(Z)$, and set +\[ + M_j=M_eM_z^{p^j}\quad(0\le j\le b),\qquad + M_0=M_z,\qquad M_b=M_e, +\] +where $b$ is the first index for which the last equality holds. +Let $Y_j$ be the normalization of $E_S$ in $M_j$, and write +\[ + \gamma_j:Y_j\longrightarrow E_S,\qquad + d_j=[M_j:M_e],\qquad + \pi_j:Y_j\longrightarrow Y_{j+1},\qquad + q_j=d_j/d_{j+1}=p^{m_j}. +\] +In particular $Y_0=Z_S$, $d_0=d$, $Y_b=E_S$, and $d_b=1$. +The defining fields satisfy +\begin{equation} + M_{j+1}=M_eM_j^p. + \label{eq:frobenius-tower-field} +\end{equation} +Thus each $\pi_j$ has exponent at most one and +\Cref{lem:height-one} applies. + +Put $T_j=(\Omega_{Y_j/\kappa})^\vee$ and +\[ + \mathcal F_j=\ker(T_j\longrightarrow\pi_j^*T_{j+1}). +\] +This is also the kernel of the composite tangent map +$T_j\to\gamma_j^*T_{E_S}$. To see this, at the function field a +$\kappa$-derivation kills $M_e$ if and only if it kills +$M_eM_j^p=M_{j+1}$, since every derivation kills $p$th powers. +The two kernels are therefore equal generically. They are equal as +sheaves: a section whose image vanishes generically has zero image +everywhere in either of the locally free, hence torsion-free, targets. + +The genus identity involves $\deg_S\mathcal F_j$. We will estimate +this degree through the image of the tangent map to $E_S$. +Choose a nonzero invariant differential $\eta$ on $A_0$. +The constant-curve identification of $E_S$ gives the splitting +\[ + \Omega_{E_S/\kappa} + \simeq\mathcal O_{E_S}\eta + \oplus(\mathcal O_{E_S}\otimes_S\Omega_{S/\kappa}). +\] +Thus its dual is a trivial bundle of rank $n+1$. In this trivial +target we can use the marked points to impose linear vanishing +conditions on the tangent-map image. + +Let $\mathcal H_j$ be the image in $\gamma_j^*T_{E_S}$, and put +$b_j^\circ=-\deg_S\mathcal H_j$. +It is a torsion-free coherent sheaf on a regular curve, hence locally +free. Its rank is $n+1-m_j$, since +$\operatorname{trdeg}_\kappa S=n$. +The exact sequence with kernel $\mathcal F_j$ and image $\mathcal H_j$, +together with \eqref{eq:height-one-genus}, gives +\[ + \nu(Y_j)=q_j\nu(Y_{j+1})+(p-1)\deg_S\mathcal F_j, + \qquad + b_j^\circ=\nu(Y_j)+\deg_S\mathcal F_j. +\] +Eliminating $\deg_S\mathcal F_j$ and using $d_j=q_jd_{j+1}$ yields +\begin{equation} + \frac{\nu(Y_j)}{d_j} + =\frac1p\frac{\nu(Y_{j+1})}{d_{j+1}} + +\left(1-\frac1p\right)\frac{b_j^\circ}{d_j}. + \label{eq:genus-recurrence} +\end{equation} +The terminal term $\nu(Y_b)$ is zero because $E_S$ has genus one. +It remains to give a lower bound for +$b_j^\circ/d_j=-\deg_S\mathcal H_j/d_j$. + +\textit{Vanishing at a ramified label.} +For each $j$ let $S_j$ be the set of labels at which $\gamma_j$ has +ramification index greater than one. For $i\in S_j$ write +$D_{j,i}=\gamma_j^*(t_i)$ and +\[ + v_i=(1,-t_i^*\eta)\in S\oplus\Omega_{S/\kappa}. +\] +Regard $v_i$ as a constant linear functional on +$\gamma_j^*T_{E_S}$. We claim that its restriction to $\mathcal H_j$ +vanishes along the entire divisor $D_{j,i}$; equivalently, its values +lie in $\mathcal O_{Y_j}(-D_{j,i})$. + +Let $a$ be a local parameter defining the rational point $t_i$ on +$E_S$. Restriction of absolute differentials along $t_i$ sends +$(\lambda,\beta)$ in the displayed splitting to +$\lambda t_i^*\eta+\beta$. Its kernel is the line spanned by $v_i$. +Since $a$ restricts to zero, $da$ belongs to this kernel modulo $(a)$. +Moreover its relative component is nonzero: $t_i$ is an $S$-rational +point on a smooth curve over $S$, and $a$ generates its relative +cotangent space. Hence +\[ + da\equiv\lambda_i v_i\pmod{a\Omega_{E_S/\kappa}} + \quad\text{for some }\lambda_i\in S^\times. +\] +At the unique point upstairs write $a=u\tau^r$, with $u$ a unit. +The nontrivial purely inseparable index $r$ is divisible by $p$; +therefore +\[ + d(\gamma_j^*a)=\tau^r\,du=(\gamma_j^*a)\,u^{-1}du. +\] +It lies in the ideal of the \emph{full} divisor $D_{j,i}$ times +$\Omega_{Y_j/\kappa}$. Pulling back the preceding congruence and +dividing by the unit $\lambda_i$ shows that the pullback of $v_i$ lies +in that same ideal times $\Omega_{Y_j/\kappa}$. +To relate this differential to the claimed functional, denote the +tangent map by $d\gamma_j:T_j\to\gamma_j^*T_{E_S}$ and write +$\gamma_j^*v_i$ for the image of the pulled-back covector in +$\Omega_{Y_j/\kappa}$. For a local section $\theta$ of $T_j$, duality +gives +\[ + v_i(d\gamma_j(\theta))=(\gamma_j^*v_i)(\theta). +\] +The divisibility just proved makes this value a section of +$\mathcal O_{Y_j}(-D_{j,i})$. Since $\mathcal H_j$ is the image of +$d\gamma_j$, the claim follows. The vanishing is along +$\gamma_j^*(t_i)$ with its full multiplicity, not just its reduced +support. + +\textit{From independent vanishing conditions to a degree bound.} +By \Cref{lem:label-rank}, among the $v_i$ for $i\in S_j$ there are +$c\ge |S_j|-h$ independent covectors over $S$. + +Complete the $c$ chosen covectors to a constant basis of the dual of +the target trivial tangent bundle. The full-divisor constraint gives +an injection +\begin{equation} + \mathcal H_j\hookrightarrow + \mathcal O_{Y_j}^{n+1-c}\oplus + \bigoplus_{i\ \mathrm{chosen}}\mathcal O_{Y_j}(-D_{j,i}). + \label{eq:label-image-injection} +\end{equation} +Let $R=n+1-m_j$ be the rank of $\mathcal H_j$. +The $R$th exterior power of the bundle on the right is a direct sum +of line bundles, each the tensor product of a choice of $R$ summands. +The induced +map from $\det\mathcal H_j$ is nonzero, so it has a nonzero component +in at least one of these lines. Only $n+1-c$ of the available +summands are trivial. The chosen line must therefore contain at least +$R-(n+1-c)=c-m_j$ negative divisor summands when this number is +positive, and at least zero otherwise. +Each $D_{j,i}$ has degree $d_j$ by finite flatness and rationality of +$t_i$. If the nonzero component uses $k$ negative summands, its target +line has degree $-kd_j$. A nonzero regular map from one line bundle +to another has an effective zero divisor, and consequently +\[ + \deg_S\mathcal H_j\le-kd_j, + \qquad + \frac{b_j^\circ}{d_j}\ge c-m_j\ge |S_j|-h-m_j. +\] +The argument also covers rank zero, with $\det(0)=\mathcal O$. + +\textit{The weighted estimate.} +Write $r_i=p^{u_i}$. Every height-one step has local index at most +$p$, as proved in \Cref{lem:height-one}. The composite +$Y_0\to Y_j$ consequently has ramification index at most $p^j$ at +this label. In particular, $u_i\leq b$. Since ramification indices +multiply, the remaining map +$Y_j\to E_S$ has index at least $p^{u_i-j}$ whenever $j-N. +\] +A multiple of $N$ strictly greater than $-N$ is nonnegative. This +proves \eqref{eq:inseparable-genus-bound}. +The sufficient threshold \eqref{eq:genus-rounding-threshold} contains +no datum from $F$, $Z$, or a tuple-class multiset. If a threshold +independent of the prime is desired, the stronger condition +\[ + 2^{s-h}>m(h+\log_2m) +\] +suffices for every $p\ge2$. This proves the stated uniformity. +\end{proof} diff --git a/preprints/The-Blockwise-Alperin-Weight-Conjecture-September-23-2026/build/sections/foundations.tex b/preprints/The-Blockwise-Alperin-Weight-Conjecture-September-23-2026/build/sections/foundations.tex new file mode 100644 index 0000000000000000000000000000000000000000..fab2dc0f06a66662ce4c235aa8aae8a564f8c8fa --- /dev/null +++ b/preprints/The-Blockwise-Alperin-Weight-Conjecture-September-23-2026/build/sections/foundations.tex @@ -0,0 +1,167 @@ +\section{Classical foundations used}\label{sec:foundations} + +We state the classical inputs to make the scope of the proof explicit. We also use elementary linear algebra, field and group theory, and the basic definitions and functorial formalism of schemes, sheaves, divisors, and K\"ahler differentials. The counting, index, inseparable-genus, valued-field, and specialization assertions particular to this proof are established in the later sections. In particular, no weight-counting theorem or conjectural reduction is an input. + +\begin{foundation}[Finite groups]\label{foundation:groups} +For a finite group $X$, its $p$-subgroups lie in Sylow subgroups of order $|X|_p$. A nontrivial finite $p$-group has nontrivial center and an element of order $p$, and each of its maximal proper subgroups is normal. We use the orbit--stabilizer formula and the usual fixed-point and conjugation actions. +\end{foundation} + +\begin{foundation}[Finite-dimensional algebras and characters]\label{foundation:algebra} +The Jacobson radical of a finite-dimensional algebra is nilpotent. Over an algebraically closed field, its semisimple quotient is a product of full matrix algebras, one for each simple module up to isomorphism. A finite-dimensional commutative algebra over an algebraically closed field is a product of local algebras with that field as residue, indexed by its primitive idempotents. Finite group algebras in characteristic zero are semisimple. Over a splitting field the simple endomorphism algebras are the scalar field, and an ordinary irreducible character satisfies +\[ + \sum_{x\in X}\chi(x)\chi(x^{-1})=|X|. +\] +Idempotents lift uniquely from the residue algebra of a finite commutative algebra over a complete discrete valuation ring. Nakayama's lemma applies to finite modules over any local ring. +\end{foundation} + +\begin{foundation}[Commutative algebra]\label{foundation:commalg} +A finite-type algebra over a Noetherian ring is Noetherian. Integral extensions satisfy lying over. Closed points of a variety over a field are dense and have finite residue fields over the ground field. For an integral variety, the local dimension at a point plus the transcendence degree of its residue field is the transcendence degree of the function field. A regular local ring is a normal domain; a part of a minimal system of generators of its maximal ideal is a regular sequence and its quotient is regular of the correspondingly smaller dimension. A one-dimensional Noetherian normal local domain is a discrete valuation ring, as is a Noetherian local domain with nonzero principal maximal ideal. Finite torsion-free modules over discrete valuation rings are free. Completion of a discrete valuation ring retains its residue field and uniformizer. +\end{foundation} + +\begin{foundation}[Fields and absolute field differentials]\label{foundation:fields} +If a finite group $D$ acts faithfully by automorphisms on a field $M$, then $M/M^D$ is Galois of degree $|D|$. Finite separable field extensions are simple. A finite field extension has a maximal separable subextension, and the remaining extension is purely inseparable; the number of embeddings in an algebraic closure is the separable degree. Multiplicative groups of finite fields are cyclic. Algebraically closed fields are classified by their characteristic and transcendence degree. If $S$ is a finitely generated field of transcendence degree $\rho$ over a perfect field $\kappa$, then +\[ + \dim_S\Omega_{S/\kappa}=\rho; + \qquad [S:S^p]=p^\rho\quad\text{in characteristic }p. +\] +\end{foundation} + +\begin{foundation}[Valuations over a complete discrete base]\label{foundation:valuation} +A valuation extends to an algebraic field extension. A complete discrete valuation on a field extends uniquely to any finite field extension; its valuation ring is the integral closure of the original ring, is finite over it, and is again a complete discrete valuation ring. The fields to which we apply the latter assertion initially have characteristic zero. No corresponding finiteness assertion for an arbitrary nondiscrete valuation ring is assumed. +See \cite[Tag 00IA]{Stacks} for prolongation and +\cite[Tags 032W, 09EM and 0325]{Stacks} for the complete +discrete-base finiteness assertions. +\end{foundation} + +\begin{foundation}[Descent and finite models]\label{foundation:descent} +A projective scheme with a semilinear descent datum under a finite Galois field extension descends if equipped with a compatible linearized ample line bundle. The same statement holds for a finite \emph{\'etale} faithfully flat Galois extension of rings and a relatively ample line bundle. Equivariant morphisms and invariant closed subschemes descend along with the scheme. Flatness, smoothness, finiteness, \emph{\'etaleness}, and relative ampleness in these settings are checked after the faithfully flat extension. Geometric integrality of a finite-type scheme over a field can be checked after an extension of that field. A finite diagram of projective schemes and morphisms of finite presentation over a directed union of fields is defined over a finite stage, including its maps to a base scheme already defined there and the identities among those maps. One may realize projective descent by descent of semilinear modules and graded algebras followed by $\Proj$. +For the projective and module descent assertions, see \cite[Tags 0CCJ, 0CDR and 0D1V]{Stacks}. +The five locality assertions are respectively +\cite[Tags 02L2, 02VL, 02LA, 02VN and 0D3C]{Stacks}; geometric +integrality is covered by \cite[Tags 038K, 0384 and 054P]{Stacks}. +The finite-stage assertion follows from \cite[Tags 01ZM and 01ZP]{Stacks}, +including chosen projective embeddings among the finite data. +\end{foundation} + +\begin{foundation}[Normalization]\label{foundation:normalization} +The normalization of an integral variety over a field in a finite extension of its function field is finite. The normalization of a finite-type $\mathbb Z$-domain in its fraction field is finite. In the integral closure of a normal domain in a finite Galois extension of its fraction field, the Galois group acts transitively on the primes above any fixed prime of the base. +The normalization finiteness assertions follow from the Nagata property in \cite[Tag 0335]{Stacks}. +\end{foundation} + +\begin{foundation}[Regularity, smoothness, and properness]\label{foundation:regularity} +Finite separable scalar extension preserves regularity and reducedness. Smooth varieties over a field are regular. A regular local ring essentially of finite type over a perfect field $\kappa$ is smooth at the corresponding point of a finite-type $\kappa$-model; its module of absolute differentials is free of rank the transcendence degree of the function field over $\kappa$. We use the valuative criterion for properness, including the unique extension of maps from fraction fields of valuation rings, and that proper maps are closed. +For regularity and smoothness over a perfect field, see \cite[Tags 00TV and 0B8X]{Stacks}. +\end{foundation} + +\begin{foundation}[Proper cohomology and ample section rings]\label{foundation:cohomology} +Over a Noetherian base, the higher direct images of coherent sheaves under a proper morphism are coherent. Over an affine base their cohomology commutes with flat affine base change. For a projective scheme over an affine Noetherian base and a relatively ample line bundle, sufficiently high powers give projective embeddings, and sufficiently high twists annihilate higher coherent cohomology. The ample section ring is finitely generated and its $\Proj$ recovers the scheme. Finite pullback preserves ampleness; tensor products of relatively ample line bundles are relatively ample, and a line bundle is relatively ample if a positive power is. Formation of $\Proj$ commutes with base change. On a projective curve over a field, coherent cohomology vanishes in degrees greater than one, and for sheaves of finite support in degrees greater than zero. Global functions on a geometrically integral projective variety over a field equal the ground field. +For proper coherence, flat cohomology base change, and ample twists, respectively, see \cite[Tags 02O5, 02KH and 0B5T]{Stacks}. +For Proj recovery and the ampleness rules, see +\cite[Tags 0EKI, 0892, 0890 and 02NN]{Stacks}. +\end{foundation} + +\begin{foundation}[Differentials, adjunction, and duality]\label{foundation:duality} +We use the transitivity sequence +\[ + \Omega_{V/U}\otimes_VW\longrightarrow\Omega_{W/U} + \longrightarrow\Omega_{W/V}\longrightarrow0 +\] +and the conormal sequence for a closed immersion. Let $S$ be a field. On $\mathbb P^b_S$, +$\det\Omega_{\mathbb P^b_S/S}=\mathcal O(-b-1)$. For a coherent sheaf $\mathcal G$ there, Serre duality identifies +\[ + H^i(\mathcal G)^\vee + \simeq\operatorname{Ext}^{b-i} + (\mathcal G,\mathcal O(-b-1)). +\] +We use the local-to-global Ext spectral sequence and the Koszul resolution of a quotient by a regular sequence. These inputs will yield the required absolute determinant-degree formula even when the curve is not smooth over $S$. +For the projective-space dualizing complex and the adjunction formalism, see \cite[Tag 0A9W and Section~48.15]{Stacks}. +The local-to-global Ext sequence is \cite[Tag 0BQP]{Stacks}. +\end{foundation} + +\begin{foundation}[Elliptic curves]\label{foundation:elliptic} +A smooth Weierstrass cubic with its point at infinity is an elliptic curve. A generalized Weierstrass equation over a discrete valuation ring with unit discriminant gives a smooth elliptic scheme, polarized by three times its origin. Its relative cotangent line is trivial with invariant differential $\eta$, and multiplication by an integer $a$ pulls $\eta$ back to $a\eta$. For a positive integer $a$, multiplication by $a$ on an elliptic curve or elliptic scheme is finite locally free of degree $a^2$. +For the relative group law, see +\cite[II, Proposition~2.7]{DeligneRapoport1973}; the unit-discriminant +Weierstrass family and the invariant cotangent description are given in +\cite[Tags 072S and 047I]{Stacks}. See also +\cite[Chapter~III]{Silverman2009} for the Weierstrass formulation. +Multiplication is finite flat and surjective by +\cite[Proposition~20.7]{MilneAV2022}; over the Noetherian bases here it +is finite locally free by \cite[Tag 02KB]{Stacks}. +\end{foundation} + +\begin{foundation}[Complex curves and covers]\label{foundation:complex} +Finite \'etale algebraic covers of a smooth complex algebraic curve +correspond to finite unramified topological covers of its analytification, +including morphisms. For a punctured projective curve, normalization of +the original projective curve in the resulting algebraic function field +gives the finite compact cover. Normal complex curves are smooth, and the +local punctured covering completes by a power map, so monodromy cycle +lengths give ramification indices. Topological covers are classified by +fundamental-group actions on finite fibers. The fundamental group of a +genus-one surface punctured at $n$ points has generators +$x,y,u_1,\ldots,u_n$ with the single relation +\[ + [x,y]u_1\cdots u_n=1. +\] +For a finite map of smooth proper curves over an algebraically closed field of characteristic zero, Riemann--Hurwitz states +\[ + 2g(C)-2=\deg(f)\bigl(2g(C')-2\bigr) + +\sum_{P\in C}(e_P-1). +\] +For the algebraic/topological cover correspondence, see \cite[Theorem~3.4]{MilneLEC2008}; for Riemann--Hurwitz, see \cite[Section~53.12, Tag 0C1B]{Stacks}. +\end{foundation} + +\begin{foundation}[Noetherian connectedness]\label{foundation:connectedness} +A proper morphism over a locally Noetherian base whose structure-sheaf direct image is canonically the structure sheaf of the base has geometrically connected fibers. This is the connected-fiber assertion of Stein factorization. We apply it only after constructing a Noetherian normal model, not directly over the nondiscrete valuation ring. +See \cite[Tag 03H0]{Stacks}. +\end{foundation} + +\subsection{Coefficient fields and scalar extension}\label{sec:coefficients} + +We justify the coefficient choice in \Cref{sec:introduction} without +requiring a mixed-characteristic lift of every algebraically closed +field. Put $\kappa=\overline{\mathbb F}_p$ and choose an embedding +$\kappa\hookrightarrow k$. For any finite group $X$, the ideal +$\rad(\kappa X)\otimes_\kappa k$ is nilpotent, and the quotient of +$kX$ by this ideal is the scalar extension of the split matrix product +$\kappa X/\rad(\kappa X)$. That quotient is again a split semisimple +algebra. The ideal is therefore exactly $\rad(kX)$, and the simple +modules correspond under scalar extension. + +The center also commutes with this extension: it is the kernel of the +finitely many linear maps $a\mapsto ax-xa$, $x\in X$, and field +extension preserves kernels. Each local factor of $Z(\kappa X)$ has +nilpotent radical and residue $\kappa$; after extension it still has +nilpotent radical, now with residue $k$, so remains local. Thus +primitive blocks correspond and their simple-module counts agree. +Coefficient restriction, restriction to centralizers in the Brauer +map, and the quotient maps for normal subgroups all commute with +scalar extension. The residue characters of the local central factors +extend as well. Consequently the central-character block assignment +in \eqref{eq:block-assignment} is preserved, for every subgroup and +subquotient involved in a weight. It suffices to prove the numerical +identity over $\kappa$. + +To construct the coefficient ring, start with $\mathbb Z_p$, lift a +tower of finite separable extensions of its residue field $\mathbb F_p$ +that exhausts $\kappa$, and complete the union. The construction at the +start of the proof of \Cref{prop:mixed-lift} shows that this gives a +complete discrete valuation ring of characteristic zero with residue +$\kappa$. +In an algebraic closure of its fraction field, choose matrix +realizations of every ordinary irreducible representation of every +subgroup of $G$. There are only finitely many such representations and +matrix entries. Adjoining their entries gives a finite field extension +that splits all those subgroups. Its valuation ring is again a complete +DVR by \Cref{foundation:valuation}; its finite residue extension is +still $\kappa$, which is algebraically closed. This gives the system +$(K,\mathcal O,\kappa)$ used in \Cref{sec:group}. Quotient +representations inflate, so the same coefficients split the required +subquotients. These coefficient fields are separate from the auxiliary +geometric fields constructed later. + +The remaining sections prove the blockwise chain identity, the marked +high-index twist, the genus estimate with exact rounding, the required +valued-field assertions, and the specialization model. The foundations +above supply their classical inputs, not the desired weight equality. diff --git a/preprints/The-Blockwise-Alperin-Weight-Conjecture-September-23-2026/build/sections/group.tex b/preprints/The-Blockwise-Alperin-Weight-Conjecture-September-23-2026/build/sections/group.tex new file mode 100644 index 0000000000000000000000000000000000000000..a3eac5562c3191f1a6adbacbf1646acce9427769 --- /dev/null +++ b/preprints/The-Blockwise-Alperin-Weight-Conjecture-September-23-2026/build/sections/group.tex @@ -0,0 +1,839 @@ +\section{From block counts to generating tuples}\label{sec:group} + +Fix a prime $p$. Write $O_p(X)$ for the largest normal $p$-subgroup +of a finite group $X$, and put $[x,y]=xyx^{-1}y^{-1}$. A +$p$-element includes the identity; a $p'$-element has order prime to +$p$; and a $p$-singular element has order divisible by $p$. +The geometric sections will establish the following precise counting +statement. Its uniformity is part of the assertion. + +\begin{theorem}[Uniform divisibility for generating tuples] +\label{thm:tuple-divisibility} +Let $J$ be a finite group with $O_p(J)=1$. For $s\geq0$, $n=p^s$, +and a multiset +$\mathcal M$ of $n$ conjugacy classes of $p$-singular elements of $J$, +let $N_{J,s}(\mathcal M)$ be the number, modulo simultaneous +$J$-conjugation, of ordered tuples +\[ + (x,y,u_1,\ldots,u_n)\in J^{n+2} +\] +satisfying +\begin{equation}\label{eq:tuple-relation} + [x,y]u_1\cdots u_n=1,\qquad + \langle x,y,u_1,\ldots,u_n\rangle=J, + \qquad \{[u_1],\ldots,[u_n]\}=\mathcal M. +\end{equation} +For every integer $h\geq1$, there is an integer $s_0=s_0(p,J,h)$ +such that +\[ + p^h\mid N_{J,s}(\mathcal M) + \qquad(s\geq s_0) +\] +for every such multiset $\mathcal M$. +\end{theorem} + +The entries $u_i$ remain ordered: prescribing their multiset of classes +does not quotient by permutations. Empty tuple sets have count zero, +so the statement includes $J=1$ and groups of order prime to $p$. +The proof of \Cref{thm:tuple-divisibility} is completed after +\Cref{prop:no-small-cover}. This section proves that it implies the +block equality, using \Cref{foundation:groups,foundation:algebra} +and proving the required counting reductions explicitly. + +\subsection{Coefficient projection and the weight transform} + +For each ambient finite group $X$, choose a splitting coefficient system +$(K,\mathcal O,k)$ as in \Cref{sec:introduction,sec:coefficients}, with +$X$ in place of $G$. Hold this system fixed throughout the calculations +with $X$ and its subquotients. +This system also splits all subquotients that occur below: a subgroup of a +quotient lifts to a subgroup by taking its preimage, and a +representation of a quotient inflates to that preimage. These +observations apply after any number of subgroup and quotient steps. +Ordinary irreducibles may therefore be realized over $K$. Their +identification with complex irreducibles, when needed, follows by +first splitting the group algebra over $\overline{\mathbb Q}$ and +then extending scalars to algebraically closed characteristic-zero +fields. + +For a central idempotent $E\in kX$, define +\begin{align*} + l(X,E)&=\#\{\text{simple $kX$-modules selected by $E$}\},\\ + k_{\mathrm{ord}}(X,E)&=\#\{\text{ordinary irreducibles selected by the lift of $E$}\},\\ + z(X,E)&=\#\{\chi\text{ selected by that lift}:\chi(1)_p=|X|_p\}. +\end{align*} +Here and throughout, modules and characters are counted up to +isomorphism. The class sums form a basis of $Z(\mathcal O X)$, and +their reductions form a basis of $Z(kX)$. Idempotent lifting over the +complete DVR thus gives a unique central idempotent lifting $E$. +Every integral central element acts on an ordinary irreducible by +a scalar in $\mathcal O$: the scalar lies in $K$ by splitting and +satisfies a monic integral equation by finiteness, and $\mathcal O$ +is integrally closed. Reduction of this scalar defines a character +of $Z(kX)$. The Artinian local decomposition of that center shows +that it is precisely $\lambda_b$, where $b$ is the block containing +the ordinary irreducible. + +For $H\leq X$, let $s_H$ denote coefficient restriction from +$Z(kX)$ to $Z(kH)$, or its integral analogue when appropriate. +For a $p$-subgroup $Q$, let $\operatorname{Br}_Q$ denote restriction +of coefficients from the $Q$-invariants $(kX)^Q$ to $kC_X(Q)$. + +\begin{lemma}[Normalizer projection]\label{lem:normalizer-projection} +The map $\operatorname{Br}_Q:(kX)^Q\longrightarrow kC_X(Q)$ is a +unital algebra homomorphism. Suppose that +\[ + Q\leq H\leq N_X(Q),\qquad C_X(Q)\leq H. +\] +Writing $\pi_Q:kH\longrightarrow k[H/Q]$ for the quotient map, one +has, for $v\in Z(kX)$, +\begin{equation}\label{eq:normalizer-projection} + \pi_Q(s_H(v))=\pi_Q(\operatorname{Br}_Q(v)), + \qquad + \lambda_b(s_H(v))=\lambda_b(\operatorname{Br}_Q(v)) +\end{equation} +for every block $b$ of $H$. In particular $\lambda_b\circ s_H$ is +the central character of a unique block of $X$. +\end{lemma} + +\begin{proof} +To compute the coefficient of $g\in C_X(Q)$ in a product of two +$Q$-invariant elements, let $Q$ conjugate the pairs $(a,b)$ with +$ab=g$. Coefficient products are constant on the orbits. Every +nontrivial orbit has size divisible by $p$, and the fixed pairs are +exactly those with $a,b\in C_X(Q)$. This proves multiplicativity; +the identity is preserved. + +Each coset $hQ$ is stable under conjugation by $Q$, because +$h\in N_X(Q)$. Applying the same orbit cancellation to the sum +of coefficients over that coset proves the first identity in +\eqref{eq:normalizer-projection}. Also +$\operatorname{Br}_Q(v)\in Z(kH)$, since its support lies in +$C_X(Q)\leq H$ and it is invariant under $H$. + +Every nonzero module for a finite $p$-group over $k$ has nonzero +invariants. For completeness, choose a central element $z$ of order +$p$; the nonzero kernel of $z-1$, whose $p$th power is zero, is a +module for the smaller group $Q/\langle z\rangle$, and induction +applies. If the module is simple for $H$, its $Q$-invariants are +$H$-stable, and hence the whole module. Thus every simple $kH$-module +factors through $H/Q$, proving the second identity. The composition +with $\operatorname{Br}_Q$ is a unital algebra character. Characters +of the finite commutative algebra $Z(kX)$ are its local residue +characters, one for each block, giving uniqueness. +\end{proof} + +For $H=N_X(Q)$ put +\[ + \overline E_Q + =\pi_Q(s_H(E)) + =\pi_Q(\operatorname{Br}_Q(E))\in Z(k[H/Q]). +\] +This is an idempotent. Its selected defect-zero quotient characters +are exactly the weights assigned to the blocks appearing in $E$. +Indeed, the lift of $\operatorname{Br}_Q(E)$ maps to the lift of +$\overline E_Q$ by uniqueness of central idempotent lifting. Its +action on the inflation of a quotient character is therefore one +exactly when the latter is selected by $\overline E_Q$. If $b$ is +the block of this inflation, the same test is +\[ + \lambda_b(\operatorname{Br}_Q(E)) + =\lambda_b(s_H(E))=1. +\] +By \Cref{lem:normalizer-projection}, this says that the induced block +is one of the blocks in $E$, with the coefficient-projection +definition in \eqref{eq:block-assignment}. Once $Q$ is fixed up to $X$-conjugacy, +there is no further identification of its quotient characters: +$N_X(Q)$ acts on $N_X(Q)/Q$ by inner automorphisms. + +For a conjugation-invariant function $g$ of such pairs $(X,E)$, +define +\begin{equation}\label{eq:weight-transform} + (Tg)(X,E)= + \sum_{\substack{10$, the binomial +formula gives +\[ + \operatorname{val}((1+a)^p-1) + \geq + \min\{\operatorname{val}(p)+\operatorname{val}(a), + p\operatorname{val}(a)\}. +\] +After normalizing the discrete valuation to integer values, each +iteration increases the positive valuation by at least one, unless +the difference has already become zero. These powers tend to one. +This argument applies also to $p=2$ and requires no finiteness of +the residue field. + +For positive-length chains the limiting trace therefore counts +all ordinary irreducibles selected by $E_{\mathcal C}$. At length +zero it omits the semisimple blocks, each of which contributes +exactly one defect-zero character by +\Cref{lem:defect-zero-detection}. There are finitely many chains +and irreducibles, so limits may be summed, proving +\eqref{eq:trace-limit}. +\end{proof} + +The next identity follows from Frobenius's commutator-counting formula +and is the weighted genus-one case of the surface formula in +\cite[Theorem~2.1 and Equation~(3.1)]{Klug2025}. Its single +denominator is important when chain orbits are replaced by actual +chains, so we verify the normalization directly. + +\begin{lemma}[Trace normalization]\label{lem:tuple-trace} +For $H\leq X$ and $w=s_H(\widehat v)$, +\begin{equation}\label{eq:tuple-trace} + \operatorname{Tr}(w^n\mid V(KH)) + =\frac1{|H|} + \sum_{\substack{x,y,u_1,\ldots,u_n\in H\\ + [x,y]u_1\cdots u_n=1}} + \prod_{i=1}^n\alpha_{u_i}. +\end{equation} +\end{lemma} + +\begin{proof} +The sum on the right before division is +$\tau(d_Hw^n)$. The regular trace of any element of $KH$ is +$|H|$ times its identity coefficient. On the other hand, +\eqref{eq:ordinary-commutator-scalar} and the regular decomposition +give +\begin{align*} + |H|\tau(d_Hw^n) + &=\sum_{\chi\in\operatorname{Irr}(H)} + \chi(1)^2\frac{|H|^2}{\chi(1)^2}\lambda_\chi(w)^n\\ + &=|H|^2\operatorname{Tr}(w^n\mid V(KH)). +\end{align*} +Dividing proves exactly the single denominator in +\eqref{eq:tuple-trace}. +\end{proof} + +For a subgroup $J\leq X$, define an integer on actual chains, +without quotienting by conjugation, +\begin{equation}\label{eq:chain-weight} + a_X(J)= + \sum_{\substack{\mathcal C\text{ a chain in }X\\ + J\leq H_{\mathcal C}}}(-1)^{|\mathcal C|}. +\end{equation} + +\begin{lemma}[Cancellation by a normal $p$-subgroup] +\label{lem:pcore-cancellation} +If $O_p(J)\ne1$, then $a_X(J)=0$. +\end{lemma} + +\begin{proof} +Put $P=O_p(J)$. In any chain normalized by $J$, each $Q_i$ is +normalized by $P\leq J$, so $Q_iP$ is a $p$-subgroup. Both +factors are normalized by $J$, making their product +$J$-normalized. Let $i$ be the last index such that $P\nleq Q_i$; +it exists since $Q_0=1$ and $P\ne1$. Toggle $Q_iP$ immediately +after $Q_i$. It strictly contains $Q_i$ and is contained in all +later members. Insertion and deletion preserve the last member +not containing $P$, so this is an involution. It is defined also +for the length-zero chain and for terminal insertion or deletion. +It preserves the property of being normalized by $J$ and reverses +the sign. The sum of actual chains in \eqref{eq:chain-weight} +therefore vanishes. +\end{proof} + +\begin{proposition}[Reduction to uniform tuple divisibility] +\label{prop:group-reduction} +If \Cref{thm:tuple-divisibility} holds for every finite group with +trivial $p$-core, then for every finite group $X$ and central +idempotent $E\in kX$ one has $l(X,E)=((1+T)z)(X,E)$. +In particular the block equality in \Cref{thm:main} follows, with +the stated block assignment and with $Q=1$ included. +\end{proposition} + +\begin{proof} +The orbit of a chain $\mathcal C$ has size +$|X|/|H_{\mathcal C}|$. Consequently +\Cref{lem:tuple-trace} changes the orbit sum +\eqref{eq:chain-trace} into the following sum over actual tuples +and actual chains: +\begin{equation}\label{eq:trace-by-generated-group} + \Phi_n=\frac1{|X|} + \sum_{\substack{x,y,u_1,\ldots,u_n\in X\\ + [x,y]u_1\cdots u_n=1}} + a_X\bigl(\langle x,y,u_1,\ldots,u_n\rangle\bigr) + \prod_{i=1}^n\alpha_{u_i}. +\end{equation} +To verify the factor, each orbit contribution has prefactor +$1/|H_{\mathcal C}|$; replacing its representative by all +$|X|/|H_{\mathcal C}|$ conjugates gives the common prefactor +$1/|X|$. For a fixed tuple the admissible chains are exactly those +normalized by its generated subgroup. This is precisely +$a_X(J)$. + +By \Cref{lem:pcore-cancellation}, only generated subgroups +$J$ with $O_p(J)=1$ contribute. By the support of $\widehat v$, +only tuples with every $u_i$ $p$-singular contribute. Fix such a +subgroup $J$ and group its generating tuples according to the +multiset $\mathcal M$ of $J$-conjugacy classes of the $u_i$. +The product $\prod_i\alpha_{u_i}$ is constant on this multiset: +the coefficients are constant on $X$-classes, hence on $J$-classes, +and their product is unchanged by reordering. Denote this integral +product by $\alpha(\mathcal M)$. A generating tuple has simultaneous +conjugation stabilizer $Z(J)$, since an element centralizing all +its entries centralizes the group they generate. Thus its number +of actual tuples is $|J|/|Z(J)|$ times its orbit count. For +$n=p^s$, the numerator in \eqref{eq:trace-by-generated-group} +is therefore +\begin{equation}\label{eq:uniform-numerator} + \sum_{\substack{J\leq X\\O_p(J)=1}} + a_X(J)\frac{|J|}{|Z(J)|} + \sum_{\mathcal M} + \alpha(\mathcal M)N_{J,s}(\mathcal M). +\end{equation} + +Fix any $h\geq1$. By \Cref{thm:tuple-divisibility}, each summand +of the inner sum belongs to $p^h\mathcal O$ for all sufficiently +large $s$, with a threshold independent of $\mathcal M$. Although +the number of multisets grows with $s$, the sum is finite for each +$s$ and remains in that same ideal. There are only finitely many +subgroups $J\leq X$, so take the largest of their thresholds. +The integer factors in \eqref{eq:uniform-numerator} preserve +divisibility. We conclude that its numerator tends to zero in the +valuation topology. The fixed denominator $|X|$ does not change +this conclusion, so $\Phi_{p^s}\longrightarrow0$. + +Combine this with \Cref{lem:trace-limit} and +\Cref{lem:alternating-subsections}. The resulting equality is +\eqref{eq:chain-inversion}; its integer terms agree in the +characteristic-zero field $K$, and hence agree as integers. +The same reasoning applies to every pair and to the splitting +subquotients arising in the transform. \Cref{lem:chain-transform} +now gives $l=(1+T)z$. The weight interpretation of +\eqref{eq:weight-transform} proves the asserted block equality. +\end{proof} diff --git a/preprints/The-Blockwise-Alperin-Weight-Conjecture-September-23-2026/build/sections/intro.tex b/preprints/The-Blockwise-Alperin-Weight-Conjecture-September-23-2026/build/sections/intro.tex new file mode 100644 index 0000000000000000000000000000000000000000..b81984ba8f9b7798f5196eac4adce8020b9b0494 --- /dev/null +++ b/preprints/The-Blockwise-Alperin-Weight-Conjecture-September-23-2026/build/sections/intro.tex @@ -0,0 +1,310 @@ +\section{Introduction}\label{sec:introduction} + +The local--global principle in modular representation theory seeks to recover +information about representations of a finite group from normalizers of its +$p$-subgroups. Alperin's weight conjecture predicts an exact numerical form +of this principle, block by block: simple modules are counted by defect-zero +ordinary characters of normalizer quotients. Alperin formulated the +conjecture at the 1986 Arcata conference; the original account appeared in +the proceedings in 1987 \cite{Alperin1987,FLZ2023}. + +Let $p$ be a prime and $G$ a finite group. For an algebraically closed field +$k$ of characteristic $p$, a block is a primitive central idempotent of +$kG$. For a block $B$, let $l(B)$ be the number of simple $kG$-modules +belonging to $B$, equivalently the number of its irreducible Brauer +characters. For character calculations we use a splitting $p$-modular +system $(K,\mathcal O,k)$: $\mathcal O$ is a complete discrete valuation +ring with characteristic-zero fraction field $K$ and residue field $k$, +and $K$ splits every subgroup of $G$. A block also denotes its unique +central lift to $\mathcal O G$. In \Cref{sec:coefficients} we justify +reduction to $k=\overline{\mathbb F}_p$ and construct such a system; +the resulting counts and block assignments are unchanged over any +algebraically closed extension of this field. + +For a positive integer $a$, write $a_p$ for its largest power-of-$p$ divisor. A \emph{$p$-weight} of $G$ is a pair $(Q,\phi)$, where $Q\leq G$ is a possibly trivial $p$-subgroup and +\[ + \phi\in\Irr(N_G(Q)/Q),\qquad + \phi(1)_p=|N_G(Q)/Q|_p. +\] +Thus $\phi$ is an ordinary irreducible character of defect zero in the quotient. The block assigned to this weight is defined using the block $b$ of the inflated character in $H=N_G(Q)$. + +We specify the block-assignment convention. For $H\leq G$, coefficient restriction is the linear map +\[ + s_H:Z(kG)\longrightarrow Z(kH),\qquad + \sum_{g\in G}a_g g\longmapsto\sum_{h\in H}a_hh. +\] +If $c$ is a block of a finite group, its central character $\lambda_c$ is the residue character of the corresponding local factor of the center. In particular it is the scalar action of the center on any simple module in $c$. We say that $b^G=B$ when +\begin{equation}\label{eq:block-assignment} + \lambda_b\circ s_H=\lambda_B. +\end{equation} +This is the central-character definition of Brauer block induction \cite[p.~338]{Carlson1971}; see also the corrigendum \cite{Carlson1973}. In the normalizer situations arising from weights, the composite is indeed a central character; we prove this directly in \Cref{sec:group}. Let $W_p(B)$ be the set of simultaneous $G$-conjugacy classes of weights assigned to $B$. + +\begin{theorem}\label{thm:main} +For every prime $p$, every finite group $G$, and every $p$-block $B$ of $G$, +\[ + l(B)=|W_p(B)|. +\] +\end{theorem} + +\Cref{thm:main} resolves the numerical blockwise Alperin weight conjecture +positively, with no restriction on the defect groups, the prime, or the +finite group. The case $Q=1$ is part of the assertion. The conclusion is +an equality of cardinalities; a canonical or equivariant correspondence +is not required for this numerical statement. + +\subsection*{Local--global consequences} + +The numerical equality also gives the following established consequences +of the weight conjecture. Write $\IBr_p(H)$ for the set of irreducible +$p$-Brauer characters of a finite group $H$. + +\begin{corollary}[Local--global and principal-block bounds]\label{cor:block-consequences} +Let $p$ be a prime, $G$ a finite group, $B$ a $p$-block with defect group +$D$, and $b$ its Brauer correspondent in $N_G(D)$. Then +\[ + l(B)\geq l(b),\qquad + D\text{ abelian}\Longrightarrow l(B)=l(b). +\] +For $P\in\operatorname{Syl}_p(G)$, one also has +\[ + |\IBr_p(G)|\geq|\IBr_p(N_G(P))|. +\] +Let $B_0$ be the principal $p$-block of $G$, and let $k_p(G)$ count the +conjugacy classes of $p$-power-order elements, including the identity. +Then +\[ + \begin{aligned} + k_p(G)=3&\Longrightarrow l(B_0)\geq (p-1)/2,\\ + p\mid |G|&\Longrightarrow l(B_0)\geq 2\sqrt{p-1}+1-k_p(G). + \end{aligned} +\] +\end{corollary} + +The local block bounds are Alperin's Consequences~1--2 +\cite{Alperin1987}; the principal-block bounds are due to Hung, +Sambale, and Tiep \cite[Propositions~3.1--3.2]{HST2024}. +Their weight-conjecture hypotheses are supplied by \Cref{thm:main}. +The complete derivation, including the passage to the block-free +inequality, appears in \Cref{sec:consequences}. + +The block-free inequality for $p=2$ was already proved by +Malle--Navarro--Tiep \cite[Theorem~B]{MNT2023}; the local inequality for +maximal-defect $2$-blocks was proved by Feng--Mart\'inez--Rossi +\cite[Theorem~D]{FMR2025}. + +The sharper classification of character-count and defect data, and the +functorial formulation of the theorem, are given in +\Cref{sec:consequences}. We first explain the historical setting and +the proof of the local--global equality itself. + +\subsection*{Prior work and the proof's ingredients} + +Alperin's formulation extends the notion of weights from the modular +representation theory of finite groups of Lie type in defining +characteristic to arbitrary finite groups \cite{AlperinFong1990}. +Early large-family cases include Alperin and Fong's work on symmetric +groups and on finite general linear groups in odd modular characteristic +\cite{AlperinFong1990}. Kn\"orr and Robinson expressed the conjecture +as alternating sums over chains of $p$-subgroups +\cite{KnorrRobinson1989}. Navarro and Tiep reduced the non-blockwise +conjecture to conditions on finite simple groups \cite{NavarroTiep2011}, +and Sp\"ath obtained a blockwise reduction \cite{Spath2013}. These +reductions organize the problem around local correspondences for simple +groups. Among the resulting advances, Feng, Li, and Zhang proved the +blockwise conjecture for finite special linear and special unitary groups +and for finite groups with abelian Sylow $3$-subgroups \cite{FLZ2023}. +Feng, Malle, and Zhang subsequently studied generic weights for finite +reductive groups under explicit hypotheses \cite{FMZ2026}. + +Other recent progress includes the inductive blockwise condition for +$F_4(q)$ at odd primes not dividing $q$, proved by An, Hiss, and L\"ubeck +\cite{AnHissLubeck2024}, and Hu and Zhou's results on inertial blocks, +including an alternative proof for $2$-blocks with abelian defect +\cite{HuZhou2025}. Huang and Y\i lmaz develop chain, Galois, and +functorial reformulations \cite{HuangYilmaz2026}. A recent preprint of +Zhang claims the inductive condition for types $\mathsf B$, $\mathsf C$ +and sporadic groups \cite{Zhang2026BC}; those classes do not exhaust +the scope of \Cref{thm:main}. Our proof instead passes from a direct +block-algebra calculation to covers of curves; it does not use a +classification of finite simple groups or an inductive weight condition. + +The algebraic part uses classical constructions with a specific counting +purpose. The subgroup-chain argument is related to the Kn\"orr--Robinson +reformulation. Powers modulo commutators belong to K\"ulshammer's theory +of generalized Reynolds ideals \cite{HHKM2005}, and conjugation averaging +is the group-algebra Higman trace \cite{Higman1955,LorenzTokoly2011}. +The sum of the $p$-elements detects defect-zero blocks as in Tsushima's +work \cite{Tsushima1971}. +The passage from central traces to tuples is a weighted genus-one form +of Frobenius's character-counting formula; see \cite{Klug2025} for the +surface-group formulation. We prove the exact versions needed here, +including the block projections and normalization of the tuple count. +The further assertion is uniform divisibility as the number of punctures +increases, not merely a character formula for a fixed surface. + +Chen's relation between Nielsen orbit counts and degrees of moduli of +elliptic covers \cite{Chen2024} informed the geometric divisibility +viewpoint. The many-marked, high-index construction and the uniform +puncture-divisibility theorem required here are developed in this paper, +not supplied by Chen's theorem. + +The geometry combines several established ideas with that divisibility +problem. High-index genus-one curves are classical objects in period--index +theory; Clark and Lacy prove arbitrary-index existence over infinite fields +finitely generated over their prime fields \cite{ClarkLacy2019}. Our +construction also prescribes marked points giving independent absolute +field differentials. The genus calculation belongs to the theory of inseparable genus +change initiated by Tate \cite{Tate1952}; see Schr\"oer +\cite{Schroer2009} for a modern treatment. Its height-one determinant +identity is a curve-level form of the canonical-bundle formula for +$p$-closed foliations \cite{Ekedahl1988}. Here the calculation must keep +additional constants and inseparable residue degrees, and the large index +turns a bounded-error estimate into an exact inequality. Finally, +maximal-immediate-extension methods of Krull and Kaplansky +\cite{Krull1932,Kaplansky1942,Poonen1993} provide a suitable valued field, +while the center-valued obstruction studied by D\`ebes and Douai +\cite{DebesDouai1997} explains the arithmetic descent issue for marked +covers. We give the required constructions and descent argument directly. + +\subsection*{The mechanism of the proof} + +Chain inversion first reduces the weight equality to an identity between +a defect-zero character count and an alternating sum of simple-module +counts in subgroup normalizers. A calculation with powers modulo +commutators, followed by cancellation of chains, replaces those simple +counts by ordinary-character counts. We then construct a central +group-algebra element supported on $p$-singular elements, with residue +scalars zero or one chosen to remove the defect-zero contribution from +the alternating sum. Lift it by class sums to characteristic zero and restrict +it to each chain normalizer. The alternating sum of traces of the +$p^s$th powers converges to the difference between the alternating +ordinary-character sum and the defect-zero count. +Expanding the traces gives weighted tuple counts. A further chain +cancellation leaves only tuples generating subgroups with no nontrivial +normal $p$-subgroup. The complete reduction is +\Cref{prop:group-reduction}. + +Here is the counting problem that remains. Fix a finite group $J$ with +$O_p(J)=1$, where $O_p(J)$ is its largest normal $p$-subgroup, and set +$n=p^s$. We count ordered tuples $(x,y,u_1,\ldots,u_n)\in J^{n+2}$ +satisfying +\[ + [x,y]u_1\cdots u_n=1,\qquad + \langle x,y,u_1,\ldots,u_n\rangle=J. +\] +Here $[x,y]=xyx^{-1}y^{-1}$, and each puncture entry $u_i$ is +$p$-singular: its order is divisible by $p$. Prescribe a multiset of +$n$ conjugacy classes for the $u_i$, and identify tuples only by +simultaneous inner conjugation in $J$, not by permutations of their +positions. For every fixed integer $h\geq1$, we must prove that all +these counts are divisible by $p^h$ for sufficiently large $s$ +(\Cref{thm:tuple-divisibility}). The threshold may depend on $p,J,h$, +but not on the multiset. This uniformity makes each weighted trace sum +tend to zero even though the number of multisets grows with $s$. + +The geometric construction supplies two properties on the same marked +genus-one curve $E/F_0$ in characteristic $p$. First, every divisor +degree on $E$ is divisible by $p^s$. Second, after a cyclic splitting +extension it becomes a constant elliptic curve with independently +varying marked points. More precisely, put +$\kappa=\overline{\mathbb F}_p$, choose an ordinary elliptic curve +$A_0/\kappa$, and take $F'=\kappa(A_0^n)$. The extension $F'/F_0$ +is cyclic Galois of degree $n$ and splits $E$. Over $F'$, the marked points are the +generic projection points $t_i\in A_0(F')$. If $\eta$ is a nonzero +invariant differential on $A_0$, then the pullbacks $t_i^*\eta$ +form a basis of $\Omega_{F'/\kappa}$. Their independence is in this +field-differential space, not in the one-dimensional space of regular +relative differentials on the elliptic curve. After a finite extension +$F/F_0$ with $[F:F_0]_p\leq p^h$, every divisor degree on $E_F$ +is still divisible by $p^{s-h}$, and at most $h$ independent +differential directions are lost over the compositum $FF'$ +(\Cref{prop:twist-index,cor:index-base-change,lem:label-rank}). + +For a purely inseparable cover of this curve, a calculation along +successive maps of exponent one gives a genus bound with an error controlled +by $h$ and the cover degree. The index property makes this bound exact: +after multiplication by the cover degree, the difference between the +genus term and the ramification expression is a multiple of $p^{s-h}$. +The differential calculation bounds that difference below by a negative +constant independent of $s$. Once $p^{s-h}$ exceeds the constant's +magnitude, the difference must be nonnegative. This is the rounding +step in \Cref{thm:inseparable-genus}. The resulting estimate is useful +independently of the block-algebra reduction: it applies to regular +curves over imperfect fields, retains inseparable residue degrees and +extra constants, and does not require geometric reducedness. + +We lift the marked genus-one curve to mixed characteristic and pass to a +spherically complete valued field $K_s$ with value group $\mathbb Q$ +and residue field $F_0$. Over an algebraic closure, the generating +tuples are precisely the monodromy data of connected $J$-covers of the +punctured genus-one curve, with the $J$-action specified. Simultaneous +inner conjugacy changes only the chosen fiber point; it does not forget +the ordering of the punctures. The lift contains all $|J|$th roots of +unity. Galois therefore preserves the local conjugacy classes as it +permutes the marked points, so it acts on the finite cover set with +the prescribed multiset. This action factors through a finite Galois +group $\Gamma$; take a Sylow $p$-subgroup of $\Gamma$. + +An orbit of size less than $p^h$ would give a cover defined over an +extension $L/K_s$ with $[L:K_s]_pm\bigl(h+\log_p m\bigr). +\end{equation} +This is a sufficient threshold for \Cref{thm:inseparable-genus}. +The relatively ample effective divisor $D$ has degree $n$ on each fiber +and is disjoint from the marked sections after their splitting extension. + +\begin{proposition}\label{prop:no-small-cover} +Suppose $O_p(J)=1$ and \eqref{eq:specialization-threshold} holds. There is no +finite extension $L/K_s$ with $[L:K_s]_p0\}=\mathfrak m_R A_j. +\end{equation} +These assertions also hold on any $L$-linear subspace of $V_j$. +At this point reduction means the map of rational functions +\[ + A_j\longrightarrow\prod_{w\in\mathcal V}K(w), + \qquad a\longmapsto(\bar a_w)_w. +\] +Its kernel is the positive ball $\mathfrak m_R A_j$, so its image +has dimension $\dim_L V_j$ over $F$. No model of $Y$ has been used. +The next lemma locates these residue-field elements in the section +spaces of the separately normalized curves $C_w$. + +\begin{lemma}\label{lem:section-specialization} +For all sufficiently large $j$, simultaneous residue gives an injection +\begin{equation}\label{eq:section-reduction} + A_j/\mathfrak m_R A_j + \lhook\joinrel\longrightarrow + \bigoplus_{w\in\mathcal V} + H^0(C_w,\mathcal O_{C_w}(jD_w)). +\end{equation} +These maps respect products. If a section vanishes on the reduced +inverse images of a selected set of marked points, its reductions +vanish on the corresponding reduced inverse images on the $C_w$, +provided the selections are defined over the respective ground fields. +The vanishing assertion may be checked after splitting the marked +points over $L'$ and $S$. +\end{lemma} + +\begin{proof} +Let $a\in A_j$. Form the polynomial using every element of $J$, +including any repeated conjugates: +\begin{equation}\label{eq:conjugate-section-polynomial} + P_a(T)=\prod_{g\in J}(T-g(a)) + =T^m+c_1T^{m-1}+\cdots+c_m. +\end{equation} +The coefficients are $J$-invariant rational functions, hence belong to +$L(Y)^J=L(X)$. At a fixed $w\in\mathcal V$ every +$g(a)$ is integral, because $g$ permutes $\mathcal V$. Thus each $c_k$ +has nonnegative Gauss value. Moreover +\[ + c_k\in H^0(X,\mathcal O_X(kjD_X)). +\] +To see the pole bound, each conjugate has poles bounded by $jD_Y$, so +the elementary symmetric function of degree $k$ has poles bounded by +$kjD_Y$. Since the coefficient is a rational function on $X$, its +pole bound on $X$ follows from that on its finite surjective pullback. + +By \Cref{lem:gauss-section-rule}, $\bar c_k$ has poles bounded by +$kjD_F$. The residue $\bar a_w\in K(w)$ satisfies the reduction of +\eqref{eq:conjugate-section-polynomial}. Let $t$ be a local equation for +$jD_F$ at any point of $E_F$. Then $t\bar a_w$ satisfies the monic +equation whose coefficient of degree $m-k$ is $t^k\bar c_k$. +These coefficients are regular there. At every point of $C_w$ above +this neighborhood, normality therefore makes $t\bar a_w$ regular. +This proves the required pole bound and hence +\eqref{eq:section-reduction}; injectivity was established in +\eqref{eq:section-lattice}. Residue maps themselves respect multiplication. + +For the last assertion, work at a selected marked point after splitting. +Every $g(a)$ vanishes at every point of its reduced inverse fiber, +because this fiber is $J$-invariant. The non-leading coefficients +$c_k$ consequently vanish at the marked point downstairs. There are +no poles there, since $D_X$ avoids all the marked points. +By \eqref{eq:gauss-evaluation}, every $\bar c_k$ vanishes at $t_i$. +The reduced monic equation at any point above $t_i$ is then +$\bar a_w^m=0$ in its residue field, so $\bar a_w$ vanishes there. +\end{proof} + +\subsection{Residue degrees and the inseparable quotient} + +For $j$ sufficiently large, the line-bundle Euler-characteristic formula +and ample vanishing turn \eqref{eq:section-reduction} into +\begin{equation}\label{eq:full-section-comparison} + jmn+\chi(Y) + \leq jn\sum_{w\in\mathcal V}m_w + +\sum_{w\in\mathcal V}\chi(C_w). +\end{equation} +Here $\deg D_X=\deg D_F=n$, and finite pullback multiplies divisor +degree by the map degree. Letting $j$ grow gives +$m\leq\sum_wm_w$. On the other hand, +\Cref{lem:simultaneous-residues}, applied to the Gauss valuation on +$L(X)$ and these distinct prolongations to the degree-$m$ extension +$L(Y)$, gives the reverse inequality. Its hypotheses hold: the value +group is $\mathbb Q$ and the residue field $F(E_F)$ is infinite. +We have proved +\begin{equation}\label{eq:residue-degree-equality} + \sum_{w\in\mathcal V}m_w=m. +\end{equation} + +Let $\Delta_w\leq J$ be the stabilizer of $w$, and let $I_w$ be the +kernel of its action on $K(w)$. The orbit has $m/|\Delta_w|$ members, +all of the same residue degree. Therefore +\begin{equation}\label{eq:decomposition-residue-degree} + m_w=|\Delta_w|. +\end{equation} +In particular this equality follows from the section comparison, before +any assertion about the degree of an inseparable quotient. + +\begin{lemma}\label{lem:inseparable-residue-quotient} +The invariant field $M_w=K(w)^{\Delta_w}$ is purely inseparable over +$F(E_F)$, of degree +\begin{equation}\label{eq:inertia-inseparable-degree} + d=[M_w:F(E_F)]=|I_w|. +\end{equation} +In particular $d$ is a power of $p$ and $d\leq m$. If $Z_w$ denotes the +normalization of $E_F$ in $M_w$, the map $C_w\to Z_w$ is generically +separable Galois of degree $m_w/d$, with group $\Delta_w/I_w$. +All these towers are isomorphic over $E_F$ as $w$ varies in $\mathcal V$. +\end{lemma} + +\begin{proof} +Take $x\in K(w)^{\Delta_w}$. Simultaneous approximation from +\Cref{lem:simultaneous-residues} gives an element $a\in L(Y)$ integral +at every member of $\mathcal V$, with residue $x$ at $w$ and zero at +the other members. The coefficients of the conjugate polynomial +$P_a(T)$ in \eqref{eq:conjugate-section-polynomial} have nonnegative +Gauss value. At $w$, its reduction is +\begin{equation}\label{eq:invariant-residue-polynomial} + \overline{P_a}(T) + =T^{m-|\Delta_w|}(T-x)^{|\Delta_w|} + \in F(E_F)[T]. +\end{equation} +Indeed, elements of $\Delta_w$ act on the residue $x$ trivially, and +each other conjugate draws its residue from a different member of +the orbit, where the prescribed residue is zero. + +Write $|\Delta_w|=p^v q$ with $p\nmid q$. In characteristic $p$, +\[ + (T-x)^{|\Delta_w|}=(T^{p^v}-x^{p^v})^q. +\] +The coefficient of $T^{m-p^v}$ in +\eqref{eq:invariant-residue-polynomial} is therefore +$-q x^{p^v}$. The scalar $q$ is nonzero in $F(E_F)$, so +$x^{p^v}\in F(E_F)$. This also covers $x=0$, for which the coefficient +is zero. Thus $M_w/F(E_F)$ is purely inseparable. + +The effective automorphism group on $K(w)$ is $\Delta_w/I_w$. +Finite fixed-field theory gives +\[ + [K(w):M_w]=|\Delta_w/I_w|. +\] +Combining this with \eqref{eq:decomposition-residue-degree} and the +tower formula proves \eqref{eq:inertia-inseparable-degree}. +The assertions about the normalizations and their separable function +fields follow. Conjugation by $J$ supplies the asserted isomorphisms +between the towers, so $d$ is independent of $w$. +\end{proof} + +\subsection{Ramification on the residue curves} + +Pass temporarily to the finite separable splitting field $S/F$. +The curve $Z_{w,S}$ remains integral: its finite map to $E_S$ is +surjective and radicial, while separable scalar extension preserves +regularity and hence reducedness. The curve $C_{w,S}$ is regular and +is a disjoint union of integral components. Each component dominates +$Z_{w,S}$: the finite flat map $C_{w,S}\to Z_{w,S}$ restricts on a +component to a nonzero finite locally free summand, of positive rank +on the connected target. The components are consequently the +normalizations in their respective function fields. These uses of +normalization and separable scalar extension are included in +\Cref{foundation:normalization,foundation:regularity}. + +Let $z_i$ be the unique point of $Z_{w,S}$ above $t_i$, and write $r_i$ +for its ramification index over $t_i$. Write $a_i$ for the ramification +index of $C_{w,S}\to Z_{w,S}$ above $z_i$. We justify that $a_i$ is +independent of the point above $z_i$, even when $C_{w,S}$ is disconnected. +Put $G_w=\Delta_w/I_w$ and $M_{w,S}=S(Z_{w,S})$. The generic algebra +of this map is +\[ + K(w)\otimes_{M_w}M_{w,S}=\prod_{\lambda=1}^t K_\lambda. +\] +It is a product of fields because $K(w)/M_w$ is finite separable. +The group $G_w$ acts on the first tensor factor, and the invariant +algebra is $M_{w,S}$: taking invariants is taking a kernel of linear +maps, which commutes with this scalar extension. An orbit of the +field factors gives an invariant idempotent. Since $M_{w,S}$ is a +field, there can be only one orbit. + +The total dimension over $M_{w,S}$ is $|G_w|$, and transitivity makes +all factor degrees equal to $|G_w|/t$. The stabilizer of a factor has +this same order. It acts faithfully on that factor, since the map +from $K(w)$ into the factor is injective. Thus the factor extension +is Galois with that stabilizer as its group. Transitivity on primes +in each Galois normalization gives transitivity within each component +above $z_i$; the action on the factors joins these componentwise +orbits. We obtain transitivity on the entire fiber of $C_{w,S}$. +Ramification indices are preserved by this action. The isomorphisms +between the towers for different $w$ give the same $r_i,a_i$ throughout +the orbit. + +Multiplicities in the tower multiply, so every point $P$ of $C_{w,S}$ +above $t_i$ has index $r_i a_i$ over $t_i$. Finite flatness gives +\[ + \sum_{P\in C_{w,S},\ P\mid t_i} + r_i a_i\,[\kappa(P):S]=m_w. +\] +Summing over $w$ and using \eqref{eq:residue-degree-equality} yields +\begin{equation}\label{eq:reduced-special-fiber-degree} + \sum_{w\in\mathcal V}\ + \sum_{P\in C_{w,S},\ P\mid t_i}[\kappa(P):S] + =\frac{m}{r_i a_i}. +\end{equation} +In particular all residue-field degrees, including inseparable ones, +are retained in the degree of the reduced fiber. + +Apply \Cref{lem:separable-different} to $C_{w,S}\to Z_{w,S}$, +component by component. At a point above $z_i$, the determinant +vanishing is at least +$a_i-1+\mathbf 1_{p\mid a_i}$. The total separable degree of the +components over $Z_{w,S}$ is $m_w/d$. The preceding local-degree +calculation gives $m_w/(r_i a_i)$ for the sum of the full residue-field +degrees above $t_i$. Hence, for each $w$, +\[ + \nu(C_w)\geq\frac{m_w}{d}\nu(Z_w) + +\sum_{i=1}^n\frac{m_w}{r_i a_i} + \bigl(a_i-1+\mathbf 1_{p\mid a_i}\bigr). +\] +Euler characteristics and divisor degrees have the same numerical +values over $F$ as after the separable extension to $S$, so this +inequality is written in the original normalization over $F$. +The hypotheses of +\Cref{thm:inseparable-genus} hold for $Z_w/E_F$: +$[F:F_0]_p\frac1{e_i}\right\}. +\] +At these labels, the degree $m/(r_i a_i)$ of the reduced special fiber +exceeds the degree $m/e_i$ of the reduced generic fiber. Imposing +vanishing there will make the section comparison detect this excess. +This selection descends on both sides. The generic index $e_i$ is +constant on the label orbits of $\operatorname{Gal}(L'/L)$, and the +special indices $r_i,a_i$ are constant on the label orbits of +$\operatorname{Gal}(S/F)$. These two permutation actions agree by +\Cref{lem:residue-permutations}. Hence the same subset of labels +defines a divisor over $L$ on the generic base and a divisor over $F$ +on the special base. + +Let $T_Y$ be the reduced inverse image on $Y$ of this selected divisor, +and let $T_w$ be its counterpart on $C_w$. These are effective divisors +on regular curves over their indicated fields. Their degrees may be +computed after the finite separable splitting extensions, which preserve +reducedness. The generic Galois cover and +\eqref{eq:reduced-special-fiber-degree} give +\begin{equation}\label{eq:selected-divisor-degrees} + \deg_L T_Y=\sum_{i\in\mathcal I}\frac m{e_i},\qquad + \sum_{w\in\mathcal V}\deg_F T_w + =\sum_{i\in\mathcal I}\frac m{r_i a_i}. +\end{equation} + +Restrict the norm to +$H^0(Y,\mathcal O_Y(jD_Y-T_Y))$. Its integral ball again has a +full-dimensional reduction by \eqref{eq:section-lattice}, and +\Cref{lem:section-specialization} places that reduction in +\[ + \bigoplus_{w\in\mathcal V} + H^0(C_w,\mathcal O_{C_w}(jD_w-T_w)). +\] +For sufficiently large $j$, ample vanishing and +\eqref{eq:residue-degree-equality} therefore give +\[ + jmn+\chi(Y)-\deg_L T_Y + \leq jmn+\sum_w\chi(C_w)-\sum_w\deg_F T_w. +\] +After rearranging and using \eqref{eq:selected-divisor-degrees}, this is +\begin{equation}\label{eq:vanishing-comparison} + \frac{\nu(Y)}m + \geq \frac{\sum_{w\in\mathcal V}\nu(C_w)}m + +2\sum_{i=1}^n + \max\left\{\frac1{r_i a_i}-\frac1{e_i},0\right\}. +\end{equation} + +The three comparisons now force equality. To display their exact +interaction, put +\[ + q_i=\frac1{r_i a_i},\qquad b_i=\frac1{e_i},\qquad + \delta_i=\mathbf 1_{p\mid a_i}. +\] +The contribution of label $i$ in \eqref{eq:residue-genus} is +$1-q_i+\delta_iq_i$. Combining that inequality with +\eqref{eq:vanishing-comparison} and subtracting +\eqref{eq:generic-hurwitz} gives +\begin{align} + 0 + &\geq \sum_{i=1}^n + \bigl(b_i-q_i+2\max\{q_i-b_i,0\}+\delta_iq_i\bigr) + \notag\\ + &=\sum_{i=1}^n\bigl(|q_i-b_i|+\delta_iq_i\bigr) + \geq0.\label{eq:genus-nonnegative-excess} +\end{align} +Every summand vanishes, and every intervening inequality is an equality. +In particular, +\begin{equation}\label{eq:sharp-specialization} + r_i a_i=e_i,\qquad p\nmid a_i\quad(1\leq i\leq n), + \qquad d=|I_w|>1, + \qquad \chi(Y)=\sum_{w\in\mathcal V}\chi(C_w). +\end{equation} +For the strict inequality $d>1$, use $p\mid e_i$ and $p\nmid a_i$: +then $r_i>1$, and $r_i$ divides the purely inseparable degree $d$ by +finite flatness at $t_i$. Also $q_i=b_i$ makes the extra term in +\eqref{eq:vanishing-comparison} zero, so equality there gives precisely +the displayed Euler-characteristic equality. + +\subsection{A finitely presented model over the valuation ring} + +\begin{proposition}\label{prop:specialization-model} +The degree equality \eqref{eq:residue-degree-equality} and the +Euler-characteristic equality in \eqref{eq:sharp-specialization} +produce a flat projective scheme $\mathcal Y$ of finite presentation +over $R$, with +\[ + \mathcal Y_L\simeq Y,\qquad + \mathcal Y_F\simeq\coprod_{w\in\mathcal V}C_w. +\] +\end{proposition} + +\begin{proof} +Form the graded $R$-algebra +\[ + A=\bigoplus_{j\geq0}A_j. +\] +Multiplication is defined because each $w$ is a valuation: +$A_iA_j\subseteq A_{i+j}$. Each piece is finite free by +\eqref{eq:section-lattice}. Moreover $A_0=R$, since $Y$ is projective +and geometrically integral and hence $H^0(Y,\mathcal O_Y)=L$. +Put +\[ + C=\coprod_{w\in\mathcal V}C_w, + \qquad B_j=\bigoplus_{w\in\mathcal V} + H^0(C_w,\mathcal O_{C_w}(jD_w)), + \qquad B=\bigoplus_{j\geq0}B_j. +\] +The two equalities in the proposition make the source and target +dimensions in \eqref{eq:section-reduction} equal for every sufficiently +large $j$. Thus there is an integer $j_0$ such that +\begin{equation}\label{eq:full-high-degree-reduction} + A_j/\mathfrak m_R A_j\simeq B_j + \qquad(j\geq j_0), +\end{equation} +compatibly with products. Equality in each high degree, rather than +only equality of leading terms, is where the Euler-characteristic +equality is used. + +We first choose a Veronese regrading for which the reduced algebra is +generated in degree one. The divisor on $C$ given by the $D_w$ is ample, +so $B$ is finitely generated by \Cref{foundation:cohomology}. Choose +positive-degree homogeneous generators $b_1,\ldots,b_t$ over $B_0$, +of degrees $d_1,\ldots,d_t$. Let $W$ be a common multiple of these +positive degrees. In a monomial in the $b_i$, remove pure powers +$b_i^{W/d_i}$, each of weight $W$, until the remaining exponent of +$b_i$ is less than $W/d_i$. The weight of the remainder is less than +$tW$. Choose $N=qW\geq j_0$ with $q\geq t$. + +If the original monomial has degree $kN$, its remainder has weight +$cW$ for an integer $0\leq ci),\qquad + \delta_i\leq\gamma_i<\delta_{i+1},\qquad + \gamma_1<\gamma_2<\cdots . +\end{equation} +The balls $B(a_i,\gamma_i)$ are nested, lie inside $B(a_i,\delta_i)$, +and still have empty intersection. + +Extend the valuation to an algebraic closure of $K_s$. +Call $b$ a \emph{pseudo-limit} of this sequence if +$v(b-a_i)=\gamma_i$ for all sufficiently large $i$. +No element of $K_s$ is a pseudo-limit, since a pseudo-limit belongs to +every ball $B(a_i,\gamma_i)$. +For any point $b$ which is not a pseudo-limit, the values $v(b-a_i)$ +are eventually constant and smaller than $\gamma_i$. +To see this, if $v(b-a_i)<\gamma_i$ at one stage, the strong triangle +law and \eqref{eq:pseudoconvergent-centers} give the same value at every +later stage. If $v(b-a_i)>\gamma_i$, the next value is $\gamma_i$, which +is smaller than $\gamma_{i+1}$, reducing to the preceding case. +Otherwise equality holds eventually, as required for a pseudo-limit. + +Suppose first that an algebraic pseudo-limit $\alpha$ exists, and +choose one of smallest degree $d$ over $K_s$. Every root $\beta$ of a +nonzero polynomial $f\in K_s[T]$ with $\deg fv(x)$. It follows that $v(c)=v(x)$, so no new values +occur. If $v(x)=0$, the same inequality says that $x$ and $c$ have +the same residue, so no new residue-field elements occur either. +Thus $K_s(\alpha)/K_s$ is a proper immediate extension, a contradiction. + +Suppose instead that no algebraic pseudo-limit exists. Factoring any +nonzero polynomial over the fixed algebraic closure shows that its +values at $a_i$ are eventually constant and finite; the same holds for +nonzero rational functions. Define, on $K_s(T)$, +\[ + w(f)=\text{the eventual value of }v(f(a_i)), + \qquad w(0)=+\infty. +\] +Evaluation proves multiplicativity and the strong triangle inequality, +so $w$ is a valuation extending that of $K_s$. +For fixed $f\ne0$ and $j>i\gg0$, every linear factor in its numerator +and denominator satisfies +$(a_j-\beta)/(a_i-\beta)\equiv1\pmod{\mathfrak m_{\overline{K_s}}}$, +where $\beta$ is its root. Indeed $v(a_i-\beta)<\gamma_i=v(a_j-a_i)$. +Consequently +\[ + v(f(a_j)-f(a_i))>v(f(a_i))=w(f). +\] +For a fixed sufficiently large $i$, the left side has an eventual +value as $j$ increases, since it evaluates the rational function +$f(T)-f(a_i)$. Thus +$w(f-f(a_i))>w(f)$ unless the difference is zero, when the assertion +is immediate. Thus every nonzero element $f$ of $K_s(T)$ admits a +scalar $f(a_i)$ with this strict approximation inequality. As in the +algebraic case, this shows that neither the value group nor the residue +field changes. This is again a proper immediate extension, +contradicting maximality. Both cases being impossible, $K_s$ is +spherically complete. +\end{proof} + +\begin{lemma}\label{lem:orthogonal-bases} +Let $K_*$ be a spherically complete field with value group $\mathbb Q$, +valuation ring $R_*$, maximal ideal $\mathfrak m_*$, and residue field +$k_*$. Let $V$ be a finite-dimensional $K_*$-vector space with a +valuation norm $\lambda:V\to\mathbb Q\cup\{+\infty\}$: it is finite +away from zero, has $\lambda(0)=+\infty$, satisfies the strong triangle inequality, and obeys +$\lambda(ax)=v(a)+\lambda(x)$ for $a\ne0$. +There is a basis $e_1,\ldots,e_d$ of norm zero such that +\begin{equation}\label{eq:orthogonal-basis} + \lambda\left(\sum_i a_i e_i\right)=\min_i v(a_i). +\end{equation} +In particular its integral and positive balls are +\begin{equation}\label{eq:orthogonal-lattice} + V_{\geq0}=\bigoplus_i R_*e_i,\qquad + V_{>0}=\bigoplus_i\mathfrak m_*e_i + =\mathfrak m_*V_{\geq0}, + \qquad \dim_{k_*}(V_{\geq0}/V_{>0})=d. +\end{equation} +The normed space $V$ is spherically complete. +\end{lemma} + +\begin{proof} +A subspace with an orthogonal basis is spherically complete: its closed +balls are products of closed balls in the finitely many coordinates, +with the radii shifted by the norms of the basis vectors. Each +coordinate nest has a common point. + +Suppose an orthogonal basis has already been constructed for a subspace +$U$, and choose $x\notin U$. For every value $\gamma$ achieved by +$\lambda(x-u)$, $u\in U$, consider +\[ + B_\gamma=\{u\in U:\lambda(x-u)\geq\gamma\}. +\] +This is a nonempty closed ball in $U$: if $u_\gamma$ belongs to it, +then it equals $\{u:\lambda(u-u_\gamma)\geq\gamma\}$. +These balls form a nest, so they have a common point $u_0$. +The finite value $\ell=\lambda(x-u_0)$ is at least every achieved value +and itself is achieved. Thus $u_0$ is a best approximation to $x$. +Put $y=x-u_0$. For $u\in U$, maximality of $\ell$ gives +$\lambda(y+u)\leq\ell$. If $\lambda(u)<\ell$, the strong triangle +law gives $\lambda(y+u)=\lambda(u)$; if $\lambda(u)\geq\ell$, it gives +$\lambda(y+u)\geq\ell$, hence equality. Therefore +\[ + \lambda(y+u)=\min\{\ell,\lambda(u)\}. +\] +Homogeneity shows that adjoining $y$ preserves orthogonality. +Induction gives an orthogonal basis of $V$. Since every basis norm is +in the base value group, scale its vectors to norm zero. +Formulae \eqref{eq:orthogonal-basis} and \eqref{eq:orthogonal-lattice} +follow, and the same coordinate argument proves spherical completeness +of $V$. +\end{proof} + +\begin{proof}[Finite-extension assertions of \Cref{prop:spherical-base}] +For any algebraic extension of a field with value group $\mathbb Q$, +every nonzero algebraic element has value in $\mathbb Q$. +Indeed, in a polynomial relation $\sum_i a_i x^i=0$, at least two +nonzero terms must have the same minimum value, so +\[ + (i-j)v(x)=v(a_j)-v(a_i)\in\mathbb Q + \quad(i\ne j). +\] +Divisibility of $\mathbb Q$ and torsion-freeness of ordered value groups +imply $v(x)\in\mathbb Q$. +Apply \Cref{lem:orthogonal-bases} to a finite extension $L/K_s$ with any +extended valuation, viewed as a normed vector space over $K_s$. +Its integral and positive balls are exactly its valuation ring and +maximal ideal. Formula \eqref{eq:orthogonal-lattice} proves +\eqref{eq:full-residue-degree}, including the full inseparable degree +of the residue extension. Orthogonal coordinates also show that $L$ +is spherically complete. The same argument applies to every finite +extension of $L$. +\end{proof} + +The preceding argument gives equality of field and residue degrees for +finite extensions of the spherical base. We also need a statement for +valued fields that are not assumed spherically complete, where several +prolongations can occur. The next lemma supplies the residue-degree bound +used for curve function fields in \Cref{sec:specialization} and completes +the uniqueness assertion for the spherical base. + +\begin{lemma}[Simultaneous residue approximation]\label{lem:simultaneous-residues} +Let $T$ be a nontrivially valued field with value group $\mathbb Q$ and +infinite residue field $k_T$. Let $U/T$ be finite and let +$w_1,\ldots,w_r$ be distinct prolongations of its valuation, with residue +fields $k_1,\ldots,k_r$. Then +\[ + \bigcap_{i=1}^r\mathcal O_{w_i}\longrightarrow + \prod_{i=1}^r k_i +\] +is surjective, and +\begin{equation}\label{eq:residue-degree-bound} + \sum_{i=1}^r[k_i:k_T]\leq[U:T]. +\end{equation} +\end{lemma} + +\begin{proof} +The preceding polynomial-relation argument puts all the values in +$\mathbb Q$, with their normalization fixed by the valuation on $T$. +For $i\ne j$, distinctness gives an element whose two values differ. +Inverting it if necessary and multiplying by an element of $T$ of a +suitable rational value produces $z\in U$ with +$w_i(z)>0>w_j(z)$. +Choose a unit $c\in T$ such that, at every $w_\ell$ for which +$w_\ell(z)=0$, the residue of $1+cz$ is nonzero. +Each such condition excludes at most one value for the residue of $c$ +in $k_T^\times$, so the infinitude of $k_T$ permits this choice. +Then +\[ + q_{ij}=\frac{1}{1+cz} +\] +is integral at every $w_\ell$, has residue $1$ at $i$, and has residue +$0$ at $j$. Indeed, a positive value of $z$ makes the denominator a +unit with residue $1$; a negative value of $z$ gives the denominator +that negative value; and value zero is covered by the choice of $c$. + +Put $q_i=\prod_{j\ne i}q_{ij}$, with $q_i=1$ if $r=1$. +It has residue $1$ at $i$ and positive value at all other places. +For a prescribed residue $b_i\in k_i$, choose any lift $\widetilde b_i$ +integral at $i$. If the lift is nonzero, choose $N_i$ so that +\[ + N_iw_\ell(q_i)+w_\ell(\widetilde b_i)>0 + \qquad(\ell\ne i). +\] +There are only finitely many conditions, and every $w_\ell(q_i)$ in +them is positive. At $i$, the factor $q_i^{N_i}$ has residue one. +Thus $q_i^{N_i}\widetilde b_i$ is integral everywhere, has residue +$b_i$ at $i$, and has residue zero at every other place. A zero lift +contributes the zero term. Summing these terms realizes any tuple $(b_i)_i$. + +For completeness, each $[k_i:k_T]$ is finite and at most $[U:T]$ by the +single-place lifting argument already used in the proof of +\Cref{prop:mixed-lift}. Choose a $k_T$-basis of each $k_i$, and use the +surjectivity just proved to lift its elements with zero residues in the +other coordinates. These lifts are linearly independent over $T$. +For if they satisfied a nonzero relation, divide it by a coefficient +of minimum value. Every coefficient then is integral and at least one +is a unit. Reducing in a coordinate which contains such a unit +contradicts the chosen residue basis there. Their total number is +therefore at most $[U:T]$, proving \eqref{eq:residue-degree-bound}. +\end{proof} + +\begin{proof}[Completion of the proof of \Cref{prop:spherical-base}] +The residue of $K_s$, and that of every finite extension $L/K_s$, is +infinite. If two distinct valuations extended the valuation of $L$ to +a finite extension $U/L$, the finite-extension argument above would +give residue degree $[U:L]$ for each. This contradicts +\eqref{eq:residue-degree-bound}. Existence of prolongations is part of +\Cref{foundation:valuation}; uniqueness on finite extensions implies +uniqueness on arbitrary algebraic extensions by restriction to finite +subextensions. + +Finally $|K_s|\leq\mathfrak c$ by the cardinality argument, while +$\mathbb Q_p\subset K_s$ gives $|K_s|\geq\mathfrak c$. +Its algebraic closure has the same cardinality. +An uncountable algebraically closed field of characteristic zero has +cardinality equal to its transcendence degree over $\mathbb Q$: the +field generated by a transcendence basis, and then its algebraic +closure, both have cardinality the maximum of that basis cardinality +and $\aleph_0$. Both $\overline{K_s}$ and $\mathbb C$ therefore have +transcendence degree $\mathfrak c$ over $\mathbb Q$. +The algebraically closed field classification in +\Cref{foundation:fields} gives the asserted abstract isomorphism. +No compatibility with either field's topology is asserted or used. +\end{proof} + +\subsection{Marked covers and a small orbit} + +Choose an embedding $K_b'\hookrightarrow\overline{K_s}$ over $K_b$ and +an abstract isomorphism $\overline{K_s}\simeq\mathbb C$. +Write $X_s=\mathcal X_{K_s}$. The embedding carries the sections +$\xi_i$ on the split model $\mathcal X_{R_b'}$ to the specified +geometric marked points on $X_s$. +A \emph{marked $J$-cover} of $\mathcal X_{\overline{K_s}}$ will mean a connected +smooth projective curve $Y$ with a finite map to $\mathcal X_{\overline{K_s}}$ +which is Galois of group $J$, together with a specified left action of +$J$ by deck transformations. Isomorphisms are over the fixed base +curve and commute with every element of this specified action. + +\begin{lemma}\label{lem:marked-cover-classification} +After choosing oriented handle and puncture generators on the complex +punctured curve, isomorphism classes of marked $J$-covers branched only at +$\xi_1,\ldots,\xi_n$ correspond bijectively to +\[ + \left\{(x,y,u_1,\ldots,u_n)\in J^{n+2}: + \begin{array}{l} + [x,y]u_1\cdots u_n=1,\\ + \langle x,y,u_1,\ldots,u_n\rangle=J + \end{array}\right\}/J, +\] +where $J$ acts by simultaneous inner conjugation. +The inertia order at $\xi_i$ is $|u_i|$. +For a prescribed multiset $\mathcal C$ of $n$ $J$-conjugacy classes of +$p$-singular elements, let $\mathcal T_{\mathcal C}$ be the finite set +of these marked-cover classes having that multiset of local classes. +Then $\mathcal T_{\mathcal C}$ is preserved by +$\operatorname{Gal}(\overline{K_s}/K_s)$. +\end{lemma} + +Thus $|\mathcal T_{\mathcal C}|=N_{J,s}(\mathcal C)$ in the notation +of \Cref{thm:tuple-divisibility}. The labels $\xi_1,\ldots,\xi_n$ +remain ordered: prescribing their class multiset does not divide out +by permutations of the labels. + +\begin{proof} +Under the chosen complex identification the base is a smooth +Weierstrass cubic. In characteristic zero, completing the square gives +an equation $y^2=f_3(x)$ with three distinct roots. Its double cover of +the projective line branches simply at those roots and at infinity. +Cut the sphere along two disjoint arcs joining pairs of these four +points and glue the two sheets crosswise. The resulting compact surface +is a torus. This identifies the topological type of our particular +base without a general analytic--algebraic genus comparison. + +Remove the $n$ marked points. Its complex fundamental group has +generators and relation +$[x,y]u_1\cdots u_n=1$ as in \Cref{foundation:complex}. +Choose a point in a fiber of a marked cover and identify the fiber with +the regular left $J$-set. Path lifting commutes with the left deck +action, so monodromy acts by right translations. With path multiplication +chosen in the order of successive traversal, these translations specify +a homomorphism from the fundamental group to $J$. The cover is +connected exactly when the monodromy action is transitive, which for +this regular action means that the homomorphism is surjective. +Changing the chosen fiber point conjugates this homomorphism by one +element of $J$. Conversely, a surjective homomorphism defines such a +fiber action and therefore a connected unramified cover. +An equivariant fiber bijection is a right translation, and compatibility +with monodromy says exactly that the two homomorphisms are simultaneously +conjugate. Thus the quotient is by inner conjugation; keeping the +specified $J$-action does not identify covers through outer automorphisms +of $J$. + +Riemann existence in \Cref{foundation:complex} algebraizes the +unramified cover of this punctured algebraic curve, with its morphisms +and specified deck action. Normalize the original projective base in +the resulting function field. Finiteness of normalization gives a +projective algebraic cover; normality makes its source smooth in +characteristic zero. The deck action and isomorphisms extend uniquely +across normalization. In a local power-map chart $z\mapsto z^e$, the +ramification index is $e$, the order of the associated monodromy +element. These extensions give precisely the branched covers and +isomorphisms in the statement. + +We describe the local conjugacy class algebraically in order to check +Galois invariance. For each divisor $e$ of $m$, use the root +$\zeta_e\in K_s$ which maps to $\exp(2\pi i/e)$ under the chosen +complex identification. All these roots belong to $K_s$ by +\Cref{prop:mixed-lift}. +Represent the based puncture loop as an approach path, a positive +circle around the puncture, and the reverse approach path. +In the chart $z\mapsto z^e$, lifting the positive circle sends $z$ to +$\zeta_e z$. Let $Q$ be the point above the puncture selected by the +lifted approach path, and let $g\in J$ be the inertia element acting +this way in its local chart. Apply $g$ to the lifted approach path. +The reverse of the resulting path is the lift of the return path +starting at the endpoint of the circle lift. Thus the complete based +loop sends the chosen fiber point to its image under $g$. +Under the preceding fiber identification, its monodromy element is +therefore $g$. The inertia group is cyclic, and $g$ is its unique +generator acting on the tangent line at $Q$ by $\zeta_e$. +Choosing another point above the puncture conjugates the element in $J$. +This recovers its conjugacy class from the algebraic tangent action. +(Reversing all local-loop conventions replaces all these generators by +their inverses consistently.) + +A Galois automorphism over $K_s$ sends the tangent action at $Q$ to +the tangent action at its conjugate point, and fixes $\zeta_e$. +Consequently it preserves the local $J$-conjugacy class at the transported +branch point. It permutes the branch points, since their union is +defined over $K_s$, and hence preserves their prescribed class multiset +and the condition that all inertia orders are divisible by $p$. +This proves invariance of $\mathcal T_{\mathcal C}$. +Its finiteness follows from the finite tuple description. +Although Galois can permute the marked points, an isomorphism of the +marked covers is over the fixed geometric base curve. The cover classes +are therefore still parametrized by simultaneous inner conjugacy, +without quotienting by those permutations. +\end{proof} + +For a cover with its group action specified, the obstruction to descent +is center-valued; compare the marked-cover analysis of +D\`ebes and Douai \cite[Section~3.2]{DebesDouai1997}. In the following +proposition we realize that obstruction as an explicit finite group +extension and split it by a Sylow argument. Thus a field of moduli is +not silently assumed to be a field of definition. + +\begin{proposition}[Descent from a small Sylow orbit]\label{prop:small-orbit-descent} +Assume $O_p(J)=1$ and fix a multiset $\mathcal C$ as in +\Cref{lem:marked-cover-classification}. There is a finite Galois +extension $M/K_s$ through which the action on $\mathcal T_{\mathcal C}$ +factors. Let $P_\Gamma$ be a Sylow $p$-subgroup of +$\Gamma=\operatorname{Gal}(M/K_s)$. +If a $P_\Gamma$-orbit has size less than $p^h$, one of its marked covers +descends to a finite extension $L/K_s$ satisfying +\begin{equation}\label{eq:small-orbit-degree} + [L:K_s]_p0$. This construction allows inseparable extensions and all +ramification indices. + +An element of $\widehat P\cap\widehat Q$ has $w$-order zero by its +$Q$-representation. Its $x$-order is therefore zero, since +multiplication by $e$ is injective on $\Zell$. This holds for every +$x$. By Lemma~\ref{prelim:curve-kernel}, the intersection embeds in +the finite-rank Tate module of $\operatorname{Pic}^0(C_P)$. +\end{proof} + +\begin{proof}[Proof of Theorem~\ref{recovery:curves}] +Let $E\subset K$ be a curve subfield, and choose $t\in E\setminus k$. +Relative algebraic closedness makes $E$ +the relative algebraic closure of $k(t)$ in $K$. Apply +Lemmas~\ref{recovery:residue-powers} and \ref{recovery:incidence} to +one representative of each $\Theta[z]\bmod\ell^j$, for +$z\in E^\times$ and $j\geq1$. We obtain fields $L\subset M$ and +$P_0\subset M$ as in the incidence lemma. + +Put $H=\Theta(\widehat E)$. We claim that its image in $\widehat M$ +lies in $\widehat P_0$. At level $\ell^j$, every element of +$\widehat E$ is represented by an actual $z\in E^\times$. +Lemma~\ref{recovery:kummer-descent} puts its image in +$P_0^\times/P_0^{\times\ell^j}$. These classes are unique because +$P_0$ is relatively algebraically closed in $M$, and their uniqueness +makes them compatible as $j$ varies. Taking inverse limits proves +the claim. Completion maps across $M/L$ are injective by +Lemma~\ref{prelim:completion}. + +Choose a finite normal extension $N/L$ containing $M$, allowing an +inseparable part, and let $P_N$ be the relative algebraic closure of +$P_0$ in $N$. It is a curve subfield of $N/l$. The image of $H$ lies +in $\widehat P_N$. Every $\sigma\in\operatorname{Aut}_L(N)$ fixes +$H$, so $H$ also lies in $\widehat{\sigma(P_N)}$. +It has infinite rank: $\widehat E/\ell$ has infinite dimension, as +the pencil argument in Lemma~\ref{recovery:residue-powers} shows. +Indeed, lifts of any finite mod-$\ell$ independent set are +$\Zell$-independent: a nonzero relation, divided by the smallest +$\ell$-power in its coefficients, would reduce to a nonzero +mod-$\ell$ relation. Torsion freeness justifies this division. +Lemma~\ref{recovery:intersection} therefore forces +$\sigma(P_N)=P_N$ for every $\sigma$. + +Set $P_L=P_N\cap L$. This field has transcendence degree one over +$l$. Indeed, for $G=\operatorname{Aut}_L(N)$, the field $P_N^G$ has +transcendence degree one and lies in $N^G$. In characteristic zero +$N^G=L$. In characteristic $p>0$, $N^G/L$ is finite purely inseparable, +so a common $p^a$-th power sends $P_N^G$ into $P_L$. Moreover $P_L$ +is relatively algebraically closed in $L$: an element of $L$ +algebraic over $P_L$ is algebraic over $P_N$, hence lies in $P_N$. +Taking the relative algebraic closure of any nonconstant rational +subfield shows that $P_L/l$ is finitely generated. + +The norm $N_{N/L}$ sends $P_N^\times$ into $P_L^\times$. In fact, +if $s$ is the inseparable degree of $N/L$, then +\[ + N_{N/L}(x)=\left(\prod_{\sigma\in G}\sigma(x)\right)^s; +\] +the right side lies in $P_N$ because $P_N$ is $G$-stable, and the norm lies in +$L$. On completions, norm composed with inclusion is multiplication +by $[N:L]$. We conclude that +$[N:L]H\subset\widehat P_L$. The latter is saturated in +$\widehat L$ by Lemma~\ref{prelim:completion}, so +$H\subset\widehat P_L$. +Here the prime-to-$\ell$ factor of $[N:L]$ is a unit of $\Zell$, +and saturation removes its remaining $\ell$-power factor. + +Apply the same argument to $\Theta^{-1}$. It gives a curve subfield +$E'\subset K$ with +\[ + \widehat E\subset\Theta^{-1}(\widehat P_L) + \subset\widehat E'. +\] +The infinite-rank intersection forces $E'=E$ by +Lemma~\ref{recovery:intersection}. Both inclusions are therefore +equalities. That lemma also proves uniqueness. Reversing $K$ and $L$ +proves the asserted bijection. +\end{proof} + +\subsection{Divisorial valuations and rational quotients} + +We can now distinguish valuations trivial on the constants. Throughout +this subsection, a \emph{divisorial valuation} means such a +quasi-prime divisor, with its discrete value group normalized to +$\mathbb Z$. + +The following criterion adapts Topaz's Lemma~A.7 +\cite[Lemma~A.7]{TopazTorelli2025} to completed multiplicative groups. +The proof below verifies the needed hypotheses in this setting. + +\begin{proposition}\label{recovery:divisorial} +A quasi-prime divisor $v$ of $K/k$ is divisorial if and only if +\begin{equation}\label{eq:recovery-divisorial} + D_v^\perp\cap\widehat E=0 +\end{equation} +for some curve subfield $E$ of $K/k$. Consequently $\varphi$ +preserves divisorial inertia and decomposition groups in both +directions. +\end{proposition} + +\begin{proof} +Suppose $v$ is trivial on $k$. Its residue field has transcendence +degree $d-1\geq1$. Choose $t$ with transcendental residue, and let +$E$ be the relative algebraic closure of $k(t)$ in $K$. The valuation +is trivial on $k(t)$ and hence on its algebraic extension $E$. +Thus $E$ embeds into $Kv$. Its relative algebraic closure in $Kv$ +is finite over $E$, so the induced completion map is injective: +factor it through that finite extension and then a relatively +algebraically closed inclusion. Residual evaluation by +$D_v/I_v=W_{Kv}$ proves \eqref{eq:recovery-divisorial}. + +Conversely, suppose $v|k$ is nontrivial, and let $E$ be any curve +subfield. Choose $x\in E\setminus k$, replacing it by $x^{-1}$ if +necessary so that $v(x)\geq0$, and choose $c\in k$ with $v(c)>0$. +Then $1+cx$ is a nonconstant principal unit. Its class in +$\widehat E$ is nonzero and is killed by $D_v$, so +\eqref{eq:recovery-divisorial} fails. The criterion is invariant +under $\Theta$ by Theorem~\ref{recovery:curves} and the local +correspondence. +\end{proof} + +For a curve subfield $E\subset K$, restriction gives a surjection +\[ + \rho_E:W_K\longrightarrow W_E. +\] +This is the continuous surjection of Lemma~\ref{prelim:completion}. + +\begin{proposition}\label{recovery:rational} +The point inertia lines of every curve subfield $E$ are precisely +the saturations of the nonzero images $\rho_E(I_v)$ for divisorial +valuations $v$ of $K/k$. The correspondence of +Theorem~\ref{recovery:curves} preserves these lines and the genus. +In particular, it matches exactly the relatively algebraically closed +rational subfields. + +If $\Theta(\widehat E)=\widehat P$, the induced continuous +isomorphism $\varphi_E:W_P\to W_E$ makes the diagram +\begin{equation}\label{eq:recovery-quotient-diagram} +\begin{tikzcd}[column sep=large] + W_L \arrow[r,"\varphi"] \arrow[d,two heads,"\rho_P"'] + & W_K \arrow[d,two heads,"\rho_E"] \\ + W_P \arrow[r,"\varphi_E"'] & W_E +\end{tikzcd} +\end{equation} +commute and matches the point inertia lines on its targets. +For rational $E$ and $P$, these are the geometric rational quotient +diagrams, with their point decomposition data. +\end{proposition} + +\begin{proof} +A divisorial valuation restricted nontrivially to $E$ is an integer +multiple of a point valuation on its smooth projective curve: its +value group is a nonzero subgroup of $\mathbb Z$, and it is trivial +on $k$. Conversely, extend any point valuation of $E$ by a Gauss +valuation over a transcendence basis for $K/E$, then prolong it over +the remaining finite extension. The resulting valuation is +divisorial on $K/k$. Its restriction is $e\operatorname{ord}_x$ for +some $e>0$. Saturation of the nonzero image recovers +$\Zell\operatorname{ord}_x$, including when $\ell$ divides $e$. + +The dual of $\Theta|_{\widehat E}$ defines $\varphi_E$. +Evaluation against elements of $\widehat E$ proves the commutativity +of \eqref{eq:recovery-quotient-diagram}. Proposition~\ref{recovery:divisorial} +and the preceding description of point lines show that $\varphi_E$ +matches them. Their generators may differ by units, which do not +change the common kernel of point-order evaluations modulo $\ell$. +Its dimension is twice the genus by Lemma~\ref{prelim:curve-kernel}. +Hence genus is preserved. A smooth projective curve over an +algebraically closed field has genus zero exactly when its function +field is rational. + +If $k(t)$ is relatively algebraically closed in $K$, the extension +$K/k(t)$ is regular. Only separability needs explanation. In +characteristic $p>0$, relative algebraic closedness implies +$t\notin K^p$. Since $k$ is perfect, $t$ can be included in a +separating transcendence basis for $K/k$, proving that $K/k(t)$ is +separably generated. Characteristic zero is immediate. Thus these +are exactly the rational subfields defined by general elements. +Finally, the residue field at a point of a rational curve is the +algebraically closed constant field, whose character group is zero. +Point decomposition therefore equals point inertia, and the diagram +also carries the required point decomposition data. +\end{proof} + +The curve correspondence has consequently recovered both the true +divisorial valuations and all geometric rational quotients from the +original datum. The final reconstruction theorem can now be applied +with these intrinsically determined objects. diff --git a/preprints/The-Bogomolov-Pop-reconstruction-theorem-September-23-2026/build/introduction.tex b/preprints/The-Bogomolov-Pop-reconstruction-theorem-September-23-2026/build/introduction.tex new file mode 100644 index 0000000000000000000000000000000000000000..a1b052285ded0ec86649397e97f8509c56fa561a --- /dev/null +++ b/preprints/The-Bogomolov-Pop-reconstruction-theorem-September-23-2026/build/introduction.tex @@ -0,0 +1,164 @@ +\section{The reconstruction problem} +\label{sec:introduction} + +Birational anabelian geometry asks how much of a field can be read from its +Galois groups. Over algebraically closed constants, the arithmetic Galois +action disappears. Bogomolov's program proposes that a particularly small +quotient still remembers every function field of dimension at least two: +the quotient in which commutators are central. We prove the Isom form of +this assertion, including surfaces and the prime $2$. + +\subsection{The exact statement} +Fix a prime $\ell$. A function field $F/\kappa$ in this paper is a finitely +generated extension of an algebraically closed field $\kappa$. +For $\operatorname{char}F\ne\ell$, let $G_F^{(\ell)}$ be the maximal +pro-$\ell$ quotient of the absolute Galois group, and set +\[ + \Pi_F^a=G_F^{(\ell)}/[G_F^{(\ell)},G_F^{(\ell)}],\qquad + \Pi_F^c=G_F^{(\ell)}/[[G_F^{(\ell)},G_F^{(\ell)}],G_F^{(\ell)}]. +\] +All commutator subgroups are closed. The kernel +$\Delta_F=\ker(\Pi_F^c\to\Pi_F^a)$ is central. Our commutator convention is +$[x,y]=x^{-1}y^{-1}xy$. Lifting elements of $\Pi_F^a$ to $\Pi_F^c$ therefore +defines a continuous alternating $\Zell$-bilinear map +\[ + [\ ,\ ]_F:\Pi_F^a\times\Pi_F^a\longrightarrow\Delta_F. +\] + +An isomorphism $\varphi:\Pi_L^a\to\Pi_K^a$ is +\emph{bracket-compatible} if it is a continuous $\Zell$-module isomorphism +and there is a continuous $\Zell$-module isomorphism +$\psi:\Delta_L\to\Delta_K$ such that +\[ + \psi([\sigma,\tau]_L)=[\varphi\sigma,\varphi\tau]_K + \quad(\sigma,\tau\in\Pi_L^a). +\] +Write $\Isom^c(\Pi_L^a,\Pi_K^a)$ for this set. The units $\Zell^\times$ +act by multiplication on $\varphi$; the corresponding map on the central +kernel can be multiplied by the square of the same unit. + +Write $F^i$ for the perfect closure of $F$; in characteristic zero this is +$F$ itself. Let $\Isom^i(K,L)$ consist of the field isomorphisms +$\alpha:K^i\to L^i$ satisfying $\alpha(k)=l$. +If the common characteristic is $p>0$, put +\[ + \Isom_F^i(K,L)=\Isom^i(K,L)/ + \bigl(\alpha\sim\Frob_L^n\circ\alpha,\ n\in\mathbb Z\bigr). +\] +Frobenius is an automorphism of a perfect field, so negative integers are +allowed. In characteristic zero take no quotient. If the +characteristics differ, set $\Isom_F^i(K,L)=\varnothing$. + +Field isomorphisms induce contravariant Galois isomorphisms. Passing to the +above quotients gives the canonical map in the following theorem. + +\begin{theorem}[Bogomolov--Pop reconstruction]\label{thm:main} +Let $k$ and $l$ be arbitrary algebraically closed fields of characteristics +different from $\ell$, and let $K/k$ and $L/l$ be function fields with +$\trdeg(K/k),\trdeg(L/l)\ge2$. Then +\[ + \Phi_{K,L}:\Isom_F^i(K,L)\longrightarrow + \Isom^c(\Pi_L^a,\Pi_K^a)/\Zell^\times, + \qquad [\alpha]\longmapsto[\alpha^*], +\] +is a bijection. +\end{theorem} + +Thus the theorem resolves the pro-$\ell$ abelian-by-central +Bogomolov--Pop reconstruction conjecture positively. Neither equality of +characteristics nor equality of relative transcendence degrees is assumed. +Both are forced by a bracket-compatible isomorphism. The input contains +no distinguished valuations, inertia groups, or curve quotients. + +\subsection{Previous work and the proof} +Bogomolov introduced the abelian-by-central approach to birational +reconstruction \cite{Bogomolov1991}. The local approach detects +valuations through commuting elements and their associated multiplicative +characters \cite{BT2002,TopazCL2017}. +Bogomolov and Tschinkel proved the surface case over algebraic closures +of finite fields \cite{BT2008}, followed by higher-dimensional +reconstruction \cite{BT2011}; Pop established a corresponding +birational reconstruction theorem \cite{PopBI2012}. +The precise Isom formulation, including its scalar and Frobenius +ambiguities, is recorded in +\cite[Introduction, Conjecture~1]{TopazRat2016}. + +Pop's global reconstruction +theorem recovers a function field from its total decomposition graph +together with suitable rational quotients +\cite{PopGraphs2012,PopBI2012}. His later reconstruction theorem with +divisorial inertia applies in relative transcendence degree greater than +two \cite[Theorem~1.1]{Pop2018}; its additional inertia datum must itself +be recovered to solve the conjecture from the group alone. +The local theory of commuting pairs supplies quasi-divisorial valuations, +which can be nontrivial on the constants +\cite{Pop2018,Topaz2019}. Pop's minimized decomposition theory also +recognizes certain residual curves over algebraic closures of finite +fields, with Topaz's appendix treating the required residue +characteristic $\ell$ case \cite{PopMin2015}. + +Related reconstruction results use additional structure. +Silberstein studies the full absolute Galois group under a +prime-field-closure hypothesis \cite{Silberstein2013}. +Topaz's integral cohomological reconstruction uses a specified +arithmetic Galois action in transcendence degree at least three +\cite[Theorems~A and~B]{Topaz2019}; his Torelli-type theorem uses +mixed Hodge structures and cup products over constants embedded in +$\mathbb C$ \cite{TopazTorelli2025}. +These results concern different input data. Here only the original +pro-$\ell$ abelian-by-central datum is supplied, and neither the +constant field nor geometric inertia is distinguished in advance. + +Our task is to recover the missing geometric information uniformly over +the constants. Write $\widehat F=\varprojlim_e F^\times/F^{\times\ell^e}$ +for the completed multiplicative group and $[f]$ for the class of a +nonzero function. Choose Kummer identifications, and dualize a +bracket-compatible isomorphism to an isomorphism +$\Theta:\widehat K\to\widehat L$ of completed multiplicative groups. +The central step proves that, for fixed $t\in K\setminus k$, the entire +pencil +\[ + h_a=\Theta[t-a],\qquad a\in k, +\] +has uniformly bounded divisor-support degree on a fixed projective model +of $L$. This is stronger than finiteness of each individual support. + +Bogomolov--Tschinkel already used bounded-genus curves and Hurwitz +estimates to prove finiteness of $\ell$-adic supports +\cite[Section~6, proof of Proposition~6.1]{BT2011}. +Here the bound is uniform over a whole pencil and is obtained by finite +tests over arbitrary constants. A finite list of Kummer classes +is specialized and restricted to a smooth residual curve of bounded +genus, preserving all relations at the chosen finite level and the +relevant divisor supports. Five fixed members of the pencil then bound +the degree of a separable map to the projective line by Riemann--Hurwitz. +The bound also controls every additional member, at every sufficiently +high $\ell$-power level. The use of five anchors works for $\ell=2$. + +The pencil detects infinitely many prime divisors, and the uniform +degree bound places them in a family of finite type. Their incidence +with the model produces a curve subfield +after a finite field extension. A normal-base-change argument allows us +to use one parameter curve for every Kummer class and every exponent; +no uncountability assumption enters. Norms and the finite-rank +intersection of distinct completed curve fields descend this subfield +to $L$. Applying the same argument to $\Theta^{-1}$ gives a bijection +of the completed relatively algebraically closed curve subfields. + +These subfields distinguish true divisorial valuations from those that +also value the constants. They also recover point inertia and genus, +and hence all relatively algebraically closed rational subfields. +Their quotient diagrams supply exactly the input to Pop's global +reconstruction theorem. The new geometric steps are the uniform +pencil bound and the incidence argument that recovers curve subfields; +the remaining steps use the local valuation theory and Pop's +reconstruction theorem described above. + +The proof follows this order. Section~\ref{sec:preliminaries} fixes the +Kummer topology and the relation between the two formulations of the +group input. Section~\ref{local:section} supplies the local correspondence. +Sections~\ref{sec:tests} and~\ref{sec:pencil} prove the residual test and +uniform support bound. Section~\ref{sec:curve-recovery} recovers curve +subfields and rational quotients. Section~\ref{sec:reconstruction} +checks compatibility on the decomposition graphs before applying Pop's +theorem and completing the Isom statement and its ambiguity checks. diff --git a/preprints/The-Bogomolov-Pop-reconstruction-theorem-September-23-2026/build/local-theory.tex b/preprints/The-Bogomolov-Pop-reconstruction-theorem-September-23-2026/build/local-theory.tex new file mode 100644 index 0000000000000000000000000000000000000000..b4d535c7425cf5ea4f4d67a562061b4a45a91145 --- /dev/null +++ b/preprints/The-Bogomolov-Pop-reconstruction-theorem-September-23-2026/build/local-theory.tex @@ -0,0 +1,494 @@ +\section{Recovering the local data} +\label{local:section} + +The commutator detects a relation between multiplicative characters. +That relation recovers quasi-prime divisors, including those that value +constants nontrivially. At this stage their residue fields may have +characteristic~$\ell$. We therefore work with character groups throughout +and use a Galois interpretation only at the top field. +For the commuting-pair approach to valuation detection and its +development via rigid elements, see +\cite{Bogomolov1991,BT2002,TopazCL2017}. +We use the character-space formulation below to include the required +residue fields. + +\subsection{Commutators and alternating characters} + +For any field $F$, put +\[ + \mathcal G_F=W_F\otimes_{\Zell}\Qell. +\] +A pair $f,g\in\mathcal G_F$ is \emph{alternating} if +\begin{equation}\label{local:alternating} + f(x)g(1-x)=f(1-x)g(x)\qquad(x\in F\setminus\{0,1\}). +\end{equation} +A subspace is alternating if each pair of its elements is alternating. +Every character in $\mathcal G_F$ kills $-1$. + +\begin{lemma}\label{local:commuting} +Let $F/\kappa$ be a function field over an algebraically closed field +of characteristic different from~$\ell$. Under Kummer duality $W_F=\Pi_F^a$, two elements of $W_F$ +are alternating if and only if their commutator in $\Delta_F$ is zero. +This assertion holds also for $\ell=2$. +\end{lemma} + +\begin{proof} +We give the finite-coefficient argument, keeping track of the +infinite rank of $W_F$. Write $G=G_F^{(\ell)}$, $A=G^{\mathrm{ab}}$ +and $R=\mathbb Z/\ell^e$, with trivial action. Kummer duality makes +$A$ a product of copies of~$\Zell$. + +First, a class in $H^2_{\mathrm{cont}}(A,R)$ is determined by its +commutator form. Indeed, a class with zero commutator defines +an abelian extension of $A$ by $R$. Such an extension splits +continuously: the Pontryagin dual of $A$ is a direct sum of copies of +$\Qell/\Zell$, hence is divisible and injective. Conversely every +continuous alternating bilinear form $A\times A\to R$ is the +commutator form of a cup-product class. To see this, continuity and +compactness give an open subgroup in the radical of the form, so it +depends on finitely many coordinates. In coordinate characters $x_i$ +it is a sum of the forms +\[ + x_i(a)x_j(b)-x_j(a)x_i(b),\qquad i0$, +then $1-x$ is a principal unit. If $v(x)<0$, the identity +$1-x=-x(1-x^{-1})$ makes the characters of $1-x$ equal those of~$x$. +If $v(x)=0$ and $v(1-x)>0$, then $x$ is a principal unit. In each +case the alternating determinant is zero. In the remaining case +$x$ and $1-x$ are units with residues different from $0,1$, and the +determinant is exactly the residual determinant. Conversely every +residual element different from $0,1$ has such a lift. +\end{proof} + +In particular, a unit representing a class modulo $\ell^e$ evaluates +on $D_v/I_v$ exactly as its residue does on $W_{Fv}$. This observation +will allow finite Kummer tests to pass through a valuation. + +\subsection{An intrinsic description of quasi-prime divisors} + +The next argument follows the character-space proof of +\cite[Fact~3.2 and Theorem~3.3]{Topaz2019}. We include it because we +need it also in residue characteristic~$\ell$, whereas that section +of the cited paper is stated in a cohomological setting with +characteristic different from~$\ell$. + +\begin{lemma}\label{local:dimension} +For $F/\kappa$ as above, in any characteristic, $s$ is the maximum +dimension of an alternating subspace of $\mathcal G_F$. +If $S$ has dimension $s$, its valuative subspace is +$\mathcal I_u$ for its associated valuation $u$, and $u$ attains +equality in \eqref{local:abhyankar}. +\end{lemma} + +\begin{proof} +For every valuation $u$, characters kill the divisible constant group, +so +\[ + \dim_{\Qell}\mathcal I_u + \leq\operatorname{rrank}(uF/u\kappa). +\] +Let $S$ be alternating and let $\Sigma$ be its valuative subspace, +with associated valuation~$u$. If $S=\Sigma$, the asserted dimension +bound follows from \eqref{local:abhyankar}. +Otherwise $\Sigma$ has codimension one in~$S$. +A character in $S\setminus\Sigma\subset\mathcal D_u$ has nonzero +residual character. Thus $Fu\ne\kappa u$, and +$\operatorname{trdeg}(Fu/\kappa u)\geq1$. Consequently +\[ + \dim S\leq\operatorname{rrank}(uF/u\kappa)+1\leq s. +\] +This also bounds arbitrary alternating subspaces, by applying the +argument to their finite-dimensional subspaces. + +There is a full discrete flag at a smooth closed point on a model +of $F/\kappa$. Its inertia space has dimension~$s$ and is +alternating by the valuation inequality for $x$ and $1-x$. +Thus the maximum equals~$s$. + +If $\dim S=s$ and $S=\Sigma$, all inequalities +\[ + s\leq\dim\mathcal I_u + \leq\operatorname{rrank}(uF/u\kappa)\leq s +\] +are equalities. If $S\ne\Sigma$, all inequalities +\[ + s-1\leq\dim\mathcal I_u + \leq\operatorname{rrank}(uF/u\kappa) + \leq s-\operatorname{trdeg}(Fu/\kappa u)\leq s-1 +\] +are equalities. In both cases $\Sigma=\mathcal I_u$ and $u$ has +no transcendence defect. +\end{proof} + +\begin{lemma}\label{local:quasiprime} +Assume $s\geq2$. A line $H\subset\mathcal G_F$ is the inertia +space of a quasi-prime divisor if and only if there are two +$s$-dimensional alternating subspaces $S_1,S_2$ with +$S_1\cap S_2=H$. The valuation is unique. If $0\ne h\in H$, its +decomposition space is +\[ + \mathcal D_v=\{f\in\mathcal G_F:(h,f)\text{ is alternating}\}. +\] +\end{lemma} + +\begin{proof} +Suppose first that $v$ is quasi-prime. The residue function field +$Fv/\kappa v$ has two independent discrete valuations of full rank +$s-1$, trivial on the constants. For example, choose two independent +prime divisors on a rational subfield on a transcendence basis, +complete each to a flag, and prolong the flags to the finite +extension~$Fv$. Independent valuation approximation +\cite[Theorem~1.2]{ADF2020} makes their inertia spaces disjoint. +Composing with $v$ gives two alternating inertia spaces of dimension +$s$ whose intersection is $\mathcal I_v$. + +Conversely, let $S_1,S_2$ be as stated. By +Lemma~\ref{local:dimension}, their valuative parts are +$\Sigma_i=\mathcal I_{u_i}$, where each $u_i$ has no transcendence +defect and $\dim\Sigma_i\geq s-1$. +We first show that $H$ is valuative. If, say, $u_1\leq u_2$, then +\[ + \Sigma_1\subset\Sigma_2, + \qquad \Sigma_1\subset S_1\cap S_2=H. +\] +Since $\dim\Sigma_1\geq1$, this proves the claim. +If the valuations are incomparable, let $w$ be their finest common +coarsening. Applying independent valuation approximation on $Fw$ +shows that the principal-unit groups of $u_1$ and $u_2$ generate +$U_w$. Indeed, approximate a unit $a$ by $b$ for the first residual +valuation and approximate $1$ by $b$ for the second. Then +$a=(a/b)b$ is a product of the two required principal units. +A character in $H$ annihilates both principal-unit groups, +and hence belongs to $\mathcal I_w$. Again $H$ is valuative. +In either case, +\[ + H=\Sigma_1\cap\Sigma_2. +\] + +Let $v=v_H$ be its coarsest associated valuation. It coarsens both +$u_i$, and +\[ + H\subset\mathcal I_v + \subset\mathcal I_{u_1}\cap\mathcal I_{u_2}=H. +\] +A coarsening of a valuation without transcendence defect also has +no transcendence defect: relative rational ranks add under +composition, and the two valuation inequalities sum to the equality +for the finer valuation. Thus $v$ has no transcendence defect. +Its relative value group is finitely generated free, and its rank +is $\dim\mathcal I_v=1$. Hence $vF/v\kappa\cong\mathbb Z$. +A coarsening with the same relative rank would have the same inertia +space, contradicting the defining minimality of~$v_H$. +Therefore $v$ is quasi-prime, and the coarsest-valuation description +also proves uniqueness. + +Every member of $\mathcal D_v$ alternates with $h$, by the calculation +in Lemma~\ref{local:residue}. For the converse apply the third part +of the local character theorem to the valuative line~$H$. +\end{proof} + +The integral groups are recovered without an index ambiguity: +\[ + I_v=\mathcal I_v\cap W_F, + \qquad D_v=\mathcal D_v\cap W_F. +\] +Indeed these are annihilators with values in the torsion-free group +$\Zell$. In particular they are saturated submodules. + +\begin{proposition}\label{local:transport} +Let $K/k$ and $L/l$ be function fields over algebraically closed +fields of characteristic different from~$\ell$, each of +transcendence degree at least two. Let +$\varphi:W_L\to W_K$ be a continuous bracket-compatible +$\Zell$-module isomorphism. Then their transcendence degrees agree, +say they are~$d$. For every $1\leq r0$, using the finite-constant +recognition above. +Remove inseparability by writing $\tau_0=\tau^{p^r}$ with $r\ge0$ +maximal. Such an $r$ is finite, as a nonzero integer point order of +$\tau_0$ cannot be divisible by arbitrarily high powers of $p$. +The function $\tau$ gives a separable morphism +$\tau:C'\to\mathbb P^1$ of some degree $n\ge1$. +Let $\beta'_j$ be the $p^r$-th root of $\beta_j$ in $kv$. +These constants are distinct, and +\[ + \tau_0-\beta_j=(\tau-\beta'_j)^{p^r}. +\] +Since $p\ne\ell$, this power does not affect whether any point order +is zero modulo $\ell$ or $q$. + +Over the value $\beta'_i$ corresponding to anchor $i$, at most $N_i$ +points have ramification index not divisible by $\ell$. +Indeed such a point contributes a nonzero order modulo $\ell$ to +$\tau_0-\beta_i$; transport to $C$ and +\eqref{tests:support} give the bound $N_i$. +Every other point of that fiber has ramification index at least +$\ell$, so there are at most $n/\ell$ of them. +If $r_i$ denotes the total number of points in the fiber, then +\[ + r_i\le N_i+n/\ell,\qquad + \sum_{P\mid\beta'_i}(e_P-1)=n-r_i + \ge(1-1/\ell)n-N_i. +\] +For a separable morphism of smooth curves, the different exponent at +$P$ is at least $e_P-1$, including in the presence of wild ramification. +The five fibers are disjoint. Riemann--Hurwitz therefore yields +\[ + 2g(C')-2+2n + \ge\sum_{i=1}^5\sum_{P\mid\beta'_i}(e_P-1) + \ge5(1-1/\ell)n-\sum_{i=1}^5N_i. +\] +Together with \eqref{pencil:genus} this gives $c_\ell n\le +2G_X-2+\sum_iN_i$, and hence $n\le B$. +This argument uses the different inequality, not the tame +Riemann--Hurwitz formula; see \cite[Section~53.12, Tag~0C1B]{Stacks}. + +\emph{Pass from finite tests to exact supports.} +For every tested $j$, all zeros and poles of $\tau-\beta'_j$ lie in two +fibers, over $\beta'_j$ and infinity. There are at most $2n$ such +points. The number of its nonzero point orders modulo $q$ is therefore +at most $2B$. +Transport this bound to $C$ and apply \eqref{tests:support} to obtain +\begin{equation}\label{pencil:finite-level-bound} + \sum_{D\in S_e(h_a)}\deg D\le2B. +\end{equation} +The right side is independent of $a$, $e$, the finite descent data and +the residual test. For each fixed $a$, the inequality holds for every +sufficiently large $e$. + +Finally $S_e(h_a)$ increases to $\operatorname{Supp}_X(h_a)$. +If the exact support had total degree greater than $2B$, a finite +subset would already have degree greater than $2B$; that subset would +lie in $S_e(h_a)$ for all sufficiently large $e$, contradicting +\eqref{pencil:finite-level-bound}. +Every prime divisor has positive integer degree, so the exact support +is finite. This proves the asserted uniform bound. +\end{proof} + +\begin{corollary}\label{pencil:finite} +For every $z\in K^\times$, the class $\Theta[z]$ has finite support on +$X$. +\end{corollary} + +\begin{proof} +If $z\notin k$, apply Theorem~\ref{pencil:uniform} to $t=z$ and $a=0$. +If $z\in k^\times$, its Kummer class is zero because $k^\times$ is +divisible. +\end{proof} diff --git a/preprints/The-Bogomolov-Pop-reconstruction-theorem-September-23-2026/build/preliminaries.tex b/preprints/The-Bogomolov-Pop-reconstruction-theorem-September-23-2026/build/preliminaries.tex new file mode 100644 index 0000000000000000000000000000000000000000..e07baac78a77123e7983501652aa2ddf5231fb20 --- /dev/null +++ b/preprints/The-Bogomolov-Pop-reconstruction-theorem-September-23-2026/build/preliminaries.tex @@ -0,0 +1,169 @@ +\section{Kummer duality and the group input}\label{sec:preliminaries} + +We first give the topological form of Kummer duality needed for selected +subfields and residue fields. In this section the residue characteristic +is allowed to equal $\ell$ unless explicitly excluded. + +For a function field $F/\kappa$ over algebraically closed constants, put +\[ + W_F=\Hom(F^\times,\Zell),\qquad + \widehat F=\varprojlim_{e\ge1} F^\times/F^{\times\ell^e}. +\] +Equip $W_F$ with pointwise convergence and $\widehat F$ with its +$\ell$-adic topology. We use additive notation for these modules and +write $[f]$ for the class of $f\in F^\times$. In particular +$[fg]=[f]+[g]$. All characters vanish on $\kappa^\times$. We abbreviate +$F^\times/F^{\times q}$ to $F^\times/q$. An annihilator $B^\perp$ +of $B\subset F^\times$ or $B\subset\widehat F$ is taken in $W_F$; +for $B\subset W_F$ it is taken in $\widehat F$, using evaluation. + +\begin{lemma}[Continuous evaluation]\label{prelim:duality} +The group $F^\times/\kappa^\times$ is free abelian. Consequently +$W_F$ is a product of copies of $\Zell$, and evaluation gives +\[ + \widehat F=\Hom_{\mathrm{cont}}(W_F,\Zell),\qquad + \widehat F/\ell^e\widehat F=F^\times/F^{\times\ell^e}. +\] +The module $\widehat F$ is torsion free, and evaluations on $W_F$ +detect every finite-level class. +\end{lemma} +\begin{proof} +Choose a normal integral projective model of $F/\kappa$. A rational +function with zero Weil divisor on this model is a global unit and hence +a constant. Thus the divisor map embeds $F^\times/\kappa^\times$ in +the free abelian group of prime divisors; a subgroup of a free abelian +group is free. + +For a basis indexed by a set $J$, the character group is $\Zell^J$. +Modulo $\ell^e$, a continuous character on this compact product depends +on finitely many coordinates. Compatible such characters are therefore +families $(c_j)_{j\in J}$ in $\Zell$ with only finitely many +$c_j\notin\ell^e\Zell$ for each $e$. This is exactly the completion +of $\bigoplus_{j\in J}\mathbb Z$. Reduction modulo $\ell^e$ and +coordinate evaluations prove the assertions. Divisibility of +$\kappa^\times$ removes the constants at every level. +\end{proof} + +When $\operatorname{char}F\ne\ell$, a choice of a compatible system of +$\ell$-power roots of unity identifies the Tate module $\Zell(1)$ with +$\Zell$. Kummer theory then identifies $\Pi_F^a$ with $W_F$. +An isomorphism $\varphi:\Pi_L^a\to\Pi_K^a$ has evaluation dual +\begin{equation}\label{eq:theta} + \Theta:\widehat K\longrightarrow\widehat L, + \qquad \Theta(h)(\sigma)=h(\varphi\sigma). +\end{equation} +Changing the Tate identifications multiplies this map by one unit. +We fix the identifications for the proof. + +\begin{lemma}[Subfields and finite extensions]\label{prelim:completion} +Let $E\subset F$ be function fields over the same algebraically closed +field $\kappa$. +\begin{enumerate} +\item If $E$ is relatively algebraically closed in $F$, then +$\widehat E\to\widehat F$ is injective, its image is closed and +$\ell$-saturated, and restriction $W_F\to W_E$ is a continuous +surjection. Here $\ell$-saturated means that $\ell h\in\widehat E$ +for $h\in\widehat F$ implies $h\in\widehat E$. +\item If $F/E$ is finite, its completion map is injective, and +inclusion followed by norm on completions is multiplication by +$[F:E]$. +\end{enumerate} +\end{lemma} +\begin{proof} +In the first case, a root in $F$ of an element of $E$ is algebraic over +$E$, so injectivity holds at every finite level. Alternatively, on a +normal integral projective $E$-model of $F$, the divisor map embeds +$F^\times/E^\times$ into a free abelian group: its kernel consists of +global units, which are algebraic over $E$ and therefore lie in $E$. +Thus $F^\times/E^\times$ is free abelian, and +\[ + 0\longrightarrow E^\times/\kappa^\times + \longrightarrow F^\times/\kappa^\times + \longrightarrow F^\times/E^\times\longrightarrow0 +\] +splits as a sequence of abelian groups. Completion preserves this +splitting. The two completed free factors are torsion free, proving +closedness and saturation. Every character on $E^\times$ extends along +the splitting, proving the asserted surjection; continuity follows from +the pointwise topologies. + +For a finite extension, the field norm is defined also in the inseparable +case and sends an element of $E$ to its $[F:E]$-th power. It induces +the stated composite on completions. Torsion freeness from +Lemma~\ref{prelim:duality} proves injectivity. +\end{proof} + +The next observation lets us apply Pop's reconstruction theorems to +the stated group input. + +\begin{lemma}[Lifting bracket-compatible isomorphisms] +\label{prelim:central-lift} +Every bracket-compatible isomorphism in Theorem~\ref{thm:main} is the +abelianization of a continuous isomorphism $\Pi_L^c\to\Pi_K^c$. +This holds also for $\ell=2$. +\end{lemma} +\begin{proof} +Use the given maps to identify the bases of the two central extensions +with one abelian pro-$\ell$ group $A$ and their kernels with one group +$B$. Let $E_1,E_2$ denote these extensions. In the fiber product +$P=E_1\times_A E_2$, quotient by the diagonal central subgroup +$\{(b,b):b\in B\}$. The resulting group $Q$ is abelian: the +commutator of two pairs has identical components by bracket +compatibility. It fits into an exact sequence of abelian pro-$\ell$ +groups +\[ + 0\longrightarrow B\longrightarrow Q\longrightarrow A\longrightarrow0. +\] +By Lemma~\ref{prelim:duality}, $A$ is a product of copies of $\Zell$. +Such products are projective in the category of abelian pro-$\ell$ +groups. Indeed Pontryagin duality changes $A$ into a direct sum of +copies of $\mathbb Q_\ell/\Zell$, a divisible, hence injective, +discrete torsion group. Consequently $Q\to A$ has a continuous +section. + +The inverse image of this section in $P$ maps isomorphically to each +$E_i$. For fixed first component, the second component can be adjusted +by a unique element of $B$ to land over the section; its uniqueness +also proves that each projection has trivial kernel. These are +continuous bijections between compact Hausdorff groups, hence +isomorphisms. The resulting map $E_1\to E_2$ induces the prescribed +base and kernel maps. The proof uses no division by $2$ and leaves +no additional power datum to specify. +\end{proof} + +\subsection{Orders on a curve} +For a smooth projective integral curve $C/\kappa$ with function field +$E$, its closed points are $\kappa$-rational. Each normalized order +$\ord_P:E^\times\to\mathbb Z$ extends to $\widehat E\to\Zell$. + +\begin{lemma}[Curve kernels]\label{prelim:curve-kernel} +If $\operatorname{char}\kappa\ne\ell$, then +\[ + \ker\left(E^\times/E^{\times\ell} + \xrightarrow{(\ord_P)_P}\bigoplus_{P\in C}\mathbb Z/\ell\right) + \simeq\Pic(C)[\ell]. +\] +Its dimension over $\Fell$ is $2g(C)$. The kernel of all exact point +orders on $\widehat E$ is the Tate module $T_\ell\Pic(C)$ and has +$\Zell$-rank $2g(C)$. +\end{lemma} +\begin{proof} +The same argument works with $q=\ell^e$. If +$\operatorname{div}(f)=qD$, send $[f]$ to the divisor class of $D$. +Changing $f$ by a $q$-th power does not change this class. Every +$q$-torsion divisor class occurs. The kernel consists of functions +$cu^q$; the constant $c\in\kappa^\times$ is a $q$-th power. This +proves the finite-level identification. The $q$-torsion of the +Jacobian over an algebraically closed field of characteristic prime to +$q$ is $(\mathbb Z/q)^{2g(C)}$ +\cite[Tag~0C1Z, Lemma~53.17.1]{Stacks}. Under reduction from $\ell^{e+1}$ +to $\ell^e$, the divisor class is multiplied by $\ell$. Taking the +inverse limit gives the exact-order kernel and its rank. +\end{proof} + +In particular, an isomorphism of the $W$-groups carrying the family of +point inertia lines bijectively onto the corresponding family preserves +the genus. A generator +of each line can change by a unit; this changes neither vanishing nor +nonvanishing of any finite-level order. We will use this observation +only after proving that the residue characteristics differ from $\ell$. diff --git a/preprints/The-Bogomolov-Pop-reconstruction-theorem-September-23-2026/build/reconstruction.tex b/preprints/The-Bogomolov-Pop-reconstruction-theorem-September-23-2026/build/reconstruction.tex new file mode 100644 index 0000000000000000000000000000000000000000..426798f9120ca1af5299bd6351237fe04e562a7c --- /dev/null +++ b/preprints/The-Bogomolov-Pop-reconstruction-theorem-September-23-2026/build/reconstruction.tex @@ -0,0 +1,134 @@ +\section{From the recovered quotients to the field} +\label{sec:reconstruction} + +We have recovered the geometric data needed for the last step. We now +specify that data and Pop's reconstruction theorems, so that their +application uses exactly the original group input. + +For a function field $F/\kappa$, a \emph{divisorial flag} is a sequence +of valuations obtained by taking a prime divisor trivial on $\kappa$, +then a prime divisor on its residue field, and continuing in this way. +The empty flag is included. The \emph{total decomposition graph} +records, at each such flag, the abelian pro-$\ell$ group of the residue +field, and at each next divisorial valuation its inertia and +decomposition subgroups and the quotient map to the next residue group. +Thus the graph retains the group maps as well as the flags. At a +one-variable field over algebraically closed constants, its terminal +data are precisely the point inertia lines: the residual groups at +closed points vanish, so point decomposition equals point inertia. + +We use the following two results of Pop. +\begin{proposition}[Pop's reconstruction theorems] +\label{reconstruction:pop} +Let $F/\kappa$ and $F'/\kappa'$ be function fields of transcendence +degree greater than one over algebraically closed fields of +characteristic different from $\ell$. +\begin{enumerate} +\item From $\Pi_F^c\to\Pi_F^a$ and the union of its true divisorial +inertia groups, one recovers the total decomposition graph, functorially +under class-two isomorphisms whose abelianizations preserve this union. +\item Suppose an isomorphism between the total decomposition graphs of +$F$ and $F'$ is compatible with all their geometric rational quotient +diagrams. Then its map on the top abelian groups is induced, up to one +unit in $\Zell$, by an isomorphism of perfect closures in the opposite +direction carrying the constant fields onto each other. That field +isomorphism is unique modulo Frobenius twists in positive characteristic, +and unique in characteristic zero. +\end{enumerate} +In (2), a geometric rational quotient is restriction to +$\kappa(u)\subset F$ with $F/\kappa(u)$ regular, together with the +point inertia and decomposition data on its target. Compatibility +means that restriction commutes with the graph isomorphisms and the +induced isomorphisms of these targets. +\end{proposition} +\begin{proof}[Reference and hypotheses] +Part (1) is \cite[Proposition~2.4]{Pop2018}; the surrounding section +assumes transcendence degree greater than one. Part (2) is the Isom +statement of \cite[Theorem~2.1(2)]{PopBI2012}, restating +\cite[Main Theorem]{PopGraphs2012}. The stated theorem requires +Bertini-type families of rational quotients; all geometric rational +quotients form such a family. More explicitly, for algebraically +independent $x,t$ with $x$ separating, the general functions +\[ + ax+t,\qquad \frac{t}{a'x+a},\qquad + \frac{a''t+a'x+a+1}{t+a'x+a} +\] +in \cite[Fact/Definition~43]{PopGraphs2012} define relatively +algebraically closed rational subfields for the prescribed cofinite +choices of constants. Every one of these subfields is among the +quotients in (2). Thus its hypotheses imply precisely the required +Bertini compatibility. No restriction to finite-field closures or to +dimension greater than two occurs in this Isom statement. +\end{proof} + +We emphasize the distinction between the two citations. The later +\cite[Theorem~1.1]{Pop2018} reconstructs rational quotients from +divisorial inertia under a dimension-greater-than-two hypothesis. +Here those quotients have already been recovered by +Section~\ref{sec:curve-recovery}, so that theorem is unnecessary. + +\begin{proof}[Proof of Theorem~\ref{thm:main}] +A field isomorphism $\alpha:K^i\to L^i$ induces a contravariant +isomorphism of absolute Galois groups, well defined up to an inner +automorphism. Purely inseparable extension does not change these +groups. Abelianization removes the inner ambiguity, and the induced +class-two map preserves the commutator. Thus it gives a member of +$\Isom^c(\Pi_L^a,\Pi_K^a)$. In positive characteristic, absolute +Frobenius on algebraic closures commutes with field automorphisms; +composition with any integer Frobenius power therefore gives the same +allowed orbit. This proves well-definedness of $\Phi_{K,L}$. + +Conversely, let $\varphi:\Pi_L^a\to\Pi_K^a$ be bracket-compatible. +Lemma~\ref{prelim:central-lift} lifts it to a class-two isomorphism. +After choosing Tate identifications, let $\Theta$ be its dual +\eqref{eq:theta}. Proposition~\ref{local:transport} recovers the common +relative transcendence degree and the quasi-divisorial pairs. +The finite residual tests and the uniform support bound prove the +completed curve-subfield correspondence in +Theorem~\ref{recovery:curves}. The recognition statements in +Section~\ref{sec:curve-recovery} then show that $\varphi$ preserves true +divisorial inertia and every relatively algebraically closed rational +quotient, including its point lines. + +Proposition~\ref{recovery:rational} gives the actual commuting +quotient diagrams in \eqref{eq:recovery-quotient-diagram}. +Their vertical maps are surjective, and their target isomorphisms +$\varphi_E$ match all point inertia lines. For rational targets these +are the entire point decomposition data. Thus the same fixed +$\varphi$ is compatible with every geometric rational quotient; +no independent rescaling of the targets is needed. + +Part (1) of Proposition~\ref{reconstruction:pop} gives the total +decomposition-graph isomorphism. The quotient squares also commute on +these graphs. Indeed, restriction along an actual regular inclusion +$E=k(t)\subset K$ is a geometric rational quotient morphism +\cite[Section~2, Embeddings and Restrictions]{PopBI2012}. +At a flag vertex its map is obtained by restricting to the decomposition +subgroup and passing to the inertia quotient +\cite[Definition/Remark~24 and Definition~25]{PopGraphs2012}. +If the flag restricts trivially to $E$, its inertia is killed, and the +top square descends to the residual square. Otherwise its nonzero +inertia image lies in a unique point line of $E$, where decomposition +equals inertia and the residue group is trivial. The same description +applies to $P\subset L$. Local images can have finite index: saturation +identifies the point line, but the actual subgroup images and maps are +retained. Applying these restriction and quotient operations successively +along each flag therefore proves +compatibility for the same fixed $\varphi$ and $\varphi_E$. + +Part (2), applied to the recovered rational quotients, +gives an isomorphism $\alpha:K^i\to L^i$ with $\alpha(k)=l$ and +$[\varphi]=[\alpha^*]$. Choices of Tate identifications are absorbed +by the one allowed global unit. This proves surjectivity. It also +proves that different characteristics cannot give a bracket-compatible +isomorphism. + +Finally, two field isomorphisms with the same Galois-side orbit give +the same compatible graph and quotient data after one global unit +adjustment. The uniqueness assertion in +Proposition~\ref{reconstruction:pop} identifies them modulo Frobenius +in positive characteristic, and identifies them outright in +characteristic zero. On perfect closures the Frobenius powers are +exactly the integer powers appearing in the definition of +$\Isom_F^i(K,L)$. This proves injectivity and completes the theorem. +\end{proof} diff --git a/preprints/The-Bogomolov-Pop-reconstruction-theorem-September-23-2026/build/references.bib b/preprints/The-Bogomolov-Pop-reconstruction-theorem-September-23-2026/build/references.bib new file mode 100644 index 0000000000000000000000000000000000000000..d296fdc2c2a1ec6140bb110ff21c506e6873ff73 --- /dev/null +++ b/preprints/The-Bogomolov-Pop-reconstruction-theorem-September-23-2026/build/references.bib @@ -0,0 +1,227 @@ +@incollection{Bogomolov1991, + author = {Bogomolov, Fedor A.}, + title = {On two conjectures in birational algebraic geometry}, + booktitle = {Algebraic Geometry and Analytic Geometry}, + publisher = {Springer}, + address = {Tokyo}, + year = {1991}, + pages = {26--52}, + doi = {10.1007/978-4-431-68172-4_2} +} + +@article{BT2011, + author = {Bogomolov, Fedor and Tschinkel, Yuri}, + title = {Reconstruction of higher-dimensional function fields}, + journal = {Moscow Mathematical Journal}, + volume = {11}, + number = {2}, + year = {2011}, + pages = {185--204}, + doi = {10.17323/1609-4514-2011-11-2-185-204}, + eprint = {0912.4923}, + archivePrefix = {arXiv} +} + +@article{MS1982, + author = {Merkurjev, Alexander S. and Suslin, Andrei A.}, + title = {{$K$}-cohomology of {Severi--Brauer} varieties and the norm residue homomorphism}, + journal = {Mathematics of the USSR-Izvestiya}, + volume = {21}, + number = {2}, + year = {1983}, + pages = {307--340}, + doi = {10.1070/IM1983v021n02ABEH001793} +} + +@article{PopMin2015, + author = {Pop, Florian}, + title = {On the minimized decomposition theory of valuations}, + journal = {Bulletin Math\'ematique de la Soci\'et\'e des Sciences Math\'ematiques de Roumanie}, + volume = {58(106)}, + number = {3}, + year = {2015}, + pages = {331--357}, + note = {With the appendix ``On the nature of base fields'' by Adam Topaz}, + url = {https://ssmr.ro/bulletin/pdf/58-3/articol_10.pdf} +} + +@misc{Topaz2019, + author = {Topaz, Adam}, + title = {Recovering function fields from their integral {$\ell$}-adic cohomology with the {Galois} action}, + year = {2019}, + eprint = {1910.03563}, + archivePrefix = {arXiv}, + primaryClass = {math.KT}, + note = {Version 1}, + url = {https://arxiv.org/abs/1910.03563} +} + +@misc{Pop2018, + author = {Pop, Florian}, + title = {Reconstruction of function fields from their pro-{$\ell$} abelian divisorial inertia}, + year = {2018}, + eprint = {1810.04768}, + archivePrefix = {arXiv}, + note = {Version 1}, + url = {https://arxiv.org/abs/1810.04768} +} + +@misc{Stacks, + author = {{The Stacks Project Authors}}, + title = {The {Stacks} Project}, + year = {2026}, + url = {https://stacks.math.columbia.edu}, + note = {Specific tags cited in the text; accessed September 23, 2026} +} + +@article{PopBI2012, + author = {Pop, Florian}, + title = {On the birational anabelian program initiated by {Bogomolov} {I}}, + journal = {Inventiones Mathematicae}, + volume = {187}, + number = {3}, + pages = {511--533}, + year = {2012}, + doi = {10.1007/s00222-011-0331-x}, + note = {Theorem 2.1; author version dated March 19, 2011, p. 6} +} + +@incollection{PopGraphs2012, + author = {Pop, Florian}, + title = {Recovering function fields from their decomposition graphs}, + booktitle = {Number Theory, Analysis and Geometry: In Memory of Serge Lang}, + editor = {Goldfeld, Dorian and Jorgenson, Jay and Jones, Peter and Ramakrishnan, Dinakar and Ribet, Kenneth and Tate, John}, + publisher = {Springer}, + address = {Boston, MA}, + year = {2012}, + pages = {519--594}, + doi = {10.1007/978-1-4614-1260-1_24}, + note = {Main Theorem and Theorem 45; Fact/Definition 43} +} + +@article{EGAIV3, + author = {Grothendieck, Alexander}, + title = {{\'E}l\'ements de g\'eom\'etrie alg\'ebrique {IV}. {\'E}tude locale des sch\'emas et des morphismes de sch\'emas, troisi\`eme partie}, + journal = {Publications Math\'ematiques de l'IH\'ES}, + volume = {28}, + year = {1966}, + pages = {5--255}, + note = {R\'edig\'es avec la collaboration de Jean Dieudonn\'e}, + url = {https://numdam.org/item/PMIHES_1966__28__5_0/} +} + +@misc{Tanaka2023, + author = {Tanaka, Hiromu}, + title = {Boundedness of regular del {Pezzo} surfaces over imperfect fields}, + year = {2023}, + eprint = {1905.00143}, + archivePrefix = {arXiv}, + note = {Version 4, Proposition 5.3}, + url = {https://arxiv.org/abs/1905.00143v4} +} + +@incollection{DeligneSGA4half, + author = {Deligne, Pierre}, + title = {Th\'eor\`emes de finitude en cohomologie {$\ell$}-adique}, + booktitle = {Cohomologie \'etale ({SGA} 4\,1/2)}, + series = {Lecture Notes in Mathematics}, + volume = {569}, + publisher = {Springer-Verlag}, + year = {1977}, + pages = {233--251}, + url = {https://publications.ias.edu/sites/default/files/Number32.pdf} +} + +@article{BT2008, + author = {Bogomolov, Fedor and Tschinkel, Yuri}, + title = {Reconstruction of function fields}, + journal = {Geometric and Functional Analysis}, + volume = {18}, + number = {2}, + year = {2008}, + pages = {400--462}, + doi = {10.1007/s00039-008-0665-8} +} + +@article{TopazRat2016, + author = {Topaz, Adam}, + title = {Reconstructing function fields from rational quotients of mod-{$\ell$} {Galois} groups}, + journal = {Mathematische Annalen}, + volume = {366}, + number = {1-2}, + year = {2016}, + pages = {337--385}, + doi = {10.1007/s00208-015-1327-4}, + eprint = {1408.5194}, + archivePrefix = {arXiv} +} + +@misc{Silberstein2013, + author = {Silberstein, Aaron Michael}, + title = {Anabelian Intersection Theory {I}: The Conjecture of {Bogomolov--Pop} and Applications}, + year = {2013}, + eprint = {1211.4608}, + archivePrefix = {arXiv}, + note = {Version 2}, + url = {https://arxiv.org/abs/1211.4608v2} +} + +@article{TopazTorelli2025, + author = {Topaz, Adam}, + title = {A {Torelli}-like theorem for higher-dimensional function fields}, + journal = {Journal of the European Mathematical Society}, + volume = {27}, + year = {2025}, + pages = {395--432}, + doi = {10.4171/JEMS/1382}, + url = {https://arxiv.org/abs/1705.01084v3}, + note = {The Lemma A.7 citation refers to arXiv:1705.01084v3; \url{https://arxiv.org/abs/1705.01084v3}} +} + +@incollection{BT2002, + author = {Bogomolov, Fedor and Tschinkel, Yuri}, + title = {Commuting elements of {Galois} groups of function fields}, + booktitle = {Motives, Polylogarithms and Hodge Theory, Part I}, + series = {International Press Lecture Series}, + volume = {3}, + publisher = {International Press}, + year = {2002}, + pages = {75--120}, + eprint = {math/0012245}, + archivePrefix = {arXiv} +} + +@article{TopazCL2017, + author = {Topaz, Adam}, + title = {Commuting-liftable subgroups of {Galois} groups {II}}, + journal = {Journal f\"ur die reine und angewandte Mathematik}, + volume = {730}, + year = {2017}, + pages = {65--133}, + doi = {10.1515/crelle-2014-0115}, + eprint = {1208.0583}, + archivePrefix = {arXiv} +} + +@article{ADF2020, + author = {Anscombe, Sylvy and Dittmann, Philip and Fehm, Arno}, + title = {Approximation theorems for spaces of localities}, + journal = {Mathematische Zeitschrift}, + volume = {296}, + year = {2020}, + pages = {1471--1499}, + doi = {10.1007/s00209-020-02516-6} +} + +@incollection{GrothHilbert1961, + author = {Grothendieck, Alexander}, + title = {Techniques de construction et th\'eor\`emes d'existence en g\'eom\'etrie alg\'ebrique {IV}: les sch\'emas de {Hilbert}}, + booktitle = {S\'eminaire Bourbaki: ann\'ees 1960/61, expos\'es 205--222}, + series = {S\'eminaire Bourbaki}, + publisher = {Soci\'et\'e math\'ematique de France}, + year = {1961}, + number = {6}, + note = {Expos\'e 221}, + pages = {249--276}, + url = {https://www.numdam.org/item/SB_1960-1961__6__249_0/} +} diff --git a/preprints/The-Bogomolov-Pop-reconstruction-theorem-September-23-2026/build/residual-tests.tex b/preprints/The-Bogomolov-Pop-reconstruction-theorem-September-23-2026/build/residual-tests.tex new file mode 100644 index 0000000000000000000000000000000000000000..f7ccec35175e605c0bf99f2a53151bf8ee967f48 --- /dev/null +++ b/preprints/The-Bogomolov-Pop-reconstruction-theorem-September-23-2026/build/residual-tests.tex @@ -0,0 +1,238 @@ +\section{Finite tests on residual curves}\label{sec:residual-tests}\label{sec:tests} + +We next construct residual curves that retain any prescribed finite collection +of Kummer relations. The curve can depend on the functions and on the +exponent. Its genus, however, is bounded in terms of one fixed projective +model. This distinction will make the pencil estimate uniform. + +Fix a normal integral projective model $X\subset\mathbb P^N_l$ of $L/l$, +and put $d=\operatorname{trdeg}(L/l)\ge2$. +For a prime divisor $D$ on $X$, let $\deg D$ denote its degree in this +embedding. For an integer $m\ge2$ and $g\in L^\times$, set +\[ + s_m(g)=\sum_{\substack{D\subset X\text{ prime divisor}\\ + \operatorname{ord}_D(g)\not\equiv0\pmod m}} + \deg D. +\] +This is a finite sum. + +\begin{lemma}[Finite residual tests]\label{tests:residual} +There is an integer $G_X\ge0$, depending only on the embedded variety $X$, +with the following property. Given $g_1,\ldots,g_r\in L^\times$ and +$q=\ell^e$, $e\ge1$, there is a quasi-prime $(d-1)$-divisor $w$ of $L/l$ +such that: +\begin{enumerate} +\item $lw$ is an algebraic closure of a finite field of characteristic + different from $\ell$, and $Lw=lw(C)$ for a smooth projective integral + curve $C/lw$ with $g(C)\le G_X$; +\item $w(g_j)=0$ for every $j$, and, for every + $(n_1,\ldots,n_r)\in(\mathbb Z/q)^r$, +\[ + \prod_jg_j^{n_j}\in L^{\times q} + \quad\Longleftrightarrow\quad + \prod_j\overline g_j^{n_j}\in(Lw)^{\times q}; +\] +\item for $m=\ell$ and $m=q$, +\begin{equation}\label{tests:support} + \#\{P\in C(lw):\operatorname{ord}_P(\overline g_j) + \not\equiv0\pmod m\}=s_m(g_j) + \qquad(1\le j\le r). +\end{equation} +\end{enumerate} +Here $\overline g_j$ is the residue of $g_j$ at $w$. +\end{lemma} + +We use the following form of Bertini irreducibility. Its hypotheses include +the finite covering maps that occur below. + +\begin{lemma}[Hyperplane sections of a quasi-finite map]\label{tests:bertini} +Let $V$ be an integral variety over an algebraically closed field, and let +$f:V\to\mathbb P^N$ be quasi-finite, with $\dim V=a\ge2$. +For a general hyperplane $H$, the inverse image $f^{-1}(H)$ is +geometrically irreducible. If $f$ factors as a finite \'etale map to a +smooth locally closed subvariety $U\subset\mathbb P^N$, this inverse image +is also smooth for a general $H$. +\end{lemma} + +\begin{proof} +We recall the incidence argument for irreducibility; see also +\cite[Lemma~37.32.3, Tag~0G4F]{Stacks}. +Write $B=(\mathbb P^N)^*$ and +\[ + I=\{(x,H)\in V\times B:f(x)\in H\}. +\] +The projection to $V$ is a projective bundle, so $I$ is integral. +In $I\times_BI$, the open set lying over pairs $(x,y)$ with +$f(x)\ne f(y)$ is a projective bundle with fiber $\mathbb P^{N-2}$ over +an irreducible open subset of $V\times V$. It is therefore irreducible +of dimension $2a+N-2$. +The locus of pairs with $f(x)=f(y)$ has dimension at most $a$, since $f$ +is quasi-finite; the incidence above it has dimension at most $a+N-1$. +This is strictly smaller than $2a+N-2$. + +Every component of the generic hyperplane section has dimension $a-1$: +a hyperplane cuts a nonzero nonunit in an integral variety of finite type +over a field. Consequently every component of its square has dimension +$2a-2$. The equal-image locus cannot supply a component of this generic +square. The generic square is thus irreducible. Equivalently, the +function field of $B$ is separably algebraically closed in that of $I$, +so the generic hyperplane section is geometrically irreducible. +Geometric irreducibility spreads to a nonempty open of $B$ +\cite[Lemma~37.27.5, Tag~0559]{Stacks}. +Finally, a general hyperplane section of the smooth embedded variety $U$ +is smooth by Bertini, and its \'etale inverse image is smooth. +\end{proof} + +\begin{proof}[Proof of Lemma~\ref{tests:residual}] +\emph{Spread the finite data.} +Let $D_1,\ldots,D_s$ be the prime divisors occurring in the divisors of +the $g_j$. There is a closed subset $B_X\subset X$ of codimension at +least two such that $X\setminus B_X$ is smooth, the +$D_i\setminus B_X$ are smooth pairwise disjoint Cartier divisors, and, +near each of them, every $g_j$ is a power of a local equation times a +unit. Off their union the $g_j$ are units. +These assertions follow by deleting the singular loci, the pairwise +divisor intersections and the proper closed exceptional subsets on +which the required local identities fail. + +Choose a smooth dense open $U\subset X$ on which all $g_j$ are units. +The simultaneous root cover +\[ + V=\operatorname{Spec}_U + \mathcal O_U[T_1,\ldots,T_r]/(T_1^q-g_1,\ldots,T_r^q-g_r) + \longrightarrow U +\] +is finite \'etale of degree $q^r$. +Let $H$ be the subgroup generated by the $[g_j]$ in +$L^\times/L^{\times q}$. Kummer theory shows that each connected +component of $V$ has degree $|H|$ over $U$. +Since $U$ is smooth, these components are integral. + +All these data descend to a finitely generated domain $R\subset l$ in +which $\ell$ is invertible. Enlarge $R$ and then localize it so that the +following properties hold in every geometric fiber: +\begin{itemize} +\item the embedded model $X_R$ is projective and flat, with normal + integral fibers and the Hilbert polynomial of $X$; +\item the $D_{i,R}$ are distinct geometrically integral divisors with + their original degrees, and the local divisor identities and + smoothness conditions above hold outside a subset of fiberwise + codimension at least two; +\item the root cover and its disjoint decomposition into components + extend, and every component has geometrically integral fibers + and finite locally free degree $|H|$ over $U_R$. +\end{itemize} +These are finite spreading conditions. Flatness makes the projective +Hilbert polynomials constant. Geometric integrality and normality +spread from the generic fiber after shrinking; the latter uses the +proper flat finite-presentation form of +\cite[Theorem~12.2.4(iv),(viii)]{EGAIV3}. +The equations $g_j=u\pi^n$, the inverses specifying units, the component +idempotents, and the finite locally free ranks are all finite algebraic +data. Spreading them preserves the exact orders and degrees, rather +than just the supports. The dimension bound on the omitted closed set +also persists after shrinking. + +For later use include a separating transcendence basis +$x_1,\ldots,x_d$ of $L/l$, a primitive element $y$ over $l(x_1,\ldots,x_d)$, +its monic minimal polynomial, and common birational opens with $X$. +After another localization, the same presentation and birational +identification hold in the geometric fibers, with the reduced polynomial +irreducible of its original degree over the rational function field. +Preserve as well the nonzero reductions of the finitely many +numerators and denominators used for the $g_j$. +This follows either by spreading those common opens and geometric +integrality, or by spreading the absolutely irreducible defining +polynomial after clearing denominators. + +\emph{Realize a good fiber as a residue field.} +Choose a closed point $s\in\operatorname{Spec}R$. +Its residue field is finite and has characteristic different from $\ell$. +A valuation ring of $l$ dominating $R_s$ exists +\cite[Lemma~10.50.2, Tag~00IA]{Stacks}. +Its residue field is algebraically closed. If that residue field is +not algebraic over a finite field, give a transcendence basis independent +values in an ordered free abelian group, prolong the resulting valuation +to the residue field, and compose. This yields a valuation $s_0$ of $l$ +with the same center $s$ and residue field $\lambda$ algebraic over a +finite field. Residues of algebraically closed fields are algebraically +closed, so $\lambda$ is an algebraic closure of a finite field. +The composition does not change the center because every nonzero +element of a finite field has value zero. +This construction uses a transcendence basis of arbitrary cardinality; +there is no restriction on the size of $l$. + +Extend $s_0$ to $l(x_1,\ldots,x_d)$ by the Gauss valuation, whose residue +field is $\lambda(x_1,\ldots,x_d)$ and whose value group is $s_0l$. +Prolong it to $L$. The element $y$ is integral and its residue satisfies +the reduced irreducible polynomial, of degree +$[L:l(x_1,\ldots,x_d)]$. The fundamental valuation inequality therefore +forces the residue degree to be that degree and the value-group index +to be one \cite[Section~15.125, Tag~0ASF]{Stacks}. +We obtain a valuation $s_X$ with +\[ + s_XL=s_0l,\qquad Ls_X=\lambda(X_s), +\] +and with the $g_j$ reducing to their specified rational functions on +$X_s$. This also explains why no extra ramification or defect is hidden +in the passage to the good fiber. + +\emph{Cut down to a curve.} +Over $\lambda$, choose a general flag of $d-1$ hyperplanes in the fixed +projective embedding. Make the final section $C$ avoid the omitted +codimension-two set and meet the $D_{i,s}$ transversely. +The intersections with different $D_{i,s}$ are disjoint. +The final section is smooth and integral. Apply +Lemma~\ref{tests:bertini} successively to the finite list of covering +components as well as to the reduced underlying base sections; each restricted covering +component remains integral over $U_s\cap C$. +At every step the varieties to which irreducibility is applied have +dimension at least two, including the last surface-to-curve step. +Smoothness of the restricted cover follows from its \'etaleness over the +smooth curve open. + +For completeness, intermediate scheme sections need only have +irreducible support and be smooth at the generic points used by the +flag. All those points lie in the original smooth locus. +Choose the hyperplanes also to avoid the finitely many associated +points at each stage, so their equations are non-zero-divisors. +The hyperplane exact sequences give +\[ + P_C(n)=\Delta^{d-1}P_X(n),\qquad + \Delta P(n)=P(n)-P(n-1). +\] +As $C$ is smooth integral, its genus is consequently fixed by the +embedded model $X$; denote it by $G_X$. +All conditions imposed on the flag are finitely many nonempty open +conditions. They can be met over the infinite algebraically closed +field $\lambda$, even when it is countable. + +Compose $s_X$ with the successive divisorial valuations of this flag. +For the first composition, the relative value group is the innermost +convex copy of $\mathbb Z$, and its residue transcendence degree drops +by one. Every proper coarsening kills this copy of $\mathbb Z$ and +therefore has all its values supplied by constants. +The first composition is thus a quasi-prime divisor. The remaining +flag valuations are prime divisors over the residue constants, giving +a quasi-prime $(d-1)$-divisor $w$. +All $g_j$ are units at every generic point in the flag, so $w(g_j)=0$, +and $Lw=\lambda(C)$. + +\emph{Check relations and orders.} +An old relation $\prod_jg_j^{n_j}=b^q$ specializes: since $w(g_j)=0$, +torsion-freeness of the value group gives $w(b)=0$. +Thus the old relation kernel in $(\mathbb Z/q)^r$ is contained in the +new one. +On the other hand, every component of the root cover restricted to +$U_s\cap C$ is integral and has degree $|H|$. +Kummer theory gives the same order $|H|$ for the subgroup generated by +the residue classes. The two finite relation kernels therefore have +the same order, so they are equal. + +Finally, $C$ meets $D_{i,s}$ in exactly $\deg D_i$ distinct transverse +points. At each such point the order of $\overline g_j$ equals the +unchanged integer $\operatorname{ord}_{D_i}(g_j)$. +There are no other zeros or poles. This proves +\eqref{tests:support}, in fact for every divisor $m$ of $q$, and +completes the construction. +\end{proof} diff --git a/preprints/The-Bogomolov-Pop-reconstruction-theorem-September-23-2026/paper.pdf b/preprints/The-Bogomolov-Pop-reconstruction-theorem-September-23-2026/paper.pdf new file mode 100644 index 0000000000000000000000000000000000000000..a4bb033d4ecfeb3a07b0c77c1dbf6498a41ed9ac --- /dev/null +++ b/preprints/The-Bogomolov-Pop-reconstruction-theorem-September-23-2026/paper.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:4d805627401fffef78e1f0d8e21a0585f6c733e8be90643d0065cd27ffaa24f2 +size 449982 diff --git a/preprints/The-Boltzmann-Grad-limit-for-stable-radial-potentials-on-regular-kinetic-intervals-September-23-2026/README.md b/preprints/The-Boltzmann-Grad-limit-for-stable-radial-potentials-on-regular-kinetic-intervals-September-23-2026/README.md new file mode 100644 index 0000000000000000000000000000000000000000..4149b4337c8a61a6481afe34cab52a2513217a9c --- /dev/null +++ b/preprints/The-Boltzmann-Grad-limit-for-stable-radial-potentials-on-regular-kinetic-intervals-September-23-2026/README.md @@ -0,0 +1,18 @@ +# [The Boltzmann–Grad limit for stable radial potentials on regular kinetic intervals](paper.pdf) + +**Author:** OpenAI + +**Date:** September 23, 2026 + +## Citation + +```bibtex +@misc{OAI:The-Boltzmann-Grad-limit-for-stable-radial-potentials-on-regular-kinetic-intervals-September-23-2026, + author = {{OpenAI}}, + title = {{The Boltzmann--Grad limit for stable radial potentials on regular kinetic intervals}}, + howpublished = {OpenAI Math Release preprint + \href{https://github.com/openai/math/blob/main/preprints/The-Boltzmann-Grad-limit-for-stable-radial-potentials-on-regular-kinetic-intervals-September-23-2026/paper.pdf}{OAI:The-Boltzmann-Grad-limit-for-stable-radial-potentials-on-regular-kinetic-intervals-September-23-2026}}, + year = {2026} +} +``` + diff --git a/preprints/The-Boltzmann-Grad-limit-for-stable-radial-potentials-on-regular-kinetic-intervals-September-23-2026/build/figures/exposure.tex b/preprints/The-Boltzmann-Grad-limit-for-stable-radial-potentials-on-regular-kinetic-intervals-September-23-2026/build/figures/exposure.tex new file mode 100644 index 0000000000000000000000000000000000000000..50bbc33f6c1ac05782e20dba0db4fb7d7afadc68 --- /dev/null +++ b/preprints/The-Boltzmann-Grad-limit-for-stable-radial-potentials-on-regular-kinetic-intervals-September-23-2026/build/figures/exposure.tex @@ -0,0 +1,60 @@ +\begin{figure}[tbp] +\centering +\begin{tikzpicture}[x=0.94cm,y=0.94cm, + visible/.style={draw=black!75,line width=1.05pt}, + hidden/.style={draw=blue!65!black,line width=1.2pt}, + every node/.style={font=\small}, + exposure point/.style={circle,fill=red!65!black,inner sep=2pt}] +\draw[-{Latex[length=2mm]},black!55] (-2.6,3.1)--(-2.6,-.05); +\node[rotate=90,anchor=south,font=\footnotesize] at (-2.72,1.5) + {backward time}; +\begin{scope} + \node[font=\small\bfseries] at (.15,3.65) {Genuine scattering}; + \fill[blue!4] (-1.65,-.15) rectangle (1.6,1.82); + \draw[blue!35,dashed,rounded corners=3pt] + (-1.65,-.15) rectangle (1.6,1.82); + \draw[visible] (0,3.15)--(0,2.1); + \draw[hidden] (0,2.1)--(-1.18,0); + \draw[hidden] (0,2.1)--(1.18,0); + \fill[black] (0,2.1) circle (1.7pt); + \node[anchor=east,font=\footnotesize] at (-.13,2.32) {launch}; + \draw[visible] (2.2,3.15) .. controls (2.2,1.85) and (1.75,1.05) + .. (.787,.7); + \node[exposure point] at (.787,.7) {}; + \node[anchor=west,font=\footnotesize] at (1.5,.38) {first exposure}; + \node[align=center,font=\footnotesize] at (-1.18,-.48) {new label}; + \node[align=center,font=\footnotesize] at (1.18,-.48) {continued parent}; + \node[blue!65!black,font=\footnotesize] at (.05,-.98) + {both descendant branches hidden}; +\end{scope} +\begin{scope}[xshift=7cm] + \node[font=\small\bfseries] at (.25,3.65) {Passage or virtual crossing}; + \fill[blue!4] (.18,-.15) rectangle (1.65,1.82); + \draw[blue!35,dashed,rounded corners=3pt] + (.18,-.15) rectangle (1.65,1.82); + \draw[visible] (0,3.15)--(0,0); + \draw[hidden] (0,2.1)--(1.18,0); + \fill[black] (0,2.1) circle (1.7pt); + \node[anchor=east,font=\footnotesize] at (-.13,2.32) {launch}; + \draw[visible] (2.2,3.15) .. controls (2.2,1.85) and (1.75,1.05) + .. (.787,.7); + \node[exposure point] at (.787,.7) {}; + \node[anchor=east,font=\footnotesize] at (-.06,-.48) {visible parent}; + \node[anchor=west,font=\footnotesize] at (.9,-.48) {new label}; + \node[blue!65!black,font=\footnotesize] at (.3,-.98) + {only the new branch hidden}; +\end{scope} +\end{tikzpicture} +\caption{The auxiliary exposure used to multiply contact savings. +Blue branches carry an unrevealed launch parameter. A genuine +scattering changes the continued parent's velocity, so both descendant +branches must be hidden. Passage leaves the parent unchanged. +Until first exposure, the black trajectories can be computed without +the hidden parameter. The construction is applied to exact signed +histories; no particles are deleted from the microscopic dynamics. +The drawing shows one selected tree. The full construction also omits +added groups whose positions have not yet been fixed relative to the +visible particles, including groups not selected for exposure; +Proposition~\ref{exp:exposure} gives the complete construction.} +\label{fig:exposure} +\end{figure} diff --git a/preprints/The-Boltzmann-Grad-limit-for-stable-radial-potentials-on-regular-kinetic-intervals-September-23-2026/build/paper.tex b/preprints/The-Boltzmann-Grad-limit-for-stable-radial-potentials-on-regular-kinetic-intervals-September-23-2026/build/paper.tex new file mode 100644 index 0000000000000000000000000000000000000000..ecf6bc139e761574bad82978ea4418311b5a7a0b --- /dev/null +++ b/preprints/The-Boltzmann-Grad-limit-for-stable-radial-potentials-on-regular-kinetic-intervals-September-23-2026/build/paper.tex @@ -0,0 +1,153 @@ +\pdftrailerid{} +\pdfinfoomitdate=1 +\pdfsuppressptexinfo=15 +\pdfgentounicode=1 +\pdfglyphtounicode{hatwider}{02C6 FE02} +\pdfglyphtounicode{mapsto}{21A6} 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\let\cref@old@refstepcounter\Hy@theorem@refstepcounter + \refstepcounter[#1]{#2}% + \let\cref@old@refstepcounter\article@saved@refstepcounter +} +\AtBeginDocument{% + \@ifundefined{Hy@theorem@refstepcounter}% + {\PackageError{article}{Deferred theorem target helper unavailable}{Check the installed hyperref version.}}{}% + \@ifundefined{cref@old@refstepcounter}% + {\PackageError{article}{Theorem counter interface unavailable}{Check the installed cleveref version.}}{}% + \patchcmd{\cref@thmnoarg}{\refstepcounter}{\Hy@theorem@refstepcounter}{}% + {\PackageError{article}{Cannot patch numbered theorem targets}{Check the installed amsthm, cleveref and hyperref versions.}}% + \patchcmd{\cref@thmoptarg}{\refstepcounter}{\article@thmrefstepcounter}{}% + {\PackageError{article}{Cannot patch shared-counter theorem targets}{Check the installed amsthm, cleveref and hyperref versions.}}% +} +\makeatother +\setlist{leftmargin=*,itemsep=3pt,topsep=5pt} +\setlength{\emergencystretch}{2em} +\allowdisplaybreaks[2] +\numberwithin{equation}{section} +\newtheorem{maintheorem}{Theorem} +\renewcommand{\themaintheorem}{\Alph{maintheorem}} +\newtheorem{maincorollary}[maintheorem]{Corollary} +\newtheorem{theorem}{Theorem}[section] +\newtheorem{lemma}[theorem]{Lemma} +\newtheorem{proposition}[theorem]{Proposition} +\newtheorem{corollary}[theorem]{Corollary} +\theoremstyle{definition} +\newtheorem{definition}[theorem]{Definition} +\newtheorem{assumption}[theorem]{Assumption} +\newtheorem{remark}[theorem]{Remark} +\newtheorem{convention}[theorem]{Convention} +\newtheorem{example}[theorem]{Example} +\theoremstyle{plain} +\newcommand{\R}{\mathbb R} +\newcommand{\N}{\mathbb N} +\newcommand{\Z}{\mathbb Z} +\newcommand{\PP}{\mathbb P} +\newcommand{\EE}{\mathbb E} +\newcommand{\1}{\mathbf 1} +\newcommand{\eps}{\varepsilon} +\newcommand{\dd}{\,d} +\DeclareMathOperator{\supp}{supp} +\DeclareMathOperator{\dist}{dist} +\DeclareMathOperator{\diam}{diam} +\DeclareMathOperator{\Lip}{Lip} +\DeclareMathOperator{\Var}{Var} +\DeclareMathOperator{\sgn}{sgn} +\title{The Boltzmann--Grad limit for stable radial potentials\\ +on regular kinetic intervals} +\author{OpenAI} +\date{September 23, 2026} +\begin{document} +\maketitle +\begin{abstract} +We derive the nonlinear Boltzmann equation from a grand-canonical +Newtonian gas with initial pair exclusion. We treat stable, finite-range +radial potentials that are $C^2$ except for an allowed repulsive singularity +at the origin, and $C^1$ initial probability +densities with spatially summable Gaussian bounds on the density and +its spatial gradient. +All fixed-order rescaled factorial marginals converge in $L^1$, +uniformly throughout every finite interval on which the classical +kinetic solution has a uniform Gaussian bound. +The result allows attractive wells and dynamically formed clusters, +without restrictions on the differential scattering cross-section. +\end{abstract} +{\small\tableofcontents} +\clearpage +\input{sections/00-introduction} +\clearpage +\input{sections/01-preliminaries} +\input{sections/02-cell-expansion} +\input{sections/03-centering} +\input{sections/04-history-estimates} +\input{sections/05-exposure} +\input{sections/06-closure} +\bibliographystyle{plain} +\bibliography{references} +\end{document} diff --git a/preprints/The-Boltzmann-Grad-limit-for-stable-radial-potentials-on-regular-kinetic-intervals-September-23-2026/build/references.bib b/preprints/The-Boltzmann-Grad-limit-for-stable-radial-potentials-on-regular-kinetic-intervals-September-23-2026/build/references.bib new file mode 100644 index 0000000000000000000000000000000000000000..bc6f08441dffc9cd4b4acaff10d75e12371bb31b --- /dev/null +++ b/preprints/The-Boltzmann-Grad-limit-for-stable-radial-potentials-on-regular-kinetic-intervals-September-23-2026/build/references.bib @@ -0,0 +1,174 @@ + +@article{PSS14, + author = {Pulvirenti, Mario and Saffirio, Chiara and Simonella, Sergio}, + title = {On the validity of the {Boltzmann} equation for short range potentials}, + journal = {Reviews in Mathematical Physics}, + volume = {26}, + number = {2}, + pages = {1450001}, + year = {2014}, + doi = {10.1142/S0129055X14500019}, + eprint = {1301.2514v2}, + archivePrefix = {arXiv}, + primaryClass = {math-ph}, + url = {https://arxiv.org/abs/1301.2514v2}, + note = {Journal article: 64 pp.; theorem locators refer to + \href{https://arxiv.org/abs/1301.2514v2}{arXiv:1301.2514v2}, + revised 4 January 2014} +} + +@misc{DHM25, + author = {Deng, Yu and Hani, Zaher and Ma, Xiao}, + title = {Long time derivation of the {Boltzmann} equation from hard sphere dynamics}, + year = {2025}, + eprint = {2408.07818v3}, + archivePrefix = {arXiv}, + primaryClass = {math.AP}, + doi = {10.48550/arXiv.2408.07818}, + url = {https://arxiv.org/abs/2408.07818v3}, + note = {Annals of Mathematics, to appear. Theorem locators refer to + \href{https://arxiv.org/abs/2408.07818v3}{arXiv:2408.07818v3}, + 18 July 2025; first posted in 2024} +} + +@misc{BGSS26, + author = {Bodineau, Thierry and Gallagher, Isabelle and Saint-Raymond, Laure and Simonella, Sergio}, + title = {Derivation of the {Boltzmann} equation from hard-sphere dynamics (after {Y. Deng, Z. Hani, and X. Ma})}, + year = {2026}, + eprint = {2602.04407v1}, + archivePrefix = {arXiv}, + primaryClass = {math.AP}, + doi = {10.48550/arXiv.2602.04407}, + url = {https://arxiv.org/abs/2602.04407v1}, + note = {Bourbaki Seminar, January 2026; + \href{https://arxiv.org/abs/2602.04407v1}{arXiv:2602.04407v1}, + preprint version 1, posted 4 February 2026} +} + +@incollection{Lanford75, + author = {Lanford, III, Oscar E.}, + title = {Time evolution of large classical systems}, + editor = {Moser, J.}, + booktitle = {Dynamical Systems, Theory and Applications}, + series = {Lecture Notes in Physics}, + volume = {38}, + pages = {1--111}, + year = {1975}, + publisher = {Springer}, + address = {Berlin, Heidelberg}, + doi = {10.1007/3-540-07171-7_1}, + url = {https://doi.org/10.1007/3-540-07171-7_1}, + note = {\href{https://doi.org/10.1007/3-540-07171-7_1}{doi:10.1007/3-540-07171-7\_1}} +} + +@incollection{Grad58, + author = {Grad, Harold}, + title = {Principles of the Kinetic Theory of Gases}, + editor = {Fl{\"u}gge, S.}, + booktitle = {Thermodynamik der Gase / Thermodynamics of Gases}, + series = {Handbuch der Physik}, + volume = {XII}, + pages = {205--294}, + publisher = {Springer}, + address = {Berlin, G{\"o}ttingen, Heidelberg}, + year = {1958}, + doi = {10.1007/978-3-642-45892-7_3} +} + +@article{Grad49, + author = {Grad, Harold}, + title = {On the kinetic theory of rarefied gases}, + journal = {Communications on Pure and Applied Mathematics}, + volume = {2}, + number = {4}, + pages = {331--407}, + year = {1949}, + doi = {10.1002/cpa.3160020403} +} + +@phdthesis{King75, + author = {King, Francis Gordon}, + title = {{BBGKY} Hierarchy for Positive Potentials}, + school = {University of California, Berkeley}, + year = {1975}, + url = {https://math.berkeley.edu/publications/bbgky-hierarchy-positive-potentials} +} + +@book{GST14, + author = {Gallagher, Isabelle and Saint-Raymond, Laure and Texier, Benjamin}, + title = {From {Newton} to {Boltzmann}: Hard Spheres and Short-range Potentials}, + series = {Z{\"u}rich Lectures in Advanced Mathematics}, + publisher = {European Mathematical Society}, + year = {2014}, + doi = {10.4171/129}, + isbn = {978-3-03719-129-3} +} + +@article{IllnerPulvirenti86, + author = {Illner, Reinhard and Pulvirenti, Mario}, + title = {Global validity of the {Boltzmann} equation for a two-dimensional rare gas in vacuum}, + journal = {Communications in Mathematical Physics}, + volume = {105}, + pages = {189--203}, + year = {1986}, + doi = {10.1007/BF01211098} +} + +@article{IllnerPulvirenti89, + author = {Illner, Reinhard and Pulvirenti, Mario}, + title = {Global validity of the {Boltzmann} equation for two- and three-dimensional rare gas in vacuum: Erratum and improved result}, + journal = {Communications in Mathematical Physics}, + volume = {121}, + pages = {143--146}, + year = {1989}, + doi = {10.1007/BF01218628} +} + +@article{Pulvirenti87, + author = {Pulvirenti, Mario}, + title = {Global validity of the {Boltzmann} equation for a three-dimensional rare gas in vacuum}, + journal = {Communications in Mathematical Physics}, + volume = {113}, + pages = {79--85}, + year = {1987}, + doi = {10.1007/BF01221398} +} + +@book{Ruelle69, + author = {Ruelle, David}, + title = {Statistical Mechanics: Rigorous Results}, + publisher = {W. A. Benjamin}, + address = {New York}, + year = {1969} +} + +@article{PulvirentiSimonella2017, + author = {Pulvirenti, Mario and Simonella, Sergio}, + title = {The {Boltzmann--Grad} limit of a hard sphere system: analysis of the correlation error}, + journal = {Inventiones Mathematicae}, + volume = {207}, number = {3}, pages = {1135--1237}, year = {2017}, + doi = {10.1007/s00222-016-0682-4}, + eprint = {1405.4676v2}, archivePrefix = {arXiv}, + url = {https://doi.org/10.1007/s00222-016-0682-4} +} + +@article{BGSSCluster2022, + author = {Bodineau, Thierry and Gallagher, Isabelle and Saint-Raymond, Laure and Simonella, Sergio}, + title = {Cluster expansion for a dilute hard sphere gas dynamics}, + journal = {Journal of Mathematical Physics}, + volume = {63}, number = {7}, pages = {073301}, year = {2022}, + doi = {10.1063/5.0091199}, + eprint = {2205.04110}, archivePrefix = {arXiv}, + url = {https://doi.org/10.1063/5.0091199} +} + +@misc{DengHani2024, + author = {Deng, Yu and Hani, Zaher}, + title = {Long time justification of wave turbulence theory}, + year = {2024}, + eprint = {2311.10082v2}, + archivePrefix = {arXiv}, + primaryClass = {math.AP}, + url = {https://arxiv.org/abs/2311.10082v2}, + note = {Version 2, 7 April 2024; first posted in 2023} +} diff --git a/preprints/The-Boltzmann-Grad-limit-for-stable-radial-potentials-on-regular-kinetic-intervals-September-23-2026/build/sections/00-introduction.tex b/preprints/The-Boltzmann-Grad-limit-for-stable-radial-potentials-on-regular-kinetic-intervals-September-23-2026/build/sections/00-introduction.tex new file mode 100644 index 0000000000000000000000000000000000000000..9882d5e0c6ad6b04ad949b9a2bd95fb724a6ade3 --- /dev/null +++ b/preprints/The-Boltzmann-Grad-limit-for-stable-radial-potentials-on-regular-kinetic-intervals-September-23-2026/build/sections/00-introduction.tex @@ -0,0 +1,214 @@ +\section{Introduction} +\label{sec:introduction} + +The Boltzmann--Grad limit describes a gas whose interaction range tends +to zero while the collision frequency remains of order one. In three +dimensions the activity is $\mu_\varepsilon=\varepsilon^{-2}$. +The microscopic dynamics is Newtonian, whereas the limiting collision +operator is determined by the actual two-body scattering of the pair +potential. We prove this limit for stable finite-range radial potentials +on every finite interval where the Boltzmann solution has the regularity +and Gaussian bound specified below. + +Write $z=(x,v)\in\R^3\times\R^3$. +For a phase-space function $h$ and $\gamma>0$, write +\begin{equation}\label{intro:norm} + \|h\|_{\mathrm{Bol},\gamma} + :=\sum_{k\in\Z^3}\; + \sup_{\substack{|x-k|\le1\\v\in\R^3}} + e^{\gamma|v|^2}|h(x,v)|. +\end{equation} +For a vector-valued function we use its Euclidean length inside this +definition. The precise potential class, grand-canonical law, Newton +equations, and scattering operator are given in +Assumption~\ref{pre:potential} and Equations~\eqref{pre:model}, +\eqref{pre:dynamics}, and \eqref{pre:collision}. +In brief, the potential is radial, +vanishes beyond unit distance, and is $C^2$, with either a finite value +or a repulsive singularity at the origin. Its only sign condition is +thermodynamic stability, in the classical sense of~\cite{Ruelle69}. +The initial law is the product activity $f_0$ +conditioned by pairwise exclusion at distance $\varepsilon$, with its +exact grand-canonical normalization. +For each fixed integer $s\ge1$, the scaled factorial density +$F_s^\varepsilon(t)$ is characterized by +\begin{equation}\label{intro:factorial-test} + \int a(Z_s)F_s^\varepsilon(t,Z_s)\,dZ_s + =\mu_\varepsilon^{-s}\EE + \sum_{\substack{1\le i_1,\ldots,i_s\le N\\ + i_1,\ldots,i_s\ \mathrm{distinct}}} + a(z_{i_1}(t),\ldots,z_{i_s}(t)) +\end{equation} +for bounded measurable tests $a$ on $(\R^3\times\R^3)^s$. +The sum runs over ordered distinct labels and is zero when $N0$. +Let $f_0\ge0$ be a $C^1$ probability density on $\R^3\times\R^3$ such that +\begin{equation}\label{intro:data-bound} + \|f_0\|_{\mathrm{Bol},2\beta} + +\|\nabla_xf_0\|_{\mathrm{Bol},2\beta}<\infty. +\end{equation} +Fix $00$. Then +\begin{equation}\label{intro:empirical} + \lim_{\varepsilon\downarrow0}\; + \sup_{0\le t\le T} + \PP\!\left( + \left|\mu_\varepsilon^{-1}\sum_{i=1}^{N}a(z_i(t)) + -\int a(z)f(t,z)\,dz\right|>\eta\right)=0. +\end{equation} +\end{maincorollary} + +The supremum in \eqref{intro:empirical} is over deterministic +observation times. The conclusion follows from the first two factorial +marginals and conservation of particle number; see +Section~\ref{sec:closure}. + +\subsection{History and scope} + +The Boltzmann--Grad scaling goes back to Grad~\cite{Grad49,Grad58}. +Lanford established the hard-sphere limit on a short time +interval~\cite{Lanford75}, and King's thesis treated positive +finite-range potentials~\cite{King75}. +Gallagher, Saint-Raymond, and Texier gave a detailed geometric analysis +of recollisions for hard spheres and a class of repulsive potentials, +including a convergence rate in the hard-sphere case~\cite{GST14}. +Pulvirenti, Saffirio, and Simonella proved a short-time theorem for the +stable radial class used here; see their Section~8, Hypothesis~1$'$ and +Theorem~1$'$~\cite{PSS14}. Their treatment permits attractive parts and +does not require a globally single-valued differential cross-section. + +The time restriction in a collision-tree expansion reflects the growth +of correlated histories. Global-time results were obtained for +sufficiently rare, spatially dispersive gases in vacuum by Illner and +Pulvirenti~\cite{IllnerPulvirenti86,Pulvirenti87,IllnerPulvirenti89}; +the last reference corrects and improves the earlier results. +For general data, extending a short-time limit requires control of the +correlations already present at the next starting time. +Pulvirenti and Simonella quantified the correlation errors remaining +after independent factors are removed~\cite{PulvirentiSimonella2017}. +Bodineau, Gallagher, Saint-Raymond, and Simonella developed a cluster +expansion directly on physical trajectories~\cite{BGSSCluster2022}, +distinguishing genuine collisions from geometric overlaps between +independently evolved components. + +Deng, Hani, and Ma proved the hard-sphere limit throughout every regular +finite kinetic interval~\cite[Theorem~1]{DHM25}. Their layered expansion +retains the earlier histories of correlated remainders while extracting +the factors approximated by the kinetic solution. This strategy has an +antecedent in the long-time wave-kinetic work of Deng and +Hani~\cite{DengHani2024}. The exposition~\cite{BGSS26} explains both the +accumulated correlations and the loss of topology that prevent +restarting a short-time chaos theorem as a black box. + +Smooth potentials require additional control because encounters last +for positive time and several particles can interact simultaneously. +Deng, Hani, and Ma discuss this extension as a further problem +in~\cite[Section~1.4.2]{DHM25}. +The present whole-component expansion retains the potential energy of +these interactions. Its stability and causal exposure estimates are +proved below; they are not supplied by the hard-sphere convergence +theorem. Similarly, the assumed classical kinetic solution is an input +to the theorem, while its additional spatial envelopes are derived in +Section~\ref{sec:preliminaries} by adapting the argument of +\cite[Proposition~A.1]{DHM25}. + +Theorem~\ref{thm:main} proves the regular-lifespan Boltzmann--Grad +statement for the full stable radial class. Its lifespan assumption +concerns the kinetic solution itself and imposes no small-data or +near-equilibrium condition. The conclusion is convergence of all fixed +factorial marginals in full phase-space $L^1$. + +\subsection{Structure of the proof} + +The argument uses a finite number of macroscopic layers of length +$b=T/L$, each divided into a much finer mesh. The fine cells permit an +exact finite expansion by whole interaction components. The coarse +layers permit centering against the known Boltzmann solution. Several +features of the construction are useful beyond ordinary binary +collision trees. +Throughout the proof, $P=1+|\log\varepsilon|$; polynomial cutoffs mean +fixed powers of $P$, chosen after the finite number $L$ of layers. + +\paragraph{Exact component identities.} +The expansion groups particles into their entire instantaneous +interaction components at every cut. A finite inclusion-exclusion +over successive generations of components represents each coefficient +by independent subsystem flows and contact indicators. This preserves +the potential energy needed for estimates even when a component has +many internal encounters. Centering acts only on isolated singleton +slots. Its cancellation requires separation of every displayed +trajectory, including the detached trajectories in the signed terms. + +\paragraph{One energy budget across all layers.} +Each formal particle lifetime contributes an independent input once +and departs once. Stability applied to intact departed components gives +a single Gaussian budget for the whole history. Repeatedly carried +velocities can then cost $L^{p/2}$ for a history with $p$ labels, while +the time factors contribute $(T/L)^{p-h_0}$ for $h_0$ top roots. +The $h_0$ roots are the labels at the upper end from which the +backward history starts. After bounding their contribution, the resulting +base +$CT/\sqrt L$ is small for a sufficiently large fixed $L$. This is the +mechanism that passes beyond the convergence interval of one +collision-tree series. + +\paragraph{Causal exposure of defects.} +The first estimates integrate all positions and velocities at the +ends of a fine time cell. They bound histories with crowded groups or +several additions in one cell. The remaining histories can be represented +by trees whose edges follow the order in which particles are added. +If extra contacts affect a positive fraction of the particle lives, +a counting argument selects many disjoint trees of bounded size. +Their fresh scattering parameters give small angular sets for contacts +with paths that are independent of those parameters. +To multiply these small factors, both outputs of an unexposed genuine +scattering must remain hidden, whereas a virtual crossing hides only +the new branch; Figure~\ref{fig:exposure} explains the distinction. +The trees used to choose integration coordinates need not coincide +with the families used in the signed cancellation identity. + +\input{figures/exposure} + +Section~\ref{sec:preliminaries} establishes the flow, initial estimates, +scattering facts, and uniform smooth-input bounds. +Section~\ref{sec:cell-expansion} constructs the exact finite expansion. +Section~\ref{sec:centering} gives the centered identities and the +short-layer approximation to the kinetic solution. +Sections~\ref{sec:history-estimates} and~\ref{sec:exposure} prove the +finite-history measure estimates and the exposure bound. +Section~\ref{sec:closure} chooses the thresholds, removes every cutoff, +and proves Theorem~\ref{thm:main} and Corollary~\ref{cor:empirical}. diff --git a/preprints/The-Boltzmann-Grad-limit-for-stable-radial-potentials-on-regular-kinetic-intervals-September-23-2026/build/sections/01-preliminaries.tex b/preprints/The-Boltzmann-Grad-limit-for-stable-radial-potentials-on-regular-kinetic-intervals-September-23-2026/build/sections/01-preliminaries.tex new file mode 100644 index 0000000000000000000000000000000000000000..e81859282c0ceba35c9f1243079235d1775a780b --- /dev/null +++ b/preprints/The-Boltzmann-Grad-limit-for-stable-radial-potentials-on-regular-kinetic-intervals-September-23-2026/build/sections/01-preliminaries.tex @@ -0,0 +1,660 @@ +\section{Microscopic flow, scattering, and kinetic envelopes} +\label{sec:preliminaries} + +The microscopic expansion uses globally defined finite-particle flows +and small initial centered correlations. We establish these facts, +identify the collision operator by two-body flux, and derive fixed +spatially summable Gaussian bounds for the assumed kinetic solution +and its spatial gradient. The scattering arguments require neither a +monotone potential nor a regular differential cross-section. + +\subsection{Potential assumptions and finite-particle flow} + +\begin{assumption}[Potential class]\label{pre:potential} +Write $\Phi(q)=\phi(|q|)$, with $\Phi(q)=0$ for $|q|\geq1$. +Assume either $\Phi\in C^2(\mathbb R^3;\mathbb R)$, or +$\Phi\in C^2(\mathbb R^3\setminus\{0\})$ with +$\Phi(q)\to+\infty$ as $q\to0$. +In the latter case set $\Phi(0)=+\infty$. +Assume thermodynamic stability: for some $B\geq0$, +\begin{equation}\label{eq:pre-stability} + \sum_{\substack{i,j\in R\\i0$ and finite label set $R$, +\eqref{pre:dynamics} has a unique global flow from every finite-energy +initial state in its domain. The flow preserves phase volume and +$\mathscr H_R$. In the singular case, every pair distance along a given +trajectory is bounded away from zero. In the nonsingular case coincident +positions are permitted. +\end{proposition} + +\begin{proof} +Taking two particles in \eqref{eq:pre-stability} gives +$\phi(r)\geq-2B$ for $r>0$. In the nonsingular case, approximation by +distinct configurations extends stability to coincident positions. +The vector field is locally Lipschitz on its domain and hence has a +unique maximal local solution. Differentiation gives energy conservation +on its existence interval. For $n=|R|$, +\begin{equation}\label{eq:pre-speed} + \frac12\sum_{i\in R}|v_i(t)|^2 + \leq\mathscr H_R(Z_R(0))+Bn. +\end{equation} +For $n\geq2$, bounding every pair except one below gives +\begin{equation}\label{eq:pre-pairupper} + \Phi_\varepsilon(x_i(t)-x_j(t)) + \leq\mathscr H_R(Z_R(0)) + +2B\left(\binom n2-1\right). +\end{equation} +In the singular case this bounds every pair distance away from zero. +Equation \eqref{eq:pre-speed} prevents escape to spatial infinity in +finite time. Thus on each bounded time interval the solution remains in +a compact subset of the vector field's domain and extends beyond any +proposed finite endpoint, forward or backward. In the nonsingular case +one can also use the globally Lipschitz gradient of the compactly +supported $C^2$ potential directly. The cases $n=0,1$ are immediate. + +The vector field has zero divergence in $(x_i,v_i)_{i\in R}$: +$\dot x_i$ is independent of $x_i$ and $\dot v_i$ is independent of +$v_i$. Its flow Jacobian therefore solves $\dot J=0$ with $J(0)=1$. +\end{proof} + +\subsection{The ensemble and external isolation} + +\begin{definition}[Exact initial ensemble] +Let $\mu=\varepsilon^{-2}$ and +\[ + {\cal D}_R^\varepsilon + =\{Z_R:|x_i-x_j|>\varepsilon\text{ for all }i\ne j\in R\}. +\] +For the prescribed $C^1$ nonnegative probability density $f_0$, set +\begin{align} + {\cal Z}_\varepsilon + &=\sum_{n=0}^\infty\frac{\mu^n}{n!} + \int f_0^{\otimes n}(Z_n) + \mathbf1_{{\cal D}_{[n]}^\varepsilon}(Z_n)\,dZ_n, + \label{eq:pre-partition}\\ + W_{0,n}(Z_n) + &={\cal Z}_\varepsilon^{-1}\mu^n f_0^{\otimes n}(Z_n) + \mathbf1_{{\cal D}_{[n]}^\varepsilon}(Z_n). + \label{eq:pre-ensemble} +\end{align} +Here $[n]=\{1,\ldots,n\}$ and empty products equal one. +The probability law on ordered coordinates is +\begin{equation}\label{pre:model} + \mathbb P(N=n,Z_n\in dZ_n)=\frac1{n!}W_{0,n}(Z_n)\,dZ_n. +\end{equation} +Thereafter the cloud evolves by \eqref{pre:dynamics}. +\end{definition} + +Since $1\leq\mathcal Z_\varepsilon\leq e^\mu$, this is a probability law +with finite $N$ almost surely. Initial exclusion gives zero potential +energy and finite kinetic energy. Proposition~\ref{pre:flow} therefore +defines its exact evolution. Let $W_{t,n}$ be the transported density. + +For $r=|R|$, the rescaled factorial density is +\begin{equation}\label{pre:factorial} + F_R^\varepsilon(t,Z_R)= + \mu^{-r}\sum_{n=0}^\infty\frac1{n!} + \int W_{t,r+n}(Z_R,Z_{[n]}^{\rm ext})\,dZ_{[n]}^{\rm ext}. +\end{equation} +External labels are distinct from those in $R$. +Symmetry makes the formula independent of their temporary names. +It equals the density defined by the ordered distinct-label sum in the +statement: in a cloud of size $r+n$, there are $(r+n)!/n!$ such selections. + +The externally isolated factorial density is +\begin{equation}\label{eq:pre-isolated} + \widetilde F_R^\varepsilon(t,Z_R)= + \mu^{-r}\sum_{n=0}^\infty\frac1{n!} + \int W_{t,r+n}(Z_R,Z_{[n]}^{\rm ext}) + \prod_{\substack{i\in R\\j\in[n]^{\rm ext}}} + \mathbf1_{\{|x_i-x_j|>\varepsilon\}}\, + dZ_{[n]}^{\rm ext}. +\end{equation} +Set $F_\varnothing^\varepsilon=\widetilde F_\varnothing^\varepsilon=1$. +External isolation imposes no condition on distances within $R$. +In particular an interacting displayed component can be externally +isolated, and $0\leq\widetilde F_R^\varepsilon\leq F_R^\varepsilon$. +A polymer is a connected component of the instantaneous graph with +edges $|x_i-x_j|\leq\varepsilon$. +The condition in \eqref{eq:pre-isolated} says exactly that the displayed +labels form a union of entire polymers of the full cloud. + +\begin{proposition}[Initial insertion and centering]\label{pre:initial} +Put $\rho_0(x)=\int f_0(x,v)\,dv$, and assume $\|\rho_0\|_\infty<\infty$. +At time zero, +\begin{equation}\label{eq:pre-insertion} + \begin{split} + \widetilde F_R^\varepsilon(0,Z_R) + &=F_R^\varepsilon(0,Z_R)\\ + &=f_0^{\otimes R}(Z_R)\mathbf1_{{\cal D}_R^\varepsilon}(Z_R) + \mathbb E\prod_{i\in R}\mathbf1_{\{N(B_i)=0\}}, + \qquad B_i=\overline B(x_i,\varepsilon). + \end{split} +\end{equation} +The expectation is over an independent cloud with the original law, +and $N(B)$ counts its positions in $B$. +Consequently $0\leq F_R^\varepsilon(0)\leq f_0^{\otimes R}$. + +For $Z_R\in{\cal D}_R^\varepsilon$, define +\begin{equation}\label{eq:pre-initialcenter} + E_R^\varepsilon(0,Z_R)= + \sum_{S\subset R}(-1)^{|R\setminus S|} + f_0^{\otimes(R\setminus S)}(Z_{R\setminus S}) + \widetilde F_S^\varepsilon(0,Z_S). +\end{equation} +There are $c_0>0,C_0<\infty$, independent of $R,\varepsilon$, such that +for sufficiently small $\varepsilon$, +\begin{equation}\label{eq:pre-centersmall} + |E_R^\varepsilon(0,Z_R)| + \leq(C_0\varepsilon^{c_0})^{|R|}f_0^{\otimes R}(Z_R). +\end{equation} +One may take $c_0=1/125$. +For fixed $r$, +\begin{equation}\label{eq:pre-mass} + 0\leq1-\int F_R^\varepsilon(0,Z_R)\,dZ_R + \leq Cr\varepsilon+Cr^2\varepsilon^3, + \qquad \int F_R^\varepsilon(t)=\int F_R^\varepsilon(0). +\end{equation} +\end{proposition} + +\begin{proof} +Insert $r$ particles into \eqref{pre:factorial} at time zero. +Their mutual exclusions give $\mathbf1_{{\cal D}_R^\varepsilon}$. +The remaining partition sum is $\mathcal Z_\varepsilon$ times the +probability that the original cloud misses $\bigcup_iB_i$. +This proves \eqref{eq:pre-insertion}. Initial exclusion already gives +external isolation. + +On $\mathcal D_R^\varepsilon$ every subset is separated, so substitution +into \eqref{eq:pre-initialcenter} gives the exact identity +\begin{equation}\label{eq:pre-occupancy} + E_R^\varepsilon(0,Z_R) + =(-1)^{|R|}f_0^{\otimes R}(Z_R) + \mathbb P\{N(B_i)\geq1\text{ for every }i\in R\}. +\end{equation} +Choose a maximal subcollection of centers with mutual distance greater +than $2\varepsilon$. Its balls $B_i$ are disjoint up to null boundaries. +A ball of radius $2\varepsilon$ contains at most $125$ original centers: +their disjoint balls of radius $\varepsilon/2$ fit inside a ball of +radius $5\varepsilon/2$. Hence the chosen subcollection has size +$m\geq |R|/125$. Disjointness ensures that occupants are distinct, and +the factorial upper bound gives +\begin{align*} + \mathbb P\{N(B_i)\geq1\text{ for every chosen }i\} + &\leq\mathbb E\prod_{\text{chosen }i}N(B_i)\\ + &=\mu^m\int_{\prod_i(B_i\times\mathbb R^3)} + F_m^\varepsilon(0)\,dZ_m\\ + &\leq\bigl(\mu\|\rho_0\|_\infty|B(0,\varepsilon)|\bigr)^m + \leq(C\varepsilon)^m. +\end{align*} +For $C\varepsilon\leq1$ this proves \eqref{eq:pre-centersmall}. + +The loss from mutual exclusion of $r$ independent inserted positions +is at most $Cr^2\varepsilon^3$, by a union bound and bounded $\rho_0$. +For each inserted position, +\[ + \mathbb P\{N(B_i)\geq1\}\leq\mathbb E N(B_i) + =\mu\int_{B_i\times\mathbb R^3}F_1^\varepsilon(0,z)\,dz + \leq C\varepsilon. +\] +A second union bound proves the first estimate in \eqref{eq:pre-mass}. +The integral of \eqref{pre:factorial} is +$\mu^{-r}\mathbb E(N)_r$, where +$(N)_r=N(N-1)\cdots(N-r+1)$. +It is finite by the initial factorial bound and conserved since $N$ is +conserved. +\end{proof} + +\subsection{Scattering flux and the collision operator} + +Use microscopic relative coordinates in this subsection. +For $g\ne0$ and $b\in g^\perp$, $|b|<1$, the incoming free trajectory +meets the interaction sphere at +\[ + n=b-\sqrt{1-|b|^2}\,\frac g{|g|},\qquad n\cdot g<0. +\] +Time translation identifies the asymptotic prescription +$q(\tau)-(b+\tau g)\to0$, $q'(\tau)\to g$ as $\tau\to-\infty$, +with $q(0)=n$, $q'(0)=g$ for $q''=-2\nabla\Phi(q)$. +Let $\tau_*(n,g)\in(0,\infty]$ be its first subsequent exit time from +the open unit ball. +Projection onto the impact disk has Jacobian +\begin{equation}\label{eq:pre-fluxdisk} + |n\cdot g|\,d\sigma(n)=|g|\,d^2b. +\end{equation} + +\begin{lemma}[Residence-time flux]\label{pre:duration} +For every $M<\infty$ there is $C_M<\infty$ such that +\begin{equation}\label{eq:pre-duration} + \int_{\substack{|g|\leq M\\n\in S^2,\ n\cdot g<0}} + \tau_*(n,g)|n\cdot g|\,d\sigma(n)\,dg\leq C_M. +\end{equation} +The incoming flux of $\{\tau_*>D,\ |g|\leq M\}$ is at most $C_M/D$. +In particular $\tau_*<\infty$ almost everywhere in incoming flux. +The entrance-to-exit scattering map preserves flux. +\end{lemma} + +\begin{proof} +Relative energy is +\[ + \tfrac14|q'|^2+\phi(|q|)=\tfrac14|g|^2. +\] +Since $\phi\geq-2B$, we have $|q'|^2\leq M^2+8B$ throughout the +encounter. Consider +\[ + (n,g,s)\longmapsto(q(s),q'(s)), + \quad n\cdot g<0,\quad |g|\leq M,\quad00$, the non-free relative incoming data $(n,g)$ +with $|g|\leq M$ +can be restricted to a rotation-invariant compact set, losing at most +$\eta$ relative incoming flux, on which the duration is bounded and +$|g|$, $|v'-v|$, and $|v_*'-v|$ have positive lower bounds. +For fixed radial scattering parameters, each of these nonzero +increments has a uniform spherical directional marginal under common +rotation. The two increments need not be independent. +\end{proposition} + +\begin{proof} +Put $G=|g|$, $\rho=|b|$, $J=G\rho$, and $E=G^2/4$. +The cases $G=0$, $J=0$, and $\rho=1$ are null in incoming flux. +For $J>0$, the effective radial potential is +\[ + W_J(r)=\phi(r)+\frac{J^2}{4r^2}. +\] +An incoming turning radius $y$ satisfies $E=W_J(y)$. +A nonsimple turning point also requires $W_J'(y)=0$. +At each fixed $J$, the critical values of the $C^2$ function $W_J$ on +$(0,1)$ are Lebesgue null, by the one-dimensional Sard theorem on a +countable compact exhaustion. The radial incoming flux is +\begin{equation}\label{eq:pre-Sardflux} + G^3\rho\,dG\,d\rho=2J\,dE\,dJ. +\end{equation} +Fubini excludes nonsimple turning energies up to zero flux. +Lemma~\ref{pre:duration} has already excluded infinite duration. + +At a simple incoming turning radius $y\in(0,1)$, +$W_J'(y)<0$ and $E>W_J(r)$ for all $r>y$. +Set $a=J^{-2}$ and +\[ + A_y(r)=y^{-2}-r^{-2},\qquad + C_y(r)=4(\phi(y)-\phi(r)),\qquad + H_{y,a}(r)=A_y(r)+aC_y(r). +\] +The half angular sweep is +\begin{equation}\label{eq:pre-halfangle} + {\cal A}_y(a)=\int_y^\infty\frac{dr}{r^2\sqrt{H_{y,a}(r)}}. +\end{equation} +Indeed $\dot r^2=J^2H_{y,a}(r)$ and the angular speed is $J/r^2$. +For admissible $a$, one has $H_{y,a}(r)>0$ for $r>y$, +$\partial_rH_{y,a}(y)>0$, and +$\lim_{r\to\infty}H_{y,a}(r)>0$. +The ratios $C_y/A_y$ and $H_{y,a}/A_y$ extend continuously to +$[y,\infty]$. At $y$ use the ratios of first derivatives; at infinity +use finite range. The second ratio is strictly positive on this compact +interval, so it has a positive minimum, while $C_y/H_{y,a}$ is bounded. +These properties also show that the admissible set of $a$ is open. + +For $a'$ near $a$, the binomial expansion in +\[ + H_{y,a'}^{-1/2} + =H_{y,a}^{-1/2} + \left(1+(a'-a)\frac{C_y}{H_{y,a}}\right)^{-1/2} +\] +converges uniformly after multiplication by the integrable majorant +$r^{-2}H_{y,a}^{-1/2}$. Thus $\mathcal A_y$ is real analytic in the +parameter $a$, although $\phi$ is only $C^2$. +Twice differentiating gives +\begin{equation}\label{eq:pre-convex} + {\cal A}_y''(a)=\frac34 + \int_y^\infty\frac{C_y(r)^2\,dr}{r^2H_{y,a}(r)^{5/2}}. +\end{equation} +Near $y$ the integrand is $O((r-y)^{-1/2})$, since $C_y(r)=O(r-y)$ +and $H_{y,a}$ has a simple root. It is also integrable at infinity. +It is strictly positive unless $\phi(r)=\phi(y)$ for every $r\geq y$. +Finite range makes this constant zero; the encountered orbit is then +a straight force-free passage. + +For a non-free encounter $\mathcal A_y$ is strictly convex. +The deflection angle is $\pi-2\mathcal A_y$ modulo $2\pi$. +Therefore $g_{\rm out}=g$ or $g_{\rm out}=-g$ requires +$\mathcal A_y\in(\pi/2)\mathbb Z$. +Each such level has at most two preimages on an admissible interval. +There are countably many levels and countably many open intervals. +On a simple turning branch, $(y,J)\mapsto(E,J)$, $E=W_J(y)$, +is a local $C^1$ diffeomorphism. +A countable cover by these charts, Fubini in $(y,a)$, and +\eqref{eq:pre-Sardflux} prove flux nullness. +These exceptional equalities are exactly the two failures of +\eqref{eq:pre-increments}. + +For the compact-set assertion, exhaust the simple-scattering non-free +radial parameter domain by compact subsets avoiding $G=0$, $J=0$, +$\rho=1$, and the zero-increment levels. +This covers all non-free data outside a null set. +On each such compact subset, continuous dependence for the $C^1$ +ODE flow and transversality of the finite exit give a continuous +exit time and scattering map. Thus durations are bounded and all the +listed nonzero quantities have positive minima. +The relative flux measure on $G\leq M$ is finite, so an exhaustion +member loses at most $\eta$ flux. Lifting with all rotations makes +the set rotation-invariant. + +Finally, write $(g,b)=(GRe_1,\rho Re_2)$, $R\in SO(3)$. +At fixed $(G,\rho)$ the orientation factor in incoming flux is a +constant multiple of Haar measure on $SO(3)$. +Central-force rotation equivariance maps each nonzero increment to +$R$ times a fixed vector. Its direction has uniform measure on $S^2$. +This is a marginal statement and leaves the joint pair correlated. +\end{proof} + +\subsection{A fixed spatial Gaussian envelope on the kinetic interval} + +Use the norm $\|\cdot\|_{\mathrm{Bol},\alpha}$ defined in +\eqref{intro:norm}, and put +\[ + \|h\|_{\infty,\alpha} + =\sup_{x,v}e^{\alpha|v|^2}|h(x,v)|. +\] +For vector-valued functions use Euclidean norm. +The following proof adapts the weighted transport estimate in +\cite[Proposition A.1]{DHM25}. Only finite impact area and kinetic +energy conservation are needed, so the argument applies to $Q_\Phi$. + +\begin{proposition}[Known-solution envelopes]\label{pre:envelopes} +Suppose $f$ is the classical nonnegative solution in the statement, and +\[ + A=\sup_{0\leq t\leq T}\|f(t)\|_{\infty,2\beta}<\infty,\qquad + B_0=\|f_0\|_{\mathrm{Bol},2\beta} + +\|\nabla_xf_0\|_{\mathrm{Bol},2\beta}<\infty. +\] +Then +\begin{equation}\label{eq:pre-envelope} + \sup_{0\leq t\leq T} + \left(\|f(t)\|_{\mathrm{Bol},\beta} + +\|\nabla_xf(t)\|_{\mathrm{Bol},\beta}\right) + \leq C(A,B_0,\beta,T)<\infty. +\end{equation} +Fix, once and for all, +\begin{equation}\label{eq:pre-beta0} + \beta_0=\beta/2. +\end{equation} +Thus all independent inputs $f(t)$ and their spatial Lipschitz +envelopes have a common bound with the fixed margin +$\beta-\beta_0$. Constants in \eqref{eq:pre-envelope} are independent +of the subsequent time-layer subdivision. +\end{proposition} + +\begin{proof} +Set +\[ + \gamma(t)=\frac{3\beta}{2}-\frac{\beta t}{2T},\qquad + c=\frac{\beta}{2T},\qquad M(v)=e^{-2\beta|v|^2}, +\] +and $(S(r)h)(x,v)=h(x-rv,v)$. +Notice $\beta\leq\gamma(t)\leq3\beta/2$. +Define a positive linear operator for $h\geq0$ by +\begin{equation}\label{eq:pre-positiveB} + \begin{split} + ({\cal B}h)(x,v)=\int_{\mathbb R^3}|v-v_*|\int_{|b|<1} + \bigl[&M(v_*')h(x,v')+M(v')h(x,v_*')\\ + &+M(v_*)h(x,v)+M(v)h(x,v_*)\bigr]\,d^2b\,dv_* . + \end{split} +\end{equation} +The disk is in $(v-v_*)^\perp$. + +We first prove that for $0\leq s\varepsilon\}}. +\] +It imposes no separation between two labels of $R$. Introduce the +positive measure kernel +\begin{equation} + \mathsf H^I_{R,A}(Z_A;\dd Y_R) + =e_I^R(A)\, + \delta_{(X_i^A(\delta),V_i^A(\delta))_{i\in R}}(\dd Y_R). + \label{eq:cell-H} +\end{equation} +The Dirac measure notation avoids introducing a density for a +lower-dimensional transport kernel. + +The only flow fact needed for the algebra is the following equivalence. +If separate groups avoid one another throughout $I$, their separate +flows are the restrictions of the flow of their union. Conversely, +different blocks of $\Pi_I(S)$ avoid one another, and their restrictions +are isolated flows. Indeed cross forces vanish in the first assertion, +so uniqueness proves it; the second follows from the definition of the +contact graph and the same uniqueness argument. + +\begin{definition}[A cell diagram] +\label{def:cell-diagram} +For $R\subset S$, with $R\ne\varnothing$, a cell diagram on $(R,S)$ is +\[ + \gamma=(A;\Gamma_1,\ldots,\Gamma_\ell). +\] +Here $R\subset A\subset S$; the nonempty sets in all the $\Gamma_g$ +together partition $S\setminus A$; and each $\Gamma_g$ is a nonempty +unordered collection of these sets. Put $\Gamma_0=\{A\}$. +Different members of the diagram are different force groups. Let +$\iota_0(\gamma)$ be the indicator that every force group in the diagram +is a union of blocks of $\pi_0(S)$. Equivalently, all pairs in different +force groups have distance greater than $\varepsilon$ at $a$. +Define +\begin{align} + w_I(\gamma;Z_S) + ={}&\iota_0(\gamma)c_I^R(A) + \prod_{g=1}^{\ell} + \left[ + \prod_{B\in\Gamma_g}c_I(B)\tau_I(\Gamma_{g-1},B) + \prod_{\{B,C\}\subset\Gamma_g}\alpha_I(B,C) + \right], \label{eq:cell-weight}\\ + \sigma(\gamma)={}&(-1)^{\sum_{g=1}^{\ell}|\Gamma_g|}. + \label{eq:cell-sign} +\end{align} +Only the root group $A$ uses a joint flow containing roots. In +particular, a contact between different generations is a condition on +separate trajectories, not a force exerted between their groups. +The empty list of generations is allowed when $S=A$. +\end{definition} + +Define the signed kernel +\begin{equation} + \mathsf K^I_{R,S}(Z_S;\dd Y_R) + =\sum_{\gamma\text{ on }(R,S)} + \sigma(\gamma)w_I(\gamma;Z_S) + \mathsf H^I_{R,A}(Z_A;\dd Y_R). + \label{eq:cell-K} +\end{equation} +The sum is finite. Set $\mathsf K^I_{\varnothing,\varnothing}=1$ and +$\mathsf K^I_{\varnothing,S}=0$ for $S\ne\varnothing$. +For a function or density $g$ on the input coordinates define +$\mathcal T^I_{R,S}g$ as a signed measure by +\begin{equation} + \langle\psi,\mathcal T^I_{R,S}g\rangle + =\int g(Z_S)\int\psi(Y_R) + \mathsf K^I_{R,S}(Z_S;\dd Y_R)\,\dd Z_S. + \label{eq:cell-T} +\end{equation} + +\begin{remark}[One additional label] +\label{rem:cell-single} +If an extra label $e$ is initially separate from all roots, its two +diagram terms are +\[ + c_I^R(R\cup\{e\})\mathsf H^I_{R,R\cup\{e\}} + \; -\; + \tau_I(R,\{e\})\mathsf H^I_{R,R}. +\] +The first follows the full force group and the second follows the +root group and a free extra particle separately. If $e$ belongs to +an initial root polymer, it is mandatory and only the full-group term +is present. No extra family-selection multiplicity is introduced +by this identity. +\end{remark} + +\subsection{Boolean inversion and the generation identity} + +\begin{proposition}[Exact cell expansion] +\label{cell:expansion} +Let $R\subset S$ and let $A_0$ be the union of the blocks of +$\pi_0(S)$ meeting $R$. For $R\ne\varnothing$, let $\mathcal P$ be +the collection of the other blocks of $\pi_0(S)$. Then +\begin{equation} + \mathsf H^I_{R,S} + =\sum_{\mathcal U\subset\mathcal P} + \mathsf K^I_{R,A_0\cup\bigcup\mathcal U}, + \label{eq:cell-subset-sum} +\end{equation} +and, as an equality of signed measures, +\begin{equation} + \mathsf K^I_{R,A_0\cup\bigcup\mathcal U} + =\sum_{\mathcal V\subset\mathcal U} + (-1)^{|\mathcal U|-|\mathcal V|} + \mathsf H^I_{R,A_0\cup\bigcup\mathcal V}. + \label{eq:cell-boolean} +\end{equation} +Every coefficient is evaluated on its displayed input coordinates. +If $m=|S\setminus R|$, the number of diagrams in +\eqref{eq:cell-K} is at most $(C(1+m))^{Cm}$, with a universal $C$. +There is no partition of the roots in this count. +\end{proposition} + +\begin{proof} +First decompose the full flow according to its unique union of contact +components meeting $R$. The flow observation preceding +Definition~\ref{def:cell-diagram} gives +\begin{equation} + \mathsf H^I_{R,S} + =\sum_{\substack{R\subset A\subset S\\ + A\text{ union of blocks of }\pi_0(S)}} + c_I^R(A)\mathsf H^I_{R,A}\Xi_I(\{A\};S\setminus A). + \label{eq:cell-root-decomposition} +\end{equation} +Precisely one term has its two component indicators nonzero: the actual +union of rooted contact components. Endpoint isolation of $R$ against +the rest of $S$ can then be checked inside $A$, because $A$ and its +complement avoid throughout the cell. + +Here is the finite identity that expands the last factor. Fix +$\mathcal X$, and consider a collection $\mathcal B$ of disjoint +nonempty subsets of $V$, each a union of the fixed initial polymers. +Put $U_{\mathcal B}=\bigcup\mathcal B$ and +\[ + d_I(\mathcal B) + =\prod_{B\in\mathcal B}c_I(B) + \prod_{\{B,C\}\subset\mathcal B}\alpha_I(B,C). +\] +Then +\begin{equation} + \Xi_I(\mathcal X;V) + =\sum_{\mathcal B} + (-1)^{|\mathcal B|}d_I(\mathcal B) + \prod_{B\in\mathcal B}\tau_I(\mathcal X,B) + \Xi_I(\mathcal B;V\setminus U_{\mathcal B}). + \label{eq:cell-generation-recursion} +\end{equation} +The term with $\mathcal B=\varnothing$ equals one. To prove +\eqref{eq:cell-generation-recursion}, expand the product +\eqref{eq:cell-Xi} over subsets of its actual components. A given +collection $\mathcal B$ consists of components of $V$ exactly when +its members are connected, mutually avoid, and avoid the remaining +$V\setminus U_{\mathcal B}$ flow. These conditions are respectively +$\prod c_I(B)$, the pair product in $d_I$, and the last $\Xi_I$ in +\eqref{eq:cell-generation-recursion}. On their support the required +flows agree. This proves the formula, including all signs. + +Apply \eqref{eq:cell-generation-recursion} repeatedly to +\eqref{eq:cell-root-decomposition}. Every nonempty selection strictly +decreases the number of remaining labels, so the process terminates. +At termination its coefficient is exactly +\eqref{eq:cell-weight}--\eqref{eq:cell-sign}; the unselected labels +have no remaining restriction. Grouping by the union of selected +initial polymers proves \eqref{eq:cell-subset-sum}. The initial +partition of such a union is the restriction of $\pi_0(S)$, so its +coefficient is exactly the kernel defined in \eqref{eq:cell-K}, not a +kernel depending on unselected coordinates. Finite Boolean inversion +proves \eqref{eq:cell-boolean}. + +There are at most $2^m$ choices of $A\setminus R$, at most $m^m$ +partitions of the remaining labels, and at most $m^m$ assignments of +their blocks to successive nonempty generations. This proves the +count, with the cases $m=0,1$ understood separately. +\end{proof} + +\subsection{Expectations and normalization} + +For any finite random cloud let its scaled factorial measure with +external isolation be denoted by $\widetilde F_R$. Its weak definition +is the usual factorial sum with the additional indicator that every +label outside the displayed set has distance greater than +$\varepsilon$ from it. This definition permits internal contacts. +For the actual evolved initial ensemble, all exponential moments of +the total particle number are finite. Indeed its initial probability +of $N=n$ is bounded by $\mu^n/n!$, and $N$ is conserved. + +\begin{corollary}[Factorial normalization] +\label{cor:cell-factorial} +For the actual microscopic cloud and $r=|R|$, the cell identity is +written after relabelling the ordered roots as $R=\{1,\ldots,r\}$: +\begin{equation} + \widetilde F_R(b) + =\sum_{m\geq0}\frac{\mu^m}{m!} + \mathcal T^I_{R,R\sqcup\{r+1,\ldots,r+m\}} + \widetilde F_{r+m}(a). + \label{eq:cell-factorial} +\end{equation} +The right side is interpreted with each finite Boolean coefficient +grouped before summing over $m$. Equivalently, one may first restrict +the ensemble to $N\leq M$, apply the finite identity, and then let +$M\to\infty$. No assertion of absolute summability of the +\emph{ungrouped} generation series is made here. +\end{corollary} + +\begin{proof} +Apply \eqref{eq:cell-subset-sum} to each ordered tuple of roots in a +realization of the cloud at $a$. A selected union of its initial +polymers is exactly an externally isolated displayed subset at $a$. +Consequently the expectation of each selected union uses +$\widetilde F_{r+m}(a)$. Its $m$ extra labels were unordered, whereas +the factorial sum orders them, giving the divisor $m!$. The scaling +of the original $r$ roots is $\mu^{-r}$, and that of the new factorial +measure is $\mu^{-(r+m)}$, giving $\mu^m$. + +For a bounded test $\psi$, Boolean inversion bounds a coefficient +in total variation by $2^{|\mathcal U|}\|\psi\|_\infty$. Summing this +over all selected polymer subsets in a fixed cloud costs at most +$3^N\|\psi\|_\infty$, and the root tuple count costs at most $N^r$. +The integrable bound $\mu^{-r}N^r3^N\|\psi\|_\infty$ justifies the +stated grouped limit. Expanding every large-order coefficient into +all its diagrams before taking absolute values would give a different, +unproved convergence assertion and is unnecessary. +\end{proof} + +\subsection{Factorization on all displayed trajectories} + +\begin{proposition}[Diagramwise factorization] +\label{cell:factorization} +Split the labels into $S=S_1\sqcup S_2$ and the roots into +$R=R_1\sqcup R_2$, where $R_i\subset S_i$. Let $\gamma_i$ be a +diagram on $(R_i,S_i)$. Define $\eta_I(\gamma_1,\gamma_2)$ to be the +indicator that every displayed particle trajectory of $\gamma_1$ +has distance greater than $\varepsilon$ throughout $I$ from every +displayed trajectory of $\gamma_2$. This includes detached groups +and includes the initial and terminal times. +For a single global diagram $\gamma$, define +$\eta_I(\gamma;S_1,S_2)$ by the same condition between its labelled +paths in $S_1$ and $S_2$, using their prescribed force-group flows. + +On this indicator, merge the two root groups, and take the union of +the two batches at each generation, padding a finished list by empty +batches. This gives a diagram on $(R,S)$. Conversely, every nonzero +diagram term satisfying $\eta_I(\gamma;S_1,S_2)=1$ splits uniquely in +this manner. Its detached groups necessarily belong to one $S_i$, +and its root flow splits over its two intersections with $S_i$. +Under this bijection the signed transport measures, with this +indicator imposed, agree with the tensor product of the two +diagram transport measures: +\begin{equation} + \sigma(\gamma)w_I(\gamma)\mathsf H^I_{R,A}\,\eta_I + =\eta_I\bigl[ + \sigma(\gamma_1)w_I(\gamma_1)\mathsf H^I_{R_1,A_1} + \otimes + \sigma(\gamma_2)w_I(\gamma_2)\mathsf H^I_{R_2,A_2} + \bigr]. + \label{eq:cell-factorization} +\end{equation} +Summation over the bijection preserves this equality. Factorial +weights split as +\begin{equation} + \binom{m_1+m_2}{m_1} + \frac{\mu^{m_1+m_2}}{(m_1+m_2)!} + =\frac{\mu^{m_1}}{m_1!}\frac{\mu^{m_2}}{m_2!}. + \label{eq:cell-factorial-split} +\end{equation} +\end{proposition} + +\begin{proof} +Cross avoidance makes the joint root flow equal to the two separate +root flows. Its rooted-component and endpoint-isolation indicators +therefore split. Initial-polymer admissibility also splits, since +all cross initial distances are greater than $\varepsilon$. +Within a merged generation, all additional cross avoidance factors +are one. A detached component in $S_i$ can touch the preceding +generation only through that generation's $S_i$ part. Hence its +touch indicator is exactly its original touch indicator. The +number of detached groups is additive, so the signs multiply. +The output Dirac measures split with the root flow. These facts +give \eqref{eq:cell-factorization}. + +For the inverse map, a connected detached group cannot contain labels +from both families: a connected graph across that partition would +have a cross contact, contrary to $\eta_I=1$. The same avoidance +splits the root group's flow by uniqueness. A nonempty generation +in one family must have +a predecessor in that same family unless it is the first generation: +otherwise its required touch indicator vanishes. There can therefore +be no gap followed by a later nonempty generation in one family. +Removing its terminal empty batches produces its unique local +generation list. This proves the asserted bijection. Formula +\eqref{eq:cell-factorial-split} is the ordinary ordered-label count. +\end{proof} + +The avoidance condition in Proposition~\ref{cell:factorization} is +\emph{diagramwise}. It is not a single condition on the flow of $S$. +For example, a third particle can deflect a prospective collision +partner away from a root in the full flow while a smaller subflow +still changes that root. Such configurations are included in the +subset coefficient and cannot be removed using full-flow separation. +For more than two factors the proposition applies successively; its +indicator must include all displayed paths from all factors. This +is the precise factorization used by the centered identities below. + +\subsection{A deterministic stopped identity} + +We give a stopped version before any probabilistic tail estimate. +Its stopping parameter $k$ counts all selected labels, including +the roots. The threshold is allowed to include labels already +selected in later cells: in that case subtract that fixed offset +from the remaining threshold in the construction below. + +Order components by the lexicographically least initial position in +each component; denote this position by $\kappa(C)\in\mathbb R^3$. +Different actual contact components have different such positions, +since particles with the same initial position belong to the same +initial polymer. This is a measurable, label-equivariant deterministic +order. The following elementary weighted product +identity is useful. For $I_1,\ldots,I_n\in\{0,1\}$, positive integer +weights $w_1,\ldots,w_n$, and an integer $d>0$, let +$w(J)=\sum_{j\in J}w_j$. Then +\begin{align} + \prod_{j=1}^n(1-I_j) + ={}&\sum_{\substack{J\subset\{1,\ldots,n\}\\w(J)\max J}(1-I_j). + \label{eq:cell-stopped-product} +\end{align} +To verify it, expand the factors in their prescribed order, stopping +a branch at its first selected factor whose accumulated weight is +at least $d$. Branches that finish without stopping are the first +sum, and stopped branches are the second. The factors remaining +on a stopped branch lie in $[0,1]$. + +Apply \eqref{eq:cell-stopped-product} to +$\Xi_I(\mathcal X;V)$, with the actual components of $V$ as its +ordered indices, $I_j=\tau_I(\mathcal X,C_j)$ and $w_j=|C_j|$. +For a selected collection $\mathcal B$, denote its last component +in this order by $B_*$ and set $V'=V\setminus U_{\mathcal B}$. +After extracting the selected components, the extra factor on +a stopped branch is +\begin{equation} + \vartheta_I(\mathcal X,\mathcal B;V') + =\Xi_I(\mathcal B;V') + \prod_{\substack{C\in\Pi_I(V')\\ + \kappa(C)>_{\mathrm{lex}}\kappa(B_*)}} + \bigl(1-\tau_I(\mathcal X,C)\bigr). + \label{eq:cell-stop-theta} +\end{equation} +It belongs to $[0,1]$. The condition that the selected components +are actual components of $V$ is represented, exactly as in +\eqref{eq:cell-generation-recursion}, by $d_I(\mathcal B)$ and the +first factor in \eqref{eq:cell-stop-theta}. On that condition the +order of the components of $V$ is the merged order of +$\mathcal B$ and $\Pi_I(V')$, which proves +\eqref{eq:cell-stop-theta}. Completed branches with +$|U_{\mathcal B}||R|$, the finite-cloud expansion equals its complete diagrams +with fewer than $k$ selected labels plus a finite signed remainder. +Each remainder term is of one of the following types. +\begin{enumerate} + \item Its root group $A$ already has $|A|\geq k$. Its transport is + $c_I^R(A)\mathsf H^I_{R,A}$ times the factor + $\Xi_I(\{A\};S\setminus A)\in[0,1]$. + \item Its completed generations and its final partial generation + select a union $U$ with $|U|\geq k$. Immediately before the last + selected component $B_*$, the number of selected labels is less + than $k$. All displayed groups satisfy the connectivity, + initial-polymer, same-generation avoidance, and preceding-generation + touch conditions of Definition~\ref{def:cell-diagram}. Its weight + is the corresponding sign and indicators times a factor + $\vartheta_I$ of the form \eqref{eq:cell-stop-theta}, lying in $[0,1]$. +\end{enumerate} +In the second case the overshoot is by at most one additional +component. In particular, if every component of every trial +subsystem formed from initial polymers has size at most $D$, then +$|A|\leq |R|D$, and a stopped union has size at most +$\max\{|R|D,k-1+D\}$. + +Replacing the sign by $+1$ and dropping the last factor gives a +pointwise positive domination of the total variation of the +remainder. Its expected version is a sum of these positive +contact expressions integrated against the initial isolated +factorial measures of the selected unions, with the factors +$\mu^m/m!$ of \eqref{eq:cell-factorial}. This is an identity and +a domination statement, not a claim that this sum tends to zero. +\end{proposition} + +\begin{proof} +Use \eqref{eq:cell-root-decomposition}. Stop a summand immediately +if its root group reaches the threshold. Otherwise apply the +weighted product expansion with $d=k-|A|$, extract its selected +components, and continue its nonempty completed selections by the +same rule with the updated remaining threshold. Empty selections +terminate a complete diagram. The explicit finite identities +\eqref{eq:cell-stopped-product} and +\eqref{eq:cell-stop-theta} prove the resulting equality at every +stage. A diagram of total size less than $k$ never triggers the +stopping rule and appears with exactly its original coefficient. +Every other branch stops in one of the two stated ways. + +Weights are positive before their displayed inclusion-exclusion +sign, and the unexpanded factors are in $[0,1]$, proving the +variation bound. All selected groups are unions of the original +initial polymers. Their union is consequently externally isolated +in the original cloud at the initial cut; removing the last factor +does not remove this initial-polymer selection condition. Taking +factorial expectations therefore gives the asserted input measures +and normalization. On the stated trial-component bound each +component of the root group contains a root and has at most $D$ +labels, proving $|A|\leq |R|D$; the other size bound follows from +the one-component overshoot. + +These expected finite stopped identities are well defined for +the grand ensemble without a tail theorem. Prior to the last +selected component there are fewer than $k$ selected labels, and +therefore at most $k$ selected groups. The root-large branch has +one group. At fixed $k$ the number of stopped descriptions in +an $N$-particle cloud is bounded by $C_k^{N+1}$: assign each label +to one of at most $k+2$ group or remainder slots, and then order +the bounded number of selected groups and generations. The +exponential moments of $N$, with its root-tuple polynomial factor, +justify expectation and the finite-$N$ limit. This argument +establishes finiteness only; it supplies no useful uniform bound +as $\varepsilon\to0$. +\end{proof} + +\subsection{Whole-cell histories} + +\begin{definition}[A prescribed whole-cell history] +\label{cell:histories} +A finite whole-cell history is a composition of the diagram kernels +above on successive cells, together with any tensor products and +singleton-slot deletions or independent insertions specified by +the exact centered identities at layer cuts. Its discrete +prescription records the following data: +\begin{enumerate} + \item every formal label and its lifetime, with a fresh label + for each independent input and no reuse of an independent source; + \item on each occupied cell, the required root labels $R$, the + full set $S$, its root force group, and its detached groups and + generations; + \item all the indicators in \eqref{eq:cell-weight} and the + endpoint isolation indicators, including the initial polymer + restrictions restored at every deleted singleton slot; + \item the signs, factorial weights, and any deterministic + stopping marks and unexpanded factors. +\end{enumerate} +On an ordinary hierarchy cell, labels in $R$ persist through its upper endpoint; +labels in $S\setminus R$ end their cell lifetime there. In the +forward direction they were present throughout the cell, not +created at a first-contact time. The root group is isolated from +its departing members at the upper endpoint by $e_I^R(A)$. +Detached groups have separate force systems; their geometric +contacts with other generations at that endpoint are unrestricted. +At the lower endpoint all force groups are unions of initial +polymers. At a layer cut a nonsingleton polymer remains intact; +independent singleton inputs are inserted only with the full +polymer-pattern restrictions of the centered identity. + +The size $p$ is the number of distinct formal label lifetimes, not +the number of their repeated appearances at cuts. All transport +maps and indicators are defined on the full coordinates of these +lifetimes. A history with a stopped factor can still depend on +the unexpanded microscopic cloud through its last factor in +$[0,1]$; its positive majorant drops that factor. +\end{definition} + +There is also a positive version needed before a stopped cell. If a +nonnegative observable of displayed labels, carrying their external +isolation indicator at the top of a cell, is evaluated along the +actual cloud, extract its actual union of +whole-cell components containing those labels, as in +\eqref{eq:cell-root-decomposition}. Retain its root endpoint +isolation requirement and drop external avoidance against the +remaining cloud for an upper bound. Repeat backwards over earlier +cells. This produces only positive root-group histories, each +connected by contacts to the preceding required set, with whole +initial polymers at each cut. The construction is an exact +decomposition followed by inequalities between nonnegative +functions. Polynomial caps or probabilities for such histories +are not presumed in this definition; they are proved in +\Cref{close:witnesses}. + +\subsection{The terminal boundary of a positive witness} + +The positive histories just described retain the usual endpoint +isolation at every cut. To detect a large component in the actual +cloud, we will also use a final block whose distinguished label may +still be interacting at the observation endpoint. The next variant +keeps that entire final force group in the integral. This boundary +condition is needed for the energy estimate as well as for the +witness construction in Section~\ref{sec:closure}. + +\begin{definition}[An unisolated terminal witness block] +\label{def:cell-terminal-witness} +A terminal witness is permitted a different upper boundary condition +from an ordinary hierarchy cell. For $R\subset A$ put +\begin{equation} + \mathsf J^I_{R,A}(Z_A;\dd Y_R) + =\delta_{(X_i^A(\delta),V_i^A(\delta))_{i\in R}}(\dd Y_R), + \label{eq:cell-J} +\end{equation} +with no endpoint isolation factor. Its exact positive root-component +decomposition is +\begin{equation} + \mathsf J^I_{R,S} + =\sum_{\substack{R\subset A\subset S\\ + A\text{ union of blocks of }\pi_0(S)}} + c_I^R(A)\mathsf J^I_{R,A}\Xi_I(\{A\};S\setminus A). + \label{eq:cell-unisolated-root-decomposition} +\end{equation} +Indeed the unique actual union of rooted contact components is still +the unique nonzero summand, and its root trajectories agree with the +full ones. This proves the identity without asserting endpoint +isolation. Integrating a nonnegative test and dropping its last +avoidance factor gives the corresponding positive bound. + +For a terminal witness block, all selected force groups end their +formal lifetimes together at the fixed upper endpoint $b$. The +distinguished labels $R$---usually a single label---serve only for +factorial normalization and spatial integration anchors. They do +not persist as a force subsystem after that endpoint. All terminal +coordinates of every selected group are retained in the integration +and energy bookkeeping before any projection in +\eqref{eq:cell-J}. In particular one keeps the full terminal group +Hamiltonians; one does not split $R$ away from close companions. +Geometric isolation of $R$ from $A\setminus R$ at $b$ is neither +required nor inferred. + +A selected block may also carry a measurable nonnegative trajectory +indicator $W_I(A;Z_A)\leq1$ specifying the witness. Such an indicator +is evaluated on its isolated group flow. If the witness was +identified at an earlier time inside the cell, keep that prefix +condition and continue the isolated group to the fixed endpoint +$b$ before using whole-cell coordinates. This is a finite +prescription with an extra indicator, not a stopping-time change +of phase-volume variables. Establishing that an actual bad event +admits a selected witness of this form belongs to +\Cref{close:witnesses}. + +At every earlier cut the ordinary rules of +Definition~\ref{cell:histories} remain in force. In particular, a +selected union is initially externally isolated and its backward +continuation across the preceding cell retains the usual endpoint +isolation indicator. Only the final witness boundary uses +\eqref{eq:cell-J}. These are formal operations in the witness +integral; the microscopic evolution is not stopped or modified. +\end{definition} + +\subsection{Contact records and coordinate refinements} +\label{cell:contact-records} + +The ownership rule at a centered cut needs a record of which contacts +connect new labels to the old array. We specify the data here; +Lemma~\ref{hist:pinning} constructs the corresponding integration +charts and proves their Jacobian and counting bounds. + +For each occupied cell and each original tensor factor, an +\emph{original contact record} is an ordered list of edges between +displayed labels. An edge records its two labels and either their +relative displacement at an endpoint overlap or their contact time +and sphere normal. Starting at the late endpoint, with all old +labels in one anchored coordinate group and the new labels in +separate groups, each edge joins two groups not previously joined. +The edge list connects every new label to the old array. +Endpoint overlaps are processed first; subsequent edges are ordered +by successive first backward contacts, with the prescribed +deterministic tie rule. All edges stay within the original tensor +factor. When there is exactly one added label across the cell and +it is separated from its own roots at the late endpoint, the record +retains its first backward contact with that factor and the old label +met there as its original parent. These groupings record contact coordinates; +they do not replace any prescribed force group. + +The algebraic generations and the geometric pinning forest +need not coincide. A diagram can be partitioned measurably according +to the first contact of a floating group with \emph{any} already +displayed path, including a path from another tensor factor. Such +a partition inserts indicators whose sum is one on the original +term, and leaves its force systems, signs, and original required +contact indicators unchanged. A cross-factor geometric pin is +therefore permitted even if it is not an edge of its algebraic +generation. All equalities above survive any such measurable +refinement. The Jacobians and causal properties of the refinements +used for estimates are proved separately in +\Cref{hist:pinning}; they are not consequences of +Boolean inversion alone. diff --git a/preprints/The-Boltzmann-Grad-limit-for-stable-radial-potentials-on-regular-kinetic-intervals-September-23-2026/build/sections/03-centering.tex b/preprints/The-Boltzmann-Grad-limit-for-stable-radial-potentials-on-regular-kinetic-intervals-September-23-2026/build/sections/03-centering.tex new file mode 100644 index 0000000000000000000000000000000000000000..4a76b54da77d4af8cee654c1b459ed2a53b20692 --- /dev/null +++ b/preprints/The-Boltzmann-Grad-limit-for-stable-radial-potentials-on-regular-kinetic-intervals-September-23-2026/build/sections/03-centering.tex @@ -0,0 +1,726 @@ +\section{Centering at the layer cuts}\label{sec:centering} + +We subtract independent kinetic factors while keeping each nonsingleton +interaction component intact. Correlation errors obtained by removing +independent factors were developed in~\cite{PulvirentiSimonella2017}; +their propagation across time layers with the earlier histories retained +is central to~\cite{DHM25}. This section proves the componentwise +centering identities needed for the present potential class. We obtain +an exact recurrence across the layers, then show that each surviving +centered root must lead to a small input or to a contact that prevents +its family from evolving independently. + +\subsection{The patternwise transform} +For a partition $\pi$ of a finite label set $R$, write +$\chi_\pi^\eps(Z_R)$ for the indicator that the connected components of +$\{\{i,j\}:|x_i-x_j|\leq\eps\}$ are exactly the blocks of $\pi$. +Let $S(\pi)$ be the set of labels in singleton blocks and let +$B(\pi)=R\setminus S(\pi)$. For $A\subset S(\pi)$, the partition +$\pi_A$ consists of the nonsingleton blocks of $\pi$ and the singletons +in $A$; its label set is $B(\pi)\cup A$. We use +$f_D(t,Z_D)=\prod_{i\in D}f(t,z_i)$, with $f_\varnothing=1$. + +\begin{definition}[Patternwise centered marginal]\label{cent:definition} +The centered marginal on pattern $\pi$ is +\begin{equation}\label{cent:definition-eq} + E_\pi^\eps(t,Z_R)=\chi_\pi^\eps(Z_R) + \sum_{A\subset S(\pi)}(-1)^{|S(\pi)\setminus A|} + f_{S(\pi)\setminus A}(t,Z_{S(\pi)\setminus A}) + \widetilde F_{B(\pi)\cup A}^\eps(t,Z_{B(\pi)\cup A}). +\end{equation} +We set $E_\varnothing^\eps=1$. A nonsingleton block is always retained +as a whole; its individual labels are not centered. +\end{definition} + +The indicator in \eqref{cent:definition-eq} belongs to the full array. +For example, deleting a singleton $i$ from a lower marginal does not +remove the conditions $|x_i-x_j|>\eps$ for the other displayed labels. +These conditions multiply the lower marginal and the independent +factor $f(t,z_i)$. + +\begin{proposition}[Centering and inversion]\label{cent:identity} +For every $\pi$ and almost every $Z_R$, +\begin{equation}\label{cent:inverse} + \chi_\pi^\eps\widetilde F_R^\eps(t) + =\chi_\pi^\eps + \sum_{A\subset S(\pi)}f_{S(\pi)\setminus A}(t) + E_{\pi_A}^\eps(t). +\end{equation} +Both formulas hold for arbitrary factorial densities, without a +smallness or factorization assumption. +\end{proposition} +\begin{proof} +On $\{\chi_\pi^\eps=1\}$ every restricted pattern $\pi_A$ has indicator +one. Substitute \eqref{cent:definition-eq} on the right of +\eqref{cent:inverse}. For $D\subset S(\pi)$, the coefficient of +$f_{S(\pi)\setminus D}\widetilde F_{B(\pi)\cup D}^\eps$ is +\[ + \sum_{D\subset A\subset S(\pi)}(-1)^{|A\setminus D|} + =(1-1)^{|S(\pi)\setminus D|}. +\] +Only $D=S(\pi)$ survives. Off that pattern both sides vanish. +\end{proof} + +\begin{corollary}[Collapsing the pattern sum]\label{cent:pattern-collapse} +For $D\subset R$ let +\begin{equation}\label{cent:singleton-deletion} + \iota_D^R(Z_R) + =\prod_{i\in D}\prod_{j\in R\setminus\{i\}} + \1_{\{|x_i-x_j|>\eps\}},\qquad + \mathcal E_R=\sum_{\pi\in\mathcal P(R)}E_\pi, +\end{equation} +where $\mathcal P(R)$ is the set of partitions of $R$. Then +\begin{align} + \mathcal E_R + &=\sum_{D\subset R}(-1)^{|D|}\iota_D^R f_D + \widetilde F_{R\setminus D}, + \label{cent:collapsed-centering}\\ + \widetilde F_R + &=\sum_{D\subset R}\iota_D^R f_D + \mathcal E_{R\setminus D}. + \label{cent:collapsed-inverse} +\end{align} +Thus a cut carries subset choices, but no additional factor equal to +the number of partitions. For arbitrary signed families $G$ and $e$ +with each $e_\pi$ supported on its pattern, write +\[ + \|G\|_{1,\leq U}=\max_{0\leq r\leq U}\|G_{[r]}\|_1, + \qquad + \|e\|_{\pi,1,\leq U} + =\max_{0\leq r\leq U}\sum_{\pi\in\mathcal P([r])}\|e_\pi\|_1. +\] +Let $\mathsf M_t$ be the transform \eqref{cent:definition-eq} and +$\mathsf I_t$ its inverse \eqref{cent:inverse}, summed over full +patterns. They satisfy +\begin{equation}\label{cent:centering-norm} + \|\mathsf M_tG\|_{\pi,1,\leq U}\leq2^U\|G\|_{1,\leq U}, + \qquad + \|\mathsf I_te\|_{1,\leq U}\leq2^U\|e\|_{\pi,1,\leq U}. +\end{equation} +\end{corollary} +\begin{proof} +At a fixed array exactly one full pattern occurs. Its singleton +labels are exactly the labels which may belong to a set $D$ with +$\iota_D^R=1$. Deleting these labels leaves the unique restricted +pattern. Summing \eqref{cent:definition-eq} and +\eqref{cent:inverse} therefore gives +\eqref{cent:collapsed-centering}--\eqref{cent:collapsed-inverse}. +For disjoint $D,A$ one can also check inversion directly from +\[ + \iota_D^R\,\iota_A^{R\setminus D}=\iota_{D\cup A}^R +\] +and the binomial cancellation in Proposition~\ref{cent:identity}. +Because pattern supports are disjoint, the sum of the patternwise +absolute values is the absolute value of their assembled density. +Taking absolute values in the collapsed formulas, dropping +$\iota_D^R\leq1$, and integrating each deleted factor using +$\int f(t)=1$ proves \eqref{cent:centering-norm}. +\end{proof} + +At time zero, the insertion formula of Proposition~\ref{pre:initial} +gives $E_\pi^\eps(0)=0$ if $B(\pi)\ne\varnothing$: every term in +\eqref{cent:definition-eq} retains all nonsingleton blocks, hence +contains a close pair on the full pattern and has zero initial +factorial density. The full-array indicator remains in place when +singleton labels are deleted. For a singleton pattern with $k$ labels, +the separated-array estimate of that proposition gives +\begin{equation}\label{cent:initial-use} + |E_\pi^\eps(0,Z_R)|\leq(C\eps^c)^k f_{0,R}(Z_R). +\end{equation} +In particular, this small initial factor is attached to each retained +singleton slot, rather than to each layer through which it is carried. + +\subsection{An explicit one-root reference sum} + +We next construct the signed binary histories against which an +independently evolving singleton family will cancel. Their construction +determines the exact pairing in Proposition~\ref{cent:cancellation}. +The separate comparison with $f(t)$ bounds the residual left by this +replacement at one fixed Gaussian exponent. + +Fix $us_1>\cdots>s_n>u, + \qquad a_i\in\{1,\ldots,i\},\quad \sigma_i\in\{+,-\}. +\] +At $s_i$ label $i+1$ is placed at +$x_{a_i}(s_i)+\eps\omega_i$, with velocity $w_i$; write +$g_i=w_i-v_{a_i}(s_i+)$. Here $\omega_i\in\mathbb S^2$ and +$g_i\cdot\omega_i>0$. The plus choice follows the isolated two-body +flow backward from this outgoing sphere crossing until the incoming +sphere crossing, then continues the two lines freely. The minus choice +continues both lines freely through the ball. Its sign is $-1$. +All other lines move freely while this pair is being resolved. +The contact time is an integration parameter, not a force-system cut: +the new label also has its outgoing free segment from $s_i$ to the +\emph{upper boundary of its insertion cell}. Its formal lifetime ends +at that cell boundary, where it is separated from the other labels. +That outgoing segment is a displayed trajectory in every avoidance +test, for both signs. The free passage has duration +$2\eps(g_i\cdot\omega_i)/|g_i|^2$; the true passage has the duration +controlled by Lemma~\ref{pre:duration}. + +Keep only parameters for which each scheduled pair passage finishes +before the earlier boundary of its insertion cell, no two births of +this tree occupy one cell, and there are no other contacts between any +of its displayed lines. At every cut at which a line starts or stops, +require separation from the other displayed lines. In particular the +array at $u$ consists of singleton blocks. Denote this indicator by +$\mathscr S_{n}^{\eps,J}$. It is zero if any scheduled passage fails to +exit. The resulting bottom array is $Z_{n+1}^{\eps}(u)$. +These are \emph{reference histories}: the free negative passage is a +term of a signed identity, not a modification of the particle process. +A one-root displayed family satisfying these conditions is called +\emph{simple}. + +The measure in this representation is +\begin{equation}\label{cent:tree-measure} + d\lambda_n + =\prod_{i=1}^n ds_i\,dw_i\,d\omega_i\, + (g_i\cdot\omega_i)_+ . +\end{equation} +Surface measure is the ordinary area measure on $\mathbb S^2$. +To see the Jacobian directly, let $c_i$ be the upper boundary of the +insertion cell. On its outgoing segment the new label's relative +position at $c_i$ is +\[ + x_{i+1}(c_i)-x_{a_i}(c_i) + =\eps\omega_i+(c_i-s_i)(w_i-v_{a_i}(s_i+)). +\] +The normal component of the derivative with respect to $s_i$ is +$-g_i\cdot\omega_i$, and the two tangential derivatives have area +factor $\eps^2$. Thus the spatial Jacobian is +$\eps^2(g_i\cdot\omega_i)_+$. Restricting to the first backward +crossing makes this parametrization one-to-one, up to the null +set of tangencies. Hamiltonian volume preservation supplies the +remaining phase coordinates for the plus sign; the minus sign uses +free transport. The factor $\eps^2$ is canceled by +$\mu=\eps^{-2}$. Birth labels are in chronological order, so there is +no additional $1/n!$ in \eqref{cent:tree-measure}; the ordered time +domain supplies the factorial in estimates. Define +\begin{equation}\label{cent:simple-sum} + \mathscr G_{u,t}^{\eps,J,K}(z) + =\sum_{0\leq n0$, depending only on the fixed +bounds in Proposition~\ref{pre:envelopes}, for which the following +holds. Suppose $b\leq b_*$, $K(\eps)\to\infty$, and +$J(\eps)\to\infty$ is a fixed power of $1+|\log\eps|$. +Then +\begin{equation}\label{cent:residual-def} + r_{u,t}^{\eps}=f(t)-\mathscr G_{u,t}^{\eps,J,K},\qquad + \sup_{\substack{0\leq u\leq t\leq T\\t-u\leq b}} + \|r_{u,t}^{\eps}\|_{\mathrm{Bol},\beta_0} + =:\eta_\eps\longrightarrow0. +\end{equation} +All the reference sums and residuals have uniformly bounded norms at +this one fixed exponent. The proof uses only short-layer tree +summability and two-body scattering. +\end{proposition} +\begin{proof} +We give the majorant and the limiting argument separately. Put +$p=n+1$ and +$\mathcal E=|v|^2+\sum_{i=1}^n|w_i|^2$. At every insertion sphere the +pair potential is zero. Elasticity consequently gives +\begin{equation}\label{cent:tree-energy} + \sum_{k=1}^p|v_k(u)|^2=\mathcal E. +\end{equation} +Intermediate pair speeds are bounded by $C(\mathcal E+p)^{1/2}$ by +stability. These bounds hold for both signs. At a birth, summing the +flux over possible parents, with unused parent choices assigned zero, +gives +\begin{equation}\label{cent:parent-sum} + \sum_{a\ \mathrm{available}}\int_{\mathbb S^2} + ((w-v_a)\cdot\omega)_+\,d\omega + \leq Cp\bigl(1+|w|^2+\mathcal E/p\bigr)^{1/2}. +\end{equation} +The inequality follows from +$\sum_a|v_a|\leq\sqrt{p\sum_a|v_a|^2}$. +It may be used successively, since its right side no longer depends +on the chosen parents or the angular variables. The product satisfies +\begin{equation}\label{cent:velocity-product} + \prod_{i=1}^n\bigl(1+|w_i|^2+\mathcal E/p\bigr)^{1/2} + \leq C^p(1+\mathcal E/p)^{p/2}. +\end{equation} +Indeed the arithmetic--geometric mean inequality bounds the product +by the $n$th power of its arithmetic mean, and +$\sum_i|w_i|^2\leq\mathcal E$. + +Let $M$ bound both the uniform Gaussian norm and the summed spatial +Gaussian norm of $f(u)$ at exponent $\beta$. Use the spatial envelope +on the continued label $1$ and the uniform bound on all other bottom +labels. Every displacement of that label has length at most +$Cb(\mathcal E+p)^{1/2}$. Translation of its summability boxes therefore +costs at most $C(1+b\sqrt{\mathcal E+p})^3$: any unit box enlarged by +that radius is covered by this number of unit boxes. To handle the velocity supremum without interchanging it with the +spatial sum, first split the root velocities into shells +$m\leq|v|0$ smaller than that margin, +\begin{equation}\label{cent:moment-bound} + \sup_v e^{-\delta|v|^2} + \int_{\R^{3n}} e^{-\delta\sum_i|w_i|^2} + (1+\mathcal E/p)^{p/2}(1+b\sqrt{\mathcal E+p})^3\,dW + \leq C^p. +\end{equation} +To verify this, use +$1+y\leq C_\delta e^{\delta y}$ with $y=\mathcal E/p$, +choose its exponent smaller than the remaining Gaussian exponent, +and absorb the fixed cubic polynomial with another fixed part of +that exponent. The resulting Gaussian integral is $C^n$; its root +supremum is finite. Its Gaussian velocity-shell version, needed in +the preceding spatial estimate, has the same $C^p$ bound. Adjusting +$C$ includes the finitely many cases $p\leq3$. + +The ordered simplex has volume $(t-u)^n/n!$, and +$p^n/n!\leq e^{2p}$. Equations +\eqref{cent:tree-energy}--\eqref{cent:moment-bound}, including the +$2^n$ sign choices, prove the absolute estimate +\begin{equation}\label{cent:geometric-tail} + \left\|\sum_{a,\sigma}\int + \mathscr S_n^{\eps,J} + f_p(u,Z_p^\eps(u))\,d\lambda_n\right\|_{\mathrm{Bol},\beta_0} + \leq C_0(C_1b)^n, +\end{equation} +where the sum inside the norm is understood with absolute integrands. +Choose $b_*$ so that $C_1b_*<1/2$. The identical estimate holds for +instantaneous scattering trees. + +Fix now $n$ and truncate the velocity and relative insertion parameters +to a compact set. The Gaussian majorant just proved makes the omitted +part arbitrarily small, uniformly in $u,t$ and $\eps$. Remove incoming +parameters whose unscaled residence time exceeds $D$, whose relative +speed is too small, or which belong to the null exceptional sets of +Proposition~\ref{pre:scattering}. Lemma~\ref{pre:duration} controls the +residence-time cutoff. The omitted flux tends to zero as +$D\to\infty$ and the other cutoffs are removed. On the retained set a +true or free passage has length $O(\eps D)$ and displacement +$O(\eps D)$. The set of insertion times lying within $C\eps D$ of a +cell boundary has measure at most $C_nJ\eps D$; two insertion times +in one cell cost $C_n/J$. Both quantities vanish, taking $D$ fixed +first. Shortened endpoint cells obey the same estimates. + +Here is the needed null-set assertion for other meetings. Parameterize +each later birth by its velocity relative to its current parent in +laboratory axes. This is a triangular translation of the $w_i$ +variables with Jacobian one. By Galilean invariance, later relative +increments do not depend on an earlier common velocity boost. +At the first divergence of two distinct rays, keep the relative +speed and the scalar scattering parameters fixed, and rotate the +relative incident velocity and impact configuration together. +The separating relative velocity has nonzero length and a spherical +direction with absolutely continuous area measure. On any subsequent +interval between scheduled births, the relative position of the two +rays has the form +\begin{equation}\label{cent:meeting-null} + (s-s_*)\,\xi+\Gamma(s), +\end{equation} +where $s_*$ is their divergence time, $\xi$ is that relative velocity, +and $\Gamma$ is a fixed affine function after the later relative +increments and times have been fixed. There are finitely many such +intervals. For $s\ne s_*$, equality to zero confines the direction of +$\xi$ to a finite union of curves on a sphere, which has area zero. +The interval immediately after divergence is free of other scheduled +births and its nonzero relative speed separates the two rays. +For a true scattering both outgoing rays must be included when +identifying this divergence; their directions are not asserted to be +independent. This argument also covers the free sign. Fubini's theorem +therefore excludes every unprescribed meeting for fixed $n$. Relative +positions and this exceptional set are independent of the root +position and root velocity, by translation and Galilean invariance. +Thus the compact exceptional estimates are uniform for the root +supremum in the spatial norm. + +Away from these exceptions, the microscopic paths converge to the +instantaneous paths, with the same velocities outside the short pair +arcs. All input positions differ by $O_n(\eps D)$, while their input +velocities agree exactly. The outgoing segments between a contact +and its upper cell boundary have length $O(b/J)$ and are also checked +for unwanted contacts. Their length tends to zero, so the same +null-set argument covers their avoidance indicators. Proposition~\ref{pre:envelopes} and the +spatial mean-value inequality give convergence of the products of +input densities in the summed spatial norm, uniformly in $u,t$. +Dominated convergence in the compact parameters, followed by removal +of the compact cutoffs, proves convergence of every fixed term. +For uniformity in the layer length, use backward elapsed times in the +fixed simplex $0<\tau_1<\cdots<\tau_n0$ and $M$ are fixed independently of the number +of layers. The factors $\eta_i$ record small initial or residual +inputs. Put $\eta_\Gamma=\prod_i\eta_i$. +One can take $g_i$ to be the Gaussian-weighted velocity supremum of +$u_i/\eta_i$ when $\eta_i>0$; if $\eta_i=0$, the term is zero. Thus +\eqref{eq:hist-source-envelope} follows from the norm used in +Propositions~\ref{pre:envelopes} and~\ref{cent:residual}. + +Before the contact changes of variables, the absolute integral for +$\Gamma$ is the integral of the product of its absolute source factors over +$\mathcal D_\Gamma$ in the $p$ source phase coordinates, multiplied by +the inherited factorial coefficients and the activity factor +$\mu^{p-h_0}$. A test of the top phase points with absolute value at most +one may be included; those phase points are integrated as well. +An initial centered input is first replaced by its pointwise product +bound. We never divide this integral by the mass of $\mathcal D_\Gamma$. + +For each label retain a position and a velocity coordinate at all times +in $[0,T]$. Before its source and after its departure freeze both +coordinates. On its lifetime evolve it with its prescribed force group. +At a fixed time this gives $p$ phase coordinates, including the frozen +ones. Denote them by $\widehat Z(t)$, and write +\begin{equation} + S=\sum_{i=1}^p|v_i^{\rm source}|^2, + \qquad \mathcal E=S+2Bp. + \label{eq:hist-source-energy} +\end{equation} +The quantity $S$ is a function of the whole history coordinates. It is +not, in general, a quadratic function of the departure velocities. + +\begin{lemma}[Augmented volume and the once-counted Gaussian] +\label{hist:gaussian} +For every fixed prescription $\Gamma$, the map from its $p$ source +coordinates to $\widehat Z(t)$ is one-to-one and volume preserving +outside null sets. On $\mathcal D_\Gamma$, +\begin{equation} + \sum_{i=1}^p|\widehat v_i(t)|^2\leq S+2Bp, + \qquad + \sup_{0\leq t\leq T}\sum_{\substack{i:\,i\text{ active at }t}}|v_i(t)|^2 + \leq S+2Bp. + \label{eq:hist-mixed-energy} +\end{equation} +In particular, with $V(t)=(\widehat v_i(t))_{i=1}^p$, +\begin{equation} + e^{-\beta_0 S} + \leq e^{\beta_0Bp} + e^{-\beta_0|V(t)|^2/2}e^{-\beta_0 S/2}. + \label{eq:hist-split-gaussian} +\end{equation} +The same assertions hold at the last time, when the mixed coordinates +consist of the top coordinates and the departure coordinates of every +other label. The exponent in \eqref{eq:hist-split-gaussian} is not +changed at a layer cut. +\end{lemma} + +\begin{proof} +Between two consecutive mesh times the augmented map is a product of +Hamiltonian maps and identity maps. Each factor is invertible and +volume preserving by Proposition~\ref{pre:flow}. Injection merely changes +which of the already present coordinates is active; departure changes +which coordinate is frozen. Both operations are identity maps on the +augmented space. A fixed prescription therefore gives a finite +composition of volume preserving maps. Coincident singular positions +at a switch form a null set; apply the argument first on compact sets +avoiding them and then exhaust the domain. + +For the energy identity, keep the Hamiltonian of every force group. +At the earlier endpoint of a cell its distinct force groups are unions +of distinct whole initial polymers. Their cross potentials are zero. +Thus passing from the preceding force partition to this partition +changes no interaction energy. At the later endpoint the root group +$A$ splits into its surviving labels $R$ and $A\setminus R$. +The indicator $e_I^R(A)$ makes their cross potentials zero. The other +force groups depart in their entirety. Store the Hamiltonian of each +departing group at that time. At a layer cut the singleton sources +are inserted on the restored polymer pattern, so their cross +potentials with each other and with all existing groups in the new +force system are zero. A nonsingleton polymer is never split by this +operation. + +At the top of a terminal witness all final groups are stored intact, +including their distinguished anchors. No split into $R$ and +$A\setminus R$ is made there, and no endpoint isolation is used. +The mixed tuple still has one terminal velocity for every label. +This proves the same identity for the positive witness variant. + +There is a relevant distinction at a late cell endpoint. A detached +group may geometrically overlap a different-generation group or the +root group. Those groups had separate Hamiltonians throughout the +cell. Their geometric overlap introduces no term into either +Hamiltonian, and they are not merged when the detached group departs. +Consequently it causes no missing cross-energy term in this calculation. + +Starting with the empty active system, conservation on open cells and +the preceding switch identities give, at every time $t$, +\begin{equation} + \frac12\sum_{i:\,\text{source}_i\leq t}|v_i^{\rm source}|^2 + =\sum_{\substack{A:\,A\text{ current at }t}}\mathscr H_A(t) + +\sum_{\substack{D:\,D\text{ departed by }t}}\mathscr H_D(\text{departure}_D). + \label{eq:hist-energy-ledger} +\end{equation} +The current and departed groups partition the injected labels. Stability +on each group therefore subtracts at most $B$ times the number of those +labels, not $B$ times that number at every cell. Adding the kinetic +energy of future sources to both sides proves +\eqref{eq:hist-mixed-energy}. Splitting the source exponential into +two equal factors and using the first inequality gives +\eqref{eq:hist-split-gaussian}. +\end{proof} + +\begin{remark}[Positive potential energy is retained] +\label{hist:positive-potential} +The reverse inequality $S\leq |V(T)|^2+Cp$ is not asserted. An +interacting group can depart with large positive potential energy and +small kinetic energy. Bounds on intermediate speeds consequently +use $\mathcal E=S+2Bp$. The last factor in +\eqref{eq:hist-split-gaussian} pays for functions of this source energy. +This distinction is necessary for singular repulsive cores and is +also useful for bounded potentials. +\end{remark} + +Here is a convenient precise way of using that last factor. Divide +the domain into the shells +\begin{equation} + rp\leq S<(r+1)p,\qquad r=0,1,\ldots,\qquad + E_r=(r+1+2B)p. + \label{eq:hist-shells} +\end{equation} +On such a shell every active or mixed squared-velocity sum is at most +$E_r$. After reserving any fixed positive part of the last exponential +in \eqref{eq:hist-split-gaussian}, the shell has an extra factor +$e^{-c rp}$. Parent choices may now be summed using the same scalar +$E_r$ on every branch. This is also a rigorous implementation of +expressions of the form +\begin{equation} + C^p\sup_{s\geq0}\{W(s)e^{-cs}\}. + \label{eq:hist-energy-payment} +\end{equation} +One must not first replace $S$ by a terminal kinetic energy, or pull its +branch-dependent value through a parent sum. + +\subsection{Translation charts on a cell} + +The following construction is used in two versions. The fine version +preserves the original tensor-factor parent of a singleton addition; +it is used for exponential counting. The coarse version is permitted +to record every geometric contact and is used when an extensive power +of $\varepsilon$ pays its larger combinatorial cost. Neither version +alters a force in the prescribed history. + +\begin{lemma}[First-contact translation charts] +\label{hist:pinning} +Consider one cell of length $\Delta$ with $q$ prescribed terminal root +labels and $m$ extra labels. Products of simultaneous cell diagrams +are allowed. Keep all root coordinates and all terminal velocities +fixed. On the domain where every extra label is linked by the +prescribed cell contacts to a root, the extra terminal positions are +covered by finitely many charts with the following properties. + +\begin{enumerate} +\item Each extra label costs one three-dimensional pin. A pin is +either a relative endpoint displacement in $B(0,\varepsilon)$, with +Jacobian one, or a contact time $t$, a normal $\omega\in\mathbb S^2$, +and the Jacobian +\begin{equation} + \varepsilon^2 + \bigl|(v_i(t)-v_j(t))\cdot\omega\bigr|\,dt\,d\omega. + \label{eq:hist-contact-jacobian} +\end{equation} +The velocities in this formula are those of the separately computed +translation components immediately before their merger. +\item The number of discrete pin charts, including a cell's generation +choices, is at most $(C(p+1))^{Cm}$. No partition of the roots is +introduced. The full spatial Jacobian is the product of the pin +Jacobians. No norm of a derivative of a many-particle flow occurs. +\item If $m=1$ globally in that cell and the extra label is separated +from its own roots at the late endpoint, its first backward contact +with its own tensor factor gives the exact chart +\begin{equation} + y=x_a(t)+\varepsilon\omega+(b_I-t)w, + \qquad + dy\,dw=\varepsilon^2|(w-v_a(t))\cdot\omega| + \dd t\,\dd\omega\,\dd w . + \label{eq:hist-single-chart} +\end{equation} +Here $b_I$ is the late endpoint, $y,w$ are the extra terminal position +and velocity, and the root trajectory is computed without that extra +label. Apart from the full or virtual choice in +Remark~\ref{rem:cell-single}, there is no multiplicity independent of +the choice of $a$. +\end{enumerate} +On the energy shell \eqref{eq:hist-shells}, multiplication by +$\mu^m$ and integration over the $m$ pin coordinates gives the coarse +bound +\begin{equation} + (Cp)^{Cm}(1+E_r)^{m/2}(\Delta+\varepsilon)^m. + \label{eq:hist-multi-cell} +\end{equation} +The convention $p+1$ can replace $p$ in this and subsequent bounds at +zero size. +\end{lemma} + +\begin{proof} +At the late endpoint initially tie all root labels into one anchored +translation component. This is a convention about fixed coordinates, +not an assertion that the roots interact. Process cross-component +endpoint overlaps, choosing a pair by a fixed label order. An overlap +fixes the relative translation by an offset of length at most +$\varepsilon$. Ties between two already anchored pieces fix no new +coordinate and are ignored. Repeat until every remaining floating +component is separated from all other components at that endpoint. + +Every component carries the original force partition restricted to its +labels. Translate the component as a whole. Its restricted dynamics +commutes with that translation. Starting at the late endpoint, evolve +all these components backward. Until their first intercomponent +geometric contact, they exert no cross force and these restricted +motions equal the original prescribed motions. At that contact fix +one relative component translation, merge the two translation +components, and continue with the restricted original force partition +on their union. Physical interactions are included when the original +partition includes them; a virtual contact only merges translation +coordinates. Interactions already internal to a component need not +be enumerated. Because the prescribed contact graph links every +extra label to roots, the algorithm eventually anchors every extra +coordinate. + +For two floating components write their paths as +$a_A+q_i^A(t)$ and $a_B+q_j^B(t)$, where all internal coordinates are +fixed. At the selected contact, +\[ + a_B-a_A=\varepsilon\omega+q_i^A(t)-q_j^B(t). +\] +The derivative in time is the relative velocity; the other two +derivatives span the tangent plane of the sphere of radius +$\varepsilon$. Their determinant is +\eqref{eq:hist-contact-jacobian}. A contact with the anchored component +uses the same formula with its position already fixed. Endpoint pins +are linear changes from a relative translation to an offset and have +determinant one. + +At each stage keep the previously chosen internal and pin coordinates +unchanged. The next change acts only on one still free relative +translation. Thus the changes are triangular in merger order. Large +derivatives with respect to earlier pin coordinates enter only +off-diagonal blocks. This proves the product assertion, including +for singular potentials by compact exhaustion of the regular flow +domains. + +There are at most $m$ mergers that remove an extra translation degree +of freedom. Choosing their pairs, representatives, orders, and +endpoint/contact types costs at most $(Cp)^{Cm}$. The extra-label +partitions and generations have the same type of bound by +Proposition~\ref{cell:expansion}. Roots have already been anchored +together and require no partition sum. On each original first-contact +domain the construction recovers the starting configuration. For an +upper bound the area formula permits summing the images of all these +charts. It does not require a uniform bound on repeated physical +contacts. Tangencies lie in zero-Jacobian portions of the contact +maps, and deterministic tie rules suffice at multiple choices. + +When there is one extra label, before its first contact with its own +factor that label is free and the old roots follow their isolated +root flow. This proves \eqref{eq:hist-single-chart}. An earlier +contact with a different tensor factor exerts no force and hence does +not invalidate this original-factor chart. Its indicators remain +part of the term. The force group and original factor are determined +by the chosen original parent and the full/virtual choice. + +Finally, on the valid energy shell every velocity at a selected contact +has magnitude at most $E_r^{1/2}$. A contact pin therefore integrates +to at most $C\varepsilon^2 E_r^{1/2}\Delta$, whereas an endpoint pin +costs $C\varepsilon^3$. After $\mu^m=\varepsilon^{-2m}$, +\eqref{eq:hist-multi-cell} follows. The energy bound is used on the +original chart domain before other restrictions are dropped; no +energy bound on an artificially reconstructed off-domain trajectory +is presumed. +\end{proof} + +\begin{remark}[Coarse global pins and relative variables] +\label{hist:coarse-chart} +For a geometric estimate one may run the same algorithm with all +currently displayed root paths, across tensor factors, in the anchored +component. A singleton then has its primary pin at its earliest +backward contact with any displayed path. If this is in a different +factor, the primary contact is virtual and its later required +original-factor contact is still an indicator. This is only a new +coordinate chart for the same force prescription. + +At a singleton pin put $g=w-v_a(t)$. All root trajectories at $t$ +are determined before this new block is introduced. The map +$w\mapsto g$ has determinant one. Combining it with the normal-to-disk +map gives +\begin{equation} + \mu\,dy\,dw=|g|\,dt\,dg\,d^2\zeta, + \qquad \zeta\in g^\perp,\quad |\zeta|<1, + \label{eq:hist-disk-chart} +\end{equation} +on the incoming hemisphere. This identity is a contact-flux identity, +not an inverse differential cross-section. A measurable orthonormal +frame identifies each disk with a fixed unit disk and preserves area. + +These substitutions are triangular in backward cell order. A whole +multiple-addition cell is one coordinate block; all its variables are +introduced before any earlier cell is computed. Consequently an +ordinary parent state is determined by preceding blocks, never by +the fresh velocity or impact variable at that pin. This remains true +with arbitrarily complicated old-root dynamics. + +On $S\leq Rp$, after fixing source summability boxes, all velocity, +root-position, and remaining relative-coordinate ranges have +polynomial size in $p$. Rescale endpoint offsets by $\varepsilon$. +The preceding determinant calculation then bounds the activity-normalized +chart density, in the product of contact blocks and remaining auxiliary +variables, by $(Cp)^{Cp}$ for a fixed cell prescription. Summing +discrete cell prescriptions replaces this by $P^{C_{A,Q,L}p}$ when +$p\leq P^A$ and $J=\lceil P^Q\rceil$. This coarse bound is used only +with an extensive geometric gain. Reassigning a primary pin across +factors can lose the original factor choice; it is not used in the +exponential parent count below. Section~\ref{sec:exposure} records the +additional exclusions that make the coarse charts causal for selected +terminal trees. +\end{remark} + +\subsection{Spatial anchors and the finite exponential bound} + +There is one independent three-dimensional velocity coordinate for +each formal label: the terminal velocity of a top or the departure +velocity of an extra. Lemma~\ref{hist:pinning} changes position +coordinates only, before the optional relative substitution. Thus +these $3p$ velocity coordinates, denoted by $V$, have a common Gaussian +majorant in all the original-factor charts. + +For spatial integration use the cubes +$Q_k=k+[-1/2,1/2)^3$ and numbers +$a_{i,k}=\sup_{|x-k|\leq1}g_i(x)$. Expand each spatial source envelope +over these cubes. Their coefficients sum to at most $M$ per label. +For a fixed list of boxes, a top label and its unique source position +satisfy +\begin{equation} + \sum_{\substack{i:\,i\text{ a top}}}|x_i^{\rm top}-x_i^{\rm source}|^2 + \leq T\int_0^T\sum_{\substack{i:\,i\text{ a top, active at }t}}|v_i(t)|^2dt + \leq T^2E_r. + \label{eq:hist-top-displacement} +\end{equation} +There is no positional jump at a change of force group. After all +extra positions have been pinned, the remaining top positions +therefore range in a ball in $\mathbb R^{3h_0}$ of squared radius at +most $C_T(E_r+h_0)$ about their source boxes. Its volume is bounded by +\begin{equation} + \mathsf S(h_0,E_r) + =\left[C_T\left(1+\frac{E_r}{h_0}\right)^{3/2}\right]^{h_0}. + \label{eq:hist-spatial-factor} +\end{equation} +The other spatial box indicators may be discarded. Summing all box +coefficients costs $M^p$. In particular, for $h_0=o(p)$ and +$E_r=O(p)$ the logarithm of \eqref{eq:hist-spatial-factor} is $o(p)$. + +Here and below cell occupancy means the total occupancy across all +simultaneous tensor factors. Let $m_*$ count labels in cells with at +least two additions, and let $o_*$ count singleton endpoint pins. +The remaining $n_*=p-h_0-m_*-o_*$ additions are called ordinary pins. +The interlayer retained-slot choices are either fixed, or retained +explicitly as their binomial selection factors. They are never +replaced by $2^{CH}$. + +\begin{proposition}[Finite-term bound with its time factors] +\label{hist:basic} +Fix $A<\infty$ and the finite number $L$ of layers. There is a +numerical exponent $d$ in the cell estimates such that, if +$J=\lceil P^Q\rceil$ with $Q>2dA$, all the following bounds hold +uniformly for $1\leq p\leq P^A$, for sufficiently small $\varepsilon$. +Put $n_j$ for the total additions in layer $j$, $h_j$ for its incoming +roots, and $q_j=h_j+n_j$. After summing cell choices, cell diagrams, +and original parents, the absolute integral on the shell +\eqref{eq:hist-shells} is bounded by +\begin{align} + &\eta_\Gamma C^p e^{-c rp}\mathsf S(h_0,E_r) + \chi_\varepsilon^{m_*+o_*}(r+2)^{(m_*+o_*)/2} + \prod_{j=1}^L\frac{b^{n_j}q_j^{n_j}}{n_j!} + \nonumber\\[-2pt] + &\hspace{22mm}\times + \int_{\mathbb R^{3p}} + \prod_{\substack{i:\,i\text{ an ordinary pin}}} + \left(1+|w_i|^2+\frac{E_r}{q_{j(i)}}\right)^{1/2} + d\gamma(V), + \label{eq:hist-finite-bound} +\end{align} +where $d\gamma$ is a normalized product Gaussian of one fixed exponent, +the integral may retain the common indicator $|V|^2\leq E_r$, +and one may take +\begin{equation} + \chi_\varepsilon + = C P^{dA}\left(J^{-1/2}+\frac{\varepsilon J}{b}\right). + \label{eq:hist-mesh-gain} +\end{equation} +Harmless increases of $d$ absorb the fixed extra powers occurring in +the finite diagrams. Constants $C,c$ in +\eqref{eq:hist-finite-bound} do not depend on $L$ or on later rarity +cutoffs. The threshold of sufficiently small $\varepsilon$ may +depend on the fixed $b=T/L$. + +In particular, after summing layer occupancies and retained slots the +total finite integral of order $p$ is at most +\begin{equation} + \eta_\Gamma C_L^p. + \label{eq:hist-coarse-exponential} +\end{equation} +Its portion with $S>Rp$ is at most +\begin{equation} + \eta_\Gamma C_L^p e^{-cRp}, + \label{eq:hist-source-tail} +\end{equation} +after changing $c>0$ and taking $R$ above a fixed constant. +These are estimates of finite independently enveloped terms, not +assertions about the actual process. +\end{proposition} + +\begin{proof} +For a set of at most $q$ old labels with squared-velocity sum at most +$E$, direct Cauchy--Schwarz gives +\begin{equation} + \sum_a|v_a-w|^2\leq2q|w|^2+2E,\qquad + \sum_a|v_a-w| + \leq Cq\left(1+|w|^2+\frac E q\right)^{1/2}. + \label{eq:hist-parent-sums} +\end{equation} +The root state at an ordinary pin is determined before its parent and +new velocity are chosen. On a fixed source-energy shell, retain the +adapted gate that this root state has squared-velocity sum at most +$E_r$. Every original term in the shell satisfies that gate. Sum +the last parent first using \eqref{eq:hist-parent-sums}, then the +preceding one, and continue. The resulting factors are independent +of the parent states. This proves the ordinary-pin product in +\eqref{eq:hist-finite-bound}. Remaining future admissibility indicators +can now be dropped. The majorant in $V$ is the fixed Gaussian of +Lemma~\ref{hist:gaussian}, so these integrations introduce no additional +per-layer exponent loss. + +For globally ordinary cells the first contact times are chronologically +ordered. Summing their disjoint cell boxes is an integral over a +subset of the corresponding ordered simplex. Hence it supplies +$b^{n}/n!$, not $J^n$ times this quantity. A maximal stretch of empty +cells simply composes the same prescribed root flows. It has one +deterministic map and no independent empty-cell choice. + +If a cell has $m\geq2$ additions, use +\eqref{eq:hist-multi-cell}, without a factorial improvement in its +internal contact times. For a total of $m_*$ such participants there +are at most $m_*/2$ occupied cells. Their location choices cost at +most $J^{m_*/2}$ in a layer. The time factors supply +$(b/J)^{m_*}$, since $\varepsilon J\leq b$ eventually. Distributing +the participants among those cells and choosing their pin/generation +data costs $p^{C m_*}$. A missing ordered-time factorial for these +participants is restored with +$n_j!/(n_j-m_j)!\leq p^{m_j}$. On the energy shell, their speed +factors cost at most $p^{m_*/2}(r+2)^{m_*/2}$ times a fixed exponential. +All these polynomial powers are absorbed in $d$ in +\eqref{eq:hist-mesh-gain}. + +A singleton endpoint pin has factor $C\varepsilon$ after its activity +normalization, instead of a contact-time integral. Its cell choice +costs at most $J$. Restoring the time simplex and any parent factor +loses at most a further fixed power of $p$ per such pin. This gives +the second term in \eqref{eq:hist-mesh-gain}. The unordered-extra +factorials of Corollary~\ref{cor:cell-factorial} cancel permutations of +new names; at a singleton no independent factor assignment remains +after choosing its original parent. At multiple cells all remaining +assignments are among the polynomial costs just counted. + +The energy and spatial bounds were proved in +\eqref{eq:hist-split-gaussian}--\eqref{eq:hist-spatial-factor}. +Combining them with these changes of variables proves +\eqref{eq:hist-finite-bound}. Notice that $\chi_\varepsilon\to0$ +uniformly for $p\leq P^A$. No rarity cutoff has entered its exponent. + +For the coarse consequence replace every parent capacity by $p$ and +use the global ordered simplex. For $n=p-h_0\leq p$, +\[ + p^n\frac{T^n}{n!}\leq e^p\max(1,T)^p. +\] +The source shell gives $\sum_i|w_i|^2\leq E_r$ on its original +domain. Alternatively integrating the product Gaussian moments gives +the same bound $C^p(r+2)^{Cp}$ for the speed product and the spatial +factor, since $h_0\leq p$ and +$h_0\log(1+p/h_0)\leq Cp$. Thus the shell sum is bounded by +\[ + C^p\sum_{r\geq0}e^{-c rp}(r+2)^{Cp}\leq C^p. +\] +All retained-slot binomials are at most +$2^{\sum_jq_j}\leq2^{(L+1)p}$, and all other fixed layer choices +cost a further $C_L^p$. This proves +\eqref{eq:hist-coarse-exponential}. Starting the shell sum at +$r\geq R-1$ proves \eqref{eq:hist-source-tail}. Polynomial costs of +special cells have already been paid by their mesh factors before +this shell sum or a later constant-cutoff estimate is used. +\end{proof} + +\subsection{Compact parameters and adapted parent counting} + +For this subsection work on $S\leq Rp$ and put $R_1=R+2B$. +The original-factor charts have an explicit auxiliary product measure. +Normalize their Gaussian velocities and disk areas, choose every +ordinary parent uniformly from a fixed list of $p$ slots, and kill +inactive or otherwise illegal choices by indicators. Root coordinates +and all auxiliary multiple-cell blocks are integrated against the +majorants already used above. A whole such block is revealed before +proceeding to an earlier ordinary cell. Fix the ordered contact times +when discussing a conditional mark. The past of pin $i$ contains all +preceding blocks, its time, and its old-root state, but not its parent +choice or its new $w_i,\zeta_i$. + +Let $G_i$ be the adapted gate that the old-root squared-velocity sum is +at most $R_1p$, together with any other adapted restrictions explicitly +retained. Write $F_i=|v_{i,I_i}-w_i|$, where $I_i$ is the uniform +parent slot. From \eqref{eq:hist-parent-sums}, +\begin{equation} + \mathbb E_{\rm ref}(G_iF_i^2\mid\text{past of }i) + \leq C(1+R_1). + \label{eq:hist-reference-square} +\end{equation} +For any prescribed set of $k$ pins whose marks have conditional +reference probability at most $\eta$, repeated conditional integration +gives a probability at most $\eta^k$. Applying one Cauchy--Schwarz +inequality to the whole product of fluxes, and integrating the last +fresh block first in each factor, proves +\begin{equation} + \mathbb E_{\rm ref}\!\left[ + \prod_iG_iF_i\,\mathbf1_{\{N_{\rm mark}\geq k\}}\right] + \leq [C(1+R_1)]^{n_*/2} + \binom{n_*}{k}^{1/2}\eta^{k/2}. + \label{eq:hist-reference-discard} +\end{equation} +Physical parent velocities need not be independent. What is used is +that they are measurable before their own new block is introduced. +Later admissibility restrictions may be dropped; this does not allow +one to drop a future dependence of a parent state. + +\begin{proposition}[Finite parameter discards] +\label{hist:irregular} +Fix $L$, $A$, $\delta>0$, and $M_0<\infty$. The source-energy cutoff +$R$ and fixed numbers +\[ + U,W<\infty,\qquad \gamma,\kappa>0 +\] +can be chosen so that, for all sufficiently small $\varepsilon$, +the total finite history integral with at least $\delta p$ marked +additions is at most $\eta_\Gamma e^{-M_0p}$. An addition is marked +if it is in a multiple-addition cell, is an endpoint pin, or is an +ordinary pin satisfying any of the following conditions: +\begin{enumerate} +\item its original parent speed exceeds $U$, its added absolute +velocity exceeds $W$, or its relative speed is below $\gamma$; +\item for a genuine non-force-free isolated continuation, either +output increment relative to the incoming parent has magnitude below +$\kappa$; +\item its isolated two-body duration in physical time exceeds +$\varepsilon^{.96}$; +\item its contact time is within $\varepsilon^{.95}$ of a cell +boundary, or it belongs to a pair of birth times separated by less +than $2\varepsilon^{.9}$. +\end{enumerate} +The contribution of $S>Rp$ has the same bound. The constants in this +proposition may depend on $L$; they do not enter the $C$ in +\eqref{eq:hist-finite-bound}. +\end{proposition} + +\begin{proof} +Choose $R$ using \eqref{eq:hist-source-tail}. Given an old-root state +with squared-velocity sum at most $R_1p$, at most $R_1p/U^2$ of its +parent slots have speed above $U$. The conditional reference +probability of choosing one is therefore at most $R_1/U^2$. A large +added absolute velocity has Gaussian probability at most +$Ce^{-cW^2}$. The ball $|w-v|<\gamma$ has reference probability at +most $C\gamma^3$, uniformly in $v$. + +On the remaining compact set let $G_*=U+W$. In the incoming disk +variables, relative velocities and impact parameters have reference +density bounded by a constant with respect to $dg\,d^2\zeta$. +On $\gamma\leq|g|\leq G_*$ this measure is equivalent to incoming +flux. Proposition~\ref{pre:scattering} therefore gives a function +$\omega_\Phi(\kappa;\gamma,G_*)\to0$ as $\kappa\downarrow0$ such +that the second mark has conditional probability at most +$C\omega_\Phi$. The bound is uniform over $|v|\leq U$: translate +$w$ to $g=w-v$, bound its Gaussian density above, and enlarge to the +fixed relative-velocity annulus. Force-free passages are crossings +and are not marked for their zero parent increment. + +Lemma~\ref{pre:duration} gives +\[ + \int_{|g|\leq G_*}\int_{|\zeta|<1} + |g|\,\mathbf1_{\{\tau(g,\zeta)>D\}}\,d^2\zeta\,dg + \leq C_{\Phi,G_*}/D. +\] +Dividing by $|g|\geq\gamma$ shows that the duration mark has +conditional probability at most +$C_{\Phi,G_*}\gamma^{-1}\varepsilon^{.04}$. +This is the duration of the isolated continuation, not an assumption +that an actual many-particle encounter is isolated. An interruption +will be an encounter defect in Section~\ref{sec:exposure}. + +Take $U,W$ large and then $\gamma,\kappa$ small to make the sum of +the first two reference-mark probabilities as small as needed. +Equation~\eqref{eq:hist-reference-discard}, +$\binom{n_*}{k}\leq2^p$, and the finite exponential normalization +give the required bound for an extensive number of these marks. +For fixed $\delta$ any prescribed exponential is obtained by making +their one-step probability sufficiently small. Multiple cells and +endpoint pins have instead the factor +$\chi_\varepsilon^{m_*+o_*}$ from +Proposition~\ref{hist:basic}, which proves their assertion. + +For the time marks use unordered uniform times before sorting in +each layer. A fixed time lies within $\varepsilon^{.95}$ of a cell +boundary with probability at most +$CJ\varepsilon^{.95}/b$. If $k$ birth times have a neighbor at +distance below $2\varepsilon^{.9}$, their closeness graph on the +line has a matching or a disjoint consecutive-pair selection of +size at least a fixed multiple of $k$. For a prescribed disjoint +pair selection the probability is at most +$(C_L\varepsilon^{.9})^{ck}$. There are at most $p^{Ck}$ selections. +Times in distinct layers which are this close lie near their common +boundary and can be charged there. Since $p\leq P^A$ and $J$ is +a power of $P$, these bounds tend to zero with a positive power of +$\varepsilon$ per fixed fraction of marked times. Apply the same +global Cauchy--Schwarz estimate to the fluxes. A finite union bound, +with thresholds reduced by the number of categories, completes the +proof. +\end{proof} + +\subsection{A two-sided volume estimate for crowding} + +Crowding here concerns the displayed paths of a finite term. In +particular, they may belong to different prescribed force groups. +The frozen extension is useful: its positions are continuous at every +fixed lifetime and force-partition switch, and on $S\leq Rp$ their +speeds are bounded by $C_R\sqrt p$. + +\begin{lemma}[A fixed-pivot forest] +\label{hist:pivot} +Fix a time $t_*$, a forest of $r$ edges on the $p$ labels, and +$\rho\geq\varepsilon$. For one fixed force/lifetime prescription +whose required contacts connect every label to a top, the source integral +before multiplication by $\mu^{p-h_0}$, on $S\leq Rp$ with +$|x_i(t_*)-x_j(t_*)|\leq\rho$ on every forest edge is at most +\begin{equation} + \eta_\Gamma C^p p^{Cp}\rho^{3r} + \varepsilon^{2(p-r-h_0)_+}. + \label{eq:hist-pivot-bound} +\end{equation} +More precisely, a stratum with $k$ final translation components has +the factor $\rho^{3r}\varepsilon^{2(p-r-k)}$, with $k\leq h_0$. +There is no factor equal to the number of cells in this geometric +bound. +\end{lemma} + +\begin{proof} +Use the augmented phase coordinates at $t_*$ from +Lemma~\ref{hist:gaussian} and first fix their velocities. Root each +component of the forest at its least label. Replace its positions +by one common translation and the edge differences along that rooted +tree. The linear map has determinant one. The $r$ relative vectors +range in balls of radius $\rho$, at cost $(C\rho^3)^r$. + +Each current translation component carries the original time-dependent +force partitions restricted to its labels, as well as the frozen +coordinates outside their lifetimes. Begin by merging any remaining +cross-component overlaps at the pivot, using endpoint offsets in +$B(0,\varepsilon)$. Next scan \emph{all} geometric contacts of the +extended paths, both forward and backward from $t_*$. This includes +contacts between different physical force systems and contacts of +inactive frozen coordinates. Choose the first remaining contact in +increasing distance $|t-t_*|$, with a deterministic rule for ties, +fix its relative translation, and merge the two translation components. +Recompute the merged component's restricted prescribed motion from +the pivot in both directions. Continue until no cross-component +contact remains anywhere in $[0,T]$. + +The reconstruction has a useful induction. Inside the already +explored two-sided time window, distinct translation components have +had no geometric contact. Hence they exert no physical force, and +their separately computed motions agree with the full prescribed +motion. A newly chosen earliest contact lies on the boundary of an +extension of that window. Merging its two components changes no +motion inside the old window. This statement holds even when +successive mergers occur on opposite sides of the pivot. Interactions +beyond the known window are recomputed with the original force +partitions after the merger. A virtual merger never adds a force. + +The contact Jacobian is again +\eqref{eq:hist-contact-jacobian}; for a frozen coordinate its position +derivative is zero. Its integral over all possible times is bounded +by $C_T\varepsilon^2\sqrt{R_1p}$. Piecewise smooth time intervals +are summed as one time integral. There is no independent endpoint +choice at each mesh time: extended positions are continuous, so an +overlap which becomes physically relevant at a switch was either +already a pivot overlap or was preceded by a geometric sphere +contact. This is why all extended geometric contacts, rather than +only currently interacting pairs, were included in the scan. + +The triangular determinant and area-formula argument of +Lemma~\ref{hist:pinning} applies verbatim. At a fixed-time kink use +the adjacent smooth time intervals; the endpoint itself has zero +three-dimensional contact-parameter measure unless it is covered by +an already counted overlap. Tangent and repeated contacts need no +separate counting. Compact exhaustion avoids singular physical +configurations. There are at most $p-1$ mergers, with at most +$p^{Cp}$ choices of pairs, directions, orders, and contact/overlap +types. + +After the first $r$ forest links, if the final number of components +is $k$, there have been exactly $p-r-k$ further pins. At termination +there are no remaining geometric contacts between these components. +Every required contact belongs to the scanned set, so each final +component contains a top and $k\leq h_0$. Every extra pivot overlap +costs $\varepsilon^3\leq\varepsilon^2$; every other pin has the +$C_T\sqrt{R_1p}\varepsilon^2$ bound just proved. + +It remains to integrate the $k$ free translations. In a final +component all of its source positions have the form $a+d_i$ and +its source velocities are independent of $a$. Choose one source +label in that component. Integrate its $g_i(a+d_i)$ using its +$L^1$ norm and bound the other spatial envelopes by their $L^\infty$ +norms. Both norms are at most $C M$ by +\eqref{eq:hist-source-envelope}. On the original stratum the +restricted component flows equal the full flows; extending these +restricted functions after discarding the stratum indicators +preserves their translation property. Thus all free translations +cost $C^p$. The mixed Gaussian integrates to $C^p$, and the speed +powers cost $p^{Cp}$. This proves the sharper stratum bound and hence +\eqref{eq:hist-pivot-bound}. +\end{proof} + +\begin{proposition}[Extensive simultaneous crowding] +\label{hist:crowding} +Fix $A,Q,L,R$ and $\delta>0$. Among independently enveloped finite +histories with $p\leq P^A$ and $S\leq Rp$, the activity-normalized integral +for which at some time at least $\delta p$ displayed labels have a +companion within $\rho=\varepsilon^{.85}$ is bounded by +\begin{equation} + \eta_\Gamma P^{C_{A,Q,L,R}p} + \varepsilon^{.275\delta p-1}. + \label{eq:hist-crowding-explicit} +\end{equation} +For $\delta p\geq8$ this is at most +$\eta_\Gamma\varepsilon^{.1\delta p}$ for sufficiently small +$\varepsilon$, uniformly in the stated range of $p$. +\end{proposition} + +\begin{proof} +The proximity graph of $m$ crowded labels has a forest of at least +$m/2$ edges: each nontrivial connected component with $s$ vertices +contributes $s-1\geq s/2$. Summing possible forests costs at most +$p^{Cp}$. Take a grid of spacing at most $\varepsilon$ in $[0,T]$. +On $S\leq Rp$, the displacement of an extended path to its closest +grid point is at most $C_R\sqrt p\,\varepsilon\leq\rho/2$ for +small $\varepsilon$. Thus an active-path crowding event gives the +same forest with radius $2\rho$ at a grid point, even across a +lifetime boundary. The grid costs at most $C_T\varepsilon^{-1}$. + +Use the sharper form of Lemma~\ref{hist:pivot}. On a stratum with +$k\leq h_0$, multiplying by +$\mu^{p-h_0}=\varepsilon^{-2(p-h_0)}$ leaves the exponent +\[ + 3(.85)r+2(p-r-k)-2(p-h_0) + =.55r+2(h_0-k)\geq .55r. +\] +Since $r\geq\delta p/2$, this gives +\eqref{eq:hist-crowding-explicit} for a fixed prescription up to a +$p^{Cp}$ factor. There are only $O_L(p)$ nontrivial cell and +retained-slot choices: empty stretches introduce no new prescription. +Even the crude count $(LJ)^{Cp}p^{C_Lp}$ of all their locations and +finite partitions is $P^{C_{A,Q,L}p}$. This proves the asserted +summed estimate. When $\delta p\geq8$, the grid loss leaves an +exponent at least $.15\delta p$; the remaining logarithmic powers +are absorbed by $.05\delta p$ for small $\varepsilon$. +\end{proof} + +\begin{corollary}[Separated original parents] +\label{hist:separated} +Given $L,A,\delta>0$ and $M_0<\infty$, one can discard finite terms +of total integral at most $\eta_\Gamma e^{-M_0p}$ so that, except at +fewer than $\delta p$ ordinary pins, the original parent is at +distance greater than $\varepsilon^{.85}$ from every other already +displayed path immediately before the insertion. This assertion is +used for large histories; fixed bounded orders are treated by their +direct geometric estimates. +\end{corollary} + +\begin{proof} +First remove $S>Rp$ with Proposition~\ref{hist:basic}. Apply +Proposition~\ref{hist:crowding} with a sufficiently small fixed +crowding fraction $\theta$, to be chosen below. On its complement, +at every ordinary pin at most $\theta p$ of the known old paths +have another known path within $\varepsilon^{.85}$. Their known +states at the original first-contact time equal the corresponding +states of the prescribed term before the insertion. Exclude the +new label itself in this count. + +In the reference measure retain the adapted gate that the known +old-path crowding count is at most $\theta p$. It contains every +term under consideration. Uniform parent selection now gives +conditional probability at most $\theta$ of choosing a crowded old +parent. Equation~\eqref{eq:hist-reference-discard} bounds the +integral with at least $\delta p$ such choices by +$C_{L,R}^p2^{p/2}\theta^{\delta p/2}$. Choose $\theta$ small enough +to make this $e^{-(M_0+1)p}$. The extensive crowding estimate with +this fixed $\theta$ is smaller than the same quantity once the +history size and $P$ are sufficiently large. This proves the +claim. Neither step presupposes a good event for the exact cloud. +\end{proof} + +All constants used for the last two propositions are chosen after +$L$ and the requested discard exponent. In contrast, the source +exponent, the parent inequalities, and the constants in +\eqref{eq:hist-finite-bound} were fixed before those choices. This +separation is what permits the few-crossing estimate in +Section~\ref{sec:closure} to retain its base $Cb\sqrt L$. diff --git a/preprints/The-Boltzmann-Grad-limit-for-stable-radial-potentials-on-regular-kinetic-intervals-September-23-2026/build/sections/05-exposure.tex b/preprints/The-Boltzmann-Grad-limit-for-stable-radial-potentials-on-regular-kinetic-intervals-September-23-2026/build/sections/05-exposure.tex new file mode 100644 index 0000000000000000000000000000000000000000..1485679e6769c82c502c1600902c84fe99e8f95e --- /dev/null +++ b/preprints/The-Boltzmann-Grad-limit-for-stable-radial-potentials-on-regular-kinetic-intervals-September-23-2026/build/sections/05-exposure.tex @@ -0,0 +1,1000 @@ +\section{Terminal trees and exposure of first defects} +\label{exp:section} +\label{sec:exposure} + +The signed terms of Definition~\ref{cell:histories} are deterministic +whole-cell transports. Randomness in this section means integration against +coordinates for such a term. In particular, none of the auxiliary omissions +below is an operation on the microscopic process. We use the fixed Gaussian +bound of Lemma~\ref{hist:gaussian}, the contact charts of +Lemma~\ref{hist:pinning}, and the exceptional-parameter and crowding estimates +of Propositions~\ref{hist:irregular} and~\ref{hist:crowding}. +Throughout the geometric applications use +\begin{equation} +\label{exp:time-scale} + \tau_\varepsilon=\frac14\varepsilon^{.95}. +\end{equation} +The existing boundary exclusion of width $\varepsilon^{.95}$ leaves no +cell or layer cut in an initial interval of length $2\tau_\varepsilon$. +The retained birth spacing $2\varepsilon^{.9}$ is stronger than needed +on this shorter interval. With arc error $C_M\varepsilon^{.96}$ the tube +ratio is $O(\varepsilon^{.01})$, and every fixed polynomial in $p$ times +$\tau_\varepsilon$ is $o(\varepsilon^{.85})$. + +\subsection{Two coordinate systems, used at different stages} + +The \emph{original forest} uses the first pin required by the factor in the +exact expansion: for a singleton addition, this is its first backward +contact with any root of that factor. All ordinary parent counts, +compact-parameter exclusions, and increasing-tree estimates below use +these original coordinates. A parent's identity determines its factor; +there is no additional choice of a factor after the parent is chosen. + +The final exposure estimate uses a second, \emph{global forest}. Its +anchored pool includes every displayed path, including paths from other +tensor factors. A floating translation is pinned at its first backward +contact with this entire pool. With several floating groups, their first +mutual contacts are used as well. Coordinate groups merge at these pins, +but forces remain those of the original whole-cell prescription. Thus a +new contact used only for this coordinate construction is virtual. All +original required-contact, isolation, and first-contact tests remain +indicators. This change of coordinates is used only after an extensive +number of fresh terminal trees has been selected, when its full +$p^{Cp}$ multiplicity is affordable. + +A singleton global pin still produces exactly one new label. A genuine +isolated scattering has both the continuing parent and the new child as +outputs. A virtual contact or force-free passage changes only the new +branch. A formerly prescribed genuine encounter following an earlier +virtual global pin remains an actual encounter, but is an additional +contact in the global forest. + +Call a contact of the free added ray with another factor \emph{premature} +if it occurs before its original pin in backward time. In the presence of +such a contact the global pin may be \emph{rerouted}. The next lemma is +proved in the original coordinates, so it incurs no extra factor choice. +Bad original pins are handled by the original exceptional-parameter +bounds, rather than being assigned an unjustified small angular measure. + +\begin{lemma}[Premature contacts in original coordinates] +\label{exp:rerouting} +Outside the original compact-parameter, crowded-parent, and multi-addition +exceptions, the contribution with a specified set of $k$ singleton cells +having a premature contact has bound +\begin{equation} +\label{exp:reroute-bound} + C_L^p\bigl(p^C\varepsilon^c\bigr)^k. +\end{equation} +Consequently an extensive number of premature contacts has arbitrarily +strong exponential suppression. The remaining premature births may be +marked irregular before selecting terminal subtrees. No global +reparenting multiplicity is included or needed in this estimate. +\end{lemma} + +\begin{proof} +There is exactly one added label across all expansions in the cell under +consideration. Let $\sigma$ be its original backward pin time, and $a$ its +original parent. The original first-pin condition says that on its +prelaunch interval the new label is free and has no contact with any root +of its own factor. The current-cell root paths of all other factors are +also independent of its free velocity $w$ and position. These paths are +known after the root states at the later cell endpoint are fixed. + +Write $v=v_a(\sigma)$ and $g=w-v$. With $u>0$ denoting elapsed time from +$\sigma$ in the prelaunch direction, the new ray is +\[ + x_a(\sigma)+\varepsilon n+u(v+g). +\] +The original parent is separated by $\varepsilon^{.85}$ from every other +already anchored path at time $\sigma$, outside the original crowding +exception. Hence, with polynomial speed bounds, a premature contact is +impossible for $u<\tau_\varepsilon$. For fixed $\sigma,n,|g|$, the remaining +angular measure is bounded by a constant times spherical area. Indeed, +$w\mapsto g$ is a translation, the Gaussian density in $w$ is bounded +above on the retained compact range, and the incoming-hemisphere +restriction and bounded flux weight preserve this upper bound. +Uniformity of the original angular law is not needed. The proof +of Lemma~\ref{exp:tube}, with time run in this prelaunch direction, bounds +the contact measure against each fixed other-factor root curve by +$p^C\varepsilon^{.01}$. The union over target lives costs only a polynomial +factor. An endpoint contact is included in the same estimate. + +Use the product Gaussian/normal reference measure in the +Cauchy--Schwarz parent comparison of Proposition~\ref{hist:basic}, retaining all +first-pin tests as indicators. Fix the entire ordered time vector and +the discrete original parent choices before this argument. Index the +remaining coordinate blocks in backward cell order, including an entire +multi-addition block whenever necessary, and let $\mathcal F_i$ contain +all fixed times and choices, the top coordinates, and the first $i$ blocks. +A complete path or prelaunch segment from a previously processed block is +then fixed. The current cell's roots and the other factors' current-cell +root paths are $\mathcal F_{i-1}$-measurable. There is no other new label +in a designated singleton cell. Its absolute velocity and normal are +fresh coordinates of block $i$. + +Define $A_i$ using only these paths and block $i$: its original parent is +separated at the pin, the paths on the tested prelaunch interval have the +required polynomial speed bound, the relative velocity has the retained +compact bounds, and the free new ray meets an already anchored path of +another factor before its original pin. In particular we do not include +any test against a future pin or condition on the eventual set of marks. +The separation and exterior-speed requirements are +$\mathcal F_{i-1}$-measurable. On an original good history the global +energy bound implies these local speed requirements; outside that domain +we may simply omit the history. The angular calculation gives, uniformly +in all past values, +\[ + A_i\in\mathcal F_i,\qquad + \mathbb E_{\mathrm{ref}}[\mathbf1_{A_i}\mid\mathcal F_{i-1}] + \le\rho_p:=p^C\varepsilon^{.01}. +\] +This is an adapted assertion, not a separate conditional bound with every +other block held fixed. For any fixed set $S$ of $k$ designated indices, +let $i$ be its largest element. The product of all other designated +indicators is $\mathcal F_{i-1}$-measurable. Conditional expectation removes +$\mathbf1_{A_i}$ at cost $\rho_p$. Iterating gives +\[ + \mathbb E_{\mathrm{ref}}\prod_{i\in S}\mathbf1_{A_i}\le\rho_p^k. +\] +All original-domain indicators not used in $A_i$ can be dropped for this +upper bound, without changing the normalized reference kernels. Finally +sum over possible fixed sets $S$; this is a union bound, not conditioning +on which indices turn out to be marked. +The square-flux comparison contributes $C_L^p$ and at worst halves the +positive exponent. Choices of target lives cost $p^{Ck}$; choices of the +$k$ cells cost at most $2^p$ when summed. This proves +\eqref{exp:reroute-bound}. Since $p$ is a fixed power of $P$, an extensive +$k$ supplies the claimed suppression. +\end{proof} + +\subsection{Increasing trees and a terminal antichain} + +Endpoint-overlap pins, including compulsory labels in a top endpoint +polymer of a positive component witness, are included among irregular +vertices. Such witnesses need only $h_0$ distinguished spatial anchors; +no isolation of those anchors at the final endpoint is used here. A +large endpoint polymer is paid for by its endpoint-volume pins before the +regular-tree count. If only a small number of endpoint participants remain, +they create only that many irregular vertices and additional roots when +cut out. Thus a witness with one distinguished anchor does not quietly +assume that its entire final component is a singleton. + +For a rooted plane binary forest $\mathcal T$, let $s(v)$ be the number of +internal vertices in the subtree rooted at an internal vertex $v$. A +continuing-parent edge is an ordinary child edge in this definition. + +\begin{lemma}[Terminal subtree selection] +\label{exp:terminal} +Fix $0<\alpha\le1$ and an exponential accuracy $A>0$, with the layer +count and polynomial size exponent $a>0$ fixed. There are constants +$\theta=\theta(\alpha,A)>0$, $M<\infty$, $p_0<\infty$, and +$\varepsilon_0>0$, independent of $\varepsilon$, such that the following +statement holds uniformly for +\[ + 0<\varepsilon<\varepsilon_0,\qquad p_0\le p\le P^a,\qquad h_0\le\theta p. +\] +In the absolute history integrals with $p$ labels and $h_0$ tops, the +following exclusions have total cost at most $e^{-Ap}$ times the +unrestricted exponential majorant, after choosing the compact and +irregularity cutoffs sufficiently strongly: +there are more than $\alpha p/8$ insertion vertices whose original descendant +subtree has more than $M$ vertices, or too many vertices influenced by an +irregular insertion. Here $M$ is a fixed constant, chosen before +$\varepsilon\downarrow0$. + +After these exclusions, if at least $\alpha p$ distinct new-particle lives +have an extra contact or an uncharged defect, there are at least $c_{\alpha,M}p$ pairwise disjoint +regular terminal trees, of size at most $M$, each containing a defect. For a +genuine launch its selected tree contains both output branches; for a +passage launch it need contain only the new branch. The selection can +additionally ensure that no outside primary pin in the global coordinates +attaches to a selected tree, and that the original and global genealogies +agree on every selected tree. +\end{lemma} + +\begin{proof} +For a fixed forest shape with $n$ internal vertices, the number of increasing +numberings is +\begin{equation} +\label{exp:hook} + \frac{n!}{\prod_{v\in\mathcal T}s(v)}. +\end{equation} +Indeed, assign the smallest available number to each tree root, distribute +the remaining numbers between its two child subtrees with the appropriate +binomial coefficient, and repeat. The resulting factorials cancel at +every internal edge and give \eqref{exp:hook}; the same multinomial +calculation distributes numbers between forest components. A plane binary +forest with $h_0$ roots has at most $C^{n+h_0}$ shapes. Ordered times in an +interval of length $T$ have volume $T^n/n!$. Dropping the layer and lifetime +restrictions only increases this volume. Consequently, if $\alpha n/8$ +vertices have $s(v)>M$, summation over their shapes and numberings has bound +\begin{equation} +\label{exp:large-subtrees} + C_T^{n+h_0} M^{-\alpha n/8}. +\end{equation} +The Cauchy--Schwarz comparison with uniform parent counting in +Proposition~\ref{hist:basic} can replace the last exponent by a fixed positive +fraction of it, which still allows arbitrary exponential suppression by +increasing $M$. Quantitatively, choose +$\theta\le\min\{1/4,\alpha/8\}$, so that $n=p-h_0\ge p/2$. +Let $\zeta>0$ be the fixed exponent left by the parent comparison. +Choose $M$ so large that +\[ + \frac{\zeta\alpha}{16}\log M + \ge A+\log C_*+2, +\] +where $C_*^p$ bounds the shape and fixed-layer constants in the comparison. +Then the large-subtree exclusion is exponentially small uniformly for +every admissible $p$, without requiring $h_0/p$ to tend to zero. + +By Lemma~\ref{exp:rerouting}, include the few remaining premature births +among the irregular vertices. This count still uses the original forest; +no global-parent multiplicity has been introduced. Irregular vertices can +be cut out before this count. Alternatively order +the pins inside an irregular cell consistently with attachment to the +anchored pool and refine them to binary vertices. The additional orderings +are bounded by the factorial in that cell's number of labels. The +multi-addition estimates in Proposition~\ref{hist:irregular} absorb these factors. +Cutting $r$ vertices creates at most $2r$ extra roots, so the shape count +remains exponential. +Here the irregular count concerns added insertion vertices and the old +lives targeted by their primary pins, not all old lives that the added +group may subsequently meet. A cell with $m$ added labels has at most +$m$ primary attachment edges, even if its motion later touches many more +old labels. Those secondary contacts remain defects on the corresponding +regular candidate trees; they are not removed as $O(m)$ contaminated +participants. Choose the allowed irregular fraction only after $M$ +is fixed. + +A vertex of a subtree with at most $M$ internal vertices has at most $M$ +ancestors whose subtrees also have at most $M$ vertices. Thus $r$ irregular +vertices contaminate at most $Mr$ candidate launch vertices. Distinct +new-particle lives have distinct birth vertices. Excluding the $h_0$ top +lives and the contaminated candidates leaves a fixed fraction of the +$\alpha p$ affected lives. Their candidate subtrees are nested or disjoint; +each contains at most $M$ candidate birth vertices. Choose an inclusion +maximal candidate, discard its at most $M$ nested candidates, and repeat. +This produces a disjoint family with at least a $1/M$ fraction of the +remaining candidates. There is one additional mark needed before this greedy selection. Form +the global pins only as combinatorial data on the current configuration. +For each rerouted singleton birth mark its new global parent life, and for +each irregular multi-pin mark the lives to which its pins attach. There +are at most a constant times the number of irregular participants such +marks. Count a target at its actual attachment segment; a life persists +through a cut along the same continuation path, so no new mark is created +merely by crossing a layer. An original bounded subtree containing a marked life is among at +most $M$ candidate ancestors of that life, with another constant to include +the life itself. Remove these candidates also. For example, take $h_0\le\alpha p/8$ +and choose the irregular participant bound so that all child and target +marks together eliminate at most $\alpha p/8$ bounded candidates; a bound +$r\le c\alpha p/(M+1)$ with a sufficiently small absolute $c$ suffices. +Together with the $\alpha p/8$ large candidates, this leaves at least +$\alpha p/2$ affected candidates. Greedy selection therefore retains at +least $\alpha p/(2(M+1))$ disjoint trees, up to integer rounding. Now an outside global pin cannot attach to a +selected tree: an unchanged singleton attachment would already be its +original descendant, a rerouted attachment has a marked target, and a +multi-pin has a marked target. No vertex within a selected tree is +rerouted. Thus both genealogies agree there. This target marking is +necessary; marking only the rerouted child's original ancestors would not +exclude attachment to an unrelated hidden parent. + +The marks and the selected antichain can depend on the phase coordinates. +No probability estimate is obtained by conditioning on this selection. +In the later exposure bound we sum over its possible labels and choices at +cost $p^{Cp}$, paid by an extensive geometric gain. Replacing a selected +passage subtree by its new-child part preserves disjointness and its +designated affected life. Choose the irregular fraction after $M$, as +specified above, and increase the requested accuracy of its finitely many +exclusions by a fixed constant to sum them. Finally take +$p_0\ge 8(M+1)/\alpha$ to absorb integer rounding; for example the number +of selected trees is then at least $\alpha p/(4(M+1))$. +All these choices are fixed before $\varepsilon\downarrow0$. +\end{proof} + +\subsection{Launch coordinates and a tube estimate} + +Histories are ordered backward, but $t$ in the geometric formulas denotes +physical time and velocities are physical velocities. At a regular singleton pin, +write $v$ for the already known parent velocity and $w$ for the free added +velocity. The change +\begin{equation} +\label{exp:relative-chart} + g=w-v +\end{equation} +has Jacobian one: before the first pin the parent trajectory is independent +of the new label. In insertion order these changes form a triangular +transformation, including when the parent velocity has been changed by +previous genuine scatterings. Later relative parameters on a selected tree +will always be fixed in laboratory axes. + +The measure on an entering sphere, or on its time reverse, is +$|g\cdot n|\,d\sigma(n)\,dg$. Equivalently, with $s=|g|$, impact radius +$\rho\in(0,1)$, and normalized Haar measure $dR$ on $SO(3)$, it is +\begin{equation} +\label{exp:rotation-measure} + 8\pi^2s^3\rho\,ds\,d\rho\,dR. +\end{equation} +This follows by writing $dg=s^2ds\,d\omega$ and the impact area as +$\rho\,d\rho\,d\varphi$. It uses no inverse of the scattering angle. On +the retained compact parameter sets, $s$ and both genuine output increments +relative to the original parent are bounded above and away from zero. +Their body-frame vectors depend on $(s,\rho)$; common rotation multiplies +both by $R$. Each nonzero vector separately has a uniform spherical +marginal. The two output vectors need not be independent. + +Galilean covariance gives a useful affine representation. Fix all relative +parameters after a launch, its launch time $t_0$, and all its radial +parameters. Up to the first defect, every descendant of output $i$ has +position +\begin{equation} +\label{exp:affine} + x_a(t)=x_0+(t-t_0)v+(t-t_0)R a_i+G_a(t)+e_a(t), + \qquad |e_a(t)|\le C_M\varepsilon^{.96}. +\end{equation} +Here $G_a$ is independent of $R$ and is Lipschitz with the retained speed +bound. At a later birth the two velocities equal the current parent +velocity plus increments depending only on that birth's relative data. +Induction proves \eqref{exp:affine} for the broken rays. Each exact +scattering arc has duration at most $\varepsilon^{.96}$ and bounded speed; +its replacement by an instantaneous broken ray changes positions by at +most a constant times that duration. There are at most $M$ such arcs. +The displacement $\varepsilon n$ at an insertion is absorbed in the same +error. This also proves the estimate during a later arc. + +For two descendants diverging at a genuine vertex, subtract their +representations at that vertex. The relevant coefficient is the nonzero +outgoing relative velocity, whose direction is again spherical under the +common rotation. If they diverge at a passage, use the nonzero incoming +relative velocity. Fixing parameters strictly below the divergence leaves +these directions available for integration. + +\begin{lemma}[A ray with an independent exterior] +\label{exp:tube} +Let $\omega$ have normalized surface measure on $\mathbb S^2$, let +$00$, let $p\le P^a$, +and retain the compact scattering, duration, birth-spacing, and boundary +gates of Proposition~\ref{hist:irregular}. Test at most polynomially many +exterior lifetime segments, independent of the tested launch rotation and +with polynomial Lipschitz speed. Each such segment must either be tested +only at elapsed times at least $\tau_\varepsilon$, or admit such an +independent extension back to the launch whose point there is at distance +greater than $\varepsilon^{.85}$ from the launch parent. The unchanged +parent of a passage is also allowed if it is free during the initial +$2\tau_\varepsilon$ interval. Prescribed scattering and passage encounters +are exempt. Under these hypotheses, a regular tree of at most $M$ vertices +has a first unprescribed internal or tested exterior contact with parameter +measure at most $p^C\varepsilon^c$, for some $c>0$, in the unconditioned +bounded-density reference measures described above. +\end{lemma} + +\begin{proof} +On $[a,2a]\cap[\tau,T]$, take a mesh of spacing +$\delta/(2(V+r_1))$. A hit implies that a mesh point satisfies the same +inequality with $2\delta$ in place of $\delta$. There are at most +$C(1+a(V+r_1)/\delta)$ points. At a fixed time the allowed directions are +empty or contained in a spherical cap with area at most +$C\delta^2/(r_0^2a^2)$. The mesh union therefore has measure at most +$C(V+r_1)\delta/(r_0^2a)$. Sum this geometric bound over +$a=2^k\tau$ to obtain \eqref{exp:tube-bound}. + +Apply the estimate to \eqref{exp:affine}, with +$\delta=C_M\varepsilon^{.96}$ enlarged to include the interaction radius, +and $\tau=\tau_\varepsilon=\varepsilon^{.95}/4$. On the earlier side of a physical +launch time use elapsed time $a=t_0-t$ and replace the spherical direction +by its negative; on the prelaunch side use $a=t-t_0$. The quotient $\delta/\tau$ is +$O(\varepsilon^{.01})$. Polynomial speed and finite choices of descendant +rays cost a factor $p^C$. +A lifetime segment tested only after $\tau_\varepsilon$ needs no +launch-separation condition: extend its independent comparison curve to +$[\tau_\varepsilon,T]$ without increasing its Lipschitz constant and apply +\eqref{exp:tube-bound}. The following initial-interval argument concerns +only segments that are tested before $\tau_\varepsilon$. +The launch parent is separated from all other +anchored paths by $\varepsilon^{.85}$, except for the already excluded +crowded-parent choices. Since $\tau p^C=o(\varepsilon^{.85})$, no hit with these other exterior +paths occurs at smaller elapsed time. For a passage tree the unchanged +launch parent remains visible and is not separated from the new child at +launch. Their prescribed passage is exempt. Separation from all other +paths and the absence of another birth or a cut in the initial +$2\tau_\varepsilon$ interval leave that parent free there. The no-cut +assertion follows from \eqref{exp:time-scale} and the original boundary +gate; consequently no component from another cell can be installed during +this interval. The nonzero relative straight ray cannot return after its +prescribed passage. For later times that parent +is included among the exterior curves in \eqref{exp:tube-bound}. +For internal contacts, use the last +common divergence and use the retained exclusion of other births within +$2\varepsilon^{.9}>2\tau_\varepsilon$ of it. The initial scattering itself is prescribed and finishes before this +interval ends. The next free outgoing relative ray cannot return to its +partner without a later change of velocity. A union over the at most +$M^2$ relevant divergence/ray pairs completes the proof. +\end{proof} + +\subsection{The joint measure and chronological exposure} + +We state the measure argument explicitly because a Gaussian upper bound +alone does not imply independence after conditioning. + +For a product of independent input functions $u_1,\ldots,u_p$, set +\begin{equation} +\label{exp:source-factor} + a_{i,k}=\sup_{|x-k|\le1,\,v}e^{\beta_0|v|^2}|u_i(x,v)|, + \qquad + \mathcal A_p=\prod_{i=1}^p\sum_{k\in\mathbb Z^3}a_{i,k} + =\prod_{i=1}^p\|u_i\|_{\mathrm{Bol},\beta_0}. +\end{equation} +Any scalar small factor multiplying these inputs is left outside the +estimate. Write $V_i^{\rm in}$ for the velocity at label $i$'s unique +independent source and define the input kinetic budget +\[ + \mathcal K=\frac12\sum_{i=1}^p|V_i^{\rm in}|^2=S/2. +\] +The once-counted energy inequality of Lemma~\ref{hist:gaussian} implies +polynomial speed bounds on the shell $\mathcal K\le Rp$, including the +current, departed, and not-yet-injected coordinate states. In the next +proposition the phrase ``source factor'' always means the explicit number +$\mathcal A_p$ in \eqref{exp:source-factor}. + +\begin{proposition}[Causal exposure for mixed prescriptions] +\label{exp:exposure} +Fix an admissible mixed whole-cell prescription with $p\le P^a$ labels and +a total energy budget $\mathcal K\le Rp$. Fix a collection of $k$ disjoint +regular terminal trees as in Lemma~\ref{exp:terminal}. Sum the absolute +activity-normalized chart integrals over the event that every selected tree has a +first defect. The result is bounded by +\begin{equation} +\label{exp:exposure-bound} + \mathcal A_p\,(pJ)^{C p}\bigl(p^C\varepsilon^c\bigr)^{k/2}, +\end{equation} +The factor $(pJ)^{Cp}$ includes cell-segment choices. Constants may +depend on the fixed layer count, compact cutoffs, $M$, and $R$. +\end{proposition} + +\begin{proof} +\emph{The unconditioned charts.} +Fix the cells, force partitions, lifetimes, and labels of the translation +pins. Inactive labels may be extended freely. The augmented map from +independent input coordinates to the current/departed coordinates preserves +volume by Lemma~\ref{hist:gaussian}. At a cell, keep the end coordinates of +all its roots anchored and decompose the added coordinates into velocities, +internal relative positions, and one common translation for each floating +group. Grow the groups backward using the common first-contact convention. +Until a merger, each group's motion is independent of the other group's +common translation. If the contact is at $t$ between labels $i$ and $j$, +its relative translation is +\begin{equation} +\label{exp:pin-jacobian} + r=\varepsilon n+X_i(t)-X_j(t),\qquad + |\det D_{(t,n)}r| + =\varepsilon^2|n\cdot(V_i(t)-V_j(t))|. +\end{equation} +Internal coordinates in this formula are held fixed. Once a relative +translation is fixed, regard its groups as a single coordinate group, +retaining the prescribed forces, including absence of force across a +virtual contact. Ordering the internal merger edges before their enclosing +edges makes the full change of variables triangular. Its Jacobian is the +product of \eqref{exp:pin-jacobian} and the endpoint-overlap volume factors. +No derivative of a many-particle scattering map is introduced. + +Earliest-contact and no-prior-hit tests are kept as indicators. Fix a +deterministic ordering of candidate contact pairs and resolve simultaneous +first contacts by that ordering; retain the resulting strata rather than +asserting that all ties are null. The area formula applied to the +translation map in \eqref{exp:pin-jacobian} gives zero spatial-translation +measure to its nontransverse images. Endpoint overlaps are represented +separately by their volume pins. For each discrete merger forest and tie +stratum the inverse reconstruction is unique outside the nontransverse +images: the pin data recover its relative translations, and the prescribed +flows recover the coordinates. There are at most $(pJ)^{Cp}$ +merger/partner choices. Singleton cells have one ordinary parent choice, +in the original coordinates. In the global coordinates there can be +additional choices even for an irregular reroute; they are included in +the present $(pJ)^{Cp}$ bound. This coarse multiplicity is used only in the present +geometric-gain estimate. The factors $\varepsilon^2$ cancel the activity +normalization. Multi-pin internal volumes and endpoint factors retain the +bounds of Lemma~\ref{hist:pinning}. + +For regular singleton vertices belonging to the selected trees make +\eqref{exp:relative-chart}, followed by \eqref{exp:rotation-measure}. +Keep the absolute terminal velocities of every other floating component +among the known coordinates $\xi$; do not replace them by relative +velocities depending on a future anchor. The parent state is known before that vertex, +so these substitutions remain triangular across cells. Partition the +source positions into summability boxes. The energy bound restricts all +velocities and displacements to polynomial ranges; source boxes contribute the weights $\prod_i a_{i,k_i}$ of +\eqref{exp:source-factor}. Every integrated top position is within a +polynomial displacement of its own source box, giving a polynomial volume +per top. The sum of the source-box weights is exactly $\mathcal A_p$. On each box +choice, bound every remaining flux and compact radial density by a +polynomial in $p$, and enlarge coordinate domains to product ranges. +Let $d|\mathfrak I_{\boldsymbol k}|$ be the absolute chart measure of +the activity-normalized integral of Section~\ref{sec:history-estimates}, +restricted to these source boxes. +It has the domination +\begin{equation} +\label{exp:product-domination} + d|\mathfrak I_{\boldsymbol k}|\le + \Bigl(\prod_i a_{i,k_i}\Bigr)(pJ)^{Cp}\,d\nu_0(\xi) + \prod_{\ell=1}^k d\nu_\ell(\theta_\ell). +\end{equation} +The reference measures are finite product measures. Put the entire +ordered \emph{global-chart} time vector in $\xi$, including all selected launch times, internal +birth times, removals, and lifetime cuts. It is fixed before exposure: +a visible parent must be removed at a known time, even while the output +velocities of its genuine launch remain hidden. The block $\theta_\ell$ +contains the relative velocities, radial impact parameters, and rotations +in tree $\ell$. The variable $\xi$ contains all other chart coordinates +and the fixed genuine/passage types. At a rerouted vertex, the original +required contact time is derived from the reconstructed flow and retained +only through its indicator; it is not an additional independent time. +Original and global pin times coincide on the selected trees. The tube estimate is uniform in the +fixed time vector, including all retained spacing restrictions. Normalize their +polynomially bounded total masses and absorb them in $(pJ)^{Cp}$. +Equation~\eqref{exp:product-domination} is applied to the entire joint +integral. We never divide by a conditional mass of an admissible sector. +In particular, a rare sector is not asserted to have bounded normalized +conditional density. + +\emph{Visible, dormant, and private states.} +Fix a list specifying the order of exposures, the one or two selected trees +revealed at each exposure, and the partner lives or internal divergences +and cell/ray segments involved. There are at most $(pJ)^{Cp}$ such +discrete choices. Each selected tree is revealed once. The list does not fix +the exposure times; these will be recovered successively from the blocks +it reveals. We first describe the state and its updates between exposures. + +Within a cell the state has three parts: +anchored visible labels with known phase points; selected trees that have +already launched and remain hidden; and components of added labels that +have not yet merged with any rooted component. Call the last components +\emph{dormant}. Every dormant component is omitted from the bath, whether +or not it contains a selected label. In particular, a nonselected +singleton or a multi-label addition is not represented in the bath by an +absolute prelaunch path whose later anchor has yet to be determined. + +For a dormant component $C$ with known parameters, choose one terminal +position as origin. Its terminal velocities, internal endpoint offsets, +and already specified internal merger pins determine relative paths +$q_i^C(t)$ and velocities $v_i^C(t)$, up to one common translation. Use the +original force partition restricted to $C$ to compute these paths; a +virtual internal merger changes the coordinate component, not the force +partition. These are functions of $\xi$ and already revealed blocks only. +A selected regular addition is a singleton in its cell. Its as yet +unknown relative velocity is kept in its hidden block and is used only +when its known parent state is available at launch. Internal additions +on an already hidden selected tree are handled privately in that tree, +using its block. They are not part of the visible bath. Its private paths +follow the scheduled isolated binary scatterings and free passages, with +all later relative parameters fixed in laboratory axes. On the enlarged +product domain these paths continue through unlisted hypothetical contacts +without adding forces. They therefore retain the affine representation +\eqref{exp:affine}. Agreement with the prescribed whole-cell dynamics is +required only up to the tree's first defect on the retained domain. + +Read the fixed global merger times in backward order. If two dormant +components merge, the contact normal and time fix their relative +translation by \eqref{exp:pin-jacobian}. Install that relative translation +inside their union and continue the restricted dynamics of the union, +retaining one free common translation. It remains dormant. No absolute +position or force from it is used in the visible evolution. Its internal +mergers precede its first rooted merger in this construction. + +Suppose next that $C$ first merges with the rooted pool at fixed time $t$, +through its label $i$ and a visible label $j$. With the normal oriented +from $j$ to $i$, set +\begin{equation} +\label{exp:dormant-install} + a_C=x_j(t)+\varepsilon n-q_i^C(t). +\end{equation} +Install every label $\ell\in C$ at +$(a_C+q_\ell^C(t),v_\ell^C(t))$ and thereafter evolve the enlarged rooted +system with its original prescribed forces. The common translation uses +only the visible anchor and the known relative state. Endpoint rooting +uses the corresponding overlap displacement in place of $\varepsilon n$. +For a selected launch this rule first computes its parent point and +velocity independently of its hidden block. A genuine launch then hides +both outputs, including the continuing parent; a passage launch hides only +the new child. No velocity of an unrevealed selected output is needed to +continue the remaining visible system. + +The selection in Lemma~\ref{exp:terminal} rules out an outside primary pin +whose first rooted target belongs to a selected hidden tree. This includes +rerouted singleton targets, multi-pin targets, and endpoint targets. An +unchanged singleton pin into a hidden branch is its original descendant +and uses that tree's private block. The launch antichain therefore ensures +that the parent of a different selected tree is visible when that tree +launches. + +At a listed exposure, reveal the indicated block or pair of blocks, +install the current private states, and resume the original prescribed +force systems. A virtual contact reveals the block without changing a +force. An internal defect reveals its tree and resumes its prescribed +internal system. Future additions whose data have just been revealed remain +dormant until their own merger times; revealing a block does not insert +all its future labels immediately. No coordinate change at the random +exposure time is made. The independent variables remain the global +whole-cell coordinates in \eqref{exp:product-domination}. + +At a cell boundary carry +the visible root states to the next cell, retain the private states of +continuing hidden lines, and initialize that cell's new additions as +dormant. A line that terminates is removed according to its prescribed +lifetime. These updates use the global time vector already fixed in +$\xi$, including the removal time of a parent hidden by a genuine launch. + +\emph{Recovering the listed exposure times.} +Suppose the first $m-1$ entries have been replayed. The visible evolution +now uses only $\xi$ and those revealed blocks. To replay entry $m$, supply +the one or two blocks named in that entry and construct their private +paths from their launch states. Starting after the preceding replayed +exposure, find the first eligible contact of the listed type on its listed +segments, in backward chronological order with a fixed order for ties. +At that time make the revelation +update just described. If there is no such contact, the list contributes +zero. The same tie order permits successive entries at the same physical +time. In this search we neither compare with a contact involving another +still-hidden block nor test that such a block has had no earlier hit. +These restrictions from the original first-exposure event are dropped. + +The replay retains a revealed tree's private path before revelation and +its resumed path afterward. They join at the installed phase point. +Thus a later test may use the whole past of a previously revealed life, +even when that life was hidden at the later-tested tree's launch. Both +this past and the replayed revelation time depend only on the previously +revealed blocks. In particular, revelation is not a new spatial placement. + +\emph{Definition on the product domain and agreement.} +The enlarged product ranges need not satisfy the original energy and +isolation tests. Choose a polynomial speed limit $V_p$ strictly larger +than the actual speed bound on $\mathcal K\le Rp$. Stop each visible +simulation at its first violation of that limit or at failure of a validity +condition for its known finite flow. Apply the same known-data rule to a +dormant relative flow needed for an installation. Keep stopped positions +constant afterward, with zero velocity for any subsequent launch. If a +prescribed visible anchor is unavailable, the list contributes zero and +its replay ends. These rules use only visible +and already revealed data. In particular, neither future contact +compatibility nor a dormant or hidden no-prior-hit test is used to continue +the simulation. Its visible portions are $V_p$-Lipschitz for every value +of the product coordinates. + +This exterior stopping rule is not a bound obtained by inspecting a hidden +tree. Its private scheduled paths instead inherit the known launch-parent +velocity and add at most $M$ compactly bounded relative increments and +isolated scattering arcs. Their speeds therefore have a polynomial bound, +uniform in the hidden rotations. The common Galilean drift cancels in the +arc-to-ray error, leaving the bound $C_M\varepsilon^{.96}$ in +\eqref{exp:affine}. Unlisted contacts do not alter those paths. Their +private pasts and the resumed visible portions thus give polynomially +Lipschitz exterior curves. The affine representation and tube estimate remain +available on the enlarged domain, without asserting an energy inequality +for its artificial trajectories. + +We verify agreement on the retained chart stratum with the specified +actual exposure list. Before a component's first rooted merger it has +no contact with an anchored path. An earlier contact with a selected +hidden path would itself be an earlier rooted merger, and its outside +hidden target would have been excluded. Omitting a dormant component +therefore changes no rooted motion before its scheduled installation. +This remains true within an original force group, since finite range +makes the cross force zero before contact. A dormant--dormant merger +recovers the relative states up to one common translation, and +\eqref{exp:dormant-install} recovers that translation at a rooted merger. + +Before the next actual first defect, the hidden private continuations +are their original isolated binary paths. Thus the next listed contact +found by the replay occurs at its actual time and installs its actual +phase points. Uniqueness then gives agreement of the resumed visible +flow. Induction over the list proves agreement at every revelation; +the energy bound ensures that the visible stop is not reached before +any tested exposure. No bound on the number of internal bath collisions +is used. Connectivity roots every added component by the earlier cell +boundary on this stratum. A hidden--visible contact at a cut is an exposure +or an excluded cut irregularity, and the intact-polymer condition prevents +a hidden force from being transferred silently through a cut. Hence the +same induction passes through all cells and macro cuts. + +We have defined curves on the entire product domain and proved that they +contain every hit event from the retained stratum. Outside that stratum +they need only have the stated dependence on revealed blocks and the +uniform speed bounds. No hidden coordinate has been conditioned on or +renormalized in this construction. + +\emph{First-hit blocks.} +A hit of two hidden trees reveals them both and uses one estimate; +every other exposure reveals one tree. Thus the fixed list has at least +$k/2$ entries. Before its next test the bath and launch anchors depend +only on $\xi$ and previously revealed blocks. Fixing the nonangular +parameters of the next block gives \eqref{exp:affine} against an +independent exterior. + +For a one-tree test of $B$, the initial-interval hypothesis of +Lemma~\ref{exp:tube} also needs verification after earlier exposures. +Fix $\xi$ and the previously revealed blocks. Their replayed paths, +including the private pasts retained at their revelations, use no parameter +of $B$. Before its own launch the continuing parent of $B$ remains +visible. On the retained domain these curves agree with the original +paths; on the enlarged domain their speed bounds were supplied above. + +The launch cell of $B$ has exactly one addition across all factors, +and its boundary gate excludes a cut in the initial +$2\tau_\varepsilon$ interval. Thus every exterior life tested before +elapsed time $\tau_\varepsilon$ was already displayed at $B$'s launch; +an earlier tree's revelation during this interval merely makes its +independent private past available. Extend the tested life back to +that launch using the paths just constructed. Except for the unchanged +parent of a passage, retain the gate that this extension starts farther +than $\varepsilon^{.85}$ from $B$'s launch parent, and declare the +enlarged early test empty if the gate fails. The gate depends only on +$\xi$ and previously revealed blocks, and it holds on the selected +good domain by separated original parents and the agreement induction. +The initial offset is $O(\varepsilon)$ and +$V_p\tau_\varepsilon=o(\varepsilon^{.85})$, so the gate rules out an +early hit. + +For the unchanged passage parent, first consider the retained domain. +At launch it is separated from every other old displayed path by +$\varepsilon^{.85}$. The original polynomial speed bound and the +absence of another birth or cut in the initial $2\tau_\varepsilon$ +interval preclude a third-path force there. After the prescribed passage +the nonzero free relative ray cannot return to that parent. Thus an +original first defect against the unchanged parent cannot occur at elapsed +time below $\tau_\varepsilon$. On the product domain we consequently make +this parent test only at elapsed times at least $\tau_\varepsilon$; +no freeness of its artificial stopped path is asserted. The tube estimate +then applies without an initial-interval gate. + +Exterior segments born later +are tested only after $\tau_\varepsilon$; the same globally singleton +cell and boundary gate exclude an earlier installation. For all tests +at elapsed times at least $\tau_\varepsilon$, apply the tube bound +without a launch-separation gate. These extensions and gates are +independent of $B$'s tested rotation. + +For a two-tree test, call $A$ the earlier-launched tree and $B$ the +later-launched tree in backward chronology; the retained spacing excludes +equal launch times. Fix all of $A$'s block first and integrate $B$'s +launch rotation. Both launch anchors were computed without either hidden +block. Retain the gate that $B$'s launch parent is at least +$\rho=\varepsilon^{.85}$ from the already anchored paths of $A$ at that +launch. The gate depends on $A$'s fixed block and the visible anchor of +$B$, not on $B$'s launch rotation. If it fails, define this enlarged +pair test to be empty. On the original selected good domain it holds by +parent separation. During elapsed time $u<\tau=\tau_\varepsilon$ after +$B$'s launch, the initial offsets and displacements are bounded by +$C\varepsilon+Cp^C\tau=o(\rho)$. The retained time spacing and boundary gate leave no +other birth or cut in this interval. Thus no $A$--$B$ hit is possible there; +$B$'s own prescribed pair is not an $A$ path. For $u\ge\tau$, apply +Lemma~\ref{exp:tube} to $B$ against the fixed paths of $A$. The bound is +uniform in $A$'s block, so its subsequent integration preserves it. +Internal +defects use their divergence parameters as in that lemma. The uniform +bound for each one- or two-tree test is +$\rho_p=p^C\varepsilon^c$. + +For a fixed list, let $B_m$ denote the one- or two-tree block revealed +by its $m$th entry, and let $A_m$ require that its replay finds the listed +contact and satisfies the gates used in the corresponding test. If the +replay has already failed, set $A_m=\varnothing$. The construction and +the uniform one-test bound give +\[ + A_m=A_m(\xi,\theta_{B_1},\ldots,\theta_{B_m}), + \qquad + \int \mathbf1_{A_m}\,d\nu_{B_m}\le\rho_p, +\] +The blocks $B_m$ are disjoint, and the retained event for this list is +contained in $\bigcap_m A_m$. The preceding no-hit requirements on +unexposed blocks have been dropped, without renormalizing their measures. +Integrate in reverse block order in +\eqref{exp:product-domination}. Each last remaining restriction gives the +factor $\rho_p$, while all earlier restrictions are independent of that +block. The resulting factor is $\rho_p^{\#\{B_m\}}$. +Summation over the guessed lists and the source-box weights, whose sum is +$\mathcal A_p$, proves +\eqref{exp:exposure-bound}. +\end{proof} + +\subsection{Extensive defects} + +\begin{proposition}[An extensive-defect gain] +\label{exp:defects} +Fix $0<\alpha\le1$, $A>0$, the layer count, and the polynomial size +exponent $a>0$. There exist fixed constants $\theta=\theta(\alpha,A)>0$, +$p_0<\infty$, and sufficiently small $\varepsilon_0>0$ such that, after +the compact cutoffs are chosen, the following bound holds uniformly for +\[ + 0<\varepsilon<\varepsilon_0,\qquad + p_0\le p\le P^a,\qquad h_0\le\theta p. +\] +For histories with at least $\alpha p$ affected lives, their absolute +integral, divided by +the source factor $\mathcal A_p$ of \eqref{exp:source-factor}, is bounded +by the sum of the exceptional-parameter contributions bounded by $e^{-Ap}$ +and a contribution bounded by +\begin{equation} +\label{exp:extensive} + P^{C_Lp}\varepsilon^{c_Lp}. +\end{equation} +The constants $c_L>0$ and $C_L$ are fixed before $\varepsilon\downarrow0$. +The assertion includes recollisions, interference with a long-lived cluster, +and contacts with detached or subtracted trajectories. +\end{proposition} + +\begin{proof} +We divide by the explicit source factor $\mathcal A_p$; if that factor +is zero the integral is zero. First make the high-energy, multi-addition, endpoint, noncompact, long +scattering, close-time, and crowded-parent exclusions of +Propositions~\ref{hist:irregular} and~\ref{hist:crowding}. Apply Lemma~\ref{exp:rerouting} to premature contacts in the original +coordinates. Next choose $M$ +and the irregular fraction in Lemma~\ref{exp:terminal}. A positive fraction of the +affected lives then yields $k\ge c_{\alpha,M}p$ regular disjoint trees. +Choose $\theta$ and $p_0$ from Lemma~\ref{exp:terminal}, decreasing $\theta$ +if the finite-participant transfer below requires a smaller affected +fraction. Every exclusion is requested at a sufficiently increased fixed +accuracy so that their sum is at most $e^{-Ap}$. +Proposition~\ref{exp:exposure} gives \eqref{exp:extensive}. Since +$p$ and $J$ are fixed powers of $P$, their logarithms are $O(\log P)=o(P)$, +and hence the expression in +\eqref{exp:extensive} is smaller than $e^{-Ap}$ for all sufficiently small +$\varepsilon$, uniformly in the indicated range of $p$: its logarithm +per label is at most $C_L\log P-c_L|\log\varepsilon|$. +For a fixed number $s$ of top roots, this applies to every +$p\ge m$ once +\[ + m\ge\max\{p_0,\lceil s/\theta\rceil\}. +\] +Thus it supplies a uniform tail in $p$, rather than a statement requiring +an $\varepsilon$-dependent limit of $s/p$. + +The affected lives used by the algebraic centering argument transfer to this +geometric count with a fixed finite-participant loss. An unchanged +algebraic extra contact marks its participating geometric lives. If a +birth is rerouted to an earlier virtual global pin, mark its new child and +the old and new parents, at most three lives. The later original required +contact is then a geometric defect of that child unless it is an already +charged irregularity. A rerouted genuine contact cannot disappear by +calling it a different primary birth: each added label has exactly one +primary geometric pin. This accounts for all discrepancies between the +two forests. The charging statement of Proposition~\ref{cent:charges} may +therefore be used with a reduced positive fraction. + +No estimate asserts that an already exposed bath has few collisions or no +bound cluster. Such a bath is allowed in Lemma~\ref{exp:tube} through its +speed bound. Its first influence on each fresh regular tree costs a +geometric factor; primary births into a small crowded bath are included in +the parent-choice exclusions. This is why the argument treats interference +with clusters without deleting clusters from the actual dynamics. +\end{proof} + +\subsection{A marked history of fixed size} + +The extensive estimate is not needed, and does not directly apply, when +the number of displayed labels is fixed. In particular, it does not cover +an array consisting only of several top roots. We record the required +finite-size statement separately. + +\begin{corollary}[Finite marked histories] +\label{exp:finite} +Fix the layer count and an integer $p_*$. Sum the absolute integrals of the +original prescribed histories with at most $p_*$ labels, integrating all +top phase points, and suppose their independent source norms in +\eqref{exp:source-factor} are uniformly bounded. The contribution with at +least one unprescribed contact, an incomplete isolated scattering at a +required cut, an interfering birth, or an endpoint-polymer defect tends to +zero as $\varepsilon\downarrow0$. This statement includes $p=h_0$ and +contacts between distinct top roots. The bounds are uniform over the +observation times and shortened endpoint cells in the fixed interval. +If instead a source has norm tending to zero, its finite-history integral +tends to zero by the same uniform finite-size majorant. +\end{corollary} + +\begin{proof} +Work throughout in the original charts, without the coarse global-chart +multiplicity. The ordinary parent/time bound of Proposition~\ref{hist:basic} +gives a finite constant depending on $p_*$ and $T$ after summing cell +locations. Empty cells introduce no extra choice. Its Gaussian bound +allows velocity truncation at a fixed large constant, with uniformly small +error. The summability-box decomposition may be retained throughout; +its weights sum to \eqref{exp:source-factor}, so no spatial truncation +depending on the observation time is required. + +Fix next $\eta>0$. Delete births whose time is within $\eta$ of another +birth and parameters with incoming relative speed or a nonzero output +increment smaller than $\eta$. The time error tends to zero with $\eta$ +by the finite-dimensional simplex measure. The parameter error tends to +zero by the flux-null statement of Proposition~\ref{pre:scattering} and +continuity from above of finite measure. There is no claim of an +algebraic rate in $\eta$. A scattering lasting more than +$\varepsilon^{.96}$ has vanishing flux measure. Births within that distance +of a cell endpoint have total time measure at most +$C_{p_*}J\varepsilon^{.96}\to0$. The multi-addition bounds give a vanishing +factor in a cell containing at least two additions, while an overlap pin +of an added label costs $\mu\varepsilon^3=\varepsilon$ in place of a +normalized flux. A close pair of already integrated top roots instead +costs an unnormalized spatial volume $O(\varepsilon^3)$. Endpoint polymers +are handled by a spatial spanning forest of these pins. Thus no isolation +of a top array has been assumed in this deletion. + +First test additional force-bearing encounters. Before the first such +encounter every factor is a finite collection of isolated binary trees, with completed scattering arcs +replaced, at a position error +$\delta_\varepsilon=C_{p_*,\eta}\varepsilon^{.96}$, by their elastic broken +rays. Purely virtual contacts do not change this reconstruction and are not +excluded at this stage. The exact prescribed dynamics after the first +force-bearing event is unrestricted. +Only its volume and Gaussian majorants are used. There are finitely many +ray pairs, divergence vertices, and birth/cut incidences to test. + +Consider first a force-bearing contact between two families issuing from different top +roots. Keep the internal collision data of both families fixed and float +one family by a common spatial translation $r$. Before their first +contact, their motions are independent and translation equivariant. +The relative position is $r+G(t)$ with $G$ uniformly Lipschitz on the +compact parameter set. The translations producing a hit within radius +$\delta$ have volume at most +\begin{equation} +\label{exp:spatial-tube} + C\bigl(\delta^3+T\,\operatorname{Lip}(G)\delta^2\bigr). +\end{equation} +Indeed, sample the time interval at spacing +$\delta/(1+\operatorname{Lip}(G))$ and cover each sampled point by a ball +of radius $2\delta$. Formula~\eqref{exp:spatial-tube} is uniform in the +locations of the source boxes. Integrating their envelope weights proves +that root--root and cross-family contacts have vanishing integral. +The same calculation controls two top roots already close at an endpoint. +This argument uses the integrated top translations; it asserts no +pointwise smallness at fixed top positions. + +For two rays within one family, use their last common divergence. Its +nonzero relative output direction is spherical, even though the two +individual output velocities can be correlated. Fix all relative data +below that divergence in laboratory axes. Their relative position then +has the form used in Lemma~\ref{exp:tube}, with elapsed time at least +$\eta$ whenever another descendant has been born. Before that time the +prescribed pair is a straight outgoing relative ray and cannot return. +Taking $\delta$ larger than +$\varepsilon+2\delta_\varepsilon$ gives a vanishing bound. This proves +smallness of internal force-bearing recollisions without restricting what +happens after the first one. Remove these vanishing physical-defect sets. +The complete remaining factor paths are then their prescribed binary +paths and free virtual rays throughout their lifetimes. We may now test +all virtual contacts on these complete paths; they introduce no new +dynamical dependence. Cross-family virtual contacts obey +\eqref{exp:spatial-tube}, and internal virtual contacts obey the same +divergence-direction argument. + +The same two tests control a parent being within +$\varepsilon^{.85}$ of a third path at a scheduled birth: use the spatial +translation for different top families and the divergence direction for +one family. The prescribed new child is exempt; all other divergences +are separated in time by the retained $\eta$. Hence the near-parent +exclusions needed for finite singleton geometry have vanishing measure +without invoking the extensive crowding estimate. For a premature +cross-factor contact of an added particle's full-cell prelaunch ray, use +its fresh incoming direction in the original chart, exactly as in +Lemma~\ref{exp:rerouting}; the comparison is with already anchored paths, +and no global reparenting is necessary. The unchanged parent of a passage +is treated as in the proof of Lemma~\ref{exp:tube}. Thus full-cell ghost +segments are included in the finite event list. + +These estimates exhaust a first extra contact or interference of the +retained binary histories. They also cover the short displacement and +cut errors, after the already removed duration and endpoint-time sets. +For fixed $p_*$, the number of combinatorial cases is finite and the time +sums are the original simplex integrals. Take +$\varepsilon\downarrow0$ with the velocity and $\eta$ cutoffs fixed, then +let $\eta\downarrow0$ and the velocity cutoff tend to infinity. The +resulting upper bounds depend on the observation time only through its +upper bound $T$; shortened cells only reduce their integration domains. +This proves uniformity. Finally, replacing one source norm in the finite +majorant by a vanishing norm gives the last assertion directly. +\end{proof} diff --git a/preprints/The-Boltzmann-Grad-limit-for-stable-radial-potentials-on-regular-kinetic-intervals-September-23-2026/build/sections/06-closure.tex b/preprints/The-Boltzmann-Grad-limit-for-stable-radial-potentials-on-regular-kinetic-intervals-September-23-2026/build/sections/06-closure.tex new file mode 100644 index 0000000000000000000000000000000000000000..3ab92884775308c1b2dde6e23243eafa71c474f1 --- /dev/null +++ b/preprints/The-Boltzmann-Grad-limit-for-stable-radial-potentials-on-regular-kinetic-intervals-September-23-2026/build/sections/06-closure.tex @@ -0,0 +1,976 @@ +\section{Summation, stopping, and convergence}\label{sec:closure} + +We now remove the size restrictions used in the finite identities. +There are two distinct inductions. The numerical cutoffs are chosen +backwards, starting with the last layer. The estimates for events of +the actual process are proved forwards, starting with the first +layer. An event in layer $j$ is bounded by a finite witness whose +input is at the beginning of that layer; only layers strictly before +$j$ are expanded to estimate that input. In particular, no +propagation-of-chaos assertion is used to bound a stopping event. + +All constants denoted by $C$ in the few-crossing calculation below +are independent of the number $L$ of layers. They may depend on +$T$, on the potential, on the fixed exponent $\beta _0$ of +Proposition~\ref{pre:envelopes}, and on the corresponding bounds for +$f$ and the initial data. Constants denoted by $C_L$ may also depend +on $L$. The compact scattering cutoffs used in +Propositions~\ref{hist:irregular} and \ref{exp:defects} will be +chosen after $L$; they do not enter the few-crossing constant $C$. + +\subsection{Completed finite histories} + +A \emph{completed history} has no unresolved stopped factor, has +been expanded through all the layers that it enters, and has +undergone the simple-family cancellations at its centered cuts. +Its inputs are independent copies of $f$, a one-body residual from +Proposition~\ref{cent:residual}, or an initial centered factor from +Proposition~\ref{pre:initial}. + +We shall need summability also when the highest expanded layer is left +uncentered. There are three applications: the finite reference +cutoff mismatch $B_j$ in \eqref{cent:reference-mismatch}; a +positive history ending in the terminal witness of +Definition~\ref{def:cell-terminal-witness}; and the completed +cells above a physical stop, together with its positive +whole-component continuation. In each case we fix the finite +upper block and expand only its lower centered inputs through +earlier layers. + +An \emph{admissible mixed uncentered upper block} lies within one, +possibly shortened, layer. It consists of the signed +completed whole-cell terms of Section~\ref{sec:cell}, any +independent reference families and residual upper slots from +Section~\ref{sec:centering}, and, when a physical stop or a witness +requires it, a positive actual-component prefix and an optional +final trial cell. All original force systems, contact indicators, +pattern restrictions, and whole-component cut rules are retained. +Only its lower centered inputs are expanded through earlier +layers, with the usual simple-family cancellations there. No +cancellation of the upper block's roots is required. + +The starting top labels are counted once. A reference root reuses +its deleted upper label and a residual is a one-body source on +that label. Choosing physical versus deleted-reference top slots +costs at most $2^{h_0}$. Bounded residual and sign alternatives +cost at most another $C^{h_0}$, included in the baseline $C^p$. +At a birth its original parent fixes its physical or reference +family. There is no additional per-birth family choice, and no +independent choice of a layer for an already chronologically +numbered birth. + +At the final endpoint of a one-root witness, ``root'' means a +distinguished label and spatial anchor only. All particles of the +final trial system terminate together there, with their full +component Hamiltonians retained. No singleton-isolation condition +is imposed on the distinguished label at that endpoint. The +positive-observable variant in +Definition~\ref{def:cell-terminal-witness} and +the terminal convention in Lemma~\ref{hist:gaussian} include +this case. Earlier cuts obey the usual whole-component rules. + +Write +\[ + p=h_0+n,\qquad n=\sum_jn_j,\qquad + q_j=h_j+n_j,\qquad H=\sum_jh_j. +\] +Here $p$ counts independent formal label lifetimes once, $h_0$ is +the number of starting upper roots, $n_j$ is the number of labels +added in layer $j$, and $h_j$ is the number of roots entering that +layer. Starting roots may be centered; after a restart they need +not be. In the charge inequality we use $h_0$ as an upper bound +for the number of uncentered restarting roots. Thus $H$ counts +entries into expanded layers, including the first one, but not +the initial inputs at time zero. Those inputs are the initial +charges in Proposition~\ref{cent:charges}. Every root position +in this section is integrated. A bounded test of those positions +and velocities is permitted. + +\begin{lemma}[Retained-slot entropy]\label{close:entropy} +Suppose a term retains $k_j$ slots among $q_j$ available slots at +successive cuts. Put $Q=\sum_jq_j$ and $H'=\sum_jk_j$. Then +\begin{equation}\label{close:slot-bound} + \prod_j\binom{q_j}{k_j} + \leq\binom{Q}{H'} + \leq\exp\left\{H'\log\frac{eQ}{H'}\right\} + \quad (H'>0). +\end{equation} +In the history expansion $Q\leq p+C H$ and $H'\leq C H$, up to a +fixed number of choices per retained slot. Consequently, when $H\leq\delta p$, +the logarithmic selection cost is at most +\begin{equation}\label{close:entropy-small} + C\delta p\left(1+\log\frac1\delta\right). +\end{equation} +For general $H\leq Lp$, its cost is at most $C_L^p$. +\end{lemma} +\begin{proof} +The coefficient of $z^{H'}$ in +$\prod_j(1+z)^{q_j}=(1+z)^Q$ is the sum of the products on the +left of \eqref{close:slot-bound} over all allocations of $H'$. +The first inequality follows. The second follows from +$\binom Qk\leq(eQ/k)^k$. A label contributes an available slot at +its first input cut, and an additional available slot each time it +is retained across another cut. This gives the asserted bound on +$Q$. A bounded number of choices per retained slot contributes +$\exp(C H)$. Bounded choices at first-input slots instead cost +$C^p$; these belong to the baseline constant in the history +estimate, not to its small retained-slot entropy. +The last two assertions follow from the displayed inequalities, +using $\binom Qk\leq2^Q$ in the general case. +\end{proof} + +\begin{lemma}[Energy and the short-layer factors]\label{close:energy} +For a completed history with the above counts, the parent, time, +and velocity factors in Proposition~\ref{hist:basic}, before +retained-slot and spatial losses, have total integral at most +\begin{equation}\label{close:energy-factor} + (C b\sqrt L)^p + \exp\{H+h_0\log_+(1/b)\}, + \qquad b\leq1. +\end{equation} +The constant $C$ is independent of $L$. A fixed part of the source +Gaussian remains available for the spatial estimates. +\end{lemma} +\begin{proof} +Use the source sum and augmented squared-speed budget of +Lemma~\ref{hist:gaussian}: +\[ + S=\sum_{\text{independent inputs }a}|v_a|^2, + \qquad \mathcal E=S+2Bp. +\] +By Lemma~\ref{hist:gaussian}, every active or mixed sum of +squared speeds is at most $\mathcal E$. The same assertion bounds +the sum of squares of the velocities of the independently added +free rays. The source sum $S$ is not replaced by terminal +kinetic energy: a departing interacting component may have positive +potential energy. Retaining its full Hamiltonian gives instead +the Gaussian comparison +\begin{equation}\label{close:source-gaussian} + e^{-\beta _0S} + \leq e^{\beta _0Bp} + e^{-(\beta _0/2)\sum_a|V_a|^2} + e^{-(\beta _0/2)S}, +\end{equation} +where $V$ is the vector of mixed terminal velocities. Further +splitting into a fixed number of portions is allowed. All these +exponents are fixed independently of the layer subdivision. + +The source energy depends on the parent prescription. We therefore +first partition its domain into shells +\[ + rp\leq S<(r+1)p,\qquad r=0,1,\ldots. +\] +On shell $r$ use the deterministic upper budget +$E_r=(r+1+2B)p$. Parent choices that cannot belong to this +shell are killed. The parent-sum bound of +Proposition~\ref{hist:basic} then applies with this common budget; +it does not require moving a prescription-dependent energy through +a sum over parents. At a birth in layer $j$, if $u\leq q_j$ paths +are present, its summed flux is bounded by +\[ + C u\left(1+|w|+\sqrt{E_r/u}\right) + \leq Cq_j\left(1+|w|+\sqrt{E_r/q_j}\right). +\] +Let $m$ count multiple-insertion participants and endpoint pins, +and let $\mathcal O$ be the ordinary births. The nonordinary +shell-speed factor in Proposition~\ref{hist:basic} is +$(r+2)^{m/2}$. Pad the speed product with $h_0$ unit factors. +AM--GM and $\sum_jn_j/q_j\leq L$ give +\begin{align} + &(r+2)^{m/2} + \prod_{i\in\mathcal O} + \left(1+|w_i|^2+\frac{E_r}{q_{j(i)}}\right)^{1/2} + \nonumber\\ + &\hspace{12mm}\leq + \left(C L(r+2)+\frac{|V|^2}{p}\right)^{p/2}. + \label{close:amgm} +\end{align} +Indeed the sum of the $p$ squared factors is at most +$p+m(r+1)+|V|^2+LE_r$. +This form of the estimate remains valid after the parent +majorization has enlarged the velocity integral to the full +product Gaussian; it does not incorrectly retain a source-shell +constraint on that enlarged integration domain. + +For a fixed product Gaussian $\gamma$, +\[ + \int\left(1+\frac{|V|^2}{p}\right)^{p/2}\,d\gamma(V) + \leq C^p. +\] +To check this, reserve half of its Gaussian exponent and bound +$e^{-c u}(1+u/p)^{p/2}$ by $C^p$ for $u\geq0$; the remaining +Gaussian integrates with an exponential-in-$p$ normalization. +Applying this to \eqref{close:amgm} gives +$[C\sqrt{L(r+2)}]^p$. A fixed portion of the source-energy +factor in \eqref{close:source-gaussian} gives $e^{-crp}$. +Thus the sum of the shell bounds is at most +\begin{equation}\label{close:shell-sum} + C^p\sum_{r\geq0}e^{-crp} + \bigl(C L(r+2)\bigr)^{p/2} + \leq (C\sqrt L)^p. +\end{equation} +For the last inequality, bound +$e^{-cr/2}\sqrt{1+C(r+1)}$ uniformly in $r$ and sum the remaining +geometric series. Reserve an additional fixed Gaussian portion +before this calculation if spatial factors are present. + +In each layer the ordered-time volume is $b^{n_j}/n_j!$. Moreover, +\[ + \frac{q_j^{n_j}}{n_j!}\leq e^{n_j+h_j}, +\] +because $n!\geq(n/e)^n$ and +$n\log(1+h/n)\leq h$. Hence the parent and time product is at most +$b^n e^{n+H}$. Combining this with \eqref{close:shell-sum}, and +using $n=p-h_0$, proves \eqref{close:energy-factor}. +\end{proof} + +There are two further summation points. First, the number of +nonnegative layer occupancies with total $n$ is +\begin{equation}\label{close:layer-count} + \binom{n+L-1}{L-1}=\exp(o(p)) +\end{equation} +for fixed $L$ and growing $p$. Births are chronologically ordered +within consecutive layer blocks; there is no independent choice +among $L$ layers for each already numbered birth. Second, the +spatial factors in Proposition~\ref{hist:basic}, allowing even a +separate localization at each cut, have logarithm at most +\begin{equation}\label{close:spatial-loss} + C H+C H\log\left(1+\frac{L\mathcal E}{H}\right), +\end{equation} +with $H$ replaced by $H+h_0$ if the upper-cut convention requires +it. This follows by the weighted concavity of the logarithm +applied to the displayed top-line displacement bound in that +proposition. On shell $r$, a reserved factor $e^{-crp}$ absorbs +the dependence on $r$ in \eqref{close:spatial-loss}. If +$H\leq\delta p$, the remaining logarithmic cost is +\begin{equation}\label{close:spatial-small} + C\delta p\left(1+\log\left(1+\frac L\delta\right)\right). +\end{equation} +For example, use +$1+a(r+c)\leq(1+ac)(1+r/c)$ with $c>0$ fixed, and reserve part +of $e^{-crp}$ for $\delta p\log(1+r/c)$. This also explains why +no independent Gaussian is needed for a carried root at each cut. + +The multiple-insertion and endpoint-overlap mesh factors in +Lemma~\ref{hist:pinning} and Proposition~\ref{hist:basic} may be bounded +by their ordinary birth factors after $J$ is chosen. Indeed their +extra factors are negative powers of $J$ or positive powers of +$\varepsilon$, whereas all their remaining per-participant costs +are fixed powers of the polynomial label cap. Choosing the power +in $J$ last makes each such extra product at most one. This step +absorbs their constants, including constants depending on $L$; +it does not introduce a new $C_L$ for every ordinary single birth. +Their unbounded shell-energy factors have already been included +in \eqref{close:amgm}, rather than absorbed by a fixed mesh power. + +\begin{proposition}[Summability of completed finite terms] +\label{close:summability} +Fix any desired number $a>0$. One can choose a finite $L$, then +the rarity cutoffs, with the following property. For every fixed +polynomial label cap $p\leq P^A$, choose $J$ to be a sufficiently +large fixed power of $P$. There exist $\eta_L>0$ and $p_L<\infty$ +such that, for sufficiently small $\varepsilon$, the sum of the +integrated absolute weights of completed histories with exactly +$p$ labels is at most +\begin{equation}\label{close:complete-tail} + e^{-ap},\qquad p\geq p_L,\qquad h_0\leq\eta_Lp. +\end{equation} +The same bound holds for an admissible mixed uncentered upper +block followed by centered lower layers, after the lower +simple-family cancellations. It is uniform over shortened endpoint +cells and layers and over tests of absolute value at most one. +For the original centered expansion, in which the upper +simple-family cancellations are also made, every fixed-size term +with a retained centered upper root tends to zero after integration. +\end{proposition} +\begin{proof} +First choose $L$ so that $b=T/L$ is within the short interval of +Proposition~\ref{cent:residual}, is at most one, and satisfies +\[ + C T/\sqrt L0$ small enough that +\eqref{close:entropy-small}, \eqref{close:spatial-small}, and the +factor $e^H$ together cost at most $e^{ap/4}$ whenever +$H\leq\delta p$. Choose $\eta_L$ still smaller so that the +upper-root factor in \eqref{close:energy-factor} and any +upper-root localization also have logarithm at most $ap/4$ for +$h_0\leq\eta_Lp$. The finite occupancy factors in +\eqref{close:layer-count} are absorbed by increasing $p_L$. +The histories with few crossings, $H\leq\delta p$, then obey +\eqref{close:complete-tail} with room to spare. + +For the complementary histories apply +Proposition~\ref{cent:charges}, after reducing $\eta_L$ to at +most $\delta/(2L)$. If $d$ is the number of distinct marked +lives, $r$ the number of residual factors, and $i$ the number of +retained initial singleton inputs, then +\[ + d+r+i\geq\theta_Lp,\qquad + \theta_L=\frac{\delta}{2L^2}>0. +\] +The fixed-factor transfer between algebraic families and the +geometric pin forest in Remark~\ref{cent:pin-forest} can be +absorbed by reducing $\theta_L$. A new geometric first pin is +still a single birth. If it reroutes a birth to another displayed +family, the original required later encounter is a defect; it is +not counted as another birth. + +The unrestricted finite-term bound is $C_L^p$ by +Proposition~\ref{hist:basic}. If at least $\theta_Lp/3$ charges +are residual or initial factors, their additional weight is at +most +\[ + \gamma_\varepsilon^{\theta_Lp/3},\qquad + \gamma_\varepsilon= + \max\{C\varepsilon^c,C\eta_\varepsilon\} + \longrightarrow0, +\] +where $\eta_\varepsilon$ is the supremum of the fixed summed +spatial Gaussian residual norms in +Proposition~\ref{cent:residual}, taking the maximum over the +finitely many choices $K_j$. Its common exponent is not weakened +through the layers. This contribution beats any fixed +$C_L^p$ once $\varepsilon$ is small. No rate for the residual is +needed. + +Otherwise an extensive number of lives is marked. The tunable +parameter, tree, and multiple-insertion discards of +Proposition~\ref{hist:irregular} and +Lemma~\ref{exp:terminal}, followed by +Proposition~\ref{exp:defects}, give a bound of the form +\begin{equation}\label{close:many-cross} + C_L^p e^{-A_1p} + +P^{C_Lp}\varepsilon^{c_Lp}. +\end{equation} +The desired $A_1$ is chosen after $L$ and $\theta_L$. Choose it +larger than all the fixed exponential costs plus $4a+20$. +Reduce $\eta_L$ to the top-root fraction supplied by +Proposition~\ref{exp:defects} for affected fraction $\theta_L/3$ +and this accuracy, and increase $p_L$ to its fixed size threshold. +These choices leave the preceding few-crossing estimates valid. +The second term is also at most $e^{-(4a+20)p}$ for small +$\varepsilon$, because $p\leq P^A$ and +$\log P=o(|\log\varepsilon|)$. The expensive exposure guesses occur +only in this geometric-gain term, not in the term controlled +solely by residual smallness. This proves +\eqref{close:complete-tail} in the many-crossing case. + +The argument used only the finite-history estimates and the +finite centered identities. An admissible mixed upper block is +allowed because the charge estimate imposes no condition on its +$h_0$ upper roots. Signed completed cells, reference families, +and a positive actual prefix are all admissible finite terms; +taking their absolute weights does not change their force rules +or contact constraints. The upper physical/reference choices +cost only $2^{h_0}$, with bounded alternatives absorbed in $C^p$. +The original parent determines a birth's factor, so these choices +introduce no extra parent power. Ordinary births across all +simultaneous factors retain the ordered-time bound; multiple +insertions have the mesh factors already estimated. Below this +one upper layer every retained input is an ordinary centered +slot, so Proposition~\ref{cent:charges} applies with precisely +the same allowance $Lh_0$. The source, spatial, and parent +bounds, and hence both crossing cases, remain unchanged. + +Finally fix the total number of labels. The path argument in +Proposition~\ref{cent:charges} makes every surviving centered +upper root lead to an initial small factor, a vanishing residual, +or a marked finite history. For the last case, first restrict +velocities and scattering parameters to compact regular sets. +Corollary~\ref{exp:finite} makes its integral tend to zero. The complement has arbitrarily small integral by the +Gaussian and flux bounds, so the compact restriction may then +be removed. This finite-size argument does not mistake a fixed +small-parameter cutoff for a quantity tending to zero. Ordered +time domination and shortened cells make the argument uniform +in the endpoint time. +\end{proof} + +\begin{remark}[Dependence on the polynomial size cap] +The rarity cutoffs, top-root fraction, and fixed size threshold in +Proposition~\ref{close:summability} can be chosen independently of +the exponent $A$ in $p\leq P^A$. +Indeed, the source-energy cutoff follows from +$C_L^p e^{-cRp}$, the compact-parameter cutoffs from their one-step +probabilities, and the terminal-tree cutoff from +$C_*^p M^{-\zeta\alpha p/16}$, with affected fraction $\alpha$ and +fixed comparison exponent $\zeta>0$ as in Lemma~\ref{exp:terminal}. +These choices depend on the fixed layer count, source bounds, affected +fraction, and requested exponential accuracy, but not on $A$. +In the geometric estimates, the losses involving $p$ and $J=\lceil P^Q\rceil$ are +polynomial powers paid by positive powers of $\varepsilon$. +For each fixed $A,Q$ and $c>0$, +\[ + P^{C(A,Q)p}\varepsilon^{cp} + =\exp\!\left\{p\bigl(C(A,Q)\log P-c|\log\varepsilon|\bigr)\right\} +\] +is smaller than any prescribed exponential in $p$ for sufficiently +small $\varepsilon$. For a fixed crowding fraction $\delta$, the grid +loss is absorbed by the fixed threshold $p\geq8/\delta$. +For the multiple-insertion mesh factor, +choose $Q>2dA$ as in Proposition~\ref{hist:basic}. +Thus only the required mesh exponent and the sufficiently-small-$\varepsilon$ +threshold depend on the polynomial cap. This permits that cap to be +calculated from the deterministic stopping schedule after the rarity +cutoffs have been fixed, and the cell mesh to be chosen last. +\end{remark} + +\subsection{Finite witnesses for actual-process events} +\label{close:witnesses} + +We give an auxiliary event slightly stronger than separate caps +on trial components and actual backward histories. It makes the +inherited-threshold issue explicit. + +Fix a full mesh of $[0,T]$ with layers $1,\ldots,L$ and cells +of length $\Delta=b/J$. At the start of a layer, the ancestry of +a label is its entire instantaneous polymer. At the end of a +cell, define its \emph{actual layer ancestry} recursively as the +union of the ancestries, at the start of that cell, of all labels +in its actual whole-cell contact component. This definition uses +the full Newtonian dynamics and resets at each layer start. +Labels in one instantaneous polymer at a mesh cut have the same +ancestry: they belonged to the same preceding whole-cell +component, or to the same initial polymer at the layer start. + +At a cell start take any trial subsystem that is a union of the +actual instantaneous polymers. Evolve that subsystem in +isolation during the cell. For each component of its cumulative +contact graph, take the union of the actual layer ancestries of +its labels at the cell start. Its \emph{ancestral size} is the +cardinality of that union. For a threshold $D$, let +$\mathcal B_j(D)$ be the event that an ancestral size exceeds +$D$ somewhere in the first $j$ layers, or that a layer-start +polymer there already exceeds $D$. All trial subsystems are +included in this definition. In particular, on its complement +every trial cell component has at most $D$ labels, and every +actual backward history within a layer has at most $D$ labels. + +\begin{lemma}[First witness at an inherited threshold] +\label{close:witness-extraction} +Let $D\geq1$ be an integer. For a finite microscopic cloud, a first occurrence of +$\mathcal B_j(D)$ admits a witness in some layer $i\leq j$ +with one distinguished integrated upper label and between $D$ +and $2D^2$ formal labels. The witness consists of positive actual +component histories before its last cell and one trial subsystem +in its last cell. Every starting or departing set respects whole +instantaneous polymers. The bound is independent of the cutoff +assigned separately to any earlier layer. The weighted ancestral +construction above in fact permits $D1$, choose +\begin{equation}\label{close:upper-cap} + U_{j-1}= + \left\lceil\left(1+U_j+W_j+ + \sum_{\ell>j}W_\ell\right)^\nu\right\rceil. +\end{equation} +The symbols $U_j$ are caps for the entire already constructed +upper block, not just for its currently retained roots. Increasing +$\nu$ once, if necessary, covers every finite composition of the +stated deterministic polynomial bounds. There are only $L$ +levels. All these quantities are fixed powers of $P$ up to +constant factors. In particular, +\begin{equation}\label{close:hierarchy} + \frac{K_j}{U_jP}\longrightarrow\infty,\qquad + \frac{D_j}{K_jP}\longrightarrow\infty, + \qquad U_i\geq W_j\quad(i\varepsilon\text{ for every }i\ne j\}, + \qquad + G_s^\varepsilon= + \mathbf1_{\mathcal D_s^\varepsilon} + \widetilde F_s^\varepsilon. +\] +The inverse centered identity in +Proposition~\ref{cent:identity}, with the pattern restrictions +restored, gives +\begin{equation}\label{close:isolated-limit} + \sup_{0\leq t\leq T} + \|G_s^\varepsilon(t)-f(t)^{\otimes s}\|_1\longrightarrow0. +\end{equation} +Indeed every nonempty retained subset contributes one of the +centered arrays just bounded. Integrating the other independent +$f$ slots costs at most their mass one; the separation indicators +only decrease this bound. The missing independent mass outside +$\mathcal D_s^\varepsilon$ is $O_s(\varepsilon^3)$, uniformly +in time, because the spatial density of $f$ is bounded by its +uniform Gaussian envelope. + +We have $0\leq G_s^\varepsilon\leq F_s^\varepsilon$. Since +$N$ is conserved, +\[ + m_s^\varepsilon:=\int F_s^\varepsilon(t) + =\mu^{-s}\mathbb E(N)_s + =\int F_s^\varepsilon(0)\longrightarrow1 +\] +by Proposition~\ref{pre:initial}. Hence +\begin{align*} + \|F_s^\varepsilon-f^{\otimes s}\|_1 + &\leq\|G_s^\varepsilon-f^{\otimes s}\|_1 + +\int(F_s^\varepsilon-G_s^\varepsilon)\\ + &\leq2\|G_s^\varepsilon-f^{\otimes s}\|_1 + +|m_s^\varepsilon-1|. +\end{align*} +Together with \eqref{close:isolated-limit}, this proves +\eqref{intro:chaos}. Thus close configurations, including +any bound clusters formed by the actual dynamics, are removed +by their nonnegative mass. They have never been excluded from +the microscopic evolution. + +Finally put +\[ + A_\varepsilon(t)=\mu^{-1}\sum_{i=1}^N a(z_i(t)), + \qquad a_f(t)=\int a(z)f(t,z)\,dz. +\] +Factorial normalization gives +\[ + \mathbb E A_\varepsilon(t)=\int aF_1^\varepsilon(t), + \qquad + \mathbb E A_\varepsilon(t)^2 + =\int a(z_1)a(z_2)F_2^\varepsilon(t) + +\mu^{-1}\int a(z)^2F_1^\varepsilon(t,z)\,dz. +\] +It follows that +\begin{align*} + \mathbb E|A_\varepsilon(t)-a_f(t)|^2 + \leq\|a\|_\infty^2\bigl(& + \|F_2^\varepsilon(t)-f(t)^{\otimes2}\|_1 + +2\|F_1^\varepsilon(t)-f(t)\|_1 + +\mu^{-1}m_1^\varepsilon\bigr). +\end{align*} +The right side tends to zero uniformly over deterministic $t$. +Chebyshev's inequality proves \eqref{intro:empirical}. +This conclusion is the stated uniformity of probabilities at +deterministic times; it makes no assertion about a supremum +inside a sample path. +\end{proof} diff --git a/preprints/The-Boltzmann-Grad-limit-for-stable-radial-potentials-on-regular-kinetic-intervals-September-23-2026/paper.pdf b/preprints/The-Boltzmann-Grad-limit-for-stable-radial-potentials-on-regular-kinetic-intervals-September-23-2026/paper.pdf new file mode 100644 index 0000000000000000000000000000000000000000..0fbd1b2548abd17d5ffd6fd93bafdcaf63b02670 --- /dev/null +++ b/preprints/The-Boltzmann-Grad-limit-for-stable-radial-potentials-on-regular-kinetic-intervals-September-23-2026/paper.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:dce9fe5976f17b8953a134a43868b1d12c988093963472be3ac5f31e9cac2066 +size 759148 diff --git a/preprints/The-Campana-Peternell-conjecture-in-dimension-six-September-25-2026/README.md b/preprints/The-Campana-Peternell-conjecture-in-dimension-six-September-25-2026/README.md new file mode 100644 index 0000000000000000000000000000000000000000..7d1a7ae9ec779c6270cc0e2f61b118813a4e130d --- /dev/null +++ b/preprints/The-Campana-Peternell-conjecture-in-dimension-six-September-25-2026/README.md @@ -0,0 +1,27 @@ +# [The Campana–Peternell conjecture in dimension six](main.pdf) + +**Author:** OpenAI + +**Date:** September 25, 2026 + +## Citation + +```bibtex +@misc{OAI:The-Campana-Peternell-conjecture-in-dimension-six-September-25-2026, + author = {{OpenAI}}, + title = {{The Campana--Peternell conjecture in dimension six}}, + howpublished = {OpenAI Math Release preprint + \href{https://github.com/openai/math/blob/main/preprints/The-Campana-Peternell-conjecture-in-dimension-six-September-25-2026/main.pdf}{OAI:The-Campana-Peternell-conjecture-in-dimension-six-September-25-2026}}, + year = {2026} +} +``` + +## Verification + +Use Python 3.12.13 with SymPy 1.14.0 and NumPy 2.3.5. From this paper directory, run the exact-arithmetic certificate with assertions enabled: + +```sh +env -u PYTHONOPTIMIZE python3 -u verification/verify.py +``` + +Successful execution prints `N done` followed by `pass`. The program checks the finite calculations documented in the paper's appendix. diff --git a/preprints/The-Campana-Peternell-conjecture-in-dimension-six-September-25-2026/build/glyph-unicode.tex b/preprints/The-Campana-Peternell-conjecture-in-dimension-six-September-25-2026/build/glyph-unicode.tex new file mode 100644 index 0000000000000000000000000000000000000000..b03d252ec7970731c1f82513cfff2eccc0bd651d --- /dev/null +++ b/preprints/The-Campana-Peternell-conjecture-in-dimension-six-September-25-2026/build/glyph-unicode.tex @@ -0,0 +1,27 @@ +\input{glyphtounicode} +\pdfgentounicode=1 +\pdfglyphtounicode{parenleftbig}{0028} +\pdfglyphtounicode{parenrightbig}{0029} +\pdfglyphtounicode{braceleftbig}{007B} +\pdfglyphtounicode{bracerightbig}{007D} +\pdfglyphtounicode{parenleftBig}{0028} +\pdfglyphtounicode{parenrightBig}{0029} +\pdfglyphtounicode{parenleftbigg}{0028} +\pdfglyphtounicode{parenrightbigg}{0029} +\pdfglyphtounicode{parenleftBigg}{0028} +\pdfglyphtounicode{parenrightBigg}{0029} +\pdfglyphtounicode{braceleftBigg}{007B} +\pdfglyphtounicode{parenlefttp}{239B} +\pdfglyphtounicode{parenrighttp}{239E} +\pdfglyphtounicode{parenleftbt}{239D} +\pdfglyphtounicode{parenrightbt}{23A0} +\pdfglyphtounicode{circleplusdisplay}{2A01} +\pdfglyphtounicode{summationtext}{2211} +\pdfglyphtounicode{integraltext}{222B} +\pdfglyphtounicode{logicalandtext}{22C0} +\pdfglyphtounicode{summationdisplay}{2211} +\pdfglyphtounicode{productdisplay}{220F} +\pdfglyphtounicode{integraldisplay}{222B} +\pdfglyphtounicode{hatwide}{0302} +\pdfglyphtounicode{tildewide}{0303} +\pdfglyphtounicode{tildewider}{0303} diff --git a/preprints/The-Campana-Peternell-conjecture-in-dimension-six-September-25-2026/build/main.tex b/preprints/The-Campana-Peternell-conjecture-in-dimension-six-September-25-2026/build/main.tex new file mode 100644 index 0000000000000000000000000000000000000000..d3d0ccb252b31063dfc5656a558ba62293f1df05 --- /dev/null +++ b/preprints/The-Campana-Peternell-conjecture-in-dimension-six-September-25-2026/build/main.tex @@ -0,0 +1,63 @@ +\documentclass[11pt]{article} +\pdfinfoomitdate=1 +\pdftrailerid{} +\pdfsuppressptexinfo=15 +\usepackage[T1]{fontenc} +\usepackage{lmodern} +\input{glyph-unicode} +\usepackage[margin=1.08in]{geometry} +\usepackage{amsmath,amssymb,amsthm,mathtools} +\usepackage{microtype} +\usepackage{tikz-cd} +\usepackage{booktabs,array} +\usepackage{xcolor} +\usepackage{listings} +\usepackage[hidelinks]{hyperref} +\hypersetup{pdftitle={The Campana--Peternell conjecture in dimension six},pdfauthor={OpenAI},bookmarksdepth=2} +\newtheorem{theorem}{Theorem}[section] +\newtheorem{proposition}[theorem]{Proposition} +\newtheorem{lemma}[theorem]{Lemma} +\newtheorem{corollary}[theorem]{Corollary} +\theoremstyle{definition} +\newtheorem{definition}[theorem]{Definition} +\theoremstyle{remark} +\newtheorem{remark}[theorem]{Remark} +\numberwithin{equation}{section} +\lstset{language=Python,basicstyle=\ttfamily\footnotesize, + columns=fullflexible,keepspaces=true,showstringspaces=false, + breaklines=true,breakatwhitespace=false,aboveskip=1em,belowskip=1em} +\setcounter{tocdepth}{1} +\title{The Campana--Peternell conjecture\protect\\ in dimension six} +\author{OpenAI} +\date{September 25, 2026} +\begin{document} +\maketitle +\begin{abstract} +We prove that every smooth connected complex projective Fano sixfold with nef +tangent bundle is rational homogeneous, resolving the Campana--Peternell +conjecture in complex dimension six. +As a consequence, a connected compact K\"ahler manifold $X$ with nef +holomorphic tangent bundle and +$\dim_{\mathbb C}X-\widetilde q(X)\leq6$ has ordinary universal cover +$F\times\mathbb C^{\widetilde q(X)}$, where $F$ is a rational homogeneous +manifold and $\widetilde q(X)$ is the maximal irregularity of a connected +finite \'{e}tale cover. +\end{abstract} +\tableofcontents +\clearpage +\input{sections/01-introduction} +\input{sections/02-minimal-curves} +\input{sections/02-p1-cohomology} +\input{sections/03-pseudoindex-four} +\input{sections/04-characteristic-classes} +\input{sections/05-arithmetic} +\input{sections/06-grassmannian} +\input{sections/07-conclusion} +\appendix +\input{sections/08-certificate} +\begingroup +\small +\bibliographystyle{plain} +\bibliography{references} +\endgroup +\end{document} diff --git a/preprints/The-Campana-Peternell-conjecture-in-dimension-six-September-25-2026/build/references.bib b/preprints/The-Campana-Peternell-conjecture-in-dimension-six-September-25-2026/build/references.bib new file mode 100644 index 0000000000000000000000000000000000000000..b11239410dc57712e8771d48a829d2ad257b6ce2 --- /dev/null +++ b/preprints/The-Campana-Peternell-conjecture-in-dimension-six-September-25-2026/build/references.bib @@ -0,0 +1,402 @@ +@article{Mok1988, + author = {Mok, Ngaiming}, + title = {The uniformization theorem for compact {K{\"a}hler} manifolds of nonnegative 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V. and Perelomov, A. M.}, + title = {Invariant {K{\"a}hler--Einstein} metrics on compact homogeneous spaces}, + journal = {Functional Analysis and Its Applications}, + volume = {20}, + number = {3}, + year = {1986}, + pages = {171--182}, + doi = {10.1007/BF01078469}, + note = {Translated from Funktsional. Anal. i Prilozhen. 20 (1986), no. 3, 1--16} +} + +@article{OcchettaSolaCondeWatanabeWisniewski2017, + author = {Occhetta, Gianluca and Sol{\'a} Conde, Luis E. and Watanabe, Kiwamu and Wi{\'s}niewski, Jaros{\l}aw A.}, + title = {{Fano} manifolds whose elementary contractions are smooth {$\mathbb{P}^1$}-fibrations: a geometric characterization of flag varieties}, + journal = {Annali della Scuola Normale Superiore di Pisa, Classe di Scienze}, + series = {5}, + volume = {17}, + number = {2}, + year = {2017}, + pages = {573--607}, + doi = {10.2422/2036-2145.201508_007}, + eprint = {1407.3658}, + archivePrefix = {arXiv}, + url = {https://journals.sns.it/index.php/annaliscienze/article/download/507/497/726} +} diff --git a/preprints/The-Campana-Peternell-conjecture-in-dimension-six-September-25-2026/build/sections/01-introduction.tex b/preprints/The-Campana-Peternell-conjecture-in-dimension-six-September-25-2026/build/sections/01-introduction.tex new file mode 100644 index 0000000000000000000000000000000000000000..63cfe3d8212552ebf18a0b728f742609a31e5cd6 --- /dev/null +++ b/preprints/The-Campana-Peternell-conjecture-in-dimension-six-September-25-2026/build/sections/01-introduction.tex @@ -0,0 +1,221 @@ +\section{Introduction} + +A smooth connected complex projective variety $X$ is \emph{Fano} if +its anticanonical line bundle $\omega_X^{-1}$ is ample. A vector bundle +$E$ is \emph{nef} if the tautological line bundle +$\mathcal O_{\mathbb P(E)}(1)$ has nonnegative degree on every curve. +We use the quotient convention +$\mathbb P(E)=\operatorname{Proj}\operatorname{Sym}E$. +A variety is \emph{rational homogeneous} if it is isomorphic to $G/P$, +where $G$ is a connected semisimple complex algebraic group and $P$ is +a parabolic subgroup. + +The tangent bundle of a rational homogeneous variety is generated by +global sections and is therefore nef. Mori's characterization of +projective space by ampleness of its tangent bundle +\cite{Mori1979} makes the corresponding question for nef tangent +bundles natural. The Campana--Peternell conjecture asks whether every +smooth complex Fano variety with nef tangent bundle is rational +homogeneous \cite{CampanaPeternell1991}. Thus it predicts that a +numerical positivity condition determines a variety with a transitive +algebraic group action. We prove the conjecture in dimension six. + +\begin{theorem}\label{thm:main} +Every smooth connected complex projective Fano variety of dimension six +with nef tangent bundle is rational homogeneous. +\end{theorem} + +The Picard number $\rho(X)$ is the rank of the group of divisor classes +modulo numerical equivalence, and the \emph{pseudoindex} is the least +anticanonical degree of a rational curve on $X$. +These two invariants organize much of the progress on the conjecture. +Campana and Peternell established it in dimensions at most three and +for fourfolds of Picard number greater than one +\cite{CampanaPeternell1991,CampanaPeternell1993}. +The remaining fourfold case was settled through Mok's and Hwang's +study of the tangent directions of minimal rational curves +\cite{Mok2002,Hwang2006}. Watanabe proved the fivefold case of Picard +number greater than one \cite[Theorem~1.2]{Watanabe2014}, and +Kanemitsu completed the fivefold classification +\cite[Theorem~0.2]{Kanemitsu2017}. + +Kanemitsu also proved that a smooth complex Fano $n$-fold with nef +tangent bundle is rational homogeneous whenever $\rho(X)>n-5$ +\cite[Theorem~0.2]{Kanemitsu2016}. For a sixfold, this covers every +$\rho(X)\geq2$, leaving only Picard number one. In that range, +standard characterization results reduce the problem to pseudoindices +four and five. Watanabe treated the pseudoindex-four sixfold case +\cite[Theorem~2.4]{Watanabe2021}. We include an independent intersection +argument for it; the principal new step is pseudoindex five, where a +characteristic-number constraint leads to the Grassmannian. + +The difficulty is to pass from positivity of $T_X$ to a specific +projective model for the tangent directions of minimal rational curves. +Nefness gives strong smoothness and deformation properties, but these +properties do not by themselves identify the threefold of directions +that occurs in pseudoindex five. We extract additional information by +using the universal family in two ways: as a family of rational curves +over their parameter space, and as a family of threefolds over $X$. +The compatibility of these two maps yields a characteristic-number +identity strong enough to determine the projective variety of tangent +directions. + +For a covering family of rational curves of least anticanonical degree, +the closure of the tangent directions to its members smooth at a general +point is called the \emph{variety of minimal rational tangents} (VMRT). +Characteristic classes and the geometry of minimal curves already +interact in Mok's work on one-dimensional varieties of minimal rational +tangents \cite{Mok2002}. Under an additional fourth-Betti-number +hypothesis, his argument uses the two rational-curve fibrations of the +universal family and characteristic-class inequalities for a rank-two +bundle on the parameter space to constrain the tangent image. +Hwang removed that hypothesis +\cite[Section~4]{Hwang2006}; the resulting classification is stated in +\cite[Theorem~15]{Mok2008}. A related structural approach studies +elementary Mori contractions, the contractions of extremal rays: +if all of these contractions of a Fano manifold are smooth +$\mathbb P^1$-fibrations, the manifold is a complete flag variety +\cite[Theorem~1.2]{OcchettaSolaCondeWatanabeWisniewski2017}. +These earlier results show how multiple curve fibrations can constrain +both characteristic classes and homogeneous structure. +The calculation here concerns a three-dimensional tangent image. It +uses the two fibrations to produce explicit linear intersection +identities, then combines them with the product identities furnished +by the projection formula. + +\subsection{The argument} + +Assume $\rho(X)=1$. Standard characterization theorems +leave pseudoindices four and five; the other possibilities are excluded +or give projective space $\mathbb P^6$ and the smooth quadric $Q^6$. +The proof is organized around the universal family of rational curves +of least degree. Its evaluation map and its projection to the parameter +space are both smooth. Along every fibre of the latter map, the +relative tangent bundle of evaluation is a sum of copies of +$\mathcal O_{\mathbb P^1}(-1)$. + +For pseudoindex four, Kanemitsu's factorization theorem reduces the +problem to a variety admitting two fibrations. We rule out the possible +fibres by Hard Lefschetz and intersection theory. The last case is an +odd-dimensional quadric bundle: its orthogonal description forces a +vanishing characteristic number, whereas the second fibration would +make an integral divisor have nonintegral degree. + +For pseudoindex five and a general point $x\in X$, the evaluation fibre +$F$ is a smooth threefold parametrizing minimal rational curves marked at +$x$. Sending a marked +curve to its tangent direction at $x$ gives a finite morphism +$F\to\mathbb P(T_x^*X)\simeq\mathbb P^5$. Its image is the variety of +minimal rational tangents, and the pullback $L$ of $\mathcal O(1)$ is +ample. At this point the degree of the map and the projective type of +its image are both undetermined. The central numerical step is +\[ + 32-20u+3u^2=0, + \qquad u=\frac{c_1(T_F)\cdot L^2}{L^3}. +\] +We derive this identity from the cohomology of the universal family, +Deligne's fixed-part theory \cite{Deligne1971}, and +Grothendieck--Riemann--Roch \cite{BorelSerre1958}. All resulting characteristic-number +constraints are written explicitly. A finite rational calculation then +extracts the quadratic relation. + +This method separates two useful ingredients. The two fibrations give +linear relations among intersection numbers; integration along evaluation +also makes selected numbers factor as products. Multiplying these +relations forces a nonlinear constraint on the polarized fibre. +The proof explains both ingredients and gives an exact certificate for +the finite elimination. + +Finally, this image spans $\mathbb P^5$. The sectional-genus +formula excludes $u=4$ and forces $L^3=3$ when $u=8/3$. +The variety of minimal rational tangents is consequently the Segre +threefold $\mathbb P^1\times\mathbb P^2\subset\mathbb P^5$. +The VMRT method of Hwang and Mok +\cite[Propositions~1.2.1--1.2.2]{HwangMok1999} supplies the +nondegeneracy criterion, and Mok's recognition theorem +\cite{Mok2008Recognition} identifies $X$ with +$\operatorname{Gr}(2,5)$, the Grassmannian of two-dimensional subspaces +of $\mathbb C^5$. + +\subsection{Consequences of the classification} + +The classification also has a consequence for compact K\"ahler manifolds +that need not be projective. A differential-geometric antecedent is +Mok's uniformization theorem for compact K\"ahler manifolds with +nonnegative holomorphic bisectional curvature \cite{Mok1988}. +For nef tangent bundles, the structure theorem of +Demailly--Peternell--Schneider relates the compact K\"ahler problem to +the Fano conjecture through the fibres of the Albanese map +\cite[Main Theorem and Theorem~3.14]{DemaillyPeternellSchneider1994}. +To state the consequence precisely, a holomorphic vector bundle +is \emph{nef in the analytic sense} when its tautological line bundle +admits smooth Hermitian metrics whose curvature is bounded below by +$-\varepsilon\omega$ for every $\varepsilon>0$, for a fixed Hermitian +form $\omega$ on its projectivization. This agrees with the preceding +numerical definition in the projective setting +\cite[Section~1.A and Definitions~1.2 and~1.9]{DemaillyPeternellSchneider1994}. +For a connected compact K\"ahler manifold $X$ with nef tangent bundle, put +\[ + \widetilde q(X)= + \max\bigl\{h^1(X',\mathcal O_{X'}): + X'\longrightarrow X\text{ is a connected finite \'{e}tale cover}\bigr\}. +\] +This maximum exists and is at most $\dim_{\mathbb C}X$ +by the Demailly--Peternell--Schneider Albanese theorem +\cite[Proposition~3.9 and Theorem~3.14]{DemaillyPeternellSchneider1994}. + +\begin{corollary}[Albanese fibres and uniformization]\label{cor:nef-tangent-uniformization} +Let $X$ be a connected compact K\"ahler manifold whose holomorphic +tangent bundle is nef in the analytic sense. Suppose +\[ + 0\leq d=\dim_{\mathbb C}X-\widetilde q(X)\leq6. +\] +Then there is a connected finite \'{e}tale cover $X'\to X$ with the +following properties. +\begin{enumerate} +\item The Albanese map + $\alpha:X'\to A=\operatorname{Alb}(X')$ is a smooth proper + surjection with connected fibres, where $A$ is a complex torus of + dimension $\widetilde q(X)$. Every fibre is rational homogeneous + of dimension $d$, with a point allowed when $d=0$. +\item There is one such fibre $F$ for which $\alpha$ is a holomorphically + locally trivial $F$-bundle. The ordinary universal cover of $X$ + is biholomorphic to + \[ + \widetilde X\simeq F\times\mathbb C^{\widetilde q(X)}. + \] + It is therefore biholomorphic to a quasi-projective variety and + is holomorphically convex. Its Remmert reduction is the projection + \[ + F\times\mathbb C^{\widetilde q(X)} + \longrightarrow\mathbb C^{\widetilde q(X)}. + \] +\end{enumerate} +In particular, these conclusions hold whenever $\dim_{\mathbb C}X\leq6$. +\end{corollary} + +The first conclusion applies the Fano classification fibrewise to the +Demailly--Peternell--Schneider fibration. The second also uses local +deformation rigidity of rational homogeneous manifolds, the +Fischer--Grauert theorem, and the universal-cover splitting theorem of +Claudon--H\"oring--Koll\'ar. The torus $A$ may be nonprojective, and the +finite-cover bundle may remain twisted: the product is asserted only +for the ordinary universal cover. We prove the corollary after the +sixfold theorem in Section~\ref{sec:conclusion}. + +The Fano classification also gives a positive K\"ahler--Einstein metric +on every Fano manifold of positive dimension at most six with nef tangent bundle; +Corollary~\ref{cor:positive-ke} records the normalization of this metric. + +\paragraph{Organization.} +Section~\ref{sec:minimal-curves} establishes the common geometry, +including the rational-curve fibration calculation in +Section~\ref{subsec:p1-cohomology}. +Section~\ref{sec:pseudoindex-four} settles pseudoindex four. +Sections~\ref{sec:characteristic-classes} and~\ref{sec:arithmetic} +derive the quadratic relation. Section~\ref{sec:grassmannian} +recognizes the Grassmannian, and Section~\ref{sec:conclusion} +combines the cases with Kanemitsu's Picard-number theorem and derives +Corollary~\ref{cor:nef-tangent-uniformization}. +Appendix~\ref{app:certificate} includes the complete executable +certificate for the finite calculation. diff --git a/preprints/The-Campana-Peternell-conjecture-in-dimension-six-September-25-2026/build/sections/02-minimal-curves.tex b/preprints/The-Campana-Peternell-conjecture-in-dimension-six-September-25-2026/build/sections/02-minimal-curves.tex new file mode 100644 index 0000000000000000000000000000000000000000..fef8f1144f93fc7a83801302f7ae66353c69daef --- /dev/null +++ b/preprints/The-Campana-Peternell-conjecture-in-dimension-six-September-25-2026/build/sections/02-minimal-curves.tex @@ -0,0 +1,147 @@ +\section{Minimal rational curves}\label{sec:minimal-curves} + +We now assume that $X$ has Picard number one. The first task is to +construct a smooth universal family of rational curves and determine its +relative tangent bundle along every member. These facts will provide the +common geometric input for the two remaining pseudoindices. + +All cohomology groups in the intersection calculations have rational +coefficients. A class in $H^{2i}$ has \emph{weight} $i$. For a smooth +morphism $a$ we write $T_a=\ker(da)$ and $-K_a=c_1(T_a)$. +Kodaira vanishing gives $H^i(X,\mathcal O_X)=0$ for $i>0$. +The exponential sequence and the Lefschetz $(1,1)$ theorem therefore give +\begin{equation}\label{eq:picard-cohomology} + H^2(X,\mathbb Q)=\mathbb Q\,c_1(T_X). +\end{equation} +We also use that smooth complex Fano manifolds are rationally connected +and simply connected; see +\cite[Theorems~2.11 and~5.3.2]{CampanaPeternell1999}. + +The \emph{pseudoindex} of $X$ is the positive integer +\[ + d=\min\{-K_X\cdot B:B\subset X\text{ is an irreducible rational curve}\}. +\] +Choose an irreducible component $V$ of the normalized space of rational +curves whose members have degree $d$. Let +\begin{equation}\label{eq:universal-family} + \begin{tikzcd}[column sep=large] + U \arrow[r,"e"] \arrow[d,"\pi"'] & X\\ + V & + \end{tikzcd} +\end{equation} +be its universal family and evaluation map. Thus a fibre of $\pi$ is the +normalization of the corresponding curve, and its map to $X$ is +birational onto that curve. + +\begin{proposition}\label{prop:minimal-family} +Let $X$ be a smooth complex Fano manifold of dimension $n$ with nef tangent +bundle, and let $V$ be a component of rational curves of minimal +anticanonical degree $d$. Then $U$ and $V$ are smooth projective varieties, +$\dim V=n+d-3$, the map $\pi:U\to V$ is a smooth $\mathbb P^1$-fibration, +and $e:U\to X$ is smooth and surjective with connected fibres of dimension +$d-2$. For every fibre $C$ of $\pi$, +\begin{equation}\label{eq:relative-tangent-splitting} + T_e|_C\simeq\mathcal O_{\mathbb P^1}(-1)^{\oplus(d-2)}. +\end{equation} +\end{proposition} + +\begin{proof} +The properties of $U,V,\pi,e$ are the minimal-family theorem recalled in +\cite[Proposition~1.6]{Kanemitsu2017}; that statement applies in arbitrary +dimension. We recall why its hypotheses hold here. For every +normalization $\varphi:\mathbb P^1\to X$, nefness implies a splitting +\[ + \varphi^*T_X=\bigoplus_{i=1}^n\mathcal O(a_i),\qquad a_i\geq0. +\] +In particular, each such map is \emph{free}, meaning that its pulled-back +tangent bundle is generated by global sections. Minimality of $d$ rules +out a degeneration into a sum of rational curves: each nonzero component +would already have anticanonical degree at least $d$. This is the +properness condition for the minimal family. Freeness makes the spaces +of maps and the evaluation smooth. Finally, the finite map in the Stein +factorization of a smooth proper evaluation is \'{e}tale; simple +connectedness of $X$ makes it an isomorphism, giving connected fibres. + +It remains to prove \eqref{eq:relative-tangent-splitting}; here it is +essential to allow \emph{every} member of $V$. Fix a fibre +$C\simeq\mathbb P^1$ and set $\varphi=e|_C$. Let $H$ be the corresponding +component of $\operatorname{Hom}_{\mathrm{bir}}(\mathbb P^1,X)$. +Since all its maps are free, $H$ is smooth, and +\[ + T_{[\varphi]}H=H^0(\mathbb P^1,\varphi^*T_X). +\] +The construction of the normalized rational-curve space gives a +principal $\operatorname{PGL}_2$-bundle $H\to V$ and a compatible map +\[ + \lambda:\mathbb P^1\times H\longrightarrow U, + \qquad e\circ\lambda(t,f)=f(t); +\] +see \cite[Sections~2.1--2.2]{BrionKannan2021}. +The normalization on the map space introduces no change because $H$ is +smooth. Over $[\varphi]$, the map $\lambda$ identifies $\mathbb P^1$ with +$C$. + +For $\xi\in T_{[\varphi]}H$, differentiating $\lambda$ in the constant +parameter direction $(0,\xi)$ produces a section of $T_U|_C$. +Its image under $de$ is precisely the section of $\varphi^*T_X$ +represented by $\xi$. Consequently +\begin{equation}\label{eq:sections-evaluation-surjective} + H^0(C,T_U|_C)\longrightarrow H^0(C,\varphi^*T_X) + \quad\text{is surjective}. +\end{equation} +The tangent sequence of $\pi$, restricted to $C$, is +\[ + 0\longrightarrow\mathcal O(2)\longrightarrow T_U|_C + \longrightarrow\mathcal O^{\oplus(n+d-3)}\longrightarrow0. +\] +It gives $h^0(T_U|_C)=n+d$ and $H^1(T_U|_C)=0$. +The nonnegative splitting of $\varphi^*T_X$ likewise gives +$h^0(\varphi^*T_X)=n+d$. Thus the map in +\eqref{eq:sections-evaluation-surjective} is an isomorphism. +The cohomology sequence of +\[ + 0\longrightarrow T_e|_C\longrightarrow T_U|_C + \longrightarrow\varphi^*T_X\longrightarrow0 +\] +now yields $H^0(C,T_e|_C)=H^1(C,T_e|_C)=0$. +Every line-bundle summand of $T_e|_C$ therefore has degree $-1$, proving +the assertion. No immersion assumption on $\varphi$ was used. +\end{proof} + +We can now isolate the cases that require further work. + +\begin{proposition}\label{prop:pseudoindex-reduction} +Let $X$ be a smooth complex Fano sixfold with $\rho(X)=1$ and nef tangent +bundle. Then either $X\simeq\mathbb P^6$, $X\simeq Q^6$, or its +pseudoindex is $4$ or $5$. +\end{proposition} + +\begin{proof} +For a nonconstant map $\varphi:\mathbb P^1\to X$, the differential +$\mathcal O(2)\to\varphi^*T_X$ is nonzero. At least one summand of the +nonnegative splitting of $\varphi^*T_X$ therefore has degree at least +$2$, so $d\geq2$. +If $d=2$, Proposition~\ref{prop:minimal-family} makes $e$ a smooth proper +map of relative dimension zero with connected fibres, hence an +isomorphism. The map $\pi$ would then give a $\mathbb P^1$-fibration +$X\to V$ with $\dim V=5$. The pullback of an ample divisor on $V$ is +nonzero and nef but has degree zero on the fibres. This contradicts +$\rho(X)=1$. + +For a general point $x\in X$, the \emph{variety of minimal rational +tangents} is the closure in $\mathbb P(T_x^*X)$ of the tangent lines at +$x$ to members of $V$ smooth at $x$. The marked family $e^{-1}(x)$ has +dimension $d-2$, and the finite tangent-map theorem gives the same +dimension for this variety; see \cite[Theorem~3.4]{Kebekus2002}. +When $d=3$, it is therefore one-dimensional. The Mok--Hwang classification +\cite[Theorem~15]{Mok2008} gives only $\mathbb P^2$, $Q^3$, and the +five-dimensional homogeneous contact manifold of type $G_2$; see also +\cite[Section~1]{Kanemitsu2017}. +None of the three resulting manifolds has dimension six. + +If $d\geq6$, every rational curve $B\subset X$ satisfies +$-K_X\cdot B\geq\dim X$. The numerical characterization of +Dedieu--H\"oring \cite[Theorem~1.3]{DedieuHoring2017} gives +$X\simeq\mathbb P^6$ or $Q^6$. +Thus the only cases left are $d=4$ and $d=5$. +\end{proof} diff --git a/preprints/The-Campana-Peternell-conjecture-in-dimension-six-September-25-2026/build/sections/02-p1-cohomology.tex b/preprints/The-Campana-Peternell-conjecture-in-dimension-six-September-25-2026/build/sections/02-p1-cohomology.tex new file mode 100644 index 0000000000000000000000000000000000000000..08fdd278315b22e0fda96179cee86793e3a6f502 --- /dev/null +++ b/preprints/The-Campana-Peternell-conjecture-in-dimension-six-September-25-2026/build/sections/02-p1-cohomology.tex @@ -0,0 +1,61 @@ +\subsection{Cohomology of a rational-curve fibration} +\label{subsec:p1-cohomology} + +The two remaining cases use integration along a smooth +$\mathbb P^1$-fibration. The following lemma expresses this operation +through an involution and a product rule, without requiring the +fibration to be the projectivization of a vector bundle. + +\begin{lemma}\label{lem:p1-involution} +Let $p:B\to D$ be a smooth projective morphism of smooth connected +projective varieties whose fibres are isomorphic to $\mathbb P^1$. +Set $c=c_1(T_p)$ and $\delta=p^*p_*$. +Every class on $B$ is uniquely $a+cb$, with $a,b$ pulled back from $D$, +and $c^2$ is pulled back from $D$. Consequently +\[ + \overline{a+cb}=a-cb +\] +defines a ring involution. On even cohomology, +\begin{align} + \delta(a+cb)&=2b,& + \bar\alpha&=\alpha-c\delta\alpha,\label{eq:p1-involution}\\ + \delta(\alpha\gamma) + &=(\delta\alpha)\gamma+\bar\alpha\,\delta\gamma. + \label{eq:p1-product} +\end{align} +For a class $h$ of weight one with fibre degree one, both $2h-c$ and +$\alpha-h\delta\alpha$ are pulled back from $D$. +If $\alpha$ has top degree on $B$, then +$\int_B\bar\alpha=-\int_B\alpha$. +\end{lemma} + +\begin{proof} +The restrictions of $1$ and $c/2$ form a basis of the cohomology of every +fibre. Leray--Hirsch therefore gives the asserted unique decomposition, +and fibre integration gives $p_*c=2$ and $p_*p^*a=0$. +The higher direct images of $\mathcal O_B$ are $p_*\mathcal O_B=\mathcal +O_D$ and $R^jp_*\mathcal O_B=0$ for $j>0$. Relative +Grothendieck--Riemann--Roch~\cite{BorelSerre1958} consequently gives +\[ + p_*\operatorname{td}(T_p)=1. +\] +Its component of weight one on $D$ is $p_*c^2/12=0$. +Writing $c^2=a+cb$ now gives $2b=0$, so $c^2$ is pulled back. This proves +that negating $c$ preserves products. The first two identities follow +from the decomposition, and multiplication of two such decompositions +proves~\eqref{eq:p1-product}. + +Write $h=a_0+c/2$. Then $2h-c=2a_0$, while +$\alpha-h\delta\alpha=a-2a_0b$ for $\alpha=a+cb$. +Finally, in top degree the summand pulled back from $D$ vanishes by +dimension. The involution negates the remaining summand. +\end{proof} + +In particular, if $\bar h=\ell-h$ and $\ell=2h-c$, repeated use of +\eqref{eq:p1-product} gives the polynomial identities +\begin{equation}\label{eq:p1-powers} + \delta(h^a)=\sum_{k=0}^{a-1}h^k(\ell-h)^{a-1-k},\qquad + \delta(c^a)=(1-(-1)^a)c^{a-1}\quad(a\ge1). +\end{equation} +These formulas are obtained by induction, so do not require division +by a cohomology class. diff --git a/preprints/The-Campana-Peternell-conjecture-in-dimension-six-September-25-2026/build/sections/03-pseudoindex-four.tex b/preprints/The-Campana-Peternell-conjecture-in-dimension-six-September-25-2026/build/sections/03-pseudoindex-four.tex new file mode 100644 index 0000000000000000000000000000000000000000..63db032ba594fe6b324586665c70e0f3ea6cae14 --- /dev/null +++ b/preprints/The-Campana-Peternell-conjecture-in-dimension-six-September-25-2026/build/sections/03-pseudoindex-four.tex @@ -0,0 +1,202 @@ +\section{Pseudoindex four}\label{sec:pseudoindex-four} + +Watanabe treats this case in \cite[Theorem~2.4]{Watanabe2021}. +We give an intersection-theoretic argument using the factorization of the +universal family established by Kanemitsu. His theorem gives either +rational homogeneity or a factorization of the evaluation map; we exclude +the latter possibility. The relative-degree calculation below reproves +Kanemitsu's Lemma~4.3, and the exclusion of projective-plane fibres +adapts the Betti-number argument in his proof of Theorem~0.2 +\cite{Kanemitsu2017}. In dimension six, the remaining quadric-bundle +case is excluded by the orthogonal-bundle calculation given here. + +Let $X$ be a smooth connected complex projective Fano sixfold with nef +tangent bundle, Picard number one and pseudoindex four. Let +$\pi:U\to V$ and $e:U\to X$ be the universal family and evaluation map +of a minimal rational component. By Proposition~\ref{prop:minimal-family}, we have +\[ + T_e|_C\simeq\mathcal O_{\mathbb P^1}(-1)^{\oplus2} + \qquad(C\text{ a }\pi\text{-fibre}). +\] +\begin{proposition}\label{prop:pseudoindex-four} +A smooth complex Fano sixfold with nef tangent bundle, Picard number one +and pseudoindex four is rational homogeneous. +\end{proposition} + +\begin{proof} +Kanemitsu's Theorem~3.3 and Proposition~4.1 +\cite{Kanemitsu2017} apply in arbitrary dimension; the latter point is +explicit in his Remark~4.2. +They show that either $X$ is rational homogeneous, or there is a diagram +\[ +\begin{tikzcd}[column sep=large,row sep=large] + U \arrow[r,"f"] \arrow[d,"\pi"'] & + W \arrow[r,"g"] \arrow[d,"q"] & X\\ + V \arrow[r] & Y & +\end{tikzcd} +\] +with $e=g\circ f$, with $f,g,\pi$ smooth $\mathbb P^1$-fibrations, +and with $q$ a smooth elementary Mori contraction, that is, a Mori +contraction of relative Picard number one. We exclude the +second possibility. Here $\dim W=7$. Applied to $f$ and $\pi$, +\cite[Remark~2.3]{Kanemitsu2017} says that all fibres of $q$ are isomorphic to +one member of the list +\[ + \mathbb P^1,\quad\mathbb P^2,\quad\mathbb P^3,\quad + Q^3,\quad Q^5,\quad K(G_2). +\] +Here $Q^r$ is a smooth quadric of dimension $r$, and $K(G_2)$ is the +five-dimensional rational homogeneous contact manifold of type $G_2$. +In particular, $\dim Y\ge2$. + +\smallskip +\noindent\emph{The two relative degrees.} +Write +\[ + H=\frac{c_1(T_X)}4,\qquad h=g^*H,\qquad c=c_1(T_g). +\] +The classes $h,c$ form a basis of $H^2(W,\mathbb Q)$: a smooth +$\mathbb P^1$-fibration has the rational Leray--Hirsch decomposition, +and $H^2(X,\mathbb Q)=\mathbb QH$. These classes are algebraic, so +the same basis describes $N^1(W)_{\mathbb Q}$, the rational divisor +classes modulo numerical equivalence. +Fix a $\pi$-fibre $C$. The map $e|_C$ is birational to its image, +so $f|_C$ is also birational to its image and $h\cdot f(C)=1$. +Set +\[ + a=c_1(T_f)\cdot C,\qquad b=c\cdot f(C). +\] +The relative tangent sequence gives $a+b=-2$. Moreover, +$T_f|_C$ is a line subbundle of $\mathcal O(-1)^{\oplus2}$, whence +$a\le-1$. Both $a$ and $b$ are integers. + +We show that $b<0$. The ray contracted by $q$ is generated by $f(C)$. +Since $\rho(W)=2$ and $q$ is elementary, $\rho(Y)=1$. Thus a +positive multiple of $c-bh$ is the pullback +of an ample rational class on $Y$. The sign is positive because +$c-bh$ has degree two on a $g$-fibre, whose numerical class is not +in the contracted ray. By Lemma~\ref{lem:p1-involution}, +\[ + g_*c=2,\qquad g_*c^2=0, +\] +and consequently +\[ + g_*(c-bh)^2=-4bH. +\] +Intersecting with any complete-intersection curve in $X$ proves +$b\le0$. + +If $b=0$, the square of $c$ has zero integral over $g^{-1}(D)$ +for every smooth curve $D\subset X$. Each $g$-fibre has positive +degree against $q^*A$, for an ample class $A$ on $Y$, and hence maps +onto a curve under $q$. Since $q$ is surjective and $\dim Y\ge2$, +two such image curves are distinct. Choose a smooth irreducible +complete-intersection curve $D\subset X$ through the corresponding +two points of $X$. The inverse image $g^{-1}(D)$ is a smooth +irreducible surface, and its image under $q$ contains both of those +distinct curves. Thus this image has dimension two. It follows that +$(q^*A)^2\cdot[g^{-1}(D)]>0$, a contradiction. We have proved +\[ + a=b=-1. +\] +Since $-K_f\cdot C=a=-1$, Kanemitsu's Remark~2.3 now reduces the +possible fibres of $q$ to +$\mathbb P^2$, $Q^3$, and $K(G_2)$. + +\smallskip +\noindent\emph{The projective-plane fibres.} +Suppose that $q$ has fibres $\mathbb P^2$, so that $\dim Y=5$. +Relative ample classes and their powers give Leray--Hirsch bases for +both $g$ and $q$. Writing $b_i(Z)=\dim H^i(Z,\mathbb Q)$, we obtain +$b_2(Y)=1$, and Poincar\'e duality +on $Y$ gives +\[ +\begin{split} + b_4(X)+1&=b_4(W)=b_4(Y)+2,\\ + b_6(X)+b_4(X)&=b_6(W)=2b_4(Y)+1. +\end{split} +\] +Hence $b_6(X)=b_4(X)-1$. This contradicts the injectivity of +cup-product by an ample class from $H^4(X,\mathbb Q)$ to +$H^6(X,\mathbb Q)$, which follows from Hard Lefschetz. + +\smallskip +\noindent\emph{The remaining fibres.} +We may therefore assume that $\dim Y\le4$. Put +\[ + z=\frac{h+c}{2},\qquad m_i=\int_W h^{7-i}z^i\quad(0\le i\le7). +\] +Since $z\cdot f(C)=0$, the class $z$ is pulled back from +$H^2(Y,\mathbb Q)$. Thus $z^5=0$, and +\[ + m_0=m_5=m_6=m_7=0,\qquad m_1=\int_X H^6>0. +\] +The rational-cohomology involution of $g$ fixes classes pulled back +from $X$ and sends $c$ to $-c$. It sends $z$ to $h-z$ and acts by +$-1$ on top cohomology. For $i=2,4,6$, therefore, +\[ + \sum_{j=0}^i(-1)^j\binom ij m_j=-m_i. +\] +In order, these identities give +\[ + m_2=m_1,\qquad m_1-2m_3+m_4=0, + \qquad9m_1-20m_3+15m_4=0. +\] +Consequently +\[ + m_1=m_2=5m_4,\qquad m_3=3m_4. +\] +In particular $m_4>0$, so $\dim Y\ge4$. Thus $\dim Y=4$ and +every fibre of $q$ is $Q^3$. + +It remains to show that this quadric fibration is incompatible with +$g$. Since $-K_W=4h+c$ has degree three on $f(C)$, write +\[ + -K_q=3h-\lambda z. +\] +The coefficient $\lambda$ is integral: a $g$-fibre has respective +degrees zero and one against $h$ and $z$, and $-K_q$ is an integral +divisor class. We will determine $\lambda$ from the moments above +using the quadric-bundle identity +\[ + q_*(-K_q)^4=0. +\] +To prove this identity, first apply Fischer--Grauert +\cite{FischerGrauert1965}: the smooth projective morphism $q$, all of +whose fibres are $Q^3$, is analytically locally trivial. Its structure +group is $\operatorname{Aut}(Q^3)=\operatorname{PGO}_5(\mathbb C)$. +In odd rank this group is isomorphic to $\operatorname{SO}_5$. +Explicitly, fix a nondegenerate symmetric $5\times5$ matrix $J$ defining +the quadric. If $A$ represents a projective automorphism and +$A^{\mathsf T}JA=\mu J$, then +\[ + \widetilde A=\frac{\mu^2}{\det A}A +\] +has determinant one and preserves $J$. This expression is unchanged +when $A$ is multiplied by a scalar; it is the unique special +orthogonal lift and respects products. The transition functions thus +define a rank-five holomorphic vector bundle $E$ equipped with a +nondegenerate symmetric form with values in $\mathcal O_Y$ and +trivial determinant. GAGA algebraizes this bundle and its form +\cite[Theorems~2--3 and Propositions~15, 18]{Serre1956}. +The resulting analytic identification of $W$ with the relative quadric +is algebraic as well. +Accordingly, in the quotient convention for projectivization, +$W\subset\mathbb P_Y(E)$ is the quadric of class $2\zeta$, where +$\zeta=c_1(\mathcal O_{\mathbb P(E)}(1))$. Self-duality identifies +the line and quotient descriptions of this quadric. If +$p:\mathbb P_Y(E)\to Y$ is the projection, adjunction gives +\[ + -K_q=3\zeta|_W, + \qquad q_*(-K_q)^4=2\cdot3^4p_*\zeta^5 + =2\cdot3^4c_1(E)=0. +\] + +Now the projection formula and $z^5=0$ give +\[ + 0=\int_W(-K_q)^4z^3=3^4m_3-4\cdot3^3\lambda m_4. +\] +Since $m_3=3m_4$ and $m_4>0$, this forces $\lambda=9/4$, +contrary to integrality. This excludes the factorization case and +proves the proposition. +\end{proof} diff --git a/preprints/The-Campana-Peternell-conjecture-in-dimension-six-September-25-2026/build/sections/04-characteristic-classes.tex b/preprints/The-Campana-Peternell-conjecture-in-dimension-six-September-25-2026/build/sections/04-characteristic-classes.tex new file mode 100644 index 0000000000000000000000000000000000000000..34b8998679ddb43717077e91910aa875084a1143 --- /dev/null +++ b/preprints/The-Campana-Peternell-conjecture-in-dimension-six-September-25-2026/build/sections/04-characteristic-classes.tex @@ -0,0 +1,388 @@ +\section{Characteristic-class constraints in pseudoindex five} +\label{sec:characteristic-classes} + +We now assume that $X$ is a Fano sixfold with nef tangent bundle, +$\rho(X)=1$, and pseudoindex five. Let +\[ + \pi:U\longrightarrow V,\qquad e:U\longrightarrow X +\] +be its smooth universal family. Thus $\dim U=9$, $\dim V=8$, +the fibres of $\pi$ are projective lines, and $e$ has relative dimension +three. The preceding geometric argument gives +$T_e|_C\simeq\mathcal O_{\mathbb P^1}(-1)^{\oplus3}$ for every +$\pi$-fibre $C$. Our goal is to constrain the polarization of an +evaluation fibre by combining the two fibrations. + +Put +\begin{equation}\label{eq:d5-generators} + h_X=\frac{c_1(T_X)}5,\qquad h=e^*h_X,\qquad + c=c_1(T_\pi),\qquad \ell=2h-c,\qquad E=T_e. +\end{equation} +The class $h$ has degree one on a $\pi$-fibre. Lemma~\ref{lem:p1-involution} +therefore shows that $\ell$ comes from $V$ and that $\bar h=\ell-h$. + +\subsection{The polarized evaluation fibre} + +\begin{lemma}\label{lem:marked-tangent} +For a general point $x\in X$, the fibre $F=e^{-1}(x)$ is a smooth +irreducible projective threefold, and the tangent direction at the marking +defines a finite morphism +\[ + \tau_x\colon F\longrightarrow\mathbb P(T_x^*X). +\] +Its image $\mathcal C_x$ is the variety of minimal rational tangents of +$V$ at $x$. The line bundle $L=\tau_x^*\mathcal O(1)$ is ample and +generated by global sections, with $c_1(L)=\ell|_F$. +\end{lemma} + +\begin{proof} +Smoothness, projectivity and connectedness follow from the established +properties of $e$. A smooth connected variety is irreducible. +Let $V_x\subset V$ be the reduced locus of curves passing through $x$. +The restriction $\pi|_F\colon F\to V_x$ is proper and has finite fibres: +each curve has only finitely many points on its normalization mapping to +$x$. It is therefore finite. Kebekus's Theorem~3.3(1) +\cite{Kebekus2002}, applied with the ample line bundle +$\omega_X^{-1}$, says that only finitely many members of $V_x$ can be +singular at a general $x$, and that even these curves are immersed at +their markings. Since $\dim V_x=3$, a general member is smooth at $x$, +so $\pi|_F$ is birational. As $F$ is normal, it is the normalization of +$V_x$. The hypotheses of Kebekus's Theorem~3.4 are satisfied: $V$ is an +irreducible covering family and $V_x$ is proper. That theorem gives the +finite tangent morphism on $F$. + +With our quotient convention for projectivization, +$\mathbb P(T_x^*X)$ parametrizes lines in $T_xX$. The differential of +$e$ is nonzero on $T_\pi$ at every marking over $x$, and identifies +$T_\pi|_F$ with the pullback of the tautological subline +$\mathcal O(-1)\subset T_xX\otimes\mathcal O$. Consequently +$c_1(L)=-c|_F=\ell|_F$. Pullback by a finite morphism preserves +ampleness, and the pullbacks of hyperplane sections generate $L$. +\end{proof} + +\subsection{Classes descending to the parameter space} + +Define $r,s,t$ to be the elementary symmetric polynomials in the three +formal roots obtained by adding $h$ to the Chern roots of $E$. +Equivalently, +\begin{align*} + r&=c_1(E)+3h,\\ + s&=c_2(E)+2hc_1(E)+3h^2,\\ + t&=c_3(E)+hc_2(E)+h^2c_1(E)+h^3. +\end{align*} +This definition uses only rational cohomology; it does not assert that +$h$ is the first Chern class of a line bundle. + +\begin{lemma}\label{lem:shifted-descent} +The classes $r,s,t$ are pulled back from $V$. +\end{lemma} +\begin{proof} +There is an actual line bundle +\[ + \mathcal A=e^*\omega_X^{-1}\otimes(T_\pi^\vee)^{\otimes2}, + \qquad c_1(\mathcal A)=5h-2c=h+2\ell. +\] +Its degree on every $\pi$-fibre is $5-4=1$. +Thus $E\otimes\mathcal A$ restricts to the trivial rank-three bundle +on each fibre. Cohomology and base change show that +$\mathcal B=\pi_*(E\otimes\mathcal A)$ is locally free of rank three. +The evaluation map $\pi^*\mathcal B\to E\otimes\mathcal A$ is an +isomorphism on every fibre, and hence is an isomorphism. +The formal roots defining $r,s,t$ are consequently the roots of +$\pi^*\mathcal B$ minus $2\ell$. Their elementary symmetric +polynomials are pulled back from $V$. +\end{proof} + +For the fibre $F$ and polarization $L$ of +Lemma~\ref{lem:marked-tangent}, the definitions and $h|_F=0$ give +\begin{equation}\label{eq:fibre-polarization} + c_1(L)=\ell|_F, + \qquad r|_F=c_1(T_F). +\end{equation} +We shall constrain the ratio +\begin{equation}\label{eq:ratio-u} + u=\frac{\int_F c_1(T_F)c_1(L)^2}{\int_F c_1(L)^3}. +\end{equation} +Its denominator is positive. +The intersection identities below will force a quadratic equation for +$u$. In Section~\ref{sec:grassmannian}, this equation and the +sectional-genus formula will determine the degree of $L$. + +\subsection{Two direct-image identities} + +We first record why the Hodge-theoretic direct images appearing in +Riemann--Roch contribute no positive-degree Chern character. + +\begin{lemma}\label{lem:constant-hodge} +Let $f:Z\to B$ be a smooth projective morphism between smooth connected +complex projective varieties, and suppose that $B$ is simply connected. +For all $p,j\ge0$, the vector bundle $R^jf_*\Omega^p_{Z/B}$ is +holomorphically trivial. In particular its positive-degree rational +Chern character vanishes. +\end{lemma} +\begin{proof} +The local systems $R^kf_*\mathbb Q$ are constant. The global invariant +cycle theorem gives a surjection +\[ + H^k(Z,\mathbb Q)\longrightarrow H^0(B,R^kf_*\mathbb Q) + \simeq H^k(Z_b,\mathbb Q) +\] +for every $b\in B$; see Deligne~\cite[Theorem~4.1.1(ii)]{Deligne1971}. +Restriction is a morphism of pure Hodge structures. Therefore each +Hodge component of $H^k(Z_b,\mathbb C)$ is spanned by restrictions of +classes of the same type on $Z$. A class on $Z$ restricts to a flat +section of the local system, and a class of fixed Hodge type keeps that +type on every fibre. Thus the Hodge decomposition, and hence the Hodge +filtration, is constant in the flat trivialization. This is also the +fixed-part statement of~\cite[Corollary~4.1.2]{Deligne1971}. +The relative Hodge--de Rham degeneration identifies +$R^jf_*\Omega^p_{Z/B}$ with the degree-$p$ graded bundle of the Hodge +filtration on $R^{p+j}f_*\mathbb C\otimes\mathcal O_B$; see +\cite[Section~2(a)--(c)]{Griffiths1970}. Since that +filtration is constant, each graded bundle is trivial. +\end{proof} + +The base $X$ is simply connected, so the lemma applies to $e$. +Relative Grothendieck--Riemann--Roch +\cite[\S7, p.~113]{BorelSerre1958} for $\mathcal O_U$ and +$\Omega^1_{U/X}=E^\vee$ gives +\begin{equation}\label{eq:grr-vanishings} + e_*\operatorname{td}_i(E)=0, + \qquad + e_*[\operatorname{td}(E)\operatorname{ch}(E^\vee)]_i=0 + \quad(i>3). +\end{equation} +Indeed these are the components of weight $i-3>0$ of +\[ + \operatorname{ch}\!\left(\sum_j(-1)^jR^je_*\mathcal O_U\right) + \quad\text{and}\quad + \operatorname{ch}\!\left(\sum_j(-1)^jR^je_*\Omega^1_{U/X}\right), +\] +respectively. + +These two choices capture the equations from all relative holomorphic +forms. Indeed, for a rank-three bundle and $p=0,1$, +$\bigwedge^{3-p}E^\vee=\det(E^\vee)\otimes\bigwedge^pE$ and +$\operatorname{td}(E^\vee)=e^{-c_1(E)}\operatorname{td}(E)$. +Replacing the Chern roots by their negatives therefore gives +\[ + [\operatorname{td}(E)\operatorname{ch}(\textstyle\bigwedge^{3-p}E^\vee)]_i + =(-1)^i + [\operatorname{td}(E)\operatorname{ch}(\textstyle\bigwedge^pE^\vee)]_i. +\] +Thus $\Omega^2_{U/X}$ and $\Omega^3_{U/X}$ repeat, up to sign in +each weight, the equations from $\Omega^1_{U/X}$ and $\mathcal O_U$. + +\subsection{Explicit polynomial representatives} + +We now turn the tangent-sequence and Riemann--Roch identities into +linear relations among weight-nine intersection numbers on $U$. The +projection formula will then identify selected normalized numbers as +products involving $u$. +Set +\[ + x_i=e^*(i!\operatorname{ch}_i(T_X)),\qquad i\ge1. +\] +Then $x_1=5h$ and $x_i=0$ for $i\ge7$, by dimension. Work in the +polynomial ring +\[ + R=\mathbb Q[h,x_2,x_3,x_4,x_5,x_6,\ell,r,s,t], +\] +with respective weights $(1,2,3,4,5,6,1,1,2,3)$. +Evaluation of these generators as cohomology classes defines a graded +homomorphism $R\to H^{2*}(U,\mathbb Q)$. + +Our unknowns are the integrals of weight-nine monomials in $R$. +We obtain linear relations among them from three sources: classes +descending to the eight-dimensional space $V$, classes pulled back +from the six-dimensional space $X$ and their $\pi$-integrals, and the +two Riemann--Roch vanishings. The tangent sequences make the +$\pi$-integrals explicit. Recall that +$\delta=\pi^*\pi_*$ lowers weight by one. The recurrences below +express these operations as polynomials, before we collect the +resulting relations into a matrix. + +Let $p_j$ be the Newton power sum of the three formal roots with +elementary symmetric functions $r,s,t$, and put $p_0=3$. +For complete specificity, if $s_1=r,s_2=s,s_3=t$, Newton's identities +read +\begin{equation}\label{eq:newton-recursion} + p_i=\sum_{j=1}^{\min(3,i)}(-1)^{j+1}s_j + \begin{cases} + i,&j=i,\\ + p_{i-j},&j0$ and define +\begin{equation}\label{eq:normalized-products} + \xi_i=\frac{\int_X M_i}{\int_Xh_X^6},\qquad + e_*G_j=D\eta_jh_X^{w_j-3}. +\end{equation} +The first expression uses the corresponding base classes on $X$. +The second defines a scalar because $w_j-3$ is zero or one and +$H^2(X,\mathbb Q)=\mathbb Qh_X$. We have +$\xi_0=\eta_0=1$ and $\eta_1=u$. + +Use the index set +\[ + J=\{0,1,2,4,5,6,7,8,9,10,12,13,15,16,17\} +\] +for the selected invariant monomials. This subcollection suffices for +the elimination in Section~\ref{sec:arithmetic} and reduces its size; +no vanishing is asserted for the omitted monomials $r^3$, $r^4$, +and $r^2s$. Let +\[ + C_1=\{(i,(j)):0\le i\le10,\ j\in J, + \ w_j=3\ \text{or}\ h\mid M_i\}. +\] +Order $C_1$ by increasing $i$ and then increasing $j$. +For each such pair the monomial $M_i/h^{w_j-3}$ is defined, and the +projection formula gives +\begin{equation}\label{eq:product-factorization} + \int_U G_j\frac{M_i}{h^{w_j-3}} + =D\int_Xh_X^6\,\xi_i\eta_j. +\end{equation} +The common factor on the right is $\int_U h^6\ell^3>0$. +We now retain all linear relations that $Av=0$ induces on these +selected coordinates. The following construction expresses them as +a matrix $N$. +If $K$ is any matrix whose columns form a rational basis of $\ker A$, +let $P$ consist of its rows indexed, with repetitions if necessary, by +the monomials on the left of~\eqref{eq:product-factorization}. +Let $N$ be the nonzero rows in the reduced row echelon form of a matrix +whose rows form a basis of $\ker(P^{\mathsf T})$. Thus $N$ is determined +by the ordered coordinates, independently of the chosen basis $K$. +Then +\begin{equation}\label{eq:N-products} + N(\xi_i\eta_j)_{(i,(j))\in C_1}=0. +\end{equation} +Indeed $v\in\ker A$, so its selected coordinates lie in the column +space of $P$, and division by the single positive scalar in +\eqref{eq:product-factorization} gives the displayed equation. +This completes the geometric derivation of the finite constraints. diff --git a/preprints/The-Campana-Peternell-conjecture-in-dimension-six-September-25-2026/build/sections/05-arithmetic.tex b/preprints/The-Campana-Peternell-conjecture-in-dimension-six-September-25-2026/build/sections/05-arithmetic.tex new file mode 100644 index 0000000000000000000000000000000000000000..23c907704acee80fc93b3d7838cd4ab2b053210d --- /dev/null +++ b/preprints/The-Campana-Peternell-conjecture-in-dimension-six-September-25-2026/build/sections/05-arithmetic.tex @@ -0,0 +1,213 @@ +\section{The numerical constraint} +\label{sec:arithmetic} + +The geometric argument has produced the bilinear equations +\eqref{eq:N-products}. We now prove that they allow only two values of +$u$. This section is a finite calculation over $\mathbb Q$: its input +is the explicitly defined matrix $A$ and its selected coordinates +$C_1$, and its output is a quadratic equation for $\eta_1$. + +\begin{lemma}\label{lem:finite-ratio} +Let $A$, $C_1$ and $N$ be the rational matrices and index set defined +in Section~\ref{sec:characteristic-classes}. Suppose that +$\xi\in\mathbb Q^{11}$ and $\eta\in\mathbb Q^{18}$ satisfy +\[ + \xi_0=\eta_0=1, + \qquad N(\xi_i\eta_j)_{(i,(j))\in C_1}=0. +\] +Then +\begin{equation}\label{eq:ratio-quadratic} + 32-20\eta_1+3\eta_1^2=0. +\end{equation} +\end{lemma} + +We give the finite certificate, including the distinction between +the modular rank bound and the rational identities needed to finish +the proof. The exact recurrences and elimination procedure are +implemented in Appendix~\ref{app:certificate}. + +\subsection{Multiplying the constraints} + +Write $y_{i,j}=\xi_i\eta_j$ for the selected bilinear coordinates. +Multiplying each equation of $Ny=0$ by $\eta_a\eta_b$, with +$a,b\in J$, gives linear equations in the cubic products +\[ + v_{i,\operatorname{sort}(j,a,b)}=\xi_i\eta_j\eta_a\eta_b, + \qquad(i,(j))\in C_1. +\] +Here $\operatorname{sort}$ arranges the three indices in nondecreasing +order, identifying equal products. +We will impose those linear equations without requiring an arbitrary +solution $v$ to have this product form. In the resulting larger +solution space, we prove that +\begin{equation}\label{eq:target-functional} + 32v_{0,(0,0,0)}-20v_{0,(0,0,1)}+3v_{0,(0,1,1)}=0. +\end{equation} +For the product vector, $\xi_0=\eta_0=1$ makes the left side +$32-20\eta_1+3\eta_1^2$. Thus a linear statement on cubic +coordinates will prove the desired quadratic relation. The following +spaces make this relaxation precise; the intermediate quadratic +space also makes its computation smaller. + +For $n=2,3$, form $C_n$ from $C_{n-1}$ by appending an index $j\in J$ +to its tuple and sorting. Repeated coordinates are identified. +Equivalently, $(i,z)\in C_n$ means that $z$ is a size-$n$ multiset +on $J$, and either $h\mid M_i$ or $z$ contains an index of weight +three. Let $\mathcal K_1=\ker N$. Recursively define +$\mathcal K_n\subseteq\mathbb Q^{C_n}$ by requiring, for every +$j\in J$, that the slice +\[ + (v_{i,\operatorname{sort}(z,j)})_{(i,z)\in C_{n-1}} + \quad\text{belong to }\mathcal K_{n-1}. +\] +Thus $\mathcal K_3$ is the kernel of a single rational matrix: +for each unordered pair $a,b\in J$, allowing $a=b$, impose $N$ on the slice +\[ + \bigl(v_{i,\operatorname{sort}(j,a,b)}\bigr)_{(i,(j))\in C_1}. +\] +There are $\binom{16}{2}=120$ such pairs. In particular, +\begin{equation}\label{eq:geometric-cubic-column} + v_{i,z}=\xi_i\prod_{j\in z}\eta_j +\end{equation} +belongs to $\mathcal K_3$: every double slice is a scalar multiple +of the given vector in $\ker N$. + +Seven monomials $M_i$ contain $h$, and six of the fifteen indices +in $J$ have weight three. Therefore +\[ + |C_n|=7\binom{14+n}{n} + +4\left(\binom{14+n}{n}-\binom{8+n}{n}\right), +\] +which gives $|C_1|=129$, $|C_2|=1140$ and $|C_3|=6820$. + +\subsection{An upper bound from finite-field elimination} + +The coefficient recurrences in Section~\ref{sec:characteristic-classes} +give $1785$ nonzero rows and $581$ columns for $A$. Exact elimination +gives a matrix $K$ of size $581\times65$ with $AK=0$. +The selected-row matrix $P$ has size $129\times65$. Choose $N$ in +reduced row echelon form. The following checks give, in particular, +complete bases rather than merely lists of relations: +\begin{center} +\begin{tabular}{c|c|c} +matrix & size & rank modulo $10007$\\\hline +$A$ & $1785\times581$ & $516$\\ +$K$ & $581\times65$ & $65$\\ +$P$ & $129\times65$ & $60$\\ +$N$ & $69\times129$ & $69$ +\end{tabular} +\end{center} +The identities $AK=0$ and $NP=0$ hold over $\mathbb Q$. +Every denominator in these matrices is prime to $10007$. +Consequently the first two ranks prove that $K$ is a basis of +$\ker A$, and the last two prove that $N$ is the full left +annihilator of $P$. +Write $\mathcal K_n(\mathbb F_{10007})$ for the spaces defined +by the same slice equations using the reduction of $N$ modulo +$10007$. + +Here is the elimination procedure used for the larger spaces. +Suppose that the columns of $Z$ are a basis for the solutions on a union +$S$ of slices, and the columns of $B$ are a basis for the solutions on the +next slice $T$. On the overlap, the two parameter vectors $a,b$ +must satisfy +\[ + Z_{S\cap T}a-B_{S\cap T}b=0. +\] +If the columns of $\binom{V_1}{V_2}$ form the kernel of this matrix, +then the columns of +\[ + \begin{pmatrix} ZV_1\\ B_{T\setminus S}V_2\end{pmatrix} +\] +parametrize all solutions on $S\cup T$. This parametrization is +injective: zero on the union is zero on each piece, and the old +parametrizations were injective. Appending a fixed index to a +multiset is itself injective, so there are no hidden identifications +within an individual slice. Ordinary Gaussian elimination and +successive application of this construction give +\begin{equation}\label{eq:finite-field-dimensions} + \dim\mathcal K_1(\mathbb F_{10007})=60, + \qquad\dim\mathcal K_2(\mathbb F_{10007})=163, + \qquad\dim\mathcal K_3(\mathbb F_{10007})=6. +\end{equation} + +For the rational conclusion, use the single double-slice matrix +defining $\mathcal K_3$. It has $69\cdot120=8280$ rows and $6820$ +columns, and its denominators are units modulo $10007$. Its modular +rank is $6814$, so its rational rank is at least $6814$. Thus +\begin{equation}\label{eq:rational-dimension-upper} + \dim_{\mathbb Q}\mathcal K_3\le6. +\end{equation} +This argument does not assume that intermediate rational kernel +bases specialize to bases. + +\subsection{Six rational columns} + +It remains to exhibit six independent rational solutions and show that +the functional giving the desired quadratic vanishes on their span. +These columns are witnesses for the linear system: no assertion that +they arise from varieties is needed. Three have product form, and +three are specified by their small supports. The relaxed system +allows both kinds without imposing the original normalization or +requiring a product representation. +Define the following rational vectors: +\begin{align*} + B(v)={}&\bigl(1,7-2v/5,45-4v,275-30v,11,65-2v,\\ + &\hspace{29mm}85,17,95-2v,25,35\bigr),\\ + B'={}&(1,11/5,5,11,-1,-1,10,-3,-1,0,14),\\ + H^a={}&\bigl(1,4,16,64,6,24,4,3,12,\\ + &\hspace{29mm}48,192,768,17,68,272,96,9,36\bigr),\\ + H^b={}&\tfrac13\bigl(3,8,21,54,9,24,6,14,37,\\ + &\hspace{29mm}97,252,648,37,97,252,96,16,42\bigr). +\end{align*} +For $(B,H)=(B(0),H^a),(B(1),H^a),(B',H^b)$, let +\[ + v(B,H)_{i,z}=B_i\prod_{j\in z}H_j. +\] +Direct rational substitution gives +$N(B_iH_j)_{C_1}=0$, so these three columns lie in $\mathcal K_3$. +Three further columns $s_1,s_2,s_3$ have the following entries and +are zero elsewhere: +\begin{center} +\begin{tabular}{c|c|rrr} +$i$ & $z$ & $s_1$ & $s_2$ & $s_3$\\\hline +$5$ & $(15,15,15)$ & $3$ & $0$ & $0$\\ +$7$ & $(15,15,15)$ & $1$ & $0$ & $0$\\ +$9$ & $(15,15,15)$ & $-5$ & $0$ & $0$\\ +$7$ & $(10,10,15)$ & $0$ & $2$ & $0$\\ +$7$ & $(10,10,10)$ & $0$ & $4$ & $1$\\ +$9$ & $(10,10,10)$ & $0$ & $25$ & $0$ +\end{tabular} +\end{center} +Applying $N$ to each of their $120$ double slices gives zero over +$\mathbb Q$. These are exact rational equalities, separate from +the finite-field calculation. + +The six columns are independent. Indeed, restrict them to the rows +\begin{gather*} + (0,(0,0,0)),\quad (0,(0,0,1)),\quad (1,(0,0,0)),\\ + (5,(15,15,15)),\quad (7,(10,10,15)),\quad (7,(10,10,10)). +\end{gather*} +The upper-right $3\times3$ block is zero. The upper-left block has +determinant $-8/15$, and the lower-right block is +$\left(\begin{smallmatrix}3&0&0\\0&2&0\\0&4&1\end{smallmatrix}\right)$, +with determinant $6$. The full minor is $-16/5$. Together with +\eqref{eq:rational-dimension-upper}, this proves that the six +displayed columns form a rational basis of $\mathcal K_3$. + +\begin{proof}[Proof of Lemma~\ref{lem:finite-ratio}] +The functional in~\eqref{eq:target-functional} vanishes on the six +basis columns. On a displayed product column its value +is $32-20H_1+3H_1^2$, with $H_1=4$ or $8/3$; the sparse columns +are zero at all three coordinates. Apply it to +\eqref{eq:geometric-cubic-column} and use $\xi_0=\eta_0=1$. +\end{proof} + +Returning to the evaluation fibre, \eqref{eq:N-products} verifies +the hypothesis of the lemma and $\eta_1=u$. Hence +\[ + (u-4)(3u-8)=0. +\] +It remains to identify which value is compatible with the tangent +map and to recognize the resulting variety of minimal rational +tangents. diff --git a/preprints/The-Campana-Peternell-conjecture-in-dimension-six-September-25-2026/build/sections/06-grassmannian.tex b/preprints/The-Campana-Peternell-conjecture-in-dimension-six-September-25-2026/build/sections/06-grassmannian.tex new file mode 100644 index 0000000000000000000000000000000000000000..23e9025bcaf5a656bc507062524715442bfc134c --- /dev/null +++ b/preprints/The-Campana-Peternell-conjecture-in-dimension-six-September-25-2026/build/sections/06-grassmannian.tex @@ -0,0 +1,167 @@ +\section{Recognizing the Grassmannian}\label{sec:grassmannian} + +We finish the pseudoindex-five case by interpreting the numerical constraint +on the fibres of the evaluation map. Throughout this section, $X$ is a +smooth complex Fano sixfold with nef tangent bundle, Picard number one and +pseudoindex five. We use its minimal rational component $V$, universal +family $\pi\colon U\to V$, and evaluation $e\colon U\to X$. +Thus $e$ is smooth with connected three-dimensional fibres. For a general +point $x\in X$, let $F=e^{-1}(x)$ and let $L$ be the tangent-map +polarization of Lemma~\ref{lem:marked-tangent}. +Our goal is to recover $X$ from the projective geometry of the tangent +directions to its minimal rational curves. +For this polarization, Lemma~\ref{lem:finite-ratio} gives +\begin{equation}\label{eq:vmrt-ratio-input} + 32-20u+3u^2=0, + \qquad u=\frac{c_1(T_F)\cdot L^2}{L^3}. +\end{equation} +We first show that the tangent image spans $\mathbb P^5$. Its degree +bound and the sectional-genus formula will then determine $L^3$ and +the degree of the tangent map. Finally, normality of the image will +let us use the smoothness of $F$ to select the Segre threefold among +the varieties of minimal degree. + +\begin{lemma}\label{lem:vmrt-nondegenerate} +For a general point $x\in X$, the threefold +$\mathcal C_x\subset\mathbb P(T_x^*X)\simeq\mathbb P^5$ is linearly +nondegenerate. +\end{lemma} + +\begin{proof} +Write $\widehat{\mathcal C}_x\subset T_xX$ for its affine cone and +$W_x=\operatorname{Span}(\widehat{\mathcal C}_x)$. Suppose that $W_x$ +is proper. Since $\mathcal C_x$ has dimension three, +$4\leq\dim W_x\leq5$. On a dense open subset of $X$, these spans +define the distribution $W\subset T_X$. + +We use two results of Hwang and Mok +\cite[Propositions~1.2.1 and~1.2.2]{HwangMok1999}. For an irreducible +variety of minimal rational tangents on a uniruled projective manifold +with $b_2=1$, a proper spanned distribution cannot be integrable. On +the other hand, it is integrable if the bivectors +\begin{equation}\label{eq:tangent-bivectors} + \alpha\wedge v,\qquad + \alpha\in\widehat{\mathcal C}_x\setminus\{0\} + \text{ a smooth point},\quad + v\in T_\alpha\widehat{\mathcal C}_x, +\end{equation} +span $\bigwedge^2W_x$ at general $x$. These results apply here: +$b_2(X)=1$ was established earlier, and $\mathcal C_x$ is irreducible +as the image of $F$. + +We verify the spanning condition directly, without assuming that +$\mathcal C_x$ is smooth. Otherwise a nonzero alternating form +$\omega\in\bigwedge^2W_x^*$ annihilates all bivectors +\eqref{eq:tangent-bivectors}. The one-form $\omega(\alpha,d\alpha)$ +vanishes on the smooth cone locus. Its exterior derivative is the +restriction of $2\omega$, so every four-dimensional tangent space +$T_\alpha\widehat{\mathcal C}_x$ is isotropic. An alternating form +of rank $2r$ on an $m$-dimensional space has isotropic subspaces of +dimension at most $m-r$. Since $m\leq5$ and $r\geq1$, we must have +\[ + 4\leq m-r\leq4, +\] +and hence $m=5$ and $r=1$. + +Projection to the two-dimensional symplectic quotient +$W_x/\ker\omega$ has differential of rank at most one on the smooth +cone locus. In characteristic zero, the Zariski closure of the image therefore +has dimension at most one. It is an irreducible cone, hence is either +the origin or a line. But $\widehat{\mathcal C}_x$ spans $W_x$, so +its projection spans the two-dimensional quotient, a contradiction. +Thus the bivectors do span $\bigwedge^2W_x$, making $W$ integrable +and contradicting the other Hwang--Mok result. +\end{proof} + +\begin{proposition}\label{prop:pseudoindex-five-recognition} +Every smooth complex Fano sixfold with nef tangent bundle, Picard number +one and pseudoindex five is isomorphic to $\operatorname{Gr}(2,5)$. +\end{proposition} + +\begin{proof} +The geometric constraints~\eqref{eq:N-products} and +Lemma~\ref{lem:finite-ratio} give $u=4$ or $u=8/3$. We first determine +the degree of $L$. By Bertini, two general members of $|L|$ meet in a smooth +curve $B$. Both successive divisors are connected: +for a smooth projective variety $Z$ of dimension at least two and an +ample divisor $D$, Kodaira vanishing and Serre duality give +$H^1(Z,\mathcal O_Z(-D))=0$, and the divisor exact sequence then gives +$H^0(D,\mathcal O_D)=\mathbb C$. Applying this first to $F$ and then +to its smooth ample surface section proves that $B$ is connected. +Adjunction gives +\begin{equation}\label{eq:vmrt-sectional-genus} + g(B)=1+\frac{(K_F+2L)\cdot L^2}{2} + =1+\left(1-\frac u2\right)L^3. +\end{equation} +Here and below $L^3$ denotes its top intersection number. + +By Lemma~\ref{lem:vmrt-nondegenerate}, $\mathcal C_x$ is a +nondegenerate threefold in $\mathbb P^5$, so its degree is at least +three. The degree formula for the finite morphism $\tau_x$ yields +\[ + L^3=\deg(\tau_x)\deg(\mathcal C_x)\geq3. +\] +If $u=4$, Equation~\eqref{eq:vmrt-sectional-genus} gives +$g(B)=1-L^3<0$. Thus $u=8/3$, and the same equation gives +$g(B)=1-L^3/3\geq0$. It follows that +\[ + L^3=3,\qquad\deg(\tau_x)=1,\qquad\deg(\mathcal C_x)=3. +\] + +The del Pezzo--Bertini classification of varieties of minimal degree +\cite[Theorem~0.1]{EisenbudGreenHulekPopescu2006} now makes +$\mathcal C_x$ a rational normal scroll, possibly a cone. Indeed, +linear spaces and quadrics have smaller degree, while a cone over the +Veronese surface has degree four. A three-dimensional scroll of +degree three has parameters $a_1,a_2,a_3\geq0$ with +$a_1+a_2+a_3=3$. + +To turn the finite degree-one tangent map into an isomorphism, we need +normality of its target. These scrolls are normal, including the cone +cases, as follows. Their homogeneous coordinate rings are generated +by the monomials +\[ + z_i s_0^j s_1^{a_i-j},\qquad + 1\leq i\leq3,\quad0\leq j\leq a_i. +\] +These monomials generate exactly the invariant subring of +$\mathbb C[z_1,z_2,z_3,s_0,s_1]$ for the multiplicative-group action +with weights $(-a_1,-a_2,-a_3,1,1)$, graded by total degree in the +$z_i$. In this grading every displayed generator has degree one. +To see that they generate the invariant ring, distribute the +$s_0$ and $s_1$ exponents of an invariant monomial among its $z_i$ +factors, each of which requires exactly $a_i$ such exponents. An +invariant subring of a normal domain is normal: an element of its +fraction field integral over it belongs to the original domain and +remains invariant. Thus the scroll is normal. + +The finite birational morphism $\tau_x\colon F\to\mathcal C_x$ is +therefore an isomorphism. In particular, $\mathcal C_x$ is smooth. +If some $a_i$ vanished, the scroll would be a nontrivial cone over a +nonlinear variety and hence singular along its vertex. Consequently +all $a_i$ are positive, so $(a_1,a_2,a_3)=(1,1,1)$. We have proved +the projective equivalence +\[ + \mathcal C_x\simeq + \mathbb P^1\times\mathbb P^2\hookrightarrow\mathbb P^5, + \qquad\text{embedded by }\mathcal O(1,1). +\] + +This is the variety of minimal rational tangents of +$S=\operatorname{Gr}(2,5)$. At a two-plane $A\subset\mathbb C^5$, +$T_AS=\operatorname{Hom}(A,\mathbb C^5/A)$, and tangent directions +to lines are exactly the rank-one maps. Their projectivization is +the displayed Segre variety. For example, in graph coordinates the +Pl\"ucker coordinates consist of the constant coordinate $1$, the entries of a map, and its +$2\times2$ minors; a direction determines a line precisely when the +quadratic minors vanish. + +Finally, Mok's recognition theorem +\cite[Main Theorem, \S2.1]{Mok2008Recognition} states that a Fano +manifold of Picard number one whose general variety of minimal +rational tangents is projectively equivalent to that of an irreducible +compact Hermitian symmetric space is isomorphic to that space. +The model $S=\operatorname{Gr}(2,5)$ has rank two, and we have +verified the required projective equivalence for the minimal component +$V$. The theorem gives $X\simeq S$. +\end{proof} diff --git a/preprints/The-Campana-Peternell-conjecture-in-dimension-six-September-25-2026/build/sections/07-conclusion.tex b/preprints/The-Campana-Peternell-conjecture-in-dimension-six-September-25-2026/build/sections/07-conclusion.tex new file mode 100644 index 0000000000000000000000000000000000000000..a1433e3667fa53070c6dcdd0441a4aac53c83f77 --- /dev/null +++ b/preprints/The-Campana-Peternell-conjecture-in-dimension-six-September-25-2026/build/sections/07-conclusion.tex @@ -0,0 +1,138 @@ +\section{The sixfold theorem and its consequences}\label{sec:conclusion} + +\begin{proof}[Proof of Theorem~\ref{thm:main}] +Let $X$ satisfy its hypotheses. Since $X$ is projective and positive +dimensional, an ample divisor has nonzero numerical class, so +$\rho(X)\geq1$. If $\rho(X)\geq2$, then +$\rho(X)>\dim X-5$. Kanemitsu's Theorem~0.2 +\cite{Kanemitsu2016} applies and gives rational homogeneity. + +Suppose now that $\rho(X)=1$. +By Proposition~\ref{prop:pseudoindex-reduction}, either $X$ is +$\mathbb P^6$ or $Q^6$, or its pseudoindex is four or five. +The first two varieties are rational homogeneous. +Proposition~\ref{prop:pseudoindex-four} settles pseudoindex four. +In pseudoindex five, the characteristic-class identities of +Section~\ref{sec:characteristic-classes} and +Lemma~\ref{lem:finite-ratio} give $32-20u+3u^2=0$. +Proposition~\ref{prop:pseudoindex-five-recognition} therefore gives +$X\simeq\operatorname{Gr}(2,5)$, which is rational homogeneous. +These cases exhaust the possibilities. +\end{proof} + +\subsection{A K\"ahler--Einstein consequence} + +\begin{corollary}\label{cor:positive-ke} +Let $X$ be a smooth connected complex projective Fano variety with nef +tangent bundle and $1\leq\dim_{\mathbb C}X\leq6$. +Then $X$ admits a positive K\"ahler--Einstein metric. More precisely, +with the Ricci-form convention +\[ + \rho(\omega)=-i\partial\bar\partial\log\det(g_{a\bar b}), +\] +there is a K\"ahler form $\omega$ such that +\[ + \rho(\omega)=\omega,\qquad [\omega]=2\pi c_1(-K_X). +\] +\end{corollary} + +\begin{proof} +In dimension six, Theorem~\ref{thm:main} makes $X$ rational homogeneous. +For $1\leq n=\dim_{\mathbb C}X\leq5$, its Picard number is at least +one and hence greater than $n-5$, so Kanemitsu's Theorem~0.2 +\cite{Kanemitsu2016} gives the same conclusion. Thus the +lower-dimensional cases already follow from the known classification. +The homogeneous metric construction of Alekseevskii and Perelomov +\cite{AlekseevskiiPerelomov1986}, recalled in +\cite[\S1, p.~285]{Demailly2018AlmostKE}, supplies a positive +K\"ahler--Einstein metric on $X$. +Write its K\"ahler form as $\sigma$, with +$\rho(\sigma)=\lambda\sigma$ and $\lambda>0$. Constant rescaling +does not change the Ricci form, so $\omega=\lambda\sigma$ satisfies +$\rho(\omega)=\omega$. Since the Ricci form represents +$2\pi c_1(-K_X)$ in this convention, the asserted class follows. +\end{proof} + +\subsection{Albanese fibres and the ordinary universal cover} + +We now combine the classification with the analytic structure theorems +for compact K\"ahler manifolds with nef tangent bundle. + +\begin{proof}[Proof of Corollary~\ref{cor:nef-tangent-uniformization}] +If $\dim_{\mathbb C}X=0$, connectedness makes $X$ a point and every +assertion follows. Assume henceforth that $\dim_{\mathbb C}X>0$. +Every finite \'{e}tale cover remains compact K\"ahler, and its tangent +bundle is analytically nef by Proposition~3.4 of +Demailly--Peternell--Schneider. +Their Proposition~3.9 +\cite{DemaillyPeternellSchneider1994}, applied to every connected finite +\'{e}tale cover, makes its Albanese map surjective; hence its +irregularity is at most $\dim_{\mathbb C}X$. These irregularities are +nonnegative integers and include that of $X$, so their maximum +$q=\widetilde q(X)$ is attained. +Choose a connected finite \'{e}tale cover $X'\to X$ attaining it. +The Main Theorem and Theorem~3.14 of +Demailly--Peternell--Schneider, together with their Proposition~3.9, +give a smooth proper Albanese surjection +\[ + \alpha:X'\longrightarrow A=\operatorname{Alb}(X') +\] +with connected fibres, where $A$ is a complex torus of dimension $q$. +Every positive-dimensional fibre is a smooth projective Fano manifold +with nef tangent bundle and has dimension $d=\dim_{\mathbb C}X-q$. +Analytic nefness agrees with numerical nefness on these projective +fibres, so the algebraic classification applies. + +If $d=6$, Theorem~\ref{thm:main} makes every fibre rational homogeneous. +For $1\leq d\leq5$, a projective fibre $F_a$ has +$\rho(F_a)\geq1>d-5$, and Kanemitsu's Theorem~0.2 +\cite{Kanemitsu2016} gives the same conclusion. This is precisely the +established lower-dimensional range recalled in the introduction. +If $d=0$, each connected fibre is a point. This proves the first +part of the corollary. + +Suppose first that $d>0$. Every rational homogeneous manifold $G/P$ satisfies +\[ + H^1(G/P,T_{G/P})=0 +\] +and is locally rigid under complex deformations +\cite[Introduction and Section~3.7]{BienBrion1996}. +Applying this at each fibre of the proper +holomorphic submersion $\alpha$ shows that the fibre's biholomorphism +type is locally constant on $A$. Since $A$ is connected, all fibres +are biholomorphic to one fixed rational homogeneous manifold $F$. +The analytic Fischer--Grauert theorem +\cite{FischerGrauert1965}, in the form recalled in +\cite[Theorem~1.1]{Poczobut2025}, now makes $\alpha$ holomorphically +locally trivial. If $d=0$, the proper submersion with one-point fibres +is a biholomorphism, so this conclusion also holds with $F$ a point. + +The universal cover of the complex torus $A$ is $\mathbb C^q$, which +is Stein and contractible even when $A$ is not projective. +Theorem~4.4 of Claudon--H\"oring--Koll\'ar +\cite{ClaudonHoringKollar2013} applies to the locally trivial proper +fibration $\alpha$ and gives +\[ + \widetilde {X'}\simeq\widetilde F\times\mathbb C^q. +\] +Here the tildes denote ordinary universal covers. A rational homogeneous +manifold $F$ is simply connected: write it using the simply connected +semisimple cover of its acting group; the corresponding parabolic +subgroup is connected, and the homotopy exact sequence of that +homogeneous fibration gives $\pi_1(F)=0$. This is also true for a point. +Thus $\widetilde F=F$. The connected finite \'{e}tale cover $X'\to X$ +has the same ordinary universal cover as $X$, proving +$\widetilde X\simeq F\times\mathbb C^q$. + +Since $F$ is projective, $F\times\mathbb A^q$ is quasi-projective and +its analytification is $F\times\mathbb C^q$. The projection of this +product to $\mathbb C^q$ is proper, surjective, and has connected compact +fibres. For every open set $U\subseteq\mathbb C^q$, a holomorphic +function on $F\times U$ is constant on each fibre and hence descends +holomorphically to $U$ by restriction to $\{f_0\}\times U$ for any +fixed $f_0\in F$. Thus the +projection has direct image of its structure sheaf equal to +$\mathcal O_{\mathbb C^q}$. Since the target is Stein, this projection +is the Remmert reduction, and the product is holomorphically convex. +The argument includes $q=0$, when the target is a point. +\end{proof} diff --git a/preprints/The-Campana-Peternell-conjecture-in-dimension-six-September-25-2026/build/sections/08-certificate.tex b/preprints/The-Campana-Peternell-conjecture-in-dimension-six-September-25-2026/build/sections/08-certificate.tex new file mode 100644 index 0000000000000000000000000000000000000000..e5be7d4935bcb3dc991a9fdd1d0458a93c351d96 --- /dev/null +++ b/preprints/The-Campana-Peternell-conjecture-in-dimension-six-September-25-2026/build/sections/08-certificate.tex @@ -0,0 +1,35 @@ +\section{Exact arithmetic certificate} +\label{app:certificate} + +The following complete Python program verifies the finite calculations +in Section~\ref{sec:arithmetic}. It requires SymPy and NumPy. Run +\texttt{python3 verification/verify.py} with assertions enabled to perform +the checks described below. The monomial order, coefficient recurrences +and projection are exactly those defined in +Section~\ref{sec:characteristic-classes}. + +The calculation has three logically separate parts. First, exact +rational identities $AK=0$ and $NP=0$, together with the four modular +ranks in Section~\ref{sec:arithmetic}, certify the kernel and +annihilator bases. Reduction of a rational entry explicitly inverts +its denominator modulo $10007$, so a noninvertible denominator would +stop the calculation. Second, the slice-merging procedure computes +the finite-field spaces and checks their dimensions. The final basis +is also tested directly against every double-slice constraint. +Third, rational substitution checks the six displayed columns, their +nonzero minor $-16/5$, and the final linear functional. + +SymPy performs the polynomial and rational operations exactly. +The finite-field arrays use signed 64-bit integers. After each +reduction, their entries lie between $0$ and $10006$. The inner +dimension of every matrix product is at most $6820$, since a basis +has no more columns than the number of coordinates of its space. +Every accumulated dot product is therefore at most +\[ + 6820\cdot10006^2=682818645520<2^{63}-1. +\] +The products and subtractions in a pivot step have smaller absolute +values. Thus no arithmetic operation can wrap around before modular +reduction. + +\lstinputlisting[language=Python]{../verification/verify.py} diff --git a/preprints/The-Campana-Peternell-conjecture-in-dimension-six-September-25-2026/main.pdf b/preprints/The-Campana-Peternell-conjecture-in-dimension-six-September-25-2026/main.pdf new file mode 100644 index 0000000000000000000000000000000000000000..6c69cd93262a3fe00a69dce77661c35c80e144d8 --- /dev/null +++ b/preprints/The-Campana-Peternell-conjecture-in-dimension-six-September-25-2026/main.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:c6c7151f9581ddd5b4d4bea3e6c1e96bf1fa776b97f6e0f2e2de1b952d5a28a1 +size 544212 diff --git a/preprints/The-Campana-Peternell-conjecture-in-dimension-six-September-25-2026/verification/verify.py b/preprints/The-Campana-Peternell-conjecture-in-dimension-six-September-25-2026/verification/verify.py new file mode 100644 index 0000000000000000000000000000000000000000..4d0593c0a70ea5f7eaa98681f09710b689ab79b7 --- /dev/null +++ b/preprints/The-Campana-Peternell-conjecture-in-dimension-six-September-25-2026/verification/verify.py @@ -0,0 +1,167 @@ +from functools import lru_cache +from math import comb, factorial, isqrt +from itertools import combinations_with_replacement +import sympy as sy +from sympy.polys.rings import ring +from sympy.polys.domains import QQ +from sympy.polys.matrices import DomainMatrix +import numpy as np + +R,h,x2,x3,x4,x5,x6,l,r,s,t=ring('h,x2,x3,x4,x5,x6,l,r,s,t',QQ) +one=R.one +base=(h,x2,x3,x4,x5,x6) +inv=(l,r,s,t) +gens=base+inv +wt=dict(zip(gens,(1,2,3,4,5,6,1,1,2,3))) +@lru_cache(None) +def mons(n,gs): + if n==0: return (one,) + if n<0 or not gs: return () + g=gs[-1]; d=wt[g] + return tuple(g**i*m for i in range(n//d+1) for m in mons(n-d*i,gs[:-1])) + +c=2*h-l +hb=l-h +S=(one,r,s,t) +p=[3*one] +for i in range(1,11): + p.append(sum((-1)**(j+1)*S[j]*(p[i-j] if i!=j else i) + for j in range(1,min(3,i)+1))) +D=[R.zero]+[(1-(-1)**i)*c**(i-1)-sum(comb(i,j)*p[j]*(-1)**(i-j) + *sum(h**k*hb**(i-j-1-k) for k in range(i-j)) for j in range(i)) + for i in range(1,11)] +dg=dict(zip(base,[one]+D[2:7])) +@lru_cache(None) +def delta(m): + if m==one: return R.zero + ex=next(iter(m)) # m a base monomial + g=base[next(i for i,a in enumerate(ex) if a)] + n=m.quo(g) + return dg[g]*n+(g-c*dg[g])*delta(n) + +q=[sum(comb(i,j)*p[j]*(-h)**(i-j) for j in range(i+1)) for i in range(10)] +beta=[0,QQ(1,2),-QQ(1,12),0,QQ(1,720),0,-QQ(1,30240),0,QQ(1,1209600),0] +td=[one] +for i in range(1,10): + td.append(QQ(1,i)*sum(beta[j]*q[j]*td[i-j] for j in range(1,i+1))) +def rows(): + for i in range(11): + for m in mons(i,base): + if i<=6: + for w in mons(9-i,inv): yield w*(m-h*delta(m)) + for w in mons(10-i,inv): yield w*delta(m) + else: + for w in mons(9-i,gens): yield w*m + for w in mons(10-i,gens): yield w*delta(m) + if i>=7: + for w in mons(10-i,gens): yield w*D[i] + if 4<=i<=9: + v=sum(QQ((-1)**j,factorial(j))*q[j]*td[i-j] for j in range(i+1)) + for w in mons(9-i,base): + yield w*td[i] + yield w*v + +M=mons(6,base) +G=mons(3,inv)+mons(4,inv) +J=[0,1,2,4,5,6,7,8,9,10,12,13,15,16,17] +C=[(i,(j,)) for i in range(len(M)) for j in J if j<=6 or M[i].degree(h)>0] +inds={next(iter(m)):i for i,m in enumerate(mons(9,gens))} +def mat(polys): + data=[{inds[m]:v for m,v in f.items()} for f in polys if f] + return DomainMatrix(dict(enumerate(data)),(len(data),len(inds)),QQ) +def make(): + A=mat(rows()) + assert A.shape==(1785,581) + KD=A.nullspace(divide_last=True).transpose() + assert (A*KD).to_Matrix().is_zero_matrix + K=KD.to_Matrix() + assert ker(red(A.to_Matrix())).shape==(581,65) + assert ker(red(K)).shape==(65,0) + P=mat(M[i].quo(h**(0 if j<=6 else 1))*G[j] for i,(j,) in C).to_Matrix()*K + N=sy.Matrix.hstack(*P.T.nullspace()).T.rref()[0] + assert ker(red(P)).shape==(65,5) + assert ker(red(N)).shape==(129,60) + assert K.shape==(581,65) and N.shape==(69,129) and (N*P).is_zero_matrix + return N + +prime=10007 +assert all(prime % k for k in range(2,isqrt(prime)+1)) +assert 6820*(prime-1)**2 < 2**63-1 +def ker(A): # columns give kernel mod prime + B=A%prime + m,n=B.shape + piv=[] + for j in range(n): + i=len(piv) + nz=np.flatnonzero(B[i:,j]) + if not len(nz): continue + u=(B[i+nz[0],:] * pow(int(B[i+nz[0],j]),-1,prime))%prime + B[i+nz[0],:]=B[i,:] + B=(B-B[:,j:j+1]*u)%prime + B[i,:]=u + piv.append(j) + if i+1==m: break + free=[j for j in range(n) if j not in piv] + K=np.zeros((n,len(free)),dtype=np.int64) + K[piv,:]=-B[:len(piv),free] + K[free,:]=np.eye(len(free),dtype=np.int64) + return K%prime +def grow(K,C): + # merge slice restrictions along J + ix={} + n=0 + Z=np.empty((0,0),dtype=np.int64) + for j in J: + cc=[(i,tuple(sorted(z+(j,)))) for i,z in C] + overlap=[k for k,a in enumerate(cc) if a in ix] + extra=[k for k,a in enumerate(cc) if a not in ix] + A=np.hstack((Z[[ix[cc[k]] for k in overlap],:],-K[overlap,:])) + V=ker(A); b=Z.shape[1] + Z=np.vstack((Z@V[:b,:] % prime, K[extra,:]@V[b:,:] % prime)) + for k in extra: + ix[cc[k]]=n + n+=1 + return Z,sorted(ix,key=ix.get) +def red(M): + fun=lambda a: int(a.p)*pow(int(a.q),-1,prime)%prime + return np.array([[fun(v) for v in row] for row in M.tolist()],dtype=np.int64) +def modN(N): + Nmod=red(N) + K=ker(Nmod) + c=C + for shape in [(1140,163),(6820,6)]: + K,c=grow(K,c) + assert K.shape==shape + assert len(c)==len(set(c))==shape[0] + lookup={a:i for i,a in enumerate(c)} + for pair in combinations_with_replacement(J,2): + rows=[lookup[(i,tuple(sorted(z+pair)))] for i,z in C] + assert not np.any((Nmod@K[rows,:]) % prime) + +def samples(N): + G1=(1,4,16,64,6,24,4,3,12,48,192,768,17,68,272,96,9,36) + G2=tuple(sy.Rational(i,3) for i in + (3,8,21,54,9,24,6,14,37,97,252,648,37,97,252,96,16,42)) + def X(a): return (1,7-sy.Rational(2,5)*a,45-4*a,275-30*a,11,65-2*a,85,17,95-2*a,25,35) + X2=(1,sy.Rational(11,5),5,11,-1,-1,10,-3,-1,0,14) + products=[(X(0),G1),(X(1),G1),(X2,G2)] + for x,y in products: + assert not any(N*sy.Matrix([x[i]*y[j] for i,(j,) in C])) + Z={(5,(15,15,15)):[3,0,0], (7,(15,15,15)):[1,0,0], (9,(15,15,15)):[-5,0,0], + (7,(10,10,15)):[0,2,0], (7,(10,10,10)):[0,4,1], (9,(10,10,10)):[0,25,0]} + for k in combinations_with_replacement(J,2): + assert not any(N*sy.Matrix([Z.get((i,tuple(sorted(z+k))),[0,0,0]) for i,z in C])) + def six_entries(i,z): + return [x[i]*sy.prod(y[j] for j in z) for x,y in products]+Z.get((i,z),[0,0,0]) + selected=[(0,(0,0,0)),(0,(0,0,1)),(1,(0,0,0)), + (5,(15,15,15)),(7,(10,10,15)),(7,(10,10,10))] + assert sy.Matrix([six_entries(i,z) for i,z in selected]).det()==-sy.Rational(16,5) + v0=six_entries(0,(0,0,0)) + v1=six_entries(0,(0,0,1)) + v2=six_entries(0,(0,1,1)) + assert all(32*a-20*b+3*c==0 for a,b,c in zip(v0,v1,v2)) + +if __name__=='__main__': + N=make(); print("N done", flush=True) + modN(N); samples(N) + print("pass") diff --git a/preprints/The-Critical-Spin-Field-of-the-Planar-XY-Model-October-5-2026/README.md b/preprints/The-Critical-Spin-Field-of-the-Planar-XY-Model-October-5-2026/README.md new file mode 100644 index 0000000000000000000000000000000000000000..759bc74b9fb562b63b57e5cb3e3418109df34ec5 --- /dev/null +++ b/preprints/The-Critical-Spin-Field-of-the-Planar-XY-Model-October-5-2026/README.md @@ -0,0 +1,15 @@ +# [The Critical Spin Field of the Planar XY Model](paper.pdf) + +OpenAI · October 5, 2026 + +## Citation + +```bibtex +@misc{OAI:The-Critical-Spin-Field-of-the-Planar-XY-Model-October-5-2026, + author = {{OpenAI}}, + title = {{The Critical Spin Field of the Planar XY Model}}, + howpublished = {OpenAI Math Release preprint \href{https://github.com/openai/math/blob/main/preprints/The-Critical-Spin-Field-of-the-Planar-XY-Model-October-5-2026/paper.pdf}{OAI:The-Critical-Spin-Field-of-the-Planar-XY-Model-October-5-2026}}, + year = {2026} +} +``` + diff --git a/preprints/The-Critical-Spin-Field-of-the-Planar-XY-Model-October-5-2026/build/figures/annular-bands.tex b/preprints/The-Critical-Spin-Field-of-the-Planar-XY-Model-October-5-2026/build/figures/annular-bands.tex new file mode 100644 index 0000000000000000000000000000000000000000..ae9748ac8c8c849e482ab5b1ddd97700fb2ea68d --- /dev/null +++ b/preprints/The-Critical-Spin-Field-of-the-Planar-XY-Model-October-5-2026/build/figures/annular-bands.tex @@ -0,0 +1,40 @@ +\begin{tikzpicture}[scale=.68, every node/.style={font=\small}] + \def\wall{(4,0)--(4,1)--(3.75,1)--(3.75,1.75) + --(3.5,1.75)--(3.5,2.25)--(3.25,2.25)--(3.25,2.75) + --(2.75,2.75)--(2.75,3.25)--(2.25,3.25)--(2.25,3.5) + --(1.75,3.5)--(1.75,3.75)--(1,3.75)--(1,4) + --(-1,4)--(-1,3.75)--(-1.75,3.75)--(-1.75,3.5) + --(-2.25,3.5)--(-2.25,3.25)--(-2.75,3.25)--(-2.75,2.75) + --(-3.25,2.75)--(-3.25,2.25)--(-3.5,2.25)--(-3.5,1.75) + --(-3.75,1.75)--(-3.75,1)--(-4,1)--(-4,-1) + --(-3.75,-1)--(-3.75,-1.75)--(-3.5,-1.75)--(-3.5,-2.25) + --(-3.25,-2.25)--(-3.25,-2.75)--(-2.75,-2.75)--(-2.75,-3.25) + --(-2.25,-3.25)--(-2.25,-3.5)--(-1.75,-3.5)--(-1.75,-3.75) + --(-1,-3.75)--(-1,-4)--(1,-4)--(1,-3.75) + --(1.75,-3.75)--(1.75,-3.5)--(2.25,-3.5)--(2.25,-3.25) + --(2.75,-3.25)--(2.75,-2.75)--(3.25,-2.75)--(3.25,-2.25) + --(3.5,-2.25)--(3.5,-1.75)--(3.75,-1.75)--(3.75,-1) + --(4,-1)--cycle} + \fill[black!14] \wall; + \fill[white] (-2.6,-2.6) rectangle (2.6,2.6); + \fill[blue!18] (-1.85,-1.85) rectangle (1.85,1.85); + \fill[white] (-1.42,-1.42) rectangle (1.42,1.42); + \fill[black!14] (-1.125,-1.125) rectangle (1.125,1.125); + \draw[black!45] (-2.6,-2.6) rectangle (2.6,2.6); + \draw[blue!60!black] (-1.85,-1.85) rectangle (1.85,1.85); + \draw[blue!60!black] (-1.42,-1.42) rectangle (1.42,1.42); + \draw[black!45] (-1.125,-1.125) rectangle (1.125,1.125); + \draw[thick] \wall; + \begin{scope}[scale=.13] + \fill[white] \wall; + \draw[thick] \wall; + \end{scope} + \node at (0,0) {\(P\)}; + \node at (0,.82) {\(H_{\mathrm{in}}\)}; + \node at (0,1.64) {\(p\)}; + \node at (0,3.2) {\(H_{\mathrm{out}}\)}; + \node[anchor=west] at (4.45,1.4) {\(\partial(\lambda P)\)}; + \draw[black!60] (4.4,1.3)--(3.75,1.3); + \node[anchor=west,align=left] at (4.45,-.7) + {free walls\\at both ends}; +\end{tikzpicture} diff --git a/preprints/The-Critical-Spin-Field-of-the-Planar-XY-Model-October-5-2026/build/figures/slit-patch.tex b/preprints/The-Critical-Spin-Field-of-the-Planar-XY-Model-October-5-2026/build/figures/slit-patch.tex new file mode 100644 index 0000000000000000000000000000000000000000..bcc9c9bd9b17ba14a95c15db0e3e55769b27a61e --- /dev/null +++ b/preprints/The-Critical-Spin-Field-of-the-Planar-XY-Model-October-5-2026/build/figures/slit-patch.tex @@ -0,0 +1,39 @@ +\begin{figure}[tb] + \centering + \begin{tikzpicture}[x=1cm,y=1cm,font=\small] + \fill[gray!8] (-3,-1.5) rectangle (3,3); + \fill[white] (-1.5,-1.5) rectangle (1.5,0); + \draw[thick] (-3,-1.5)--(-3,3)--(3,3)--(3,-1.5); + \draw[thick] (-1.5,-1.5)--(-1.5,0)--(1.5,0)--(1.5,-1.5); + \fill[blue!15] (-.78,0) rectangle (-.55,3); + \fill[blue!15] (.55,0) rectangle (.78,3); + \fill[orange!25] (-.3,0) rectangle (-.12,3); + \fill[orange!25] (.12,0) rectangle (.3,3); + \draw[densely dashed] (-1.05,0)--(-1.05,3); + \draw[densely dashed] (1.05,0)--(1.05,3); + \draw[dotted,thick] (-.21,0)--(-.21,3); + \draw[dotted,thick] (.21,0)--(.21,3); + \node at (0,-.55) {hole}; + \node at (2,2.5) {\(U\)}; + \node at (0,1.3) {\(S\)}; + \node at (-.665,2.5) {\(p\)}; + \node at (.665,2.5) {\(p\)}; + \draw[->] (-2,1.25)--(-.3,1.25); + \node[anchor=east] at (-2,1.25) {\(H\)}; + \draw[->] (2,1.7)--(1.05,1.7); + \node[anchor=west] at (2,1.7) {\(\partial W\)}; + \draw[<->] (-1.05,3.35)--(1.05,3.35); + \node[above] at (0,3.35) {\(W\)}; + \end{tikzpicture} + \caption{A straight part of a strip \(S\) joining a hole to the + outer free wall. Dotted lines indicate the interfaces cut to isolate + \(S\). The pin region \(H\), shown along both interfaces, contains + every endpoint of every lattice edge crossing either interface. + The two + guard bands \(p\) separate them from the artificial boundary of the + wider patch \(W\). All bands have positive widths and positive + intervening gaps. Parallel collars give the same separation at + rectilinear turns. The bottom ends of the outer boundary and hole + are omitted from the schematic.} + \label{sectors:slit-figure} +\end{figure} diff --git a/preprints/The-Critical-Spin-Field-of-the-Planar-XY-Model-October-5-2026/build/main.tex b/preprints/The-Critical-Spin-Field-of-the-Planar-XY-Model-October-5-2026/build/main.tex new file mode 100644 index 0000000000000000000000000000000000000000..8fff4ec86ee05ef9e0c7b7f1b0bfbaf987c11513 --- /dev/null +++ b/preprints/The-Critical-Spin-Field-of-the-Planar-XY-Model-October-5-2026/build/main.tex @@ -0,0 +1,73 @@ +\documentclass[11pt]{article} +\usepackage[T1]{fontenc} +\usepackage[utf8]{inputenc} +\usepackage{lmodern} +\usepackage[margin=1.05in]{geometry} +\usepackage{amsmath,amssymb,amsthm,mathtools} +\numberwithin{equation}{section} +\usepackage{microtype} +\usepackage{enumitem,booktabs,array} +\usepackage{needspace,etoolbox} +\usepackage{tikz} +\usetikzlibrary{arrows.meta,positioning,calc,patterns} +\usepackage{xcolor} +\usepackage[hypertexnames=false,colorlinks=true,linkcolor=blue!45!black,citecolor=green!35!black,urlcolor=blue!45!black]{hyperref} +\hypersetup{pdftitle={The Critical Spin Field of the Planar XY Model},pdfauthor={OpenAI}} +\newtheorem{theorem}{Theorem}[section] +\newtheorem{lemma}[theorem]{Lemma} +\newtheorem{proposition}[theorem]{Proposition} +\newtheorem{corollary}[theorem]{Corollary} +\theoremstyle{definition} +\newtheorem{definition}[theorem]{Definition} +\theoremstyle{remark} +\newtheorem{remark}[theorem]{Remark} +\BeforeBeginEnvironment{theorem}{\Needspace{6\baselineskip}} +\BeforeBeginEnvironment{lemma}{\Needspace{6\baselineskip}} +\BeforeBeginEnvironment{proposition}{\Needspace{6\baselineskip}} +\BeforeBeginEnvironment{corollary}{\Needspace{6\baselineskip}} +\BeforeBeginEnvironment{definition}{\Needspace{6\baselineskip}} +\preto\subsection{\Needspace{9\baselineskip}} +\newcommand{\R}{\mathbb R} +\newcommand{\Z}{\mathbb Z} +\newcommand{\C}{\mathbb C} +\newcommand{\N}{\mathbb N} +\newcommand{\E}{\mathbb E} +\newcommand{\Pbb}{\mathbb P} +\DeclareMathOperator{\Var}{Var} +\DeclareMathOperator{\Cov}{Cov} +\DeclareMathOperator{\supp}{supp} +\DeclareMathOperator{\diam}{diam} +\DeclareMathOperator{\dist}{dist} +\DeclareMathOperator{\Tr}{Tr} +\setlist{topsep=4pt,itemsep=3pt,parsep=0pt} +\setlength{\emergencystretch}{1.5em} +\clubpenalty=10000 +\widowpenalty=10000 +\displaywidowpenalty=10000 +\title{The Critical Spin Field of the Planar XY Model} +\author{OpenAI} +\date{October 5, 2026} +\begin{document} +\maketitle +\begin{abstract} +Using the companion critical Bessel-height and pin-limit theorems, we +prove that at the mass-defined critical inverse temperature of the +nearest-neighbor XY model on the square lattice, the spin field on a +square with boundary angles fixed to zero converges along the full +sequence to the full-variance imaginary exponential of a zero-Dirichlet +Gaussian free field with stiffness \(2/\pi\). The normalization uses +the exact center magnetization and the lattice Green-function factor. +Convergence holds in law in \(H^{-3}_{\mathrm{loc}}((-1,1)^2)\); +all mixed moments and joint laws of fields smeared against smooth +compactly supported test functions in \((-1,1)^2\) converge as well. +\end{abstract} +\tableofcontents +\input{sections/01-introduction} +\input{sections/02-heights} +\input{sections/03-sectors} +\input{sections/04-annuli} +\input{sections/05-kernels} +\input{sections/06-correlations} +\input{sections/07-fields} +\input{references} +\end{document} diff --git a/preprints/The-Critical-Spin-Field-of-the-Planar-XY-Model-October-5-2026/build/references.tex b/preprints/The-Critical-Spin-Field-of-the-Planar-XY-Model-October-5-2026/build/references.tex new file mode 100644 index 0000000000000000000000000000000000000000..f8f227ee3c2724c5d97f30a9734f2f6ed4b916b2 --- /dev/null +++ b/preprints/The-Critical-Spin-Field-of-the-Planar-XY-Model-October-5-2026/build/references.tex @@ -0,0 +1,145 @@ +\begin{thebibliography}{99} +\raggedright + +\bibitem{Berezinskii} +V.~L.~Berezinskii. +Destruction of long-range order in one-dimensional and two-dimensional +systems having a continuous symmetry group. I.~Classical systems. +\emph{Soviet Physics JETP} \textbf{32}(3) (1971), 493--500. +Russian original: \emph{Zh. Eksp. Teor. Fiz.} \textbf{59} +(1970), 907--920. +\href{https://www.jetp.ras.ru/cgi-bin/dn/e_032_03_0493.pdf} +{English translation}. + +\bibitem{DunlopNewman1975} +F.~Dunlop and C.~M.~Newman. +Multicomponent field theories and classical rotators. +\emph{Communications in Mathematical Physics} +\textbf{44} (1975), 223--235. +\href{https://doi.org/10.1007/BF01609827} +{doi:10.1007/BF01609827}. + +\bibitem{FS} +J.~Fr\"ohlich and T.~Spencer. +The Kosterlitz--Thouless transition in two-dimensional Abelian +spin systems and the Coulomb gas. +\emph{Communications in Mathematical Physics} +\textbf{81}(4) (1981), 527--602. +\href{https://doi.org/10.1007/BF01208273} +{doi:10.1007/BF01208273}. + +\bibitem{JSW2020} +J.~Junnila, E.~Saksman, and C.~Webb. +Imaginary multiplicative chaos: Moments, regularity and connections +to the Ising model. +\emph{Annals of Applied Probability} +\textbf{30}(5) (2020), 2099--2164. +\href{https://doi.org/10.1214/19-AAP1553} +{doi:10.1214/19-AAP1553}. +\href{https://arxiv.org/abs/1806.02118v2}{arXiv:1806.02118v2}. + +\bibitem{Kosterlitz1974} +J.~M.~Kosterlitz. +The critical properties of the two-dimensional XY model. +\emph{Journal of Physics C: Solid State Physics} +\textbf{7}(6) (1974), 1046--1060. +\href{https://doi.org/10.1088/0022-3719/7/6/005} +{doi:10.1088/0022-3719/7/6/005}. + +\bibitem{BKT1973} +J.~M.~Kosterlitz and D.~J.~Thouless. +Ordering, metastability and phase transitions in two-dimensional systems. +\emph{Journal of Physics C: Solid State Physics} +\textbf{6}(7) (1973), 1181--1203. +\href{https://doi.org/10.1088/0022-3719/6/7/010} +{doi:10.1088/0022-3719/6/7/010}. + +\bibitem{Lammers2023} +P.~Lammers. +Bijecting the BKT transition. +Preprint (2023). +\href{https://arxiv.org/abs/2301.06905v2}{arXiv:2301.06905v2}. + +\bibitem{MS} +O.~A.~McBryan and T.~Spencer. +On the decay of correlations in SO($n$)-symmetric ferromagnets. +\emph{Communications in Mathematical Physics} +\textbf{53}(3) (1977), 299--302. +\href{https://doi.org/10.1007/BF01609854} +{doi:10.1007/BF01609854}. + +\bibitem{MW} +N.~D.~Mermin and H.~Wagner. +Absence of ferromagnetism or antiferromagnetism in one- or +two-dimensional isotropic Heisenberg models. +\emph{Physical Review Letters} \textbf{17}(22) (1966), 1133--1136. +\href{https://doi.org/10.1103/PhysRevLett.17.1133} +{doi:10.1103/PhysRevLett.17.1133}. + +\bibitem{NK} +D.~R.~Nelson and J.~M.~Kosterlitz. +Universal jump in the superfluid density of two-dimensional superfluids. +\emph{Physical Review Letters} \textbf{39}(19) (1977), 1201--1205. +\href{https://doi.org/10.1103/PhysRevLett.39.1201} +{doi:10.1103/PhysRevLett.39.1201}. + +\bibitem{Newman1975} +C.~M.~Newman. +Inequalities for Ising models and field theories which obey the +Lee--Yang theorem. +\emph{Communications in Mathematical Physics} +\textbf{41} (1975), 1--9. +\href{https://doi.org/10.1007/BF01608542} +{doi:10.1007/BF01608542}. + +\bibitem{NewmanWu2018} +C.~M.~Newman and W.~Wu. +Gaussian fluctuations for the classical XY model. +\emph{Annales de l'Institut Henri Poincar\'e, Probabilit\'es et Statistiques} +\textbf{54}(4) (2018), 1759--1777. +\href{https://doi.org/10.1214/17-AIHP854} +{doi:10.1214/17-AIHP854}. +\href{https://arxiv.org/abs/1605.01297}{arXiv:1605.01297}. + +\bibitem{NewmanWu2019} +C.~M.~Newman and W.~Wu. +Lee--Yang property and Gaussian multiplicative chaos. +\emph{Communications in Mathematical Physics} +\textbf{369} (2019), 153--170. +\href{https://doi.org/10.1007/s00220-019-03453-0} +{doi:10.1007/s00220-019-03453-0}. +\href{https://arxiv.org/abs/1708.08820v3}{arXiv:1708.08820v3}. + +\bibitem{B} +OpenAI. +BKT universality for height and planar spin fields. +OpenAI Math Release preprint +\href{https://github.com/openai/math/blob/main/preprints/BKT-universality-for-height-and-planar-spin-fields-September-24-2026/paper.pdf}{OAI:BKT-universality-for-height-and-planar-spin-fields-September-24-2026}, +2026. + +\bibitem{C} +OpenAI. +The critical correlation exponent of the planar XY model. +OpenAI Math Release preprint +\href{https://github.com/openai/math/blob/main/preprints/The-critical-correlation-exponent-of-the-planar-XY-model-September-24-2026/paper.pdf}{OAI:The-critical-correlation-exponent-of-the-planar-XY-model-September-24-2026}, +2026. + +\bibitem{RSD} +O.~Regev and N.~Stephens-Davidowitz. +An inequality for Gaussians on lattices. +\emph{SIAM Journal on Discrete Mathematics} +\textbf{31}(2) (2017), 749--757. +\href{https://doi.org/10.1137/15M1052226} +{doi:10.1137/15M1052226}. +\href{https://arxiv.org/abs/1502.04796v3}{arXiv:1502.04796v3}. + +\bibitem{vEL} +D.~van Engelenburg and M.~Lis. +An elementary proof of phase transition in the planar XY model. +\emph{Communications in Mathematical Physics} +\textbf{399}(1) (2023), 85--104. +\href{https://doi.org/10.1007/s00220-022-04550-3} +{doi:10.1007/s00220-022-04550-3}. +\href{https://arxiv.org/abs/2110.09465v2}{arXiv:2110.09465v2}. + +\end{thebibliography} diff --git a/preprints/The-Critical-Spin-Field-of-the-Planar-XY-Model-October-5-2026/build/sections/01-introduction.tex b/preprints/The-Critical-Spin-Field-of-the-Planar-XY-Model-October-5-2026/build/sections/01-introduction.tex new file mode 100644 index 0000000000000000000000000000000000000000..5a263fff7a358c6dca8eb9a0b8a7aa4460c57355 --- /dev/null +++ b/preprints/The-Critical-Spin-Field-of-the-Planar-XY-Model-October-5-2026/build/sections/01-introduction.tex @@ -0,0 +1,269 @@ +\section{Introduction}\label{sec:intro} + +The two-dimensional XY model has a massless phase without ordinary +long-range order. Its expected large-scale description is Gaussian, but +the spin is an exponential of an angular variable, not the angular +variable itself. Identifying a height-field limit therefore does not by +itself identify the spin field. At criticality a further difficulty is +that a scale-dependent scalar factor may survive between the microscopic +spin and its continuum normalization. + +We prove a critical spin-field limit on a square with fixed boundary +spins. The scalar normalization is the actual magnetization at the +center of that square. This choice cancels the microscopic amplitude +without assuming a power law, a logarithmic correction, or the existence +of an asymptotic amplitude. The remaining spatial normalization is fixed +by the lattice Green function. + +\subsection{The physical critical parameter} + +We first specify the critical inverse temperature using an infinite-volume +quantity, independently of the field limit. For \(R\in\N\), let +\[ + \Lambda_R=[-R,R]^2\cap\Z^2,\qquad + E_R=\{\{x,y\}\subset\Lambda_R:|x-y|=1\}. +\] +Every unordered edge occurs once. At inverse temperature \(b>0\), the +free-box law is +\begin{equation}\label{eq:free-spin-law} + \frac1{Z_{R,b}} + \exp\!\left\{b\sum_{\{x,y\}\in E_R}\cos(\theta_x-\theta_y)\right\} + \prod_{x\in\Lambda_R}\frac{d\theta_x}{2\pi}, + \qquad \theta_x\in\R/(2\pi\Z). +\end{equation} +There are no exterior edges or fixed spins in this definition. Write +\begin{equation}\label{eq:critical-parameter} + \begin{split} + C_b(r)&=\lim_{\substack{R\to\infty\\R\ge r}} + \E_{R,b}\cos(\theta_0-\theta_{re_1}),\qquad e_1=(1,0),\\ + m(b)&=\lim_{r\to\infty}-\frac1r\log C_b(r),\qquad + b_c=\inf\{b>0:m(b)=0\}. + \end{split} +\end{equation} +The free-box correlation and mass limits exist, +\(00. +\end{equation} +Angle reflection proves the equality, and the positive Fourier-current +expansion proves strict positivity. Thus \(a_n\) is a specified +observable of \eqref{eq:pinned-spin-law}, not a sequence chosen to fit +a limit. + +On functions zero at the boundary let +\[ + (L_nu)(x)=\sum_{\substack{y\in D_n\\|x-y|=1/n}} + (u(x)-u(y)),\qquad G_n=L_n^{-1}. +\] +The Laplacian is unscaled: it contains no factor \(n^2\). +Put \(K_*=2/\pi\), and define the complex random distribution +\begin{equation}\label{eq:spin-field} + S_n=\frac1{n^2}\sum_{x\in D_n^\circ} + a_n^{-1}\exp\!\left\{\frac{G_n(x,x)-G_n(0,0)}{2K_*}\right\} + e^{i\theta_x}\delta_x, + \qquad \delta_x(f)=f(x). +\end{equation} +The center factor and the Green correction in this definition are +the same for every test function and every correlation order. + +Let \(\Phi_D\) be the real zero-Dirichlet Gaussian free field with +covariance \((-\Delta_D)^{-1}\). Its kernel has the convention +\[ + G_D(z,w)=\frac1{2\pi}\log\frac1{|z-w|}+R_D(z,w), +\] +where \(R_D\) is smooth across the interior diagonal. For circle +averages \(\Phi_{D,\epsilon}\), taken where the circles lie in \(D\), +let +\begin{equation}\label{eq:target-field} + V_*=\lim_{\epsilon\downarrow0} + \exp\!\left\{\frac{i\Phi_{D,\epsilon}}{\sqrt{K_*}} + +\frac{\E\Phi_{D,\epsilon}^2}{2K_*}\right\}. +\end{equation} +The limit is understood as a complex random distribution tested +against compactly supported smooth functions. This is the +full-variance imaginary Gaussian chaos: its counterterm uses the +entire variance, including its finite spatial part. +In this convention +\[ + \E V_*(f)=\int_D f(z)\,dz,\qquad + \E\bigl[V_*(z)\overline{V_*(w)}\bigr] + =\exp\{G_D(z,w)/K_*\}, +\] +where the second expression denotes the off-diagonal moment kernel. +Its singularity has exponent \(1/4\). Existence and the precise +mixed-moment interpretation are recalled in +Lemma~\ref{fields:circle-chaos}; see also \cite{JSW2020,B}. + +\begin{theorem}\label{thm:main} +For the fixed-boundary critical XY law \eqref{eq:pinned-spin-law}, +the distributions \(S_n\) in \eqref{eq:spin-field} converge in law +along the full sequence \(n\to\infty\) to \(V_*\) in +\(H^{-3}_{\mathrm{loc}}(D)\). +Moreover, for every \(k\ge1\), every +\(f_1,\ldots,f_k\in C_c^\infty(D;\C)\), and every collection of +nonnegative integers \(p_1,q_1,\ldots,p_k,q_k\), +\begin{equation}\label{eq:main-moment-convergence} + \E_n^0\prod_{j=1}^k + S_n(f_j)^{p_j}\overline{S_n(f_j)}^{\,q_j} + \longrightarrow + \E\prod_{j=1}^k + V_*(f_j)^{p_j}\overline{V_*(f_j)}^{\,q_j}. +\end{equation} +The joint laws of these complex smeared fields converge as well. +\end{theorem} + +Here \(H^{-3}_{\mathrm{loc}}(D)\) consists of distributions \(T\) +such that \(\chi T\), extended by zero, belongs to +\(H^{-3}(\R^2)\) for every \(\chi\in C_c^\infty(D)\), with the +usual local topology. The theorem includes nonneutral moments: +there is no requirement that the total number of fields equal +the total number of conjugate fields. + +The critical input is the Bessel-height coefficient and the height +and pin limit theorems of \cite{C}. Their exact statements and +conventions are recorded in Section~\ref{sec:heights}; the critical +coefficient there is \(a(b_c)=8\pi\), giving +\(K_*=a(b_c)/(4\pi^2)\). We also use the finite comparisons and field +lemmas identified in \cite{B}. We do not deduce the result by +continuing a low-temperature spin theorem to \(b_c\). +Nor does Theorem~\ref{thm:main} assert an asymptotic formula for +\(a_n\), or by itself transfer the result to the free-boundary +infinite-volume correlation. Such a transfer would require +additional amplitude and boundary-to-bulk information. + +\subsection{Historical context and related work} + +The distinction between absence of magnetization and slowly decaying +correlations is central to the two-dimensional XY model. Mermin and +Wagner~\cite{MW} established the low-dimensional obstruction to +conventional order for isotropic Heisenberg systems. For classical +planar spins, McBryan and Spencer~\cite{MS} obtained quantitative +power-law upper bounds on correlations. Berezinskii~\cite{Berezinskii} +developed the algebraic low-temperature picture and argued for a +phase transition. Kosterlitz and Thouless~\cite{BKT1973} proposed +the transition mechanism through the behavior of topological defects. +The ensuing renormalization-group theory predicts +the critical exponent \(1/4\) and renormalized stiffness +\(2/\pi\)~\cite{Kosterlitz1974,NK}. + +Fr\"ohlich and Spencer~\cite{FS} proved a low-temperature phase with +power-law lower bounds for the rotator model and related systems. +Van Engelenburg and Lis~\cite[Theorem 1]{vEL} established a sharp +exponential-versus-polynomial dichotomy for free infinite-volume XY +correlations, including a polynomial lower bound at the threshold. +Lammers~\cite[Theorem 1]{Lammers2023} related the spin and dual-height +correlation lengths and identified the coincidence of the XY +transition with dual-height delocalization. These results identify +the physical threshold in Equation~\eqref{eq:critical-parameter}; +they do not supply the critical spin-field limit. + +Gaussian descriptions of angular fluctuations also have rigorous +precedents. Newman and Wu~\cite[Theorem 2]{NewmanWu2018} proved +Gaussian limits for angular-gradient observables in a joint +zero-temperature and continuum regime. The companion +\cite[Theorem 7.6]{B} treats the exponential spin field at sufficiently +low fixed temperature. The critical coefficient and fixed-geometry +Bessel-height inputs used here come instead from +\cite[Section 6 and Theorems 2.4, 3.1, 3.2]{C}. +Our task is to pass from these critical inputs to the spin field, +retaining the actual center magnetization as normalizer and treating +all signed correlations, including nonneutral ones. + +The proof begins with integer-current and planar-height +duality~\cite[Section 2]{vEL}. The finite comparisons descend from +the lattice-Gaussian inequalities of Regev and +Stephens-Davidowitz~\cite{RSD}, in their Bessel-height form +in~\cite{C}. The target belongs to the theory of imaginary Gaussian +multiplicative chaos developed by Junnila, Saksman and +Webb~\cite{JSW2020}. To pass from correlations to field laws and +moments, we use the Lee--Yang theorem for planar +rotators~\cite{DunlopNewman1975} and its moment-generating-function +inequalities~\cite{Newman1975}, in the finite-volume formulation +of Newman and Wu~\cite{NewmanWu2019}. The sharp comparison for holes +of positive radius and the control of arbitrarily many annular +interfaces are developed below. + +\subsection{Proof strategy} + +Fourier expansion turns a product of spins into a ratio of height +partition functions. A spin insertion prescribes an integral winding +of the dual height around its location. We call a height law with +such prescribed windings a magnetic sector. Fixed spin boundary +conditions give free dual heights, modulo one common integer +translation. + +The height limit does not directly determine a magnetic-sector +ratio when a puncture has the size of one lattice spacing. We first +keep every puncture at a positive macroscopic radius. For such +fixed domains, Section~\ref{sec:sectors} proves the sharp Gaussian +variational formula. Its upper bound uses divergence-free tests. +Its lower bound introduces quadratic penalties on box averages, +uses separated-pin estimates to compare pieces, and refines the +observation cells to recover the Dirichlet energy. All height limits +occur at fixed geometry. + +To return to microscopic punctures, we surround each spin insertion +by a sequence of annuli. At an interface, the gluing kernel is the +partition-function factor for restoring the deleted bonds, conditional +on all height differences in a separating band. The integer constant +between two pieces is still summed. This relative-data convention +is essential both for exact factorization and for the treatment of +the free constant mode. + +Sections~\ref{sec:annuli} and~\ref{sec:kernels} give two different +controls on these kernels. After division by their centered +zero-data value, they lie between zero and one for every datum. +Their average loss from one is small on long, approximately circular +annuli, including in a unit magnetic sector. These estimates are +not multiplied into an error proportional to the number of scales. +Instead, a contraction on mean-zero functions shows that an +arbitrary fixed microscopic core eventually produces a last-band +density bounded by two relative to the free annulus law. The core +may have an arbitrarily poor initial gluing factor; positivity is +enough. + +We then attach all the annular sequences to one exterior region. +The exterior has positive-radius holes and is governed by the +variational formula already proved. The entire microscopic +contribution is the same for every unit insertion and for the +single insertion defining \(a_n\). Subtracting the corresponding +logarithms cancels that contribution exactly. Section~\ref{sec:correlations} +computes the remaining Green-function interaction and proves +uniform convergence on separated compact configurations. + +The final step needs more than separated correlations. A weaker, +integrable two-point bound controls collisions uniformly in \(n\). +Together with positivity and the Lee--Yang moment estimate, it +removes every collision region from every mixed moment, without +requiring a sharp pointwise bound for higher correlations. +Section~\ref{sec:fields} then identifies the moments, proves +moment determinacy, and obtains local Sobolev tightness. + +The sharp fixed-hole comparison and the positive-kernel stability +argument are the transferable parts of the proof. They separate +fixed-geometry Gaussian information from the microscopic +normalization, and use averaged control of conditional data where +a uniform conditional limit is unavailable. diff --git a/preprints/The-Critical-Spin-Field-of-the-Planar-XY-Model-October-5-2026/build/sections/02-heights.tex b/preprints/The-Critical-Spin-Field-of-the-Planar-XY-Model-October-5-2026/build/sections/02-heights.tex new file mode 100644 index 0000000000000000000000000000000000000000..cc91fc62a90482fba05053ed8a29d0095d7cfa73 --- /dev/null +++ b/preprints/The-Critical-Spin-Field-of-the-Planar-XY-Model-October-5-2026/build/sections/02-heights.tex @@ -0,0 +1,491 @@ +\section{Free heights and finite comparisons}\label{sec:heights} + +The fixed spin boundary becomes a free boundary after duality. We first +establish this correspondence, including nonneutral charges, and then +record the height limits that we use from \cite{C}. The finite comparisons +must keep track of independent additive constants and prescribed data +consisting only of height differences. We spell out how the cited +inequalities apply with these conventions. + +\subsection{Partition sums and fixed-spin duality} + +Choose one orientation of every edge of a finite nearest-neighbor graph +\(G=(V,E)\), and write \(\nabla_eh=h_y-h_x\) for \(e=(x,y)\). +We call its vertices and edges primary, to distinguish them from the +auxiliary subdivisions used below. +An \emph{integral connection} is an antisymmetric edge function +\(\alpha\) with values in \(2\pi\Z\). The height variables lie in +\(2\pi\Z\), and their edge weights are +\begin{equation}\label{heights:weights} + p_b(j)=e^{-b}I_j(b),\qquad + \prod_{e\in E}p_b\left(\frac{\nabla_eh+\alpha_e}{2\pi}\right). +\end{equation} +Here \(I_j\) is the modified Bessel function. The symmetry +\(p_b(j)=p_b(-j)\) makes the chosen orientations immaterial. We will +occasionally include an observation penalty +\(\exp\{-\|Bh-v\|^2/2\}\), where \(B\) is a finite real linear map +and \(v\) is a prescribed real vector. Retained observation penalties +are sums of component-local forms in every cut topology considered: +each row of \(B\) is supported in one component. Within each component +they either annihilate its constant or confine it. + +A component with no absolute constraint or confining observation is +summed modulo one common translation by \(2\pi\Z\). On a component whose +constant is confined by observations, all translation copies are summed. +An absolute height pin fixes that constant instead. These conventions +also apply after an edge cut produces several components. Thus fixing a +representative on each cut piece does not fix the relative translations +when the pieces are subsequently joined. + +Write \(s\) for the collection of shifts, including \(\alpha\) and any +observation centers, and let \(Z_G^s\) be the corresponding partition +sum with counting measure and the preceding constant convention. The +superscript \(0\) means zero connection and zero observation centers, +with all cuts and quadratic penalties retained. Set +\begin{equation}\label{heights:sector-ratio} + F_G^s=\frac{Z_G^s}{Z_G^0}. +\end{equation} +If data \(u\) are prescribed on a set \(p\), then \(Z_G^s(u)\) +denotes the partition sum with those data imposed, not their probability. +Data can specify every height on \(p\), or every difference from a +chosen site of \(p\). In the second case the common height on \(p\) +is still free. Each such relative-data set lies in one component of +every topology being compared; data on several components are +specified separately. We prescribe only compatible lattice data. + +Changing \(\alpha\) by the gradient of a \(2\pi\Z\)-valued function +and changing the height variable by the opposite function is an exact +reindexing of the sum. Observation centers and prescribed data must be +changed at the same time. We refer to this reindexing as an integral +gauge change. A connection is \emph{flat} on a region if its circulation +around every contractible cycle there is zero; on a multiply connected +region its remaining circulations are called its periods. + +We use the classical Fourier-current duality in the formulation of +\cite[Section~2]{vEL}, with the fixed-spin boundary convention checked +in the following proof. + +\begin{lemma}[Duality for the fixed spin boundary]\label{heights:duality} +Let \(\mathcal G_n\) have as vertices the centers of the lattice squares in +\([-1,1]^2\), and join centers of neighboring squares. This is the free +dual square grid. Given integers \(l_x\), indexed by the interior primal +vertices, choose an integral connection \(s\) on \(\mathcal G_n\) whose +circulation around \(x\) is \(2\pi l_x\), with a consistent choice of +orientation. Then +\[ + \E_n^0\exp\left(i\sum_xl_x\theta_x\right)=F_{\mathcal G_n}^s. +\] +Such a connection exists without a neutrality assumption. If holes +containing the charged vertices are removed, its restriction to the +remaining graph is flat, with periods specified by the enclosed charges. +On that graph the bare partition ratio depends only on those periods. +\end{lemma} + +\begin{proof} +Edges lying along the spin boundary have both endpoints fixed at zero, +so their interaction factors cancel from the expectation. Expand every +other interaction as +\[ + e^{b\cos(\theta_x-\theta_y)} + =\sum_{j\in\Z}I_j(b)e^{ij(\theta_x-\theta_y)}. +\] +Integration imposes divergence conditions only at interior vertices. +For the numerator these are the prescribed integer charges, with their +sign determined by the current orientation; the denominator has zero +divergence. There is no condition at a boundary vertex. Consequently +one can construct a current carrying each charge along a path to the +boundary, whether or not the total charge vanishes. + +Subtract that fixed current in the numerator and rotate the remaining +edge variables onto the dual edges. Every retained primal edge crosses +exactly one edge of \(\mathcal G_n\). Every cycle of \(\mathcal G_n\) is generated by the +elementary cycles around interior primal vertices, so the rotated +zero-divergence current is a closed integral form. Integration along +dual paths represents it as \(\nabla h/(2\pi)\), uniquely modulo one +common height translation. The subtracted current gives the connection +\(\alpha\). The factors \(e^b\) needed to replace \(I_j(b)\) by \(p_b(j)\) +cancel. This proves the identity and also its positivity. + +After holes are removed, any contractible cycle encloses no remaining +charge, while a cycle winding around a hole has circulation equal to +the total charge in that hole. Two integral flat connections with the +same periods differ by an integral gradient: integrate their difference +from a fixed vertex, using vanishing circulation for path independence. +The last assertion follows by gauge change. +\end{proof} + +\subsection{The cited height limits} + +For a finite graph \(G\), let \(A_G\) be its unscaled, unit-conductance +Laplacian, so that +\[ + \langle g,A_Gg\rangle=\sum_{\{x,y\}\in E(G)}(g_x-g_y)^2. +\] +Pairings here use counting measure. On a free square of side \(m\) in +lattice units, set \(g_m(x)=x_1/m\). The coefficient of \cite{C} is +\[ + a(b)=\lim_{m\to\infty} + \Var\langle h,A_Gg_m\rangle. +\] +The existence of this limit and the limit theorems below are cited +results about the Bessel weights \eqref{heights:weights}. + +\begin{theorem}[Critical height coefficient {\cite[Section~6]{C}}] +\label{heights:critical-input} +For the free-box spin mass threshold \(b_c\) of Section~\ref{sec:intro}, +the Bessel height coefficient satisfies \(a(b_c)=8\pi\). +\end{theorem} + +The parameter and normalization in this theorem agree with ours: +\cite{C} uses the weight \(e^{-b}I_j(b)\), height spacing \(2\pi\), +and one copy of each unoriented edge. Its threshold is defined by the +same free-square spin mass. We use this as an independent theorem of +\cite{C}; it is not a consequence of the low-temperature spin theorem +of \cite{B}. In the remainder of the paper, +\[ + b=b_c,\qquad a=8\pi,\qquad + K_*=\frac{a}{4\pi^2}=\frac2\pi. +\] + +The following limit statements use three types of height observations. +A box average means +\[ + h_Q=\frac1{|Q_n|}\sum_{x\in Q_n}h_x, +\] +where \(Q_n\) is the set of primary vertices in the discretized box. +A smear \(\langle h,f_n\rangle=\sum_xh_xf_n(x)\) on a free component +is defined only when \(\sum_xf_n(x)=0\) exactly. A continuum +mean-zero density \(f\) is represented by such profiles \(f_n\) +whose piecewise constant scaled densities \(n^2f_n\) converge to +\(f\) in \(L^2\); sampled profiles can be centered by subtracting +their discrete average. +For a free component with vertex set \(V_n\), its fractional mean is +\((|V_n|^{-1}\sum_{x\in V_n}h_x)\bmod 2\pi\), which is well-defined +on the height quotient. + +\begin{theorem}[Free rectangle limits {\cite[Theorem~2.4]{C}}] +\label{heights:free-input} +Let \(U_n\) be free axis-parallel rectangles at mesh \(n^{-1}\), with +endpoints converging to those of a nondegenerate rectangle \(U\). +For any finite collection of smooth functions defined near +\(\overline U\), the observations +\(\langle h,A_{U_n}g\rangle\) converge jointly, in Laplace transforms +and all moments, to centered Gaussians with covariance +\[ + a\int_U\nabla g\cdot\nabla g'. +\] +For bounded piecewise continuous mean-zero densities and the exactly +centered profiles just specified, height smears converge jointly in +Laplace transforms and all moments, with covariance +\(a(-\Delta_{U,N})^{-1}\), where the inverse is on mean-zero functions. +\end{theorem} + +We next specify the domains for the mixed version of this theorem. +This will let us use rectangular partitions and cut regions without +requiring a limit theorem for arbitrary varying domains. + +\begin{definition}[Fixed mixed geometry]\label{heights:geometry} +A fixed mixed geometry is a finite union of bounded axis-parallel +rectangular pieces with nondegenerate widths and faces. Pieces belonging +to one component meet along full interfaces, which may be subdivided +into segments. Components are polygonal Lipschitz domains, and no two +parts are joined solely through a point. A pin region is an aligned +finite union of positive-area boxes; pinning it means setting every +primary lattice height there to zero. + +Discretizations use the nearest-neighbor graph at mesh \(n^{-1}\). +Boundaries may be rounded by a bounded number of lattice steps, +preserving the faces, corners, and positive-width passages. Observation +boxes lie within their respective components. A translated grid, such +as the grid of dual square centers, uses the same relative alignment +and rounding conventions. +\end{definition} + +\begin{theorem}[Mixed height limits {\cite[Theorem~3.1]{C}}] +\label{heights:mixed-input} +In any fixed mixed geometry, the centered massless Bessel height law +with primary hard pins has Gaussian Laplace and moment limits for each +finite collection of bounded piecewise continuous density smears. +On a component \(U\) meeting a pin \(p\), the covariance is the inverse +of the form +\[ + a^{-1}\int_U|\nabla u|^2 + \quad\hbox{on}\quad + \{u\in H^1(U):u=0\text{ almost everywhere on }p\}. +\] +Free walls have the variational Neumann condition. On an unpinned +component, the assertion concerns mean-free smears; the fractional mean +modulo \(2\pi\) tends to an independent uniform variable. + +A fixed finite-rank nonnegative quadratic penalty in box averages gives +the corresponding Gaussian limit. When the penalty controls every +unpinned component constant, lattice translation copies are summed and +the limiting constants are integrated. The assertions hold along +every admissible sequence of rounded discretizations. +\end{theorem} + +The next input concerns an interaction of rare pin events, not their +individual probabilities. It is this ratio that will control the cost +of restoring cut edges. + +\begin{theorem}[Separated primary pins {\cite[Theorem~3.2]{C}}] +\label{heights:pin-input} +In a fixed mixed geometry, start with no hard pins and add the centered +penalty \(\|Bh\|^2/2\), where \(B\) is a fixed finite list of +component-local box averages controlling every component constant. +For positively separated pin regions \(H,p\), put +\[ + D_n(H,p)= + \log\frac{\Pbb(h_H=0\mid h_p=0)}{\Pbb(h_H=0)}. +\] +Then \(D_n(H,p)\ge0\) and it converges to the Gaussian pin interaction +for the Hilbert-space inner product associated with +\[ + \mathcal T(u)=a^{-1}\sum_U\int_U|\nabla u|^2+\|Bu\|^2, + \qquad u\in\bigoplus_U H^1(U). +\] +More precisely, with orthogonality and projections taken for +\(\mathcal T\), +\begin{equation} + \begin{gathered} + V_Q=\{u:u=0\text{ almost everywhere on }Q\}^{\perp_{\mathcal T}}, + \qquad \Pi_Q=\operatorname{proj}_{V_Q},\qquad + C=\Pi_H|_{V_p},\\ + D_n(H,p)\longrightarrow + -\frac12\log\det(1-C^*C). + \end{gathered} + \tag{1}\label{eq:source-1} +\end{equation} +The operator \(C\) is Hilbert--Schmidt and has norm strictly below one. +\end{theorem} + +The determinant formula is the Gaussian pin formula used in +\cite[Theorem~3.2]{C}; its mixed-domain formulation is given in +\cite[Proposition~3.14]{B}. Each use below fixes the geometry, the +observation boxes, and the positive separation before refining the +lattice. Further limits in cell size or geometric separation are +taken afterward. No conclusion uniform over all microscopic pin data +is included in these inputs. + +\subsection{Affine comparisons and edge restoration} + +We now pass from these limit inputs to comparisons on individual finite +graphs. The affine inequality of \cite[Lemma~2.2]{C} already includes +real observation centers and real numerator tilts. We recall its +Gaussian-chain proof before spelling out its application to cut graphs, +relative data, and independently released constants. All comparisons +in this subsection take place on fixed finite graphs. + +\begin{lemma}[Finite comparison rules] +\label{heights:finite-comparisons} +Allow nonnegative quadratic observation penalties and compatible +linear equality constraints on a finite primary height graph. +The following comparisons hold. +\begin{enumerate}[label=\textup{(\roman*)}] +\item Increasing centered precision decreases centered covariance. +Consequently, in a proper centered law, the probability of an additional +centered linear equality increases under a centered precision increase. +The same assertion holds in a quotient law for equalities invariant +under its released constants. +\item A shifted-to-centered partition ratio is at most one and +decreases when the same nonnegative precision is added in the affine +and centered coordinates. Shifts may include integral connections, +nonzero prescribed data, and arbitrary real observation centers. +The monotonicity persists with a fixed real linear statistic in the +numerator exponent; the bound by one is not asserted for that +tilted ratio. +\item Equality constraints are obtained by increasing their quadratic +penalties to infinity. Free constants can be released when all data +and real tests in question are invariant under those constants. +\end{enumerate} +\end{lemma} + +\begin{proof} +We first give the Gaussian comparison that will implement an edge +restoration. Let \(\Lambda\) be a lattice in a real Euclidean space, +let \(z+\Lambda\) be a translate, and let \(T\) be positive definite. +For \(Q\ge0\), \(t\ge0\), and a fixed real linear functional \(\ell\), +put +\[ + R(t)= + \frac{\displaystyle\sum_{x\in z+\Lambda} + e^{-\langle x,(T+tQ)x\rangle/2+\ell(x)}} + {\displaystyle\sum_{x\in\Lambda} + e^{-\langle x,(T+tQ)x\rangle/2}}. +\] +The relevant lattice-Gaussian inequalities originate in the work of +Regev and Stephens-Davidowitz \cite[Theorem~2.1 and Corollary~3.2]{RSD}. +The covariance under the numerator law dominates that under the +centered denominator law; see \cite[Lemma~2.1]{B}. If \(m_{z,\ell}\) +and \(\Cov_{z,\ell}\) denote the numerator mean and covariance, +and \(\Cov_0\) the denominator covariance, differentiation gives +\[ + \frac{d}{dt}\log R(t) + =-\frac12\Tr Q\left( + \Cov_{z,\ell}-\Cov_0+ + m_{z,\ell}m_{z,\ell}^{\mathsf T}\right)\le0. +\] +For a proper affine subspace, first complete the square in its +parallel span. The remaining offset is orthogonal for the current +precision \(T+tQ\); the covariance on the parallel span is that of a +translated, tilted lattice Gaussian with the restricted precision. +Thus its covariance still dominates the centered covariance, while +the raw mean \(m_{z,\ell}\) includes the fixed offset. +The centered maximum of a Gaussian lattice sum gives the untilted +bound \(R(t)\le1\). + +To obtain the Bessel law, replace each primary edge by \(N\) +independent integer increments with normalized weight \(q_N^{j^2}\), +where \(q_N=b/(2N)\); their physical height increments are \(2\pi j\). +Uniformly for \(u\) in compact real sets, +\[ + \frac{\sum_{j\in\Z}q_N^{j^2}e^{uj}}{\sum_{j\in\Z}q_N^{j^2}} + =1+\frac bN(\cosh u-1)+O_u(N^{-2}). +\] +Their sum tends to the probability weight \(p_b\), since its limiting +moment generating function is \(\exp\{b(\cosh u-1)\}\). +Matching endpoints, cycles, periods, and prescribed data gives +parallel affine lattices for the numerator and denominator. +Appending the residual coordinates \(Bh-v\) allows arbitrary real +observation centers. Thus adding a quadratic penalty or a matching +constraint is precisely the precision increase just considered. +Centered covariance decreases under such increases by the same +finite Gaussian comparisons; see \cite[Lemma~2.1]{B}. + +All limits just described are on a fixed primary graph. A spanning +forest expresses every height difference as a sum of full-chain +increments, which have uniformly bounded exponential moments at +each fixed real argument. The other normalized edge weights are at +most one. This bounds the partition sums, their fixed real tilts, +and any polynomial factors. A confined component constant is +summed against its Gaussian penalty. Dominated convergence therefore +passes the comparisons to \(p_b\), as in \cite[Lemma~2.2]{C}. +Equality limits use monotone convergence of the penalties; only +nonempty affine fibers are used. +For centered pin monotonicity, first use a finite pin penalty. +The derivative of the logarithm of its expectation under an +additional centered precision \(Q\) is +\(\frac12\Tr Q(\Cov_{\rm unpinned}-\Cov_{\rm pinned})\ge0\). +Then increase the pin penalty to infinity. +The release of unconfined constants, including across cuts, is +justified next. +\end{proof} + +\begin{lemma}[Cut comparison with prescribed data]\label{heights:cut-comparison} +Let \(G^{\rm cut}\) be obtained from \(G\) by deleting edges, with +the same observation penalties and compatible prescribed data on +both graphs. Each retained observation and each relative-data set +is component-local in \(G^{\rm cut}\). Whenever the free constants are left free by those +data, +\[ + \frac{Z_G^s(u)}{Z_G^0(0_p)} + \le + \frac{Z_{G^{\rm cut}}^s(u)} + {Z_{G^{\rm cut}}^0(0_p)}. +\] +The conclusion also holds with absolute data fixing some constants +and with observations confining others. It holds for a fixed real +numerator tilt on affine edge increments if each deleted increment +is retained independently, with its tilt, on the cut side. Every +real test on a released component must annihilate its constant. +\end{lemma} + +\begin{proof} +For a deleted edge \(e=(x,y)\), retain an independent chain with total +increment \(J_e\in2\pi\Z\). Its normalized sum without matching is +one. Restoring that edge imposes the equality +\[ + J_e-h_y+h_x-\alpha_{xy}=0; +\] +the centered equality has \(\alpha_{xy}=0\). +On the enlarged affine coordinates this amounts to increasing the +coefficient of the square of the displayed residual to infinity. +The prescribed-data fibers in numerator and denominator are parallel +lattices. This remains true for relative data +\(h_z-h_{z_0}=u_z\), since these are linear equality constraints. +Lemma~\ref{heights:finite-comparisons} gives the comparison. +Integration of the matched chain contributes +\(p_b((h_y-h_x+\alpha_{xy})/(2\pi))\), and all independent-chain +normalizers cancel. For a real edge tilt, use \(J_e\) on the enlarged +graph. On matching it is precisely the affine edge increment in +the restored graph, so the tilted comparison applies with no +additional scalar. + +Here is the promised treatment of constants. Before comparing, add +an auxiliary site penalty \(\delta\sum_xh_x^2/2\) and sum all +translations. For a free component with \(m\) vertices, a translation +orbit represented by \(h\), with mean \(\bar h\), contributes +\[ + e^{-\frac\delta2\sum_x(h_x-\bar h)^2} + \sum_{k\in\Z}e^{-\frac{m\delta}{2}(\bar h+2\pi k)^2}. +\] +Divide its translation sum by +\[ + c_{m,\delta} + =\frac1{2\pi}\sqrt{\frac{2\pi}{m\delta}}. +\] +Poisson summation shows that the quotient of the sum by +\(c_{m,\delta}\) tends uniformly in \(\bar h\) to one and is +uniformly bounded for small \(\delta\). The first factor is at +most one and tends pointwise to one. Forest exponential-moment +bounds again give domination, also for fixed real tilts invariant +under the released constant. Thus the regularized sum divided by +one \(c_{m,\delta}\) per free component tends to the quotient sum. +Within each shifted-to-centered ratio, its component factors cancel. +They cancel separately on the cut and restored sides, although +those graphs can have different numbers of components. + +On a component already fixed by absolute data or confined by +observations, the auxiliary penalty is removed by ordinary dominated +convergence, with no divergent factor. This also covers comparison +of an unconfined ratio with a ratio in which new observations confine +every component: cancellation is performed separately in each ratio. +Semidefinite intermediate forms cause no difficulty, since the +auxiliary penalty makes every sum proper before the limit. Finally +take the chain limit at fixed primary graph as in +Lemma~\ref{heights:finite-comparisons}. +\end{proof} + +In particular, the bare ratios satisfy +\begin{equation} + 01\) to the constant-vector +part. Bound the transform of the error, at the conjugate exponent, +by freeing all Bessel increments. For every fixed tilt multiple, +\[ + \log\E\exp\!\left\{\sum_e r_e J_e\right\} + \le C\sum_e r_e^2 +\] +when the \(J_e\) are independent increments of law \(p_{b_c}\) in +height units and the \(r_e\) are uniformly bounded. +This follows from +\(\E e^{tJ_e}=\exp\{b_c(\cosh(2\pi t)-1)\}\). +First let \(n\to\infty\), then \(d\downarrow0\), and finally +\(p\downarrow1\). We obtain +\begin{equation} + \limsup_{n\to\infty}\log F_{U,n}^s + \le -\int_Uw\cdot dv_0+\frac a2\int_U|w|^2. + \label{sectors:dual-bound} +\end{equation} + +The fields \(w\) just used are dense in the orthogonal complement of +gradients in \(L^2(U;\R^2)\). Indeed, extend a field in that complement +by zero outside \(U\). Its distributional divergence vanishes, so in +a larger simply connected box it has an \(H^1\) stream function. +This function is constant on every complementary component of \(U\). +Subtract a smooth function having the same constants near these +components. The remainder has zero trace on \(\partial U\), and +\(H^1_0(U)\) approximation now proves the density assertion. +The gradient space is closed by the Poincar\'e inequality modulo +constants. If \(P\) is the orthogonal projection onto its complement, +then \(\mathcal M(U)=\|P\,dv_0\|_2^2\). +Optimizing Equation~\eqref{sectors:dual-bound} over the dense class, +with optimizer \(a^{-1}P\,dv_0\), gives the required upper bound. +\end{proof} + +\subsection{Confinement suppresses separated pin interactions} + +We next add one quadratic observation for each small cell. Although +each observation has coefficient one, their number grows as the cell +size decreases. This gives the lower-bound construction its spatial +localization. + +Fix one of the polygonal geometries of +Definition~\ref{heights:geometry} whose rectangular pieces have +dyadic coordinates, possibly with finitely many components or cuts. +Partition it into dyadic squares \(Q\) of side +\(d\), aligning all boundaries and cuts, and write +\[ + u_Q=\frac1{|Q|}\int_Q u,\qquad + \mathcal T_d(u)=a^{-1}\int|\nabla u|^2+\sum_Q u_Q^2 . +\] +The lattice observation on \(Q\) is the average over its rounded array +of vertices. Its penalty is one half of the square of that average. +These observations control the constant on every component. +For two positively separated unions of positive-area boxes \(H,p\) +with dyadic coordinates, +let \(D_{n,d}(H,p)\) be the centered absolute-zero pin interaction of +Theorem~\ref{heights:pin-input} for this law. + +\begin{lemma}[Localization under cell observations] +\label{sectors:localization} +For every fixed dyadic mixed geometry and fixed dyadic pin regions +\(H,p\) as above, +\begin{equation} + \lim_{d\downarrow0}\limsup_{n\to\infty}D_{n,d}(H,p)=0 . + \tag{4}\label{eq:source-4} +\end{equation} +Here \(d\) runs through dyadic values aligned with every fixed +geometric division. The mesh limit is taken at each fixed \(d\). +\end{lemma} + +\begin{proof} +At each fixed \(d\), Theorem~\ref{heights:pin-input} identifies the +limit with the Gaussian determinant in Equation~\eqref{eq:source-1}. +We prove that this determinant tends to one. +Equip the component \(H^1\) spaces with the norm +\(\|u\|_d^2=\mathcal T_d(u)\), and write +\[ + V_Q=\{u:u=0\text{ on }Q\}^{\perp_{\mathcal T_d}},\qquad + C_d=\Pi_H|_{V_p}, +\] +where \(\Pi_H\) is orthogonal projection onto \(V_H\). +The letter \(Q\) in this definition can denote a pin region; +the observation cells remain the squares in the sum defining +\(\mathcal T_d\). + +Rescale space by \(d^{-1}\). The observation cells are now unit +squares. In dimension two the gradient energy is unchanged by this +rescaling, and cell Poincar\'e gives, with constants independent of \(d\), +\begin{equation} + c\|u\|_{H^1}^2\le \mathcal T_d(u)\le C\|u\|_{H^1}^2 + \quad\hbox{in these rescaled coordinates}. + \label{sectors:cell-coercivity} +\end{equation} +There are \(N_d=O(d^{-2})\) cells. +If \(f\in V_p\), then +\(\mathcal T_d(f,\phi)=0\) for every \(\phi\) vanishing on \(p\). +In particular \(f\) is form-harmonic off \(p\). + +We record the cutoff estimate, including the observation terms. +Take a Lipschitz cutoff \(\chi\) vanishing near \(p\), with transition +width and Lipschitz constant bounded in cell units, and test with +\(\chi^2f\). The gradient part satisfies +\[ + \int\nabla f\cdot\nabla(\chi^2f) + \ge \frac12\int\chi^2|\nabla f|^2 + -C\int_{\{\nabla\chi\ne0\}}f^2. +\] +On cells where \(\chi=1\), the observation contribution is \(f_Q^2\). +On transition cells its absolute value is at most +\(C\|f\|_{L^2(Q)}^2\), by Cauchy--Schwarz. +Thus the energy on the plateau is bounded by the \(L^2\) mass in +a fixed enlargement of the transition layer. The test is also +admissible at a free side, where no boundary value is prescribed. + +To iterate the estimate, let \(E_r\) be the sum of cell energies at +cell-center distance at least \(r\) from \(p\). Choose \(\chi\) zero +up to distance \(r+2\) and one beyond \(r+6\). +By Equation~\eqref{sectors:cell-coercivity}, enlarging the layer by +a fixed number of cells gives +\[ + E_{r+10}\le C(E_r-E_{r+10}). +\] +Iteration gives exponential decay in cell distance. The fixed positive +separation of \(H\) and \(p\) is of order \(d^{-1}\) in these +coordinates. Multiplying \(f\) by a cutoff equal to one near \(H\) +and supported away from \(p\) therefore produces an extension of +its \(H\)-data of energy at most \(Ce^{-c/d}\|f\|_d^2\). +Orthogonal projection gives the least-energy extension of these data, +so, after adjusting the constants, +\begin{equation} + \|C_d\|\le \delta_d,\qquad \delta_d=Ce^{-c/d}. + \label{sectors:operator-decay} +\end{equation} + +An operator-norm bound alone does not control the determinant. +We also need a summable singular-value bound. For an integer \(m\ge1\), +restrict \(f\in V_p\) by requiring its Neumann cosine coefficients +on every unit cell, for frequencies \(0\le k_1,k_2\le m\), to vanish. +The codimension is at most \(C N_dm^2\), and the cell spectral +inequality gives +\begin{equation} + \|f\|_{L^2(Q)}\le C m^{-1}\|\nabla f\|_{L^2(Q)}. + \label{sectors:cell-high-modes} +\end{equation} +Choose nested neighborhoods +\(\mathcal N_1\subset\mathcal N_2\subset\mathcal N_3\) of \(H\), +each a union of cells and separated from \(p\). +Leave fixed-width buffers, in cell units, between successive +neighborhoods and between \(H\) and the relative boundary of +\(\mathcal N_1\). The neighborhoods may meet free walls. +Define +\[ + E_j(f)=\sum_{Q\subset\mathcal N_j} + \left(a^{-1}\int_Q|\nabla f|^2+f_Q^2\right). +\] +Use cutoffs equal to one on \(\mathcal N_j\), supported in +\(\mathcal N_{j+1}\), and with uniformly bounded gradients. +In both harmonic tests the function is the original \(f\). +The cutoff estimate and Equation~\eqref{sectors:cell-high-modes} +give +\[ + E_1(f)\le C m^{-2}E_2(f) + \le C m^{-4}E_3(f) + \le C m^{-4}\|f\|_d^2 . +\] +Finally choose \(\eta=1\) near \(H\), supported in +\(\mathcal N_1\), with bounded gradient. Cell coercivity bounds the +cutoff extension by \(\|\eta f\|_d^2\le C E_1(f)\). +Minimality of \(\Pi_Hf\) then gives +\(\|\Pi_Hf\|_d\le C m^{-2}\|f\|_d\). +The fixed positive physical separation leaves room for these +buffers when \(d\) is small. + +The min-max characterization of singular values consequently gives +\[ + s_j(C_d)\le \frac{C N_d}{j}\quad (j\ge C N_d). +\] +Together with Equation~\eqref{sectors:operator-decay}, splitting the +sum at an index of order \(N_d/\delta_d\) yields +\[ + \|C_d\|_{\mathrm{HS}}^2 + \le C N_d\delta_d\longrightarrow0 . +\] +For small \(d\), \(\|C_d\|<1/2\), and hence +\[ + -\frac12\log\det(1-C_d^*C_d) + \le \|C_d\|_{\mathrm{HS}}^2\longrightarrow0. +\] +Components contribute independently, so the proof also covers all +the fixed cut geometries in the statement. +\end{proof} + +\subsection{The confined sector and single-valued lifts} + +Return to the domain \(U\) of Theorem~\ref{sectors:positive-radius}. +Fix \(v=v_0+g\), where \(g\) is smooth near \(\overline U\). +Choose real vertex values \(v_n\), in cut coordinates compatible with +\(\alpha_n\), so that +\[ + v_n(y)-v_n(x)+\alpha_n(x,y)=\int_{[x,y]}dv . +\] +These values exist because the difference of the two edge forms has +zero cycle periods. We choose them by fixing branch cuts and local +branches of \(v\) first, then replacing \(\alpha_n\) by the +gauge-equivalent connection with the matching integral jumps across +these cuts. This gives samples of fixed branches, rather than +mesh-dependent choices of their additive constants. +On any piece where \(\alpha_n\) is gauged to zero, +the simultaneous change of \(v_n\) gives samples of a smooth +single-valued lift of \(v\). Add the cell penalties of the preceding +subsection, using +\[ + \frac12\sum_Q\bigl(h_Q-(v_n)_Q\bigr)^2 + \quad\hbox{in the sector numerator},\qquad + \frac12\sum_Qh_Q^2 + \quad\hbox{in the centered denominator}. +\] +Here the subscripts denote lattice averages. All common translation +copies are summed. Denote the resulting ratio by \(R_{n,d}^s(v)\). +The affine precision comparison of +Lemma~\ref{heights:finite-comparisons} gives +\begin{equation} + 00\), and choose a closed rectilinear polygon +\(P\), homeomorphic to a closed disk, with dyadic vertices and +\[ + B(0,1-\varepsilon)\subset P\subset B(0,1+\varepsilon). +\] +One may use a sufficiently fine staircase approximation to the circle, +chosen symmetrically in the four quadrants. All side lengths and +passages in the resulting fixed geometry are nondegenerate. Write +\(Q_r=(-r,r)^2\). Choose \(\lambda\) to be a sufficiently large power +of \(4\), and put +\[ + \begin{gathered} + A_\lambda=\lambda P\setminus P^\circ,\qquad + p=\overline{Q_{2\sqrt\lambda}}\setminus Q_{\sqrt\lambda},\\ + H_{\mathrm{in}}=A_\lambda\cap\overline{Q_4},\qquad + H_{\mathrm{out}}=A_\lambda\setminus Q_{\lambda/4}. + \end{gathered} +\] +Sobolev spaces on \(A_\lambda\) refer to its interior. +The guard \(p\) is a full square band separating the two ends. +Each end region contains all vertices next to its annular wall after +fine discretization. The guard and either end are positively separated. +They are finite unions of positive-area boxes in the polygonal +geometry of Section~\ref{sec:heights}. Discretizations use the coherent +array divisions specified there. In particular, a boundary division +never identifies pieces merely at a point. +Figure~\ref{annuli:figure} displays these regions. + +\begin{figure}[tb] + \centering + \input{figures/annular-bands.tex} + \caption{A polygonal annulus with its full square guard \(p\) and the + two possible end regions. The inner and outer walls are rectilinear + approximations to circles; the square boundaries belong to the + bands, not to the annular walls. Radial distances are compressed: + the actual walls are \(P\) and \(\lambda P\), the guard is at scale + \(\sqrt\lambda\), and either end is separated from it by a round + annulus of logarithmic length + \(\frac12\log\lambda-O(1)\). The two end regions are displayed + together, but the pin estimate is applied to one end at a time.} + \label{annuli:figure} +\end{figure} + +Consider the centered, free height law \(\mu\) on this annulus, with +one common translation quotiented out. For a nonempty vertex set \(Q\), +write +\[ + [Q]=\{h_x=h_y\text{ for every }x,y\in Q\}. +\] +Thus \([Q]\) pins all differences within \(Q\), without pinning its +common height. For \(H=H_{\mathrm{in}}\) or \(H_{\mathrm{out}}\), set +\[ + \overline D(H,p) + =\log\frac{\mu([H]\mid[p])}{\mu([H])}. +\] + +\begin{proposition}[Separated flat pins]\label{annuli:flat-interaction} +Fix \(P\) as above. For either choice of \(H\), the interaction +\(\overline D(H,p)\) is nonnegative and +\begin{equation} + \limsup_{\mathrm{mesh}\to0}\overline D(H,p) + \quad\text{can be made arbitrarily small by taking \(\lambda\) large.} + \tag{7}\label{eq:source-7} +\end{equation} +The mesh limit is taken at each fixed polygonal geometry. +\end{proposition} + +The proof of Proposition~\ref{annuli:flat-interaction} first estimates +the continuum interaction with one confining +box average. It then weakens that confinement and passes from absolute +pins to flat pins. The constant mode is retained throughout. + +\subsection{The continuum estimate with confinement} + +Choose a dyadic box inside \(p\) whose side lengths are fixed positive +multiples of \(\sqrt\lambda\), and let \(B_0\) be its probability +average. Sum all common height translations and add the penalty +\(\gamma(B_0h)^2/2\), where \(\gamma>0\). Let \(D_\gamma(H,p)\) +be the interaction of the absolute events \(h_H=0\) and \(h_p=0\) +under this confined law. + +Start with \(\gamma=1\). By Theorem~\ref{heights:pin-input}, +\(D_1\) converges to the Gaussian +interaction for +\[ + \mathcal T(u)=a^{-1}\int_{A_\lambda}|\nabla u|^2+(B_0u)^2. +\] +Write \(\|u\|_{\mathcal T}^2=\mathcal T(u)\). +As in Equation~\eqref{eq:source-1}, let +\[ + V_Q=\{u\in H^1(A_\lambda):u=0\text{ on }Q\}^{\perp_{\mathcal T}}, + \qquad C=\Pi_H|_{V_p}. +\] +We claim that \(\|C\|_{\mathrm{HS}}\to0\) as \(\lambda\to\infty\). + +Between \(p\) and \(H\) there is a full round annulus which is a +conformal cylinder +\begin{equation}\label{annuli:cylinder} + [0,T]\times\bigl(\R/(2\pi\Z)\bigr),\qquad + T=\tfrac12\log\lambda-O(1). +\end{equation} +The end \(s=0\) faces \(p\). For the inner end, for example, one may +take radii \(8\) and \(\sqrt\lambda/2\), with the logarithmic coordinate +reversed. For the outer end one may take radii \(4\sqrt\lambda\) and +\(\lambda/8\). Enlarging the lower bound on \(\lambda\) makes both +choices available, with fixed clearances from the bands. + +For \(f\in V_p\), the trace at \(s=0\) satisfies +\begin{equation}\label{annuli:trace} + \|f|_{s=0}\|_{H^{1/2}(\mathbb S^1)} + \le C_0\|f\|_{\mathcal T}, +\end{equation} +with \(C_0\) independent of large \(\lambda\). Indeed, after scaling +lengths by \(\sqrt\lambda\), this circle, the guard, and the observation +box lie in a fixed connected annular region. Poincare's inequality +with the box mean controls the \(H^1\) norm there, including constants; +the trace theorem then gives Equation~\eqref{annuli:trace}. +On the side of this circle containing \(H\), the function \(f\) +minimizes the gradient energy with its given trace and free walls. +There is no observation term on that side. + +Let \(A\) map a trace at \(s=0\) to the restriction to \(H\) of this +energy minimizer, with the target norm equal to the least +\(\mathcal T\)-norm of an extension of that restriction to the whole +annulus. This target is naturally identified with \(V_H\). +We estimate \(A\) on a normalized Fourier basis of \(H^{1/2}(\mathbb S^1)\). +For the constant trace, the minimizer on the \(H\)-side is constant. +An extension of its \(H\)-data can be chosen constant beyond \(s=T\), +linear across the cylinder, and zero before \(s=0\). It vanishes on +the observation box and has energy \(O(T^{-1})\). Thus the norm of +the image of the constant mode is \(O(T^{-1/2})\). + +For a sine or cosine mode of frequency \(j\ge1\), normalization in +\(H^{1/2}\) makes its amplitude \(O((1+j)^{-1/2})\). +The Neumann minimum at \(s=T\), on the cylinder alone, is a lower +bound for the actual minimum. The Dirichlet-zero minimum at \(s=T\), +extended by zero beyond that circle, is an admissible upper trial. +The solutions of \(u''=j^2u\) give, respectively, the factors +\(\tanh(jT)\) and \(\coth(jT)\) in their energies. Their difference +is therefore \(O(e^{-2jT})\). Orthogonality of the actual minimizer +to zero-trace variations shows that the difference between it and +the upper trial has gradient norm \(O(e^{-jT})\). This difference +has zero trace at \(s=0\), so it extends by zero toward \(p\), pays no +observation penalty, and has, up to sign, the required restriction +on \(H\). Consequently +\[ + \|A\|_{\mathrm{HS}}^2 + \le \frac{C_1}{T}+C_1\sum_{j\ge1}e^{-2jT} + \longrightarrow0. +\] +All extensions are in \(H^1\): across each interior circle their +traces agree. Composing \(A\) with the uniformly bounded trace map +in Equation~\eqref{annuli:trace} proves the claim for \(C\). +The determinant in Equation~\eqref{eq:source-1} now tends to one, +since \(\|C\|\le\|C\|_{\mathrm{HS}}\to0\). We have proved +\begin{equation}\label{annuli:unit-confinement} + \lim_{\lambda\to\infty}\lim_{\mathrm{mesh}\to0}D_1(H,p)=0. +\end{equation} + +\subsection{From absolute pins to flat pins} + +We next establish Equation~\eqref{annuli:unit-confinement} with any +fixed \(\gamma>0\) in place of \(1\). Consider the four partition +functions defining the absolute-pin interaction. When \(p\) is +pinned, \(B_0h=0\), so changing \(\gamma\) has no effect. With neither +region pinned, the ratio of partition functions at precisions +\(\gamma\) and \(1\) tends to \(\gamma^{-1/2}\). This follows from +Theorem~\ref{heights:mixed-input}: the fractional mean becomes uniform and +independent of the mean-free field, and the translation sums of the +Gaussian penalties are bounded periodic functions. + +With \(H\) alone pinned, let \(x=x(\lambda)\) be the variance of \(B_0\) +in the massless continuum Gaussian law with that pin. Applying the +hard-pin height limit in Theorem~\ref{heights:mixed-input} to the two bounded weights +\(\exp(-\gamma(B_0h)^2/2)\) and \(\exp(-(B_0h)^2/2)\) gives the ratio +\[ + \left(\frac{1+x}{1+\gamma x}\right)^{1/2}. +\] +Moreover \(x\to\infty\). To see this from the variational formula for +the variance, take a function equal to one on the \(p\)-side, zero +on the \(H\)-side, and linear across the cylinder +in Equation~\eqref{annuli:cylinder}. Its box average is one and +its gradient energy is \(O(T^{-1})\), so \(x\ge cT\). +Combining the four partition functions yields, at fixed geometry, +\begin{equation}\label{annuli:change-confinement} + D_\gamma-D_1\longrightarrow + \frac12\log\frac{1+\gamma x}{\gamma(1+x)}. +\end{equation} +The right side tends to zero as \(\lambda\to\infty\). The absolute-pin +interaction thus tends to zero for every fixed positive \(\gamma\). + +It remains to compare these absolute pins with the desired flat +events. In a centered confined law, the logarithmic interaction of +two centered quadratic penalties increases when either precision +increases. In fact, if the first precision changes by \(tQ\), +differentiating the four log partition functions gives +\[ + \frac12\Tr Q\bigl(\Cov_{\text{first penalty}} + -\Cov_{\text{both penalties}}\bigr)\ge0 +\] +by centered covariance comparison. Taking equality limits proves +the same assertion for nested centered constraints. Absolute zero +pins are obtained from flat pins by also pinning one reference +height in each set after the within-set constraints are imposed. +Hence the interaction of \([H]\) and \([p]\) +in the confined law is between zero and \(D_\gamma(H,p)\). + +Finally, project the confined law onto common-translation orbits. +If \(t=B_0h\) in any representative, the orbit receives the factor +\[ + w_\gamma(t)=\sum_{k\in\Z} + \exp\{-\tfrac{\gamma}{2}(t+2\pi k)^2\}. +\] +Poisson summation gives the uniform estimate +\[ + \sup_t\left|\sqrt{2\pi\gamma}\,w_\gamma(t)-1\right| + \le 2\sum_{j\ge1}e^{-j^2/(2\gamma)} + =:\delta_\gamma\longrightarrow0 + \qquad(\gamma\downarrow0). +\] +For \(\gamma\) sufficiently small that \(\delta_\gamma<1\), the +normalized projected density relative to the free +orbit law is between +\((1-\delta_\gamma)/(1+\delta_\gamma)\) and its reciprocal. +This multiplicative estimate applies to the individual pin events +and their intersection, however small their probabilities. Their +log interactions differ by \(O(\delta_\gamma)\), uniformly in the +mesh and the annulus. + +Given a tolerance, first choose \(\gamma>0\) so that this error is +small. Then choose \(\lambda\) large so that the limiting absolute-pin +interaction is small, and finally take the mesh limit at that +fixed geometry. This proves Equation~\eqref{eq:source-7}. +Nonnegativity in the free law follows either from centered +precision comparison directly or by sending \(\gamma\) to zero +at each fixed finite graph. +\qed diff --git a/preprints/The-Critical-Spin-Field-of-the-Planar-XY-Model-October-5-2026/build/sections/05-kernels.tex b/preprints/The-Critical-Spin-Field-of-the-Planar-XY-Model-October-5-2026/build/sections/05-kernels.tex new file mode 100644 index 0000000000000000000000000000000000000000..e12aa78019319330190db142beed399fbce21c4b --- /dev/null +++ b/preprints/The-Critical-Spin-Field-of-the-Planar-XY-Model-October-5-2026/build/sections/05-kernels.tex @@ -0,0 +1,361 @@ +\section{Gluing kernels and stability from a fixed core}\label{sec:kernels} + +We now return to the bare height weights, with no confining penalty. +The flat-pin estimate gives an average bound for restoring an interface +between annuli. A uniform upper bound for the same kernel will make +that estimate stable through any number of annuli. + +\subsection{Guard data and interface kernels} + +Use the polygon \(P\), the guard, and the end regions of +Section~\ref{sec:annuli}. In lattice units, an annular block has walls +\(RP\) and \(\lambda RP\). We take \(R\) large enough and dyadically +aligned so that all divisions lie between rows of dual vertices. +Every block is connected. The bonds between adjacent blocks are +precisely the bonds to be restored. + +Choose one guard vertex in each annulus. Its \emph{guard datum} +\(u\) is the list of all height differences from that vertex to the +other guard vertices, in the chosen connection gauge. Every +compatible datum has positive probability. Denote its law in block +\(i\), in sector \(s\), by \(m_i^s(du)\). Fixing the representative +whose chosen height is zero turns \(u\) into full prescribed height +data on the guard. The configurations on the two sides are then +conditionally independent, since the guard is a full separating +band and the energy is a sum of nearest-neighbor terms. + +For two adjacent annuli, with data \(u_i,u_{i+1}\), define +\[ + T_i^s(u_i,u_{i+1}) + =\frac{Z_{\mathrm{restored}}^s(u_i,u_{i+1})} + {Z_{\mathrm{cut}}^s(u_i,u_{i+1})}. +\] +The numerator is summed modulo one common translation; the +denominator is a product of sums modulo one translation in each +block. Thus the numerator sums the relative integer translation +of the two blocks. + +We use the same definition in two other cases. A \emph{core} is +a connected block filling the inner hole of the first annulus; +it may contain a magnetic source and carries no prescribed guard. +Its interface kernel is a function \(T_0^s(u_1)\). At the other +end, several disjoint last annuli may be attached to one connected +exterior, with no direct bonds between these annuli. The exterior +carries no prescribed data. Its one joint kernel is a function of +the incident annular data. In every case let +\[ + d_i=T_i^0(0), +\] +where \(0\) means zero guard differences in the centered sector. +No size estimate for this positive scalar is required. + +\begin{lemma}[Upper bound and centered average]\label{kernels:bounds} +For each of the kernels just defined, every sector and every +compatible guard datum satisfy +\begin{equation} + 00\), first choose \(\varepsilon\) sufficiently small +and fix a polygon \(P\) satisfying its disk inclusions. Then choose +\(\lambda\) sufficiently large. There is a fixed aligned +threshold \(R_0\) such that every regular interface with smaller +inner scale at least \(R_0\) satisfies +\begin{equation} + \int(1-T_i^s/d_i)\prod_\ell dm_\ell^s\le\xi + \tag{11}\label{eq:source-11} +\end{equation} +in each of the two unit-period sectors and in the centered sector. +The individual flat-pin interactions in +Equation~\eqref{eq:source-9} can also be made arbitrarily small +with these choices. +\end{proposition} + +\begin{proof} +The integral of \(T_i^s\) under the independent cut-block laws is +\(Z_{\mathrm{restored}}^s/Z_{\mathrm{cut}}^s\). Dividing this +identity by its centered counterpart proves +Equation~\eqref{eq:source-10}. The same identity is valid for a +joint exterior kernel, without a claim yet about its average loss. + +For a circular annulus of radii \(1\) and \(\lambda\), the minimum +energy with unit period \(2\pi\) is \(2\pi\log\lambda\). +The angular function attains it, and integration on concentric +circles proves the matching lower bound. Restriction of admissible +forms, together with +\(B(0,1-\varepsilon)\subset P\subset B(0,1+\varepsilon)\), gives +\[ + \mathcal M(\lambda P\setminus P^\circ) + =2\pi\log\lambda+O(\varepsilon), +\] +where the constant is independent of large \(\lambda\). For example, +the circular annulus with radii \(1+\varepsilon\) and +\(\lambda(1-\varepsilon)\) is contained in the polygonal one, +which in turn is contained in the circular annulus with radii +\(1-\varepsilon\) and \(\lambda(1+\varepsilon)\). +The same bounds apply with \(\lambda^2\) to the union of two blocks. +It follows that +\[ + 0\le + \mathcal M(\lambda^2P\setminus P^\circ) + -2\mathcal M(\lambda P\setminus P^\circ) + \le C\varepsilon. +\] +The lower bound follows by restriction to the two cut annuli; +their energies agree by scale invariance in dimension two. +Theorem~\ref{sectors:positive-radius}, applied at this fixed geometry, +therefore makes the last factor in Equation~\eqref{eq:source-10} +converge to a number in \([e^{-C\varepsilon/(2a)},1]\). +Changing the unit period's sign does not change its energy. + +Choose \(\varepsilon\) to make this error small, and then choose +\(\lambda\) so that Proposition~\ref{annuli:flat-interaction} makes +the two centered pin interactions small. +Equation~\eqref{eq:source-9} and +Equation~\eqref{eq:source-10} now give the desired small loss in +the mesh limit. With \(P\) and \(\lambda\) fixed, every regular +interface is a refinement of one fixed geometry. The convergence +theorems hold along every admissible mesh sequence, so there is +one threshold \(R_0\) beyond which the inequalities hold at every +aligned scale. There are only three sectors to consider. +This proves Equation~\eqref{eq:source-11}. +\end{proof} + +\subsection{A positive-kernel stability lemma} + +The preceding estimate controls the average loss under independent +guard laws. Actual guard data in a nest have been influenced by +all the inner blocks. The following lemma bounds that change of law +without accumulating a fixed error at every interface. + +\begin{lemma}[Stability from a positive initial weight] +\label{kernels:stability} +Let \((X_i,m_i)\), \(i\ge1\), be probability spaces, and let +\(t_i:X_i\times X_{i+1}\to(0,1]\) be measurable kernels with +\[ + \int(1-t_i)\,d(m_i\otimes m_{i+1})\le\xi,\qquad + 0<\xi<1/64. +\] +Fix a measurable \(g:X_1\to(0,1]\), positive \(m_1\)-almost +everywhere. For each \(N\), give \(X_1\times\cdots\times X_N\) +density proportional to +\[ + g(u_1)\prod_{i=1}^{N-1}t_i(u_i,u_{i+1}) +\] +with respect to \(\prod_i m_i\). For all sufficiently large \(N\), +its last-coordinate marginal has density at most \(2\) with respect +to \(m_N\). No positive essential lower bound on \(g\) is required. +\end{lemma} + +\begin{proof} +Define +\[ + L_i f(v)=\int t_i(u,v)f(u)\,dm_i(u). +\] +First propagate the constant density: +\[ + q_1=1,\qquad + \beta_i=\int L_iq_i\,dm_{i+1},\qquad + q_{i+1}=\beta_i^{-1}L_iq_i. +\] +Inductively \(q_i\le2\), because +\[ + \beta_i + =1-\int(1-t_i)q_i\,d(m_i\otimes m_{i+1}) + \ge1-2\xi,\qquad + q_{i+1}\le M:=(1-2\xi)^{-1}<2. +\] +The same argument, with the kernels reversed, bounds by \(2\) any +normalized density obtained by propagating \(1\) backwards through +a finite number of them. + +Propagate \(g\) using these same scalars, rather than renormalizing +it at each step: +\[ + f_1=g,\qquad f_{i+1}=\beta_i^{-1}L_if_i,\qquad + c_i=\int f_i\,dm_i,\qquad E_i=f_i-c_iq_i. +\] +The scalar \(c_i\) is the ratio of the two integrals obtained by +starting with \(g\) and with \(1\). Equivalently, it is the +expectation of \(g\) under a normalized backwards density on +\(X_1\), which is at most \(2\). Choose \(\delta>0\) with +\(m_1(g<\delta)<1/4\). Every such density assigns at least half +its mass to \(\{g\ge\delta\}\), so +\begin{equation}\label{kernels:core-mass} + c_i\ge\delta/2 + \quad\text{for every }i. +\end{equation} + +For a mean-zero function \(E\), replace \(t_i\) in \(L_iE\) by +\(t_i-1\). Since \(0\le(1-t_i)^2\le1-t_i\), the resulting +operator has \(L^2(m_i)\)-to-\(L^2(m_{i+1})\) norm at most +\(\sqrt\xi\). The exact recurrence for the centered remainder is +\[ + E_{i+1} + =\frac{L_iE_i-\bigl(\int L_iE_i\,dm_{i+1}\bigr)q_{i+1}} + {\beta_i}. +\] +Using \(\|q_{i+1}\|_2\le2\), we obtain +\begin{equation}\label{kernels:contraction} + \|E_{i+1}\|_2 + \le\frac{3\sqrt\xi}{1-2\xi}\|E_i\|_2. +\end{equation} +The factor is strictly smaller than one. Also \(t_i\le1\) gives +\(\|L_iE_i\|_\infty\le\|E_i\|_2\), and hence +\[ + \|E_{i+1}\|_\infty + \le\frac{3}{1-2\xi}\|E_i\|_2. +\] +Thus the remainders tend to zero in the last displayed norm. +The last-coordinate density in the lemma is +\[ + f_N/c_N=q_N+E_N/c_N. +\] +Equation~\eqref{kernels:core-mass} and the strict inequality +\(M<2\) show that it is eventually bounded by \(2\). +\end{proof} + +\subsection{Application to a nest of annuli} + +Fix the parameters in Proposition~\ref{kernels:average-loss} with +\(\xi<1/64\), and increase the aligned \(R_0\) if necessary. Set +\[ + R_j=R_0\lambda^j,\qquad j\ge0. +\] +Let annulus \(i\) have walls \(R_{i-1}P\) and \(R_iP\), and fill +the inner hole by the fixed core in \(R_0P\). The core is a fixed +finite graph in lattice units. In a unit magnetic sector it +contains the source; all regular annuli carry its period. +Fix the core and the connection on the increasing family of blocks +once and for all, so restrictions agree as the depth \(N\) increases. +A fixed straight connection cut from the source gives such a choice. +Thus the first guard law and its core interface kernel do not vary +with \(N\). + +\begin{corollary}[The last guard law]\label{kernels:last-guard} +For these fixed choices, the last-guard law in a nest of \(N\) +annuli has density at most \(2\) relative to \(m_N^s\) for all +sufficiently large \(N\), in the centered sector and in either +unit-period sector. The conclusion holds for identical lattice +translates of the nest with the same threshold in \(N\). +\end{corollary} + +\begin{proof} +Put \(t_i=T_i^s/d_i\) for regular interfaces and +\(g=T_0^s/d_0\) for the core interface. Lemma~\ref{kernels:bounds} +gives \(00\) is small enough that +these disks are disjoint and contained in \(D\). For affine gradients +with periods \(2\pi\sigma_i\) about the holes, the minimum energy of +Equation~\eqref{eq:source-3} satisfies +\begin{equation} + \mathcal M + =2\pi k\log(1/\rho) + +4\pi^2H_D(\mathbf z,\boldsymbol\sigma)+O(\rho). + \tag{12}\label{eq:source-12} +\end{equation} +The error constant may depend on the separated interior configuration, +but is uniform for all sufficiently small \(\rho\). +\end{lemma} + +\begin{proof} +Set \(U(y)=\sum_i\sigma_iG_D(y,z_i)\), and rotate +\(-2\pi\nabla U\) by a right angle, with the orientation chosen to give +the indicated periods. Denote the resulting vector field by \(v\). +It is a smooth affine gradient on the punctured square, is divergence +free there, and is tangent to the outer boundary because \(U\) has +zero Dirichlet trace. Thus \(v\) is an admissible trial field. + +Near \(z_i\), write +\[ + U(y)=\frac{\sigma_i}{2\pi}\log\frac1{|y-z_i|}+u_i(y), + \qquad + u_i(z_i)=\sigma_iR_D(z_i,z_i) + +\sum_{j\ne i}\sigma_jG_D(z_i,z_j). +\] +The functions \(u_i\) are smooth and harmonic on fixed disjoint +neighborhoods of the sources. On the boundary of a hole, the radial +singularity contributes no normal component to \(v\). Its remaining +normal component is bounded uniformly in \(\rho\) and has integral +zero, since it is a tangential derivative of \(u_i\). + +Let \(g\in H^1\) on the punctured square. Integration by parts, +followed by subtraction of the mean of \(g\) on each inner circle, +gives +\[ + \left|\int v\cdot\nabla g\right| + \leq C\rho\,\|\nabla g\|_2. +\] +Indeed, the \(L^2\) norm of \(v\cdot\nu\) on such a circle is +\(O(\rho^{1/2})\), whereas the trace and Poincar\'e inequalities on +the adjacent annulus of radii \(\rho\) and \(2\rho\) give +\[ + \|g-\text{its circle mean}\|_{L^2(\partial B(z_i,\rho))} + \leq C\rho^{1/2} + \|\nabla g\|_{L^2(B(z_i,2\rho)\setminus B(z_i,\rho))}. +\] +The collars are disjoint. Minimizing +\(\|v+\nabla g\|_2^2\) therefore changes \(\|v\|_2^2\) +by at most \(O(\rho^2)\). + +It remains to calculate +\(\|v\|_2^2=(2\pi)^2\int|\nabla U|^2\). +The outer-boundary contribution to Green's identity vanishes. +On the circle about \(z_i\), with \(\nu\) pointing into the hole, +the integral of \(U\partial_\nu U\) is +\[ + \frac1{2\pi}\log(1/\rho)+\sigma_i u_i(z_i)+O(\rho). +\] +The integral of the normal derivative of the regular harmonic part +is zero; in particular it produces no logarithmically enhanced +error. Summing over the circles gives +Equation~\eqref{eq:source-12}. These integrations may also be +performed with weak Dirichlet data on the outer square. Alternatively, +reflection across its sides and corners, away from the sources, +justifies the boundary calculation directly. +\end{proof} + +We shall apply this calculation to the rectilinear holes required by +the lattice decomposition. Let \(P\) be the polygon chosen in +Section~\ref{sec:annuli}, so that +\[ + B(0,1-\varepsilon)\subset P\subset B(0,1+\varepsilon). +\] +Here and in the next lemma, radii are in the scaled coordinates of +\(D\), rather than in lattice units. + +\begin{lemma}[Moving lattice holes]\label{correlations:exterior-lattice} +Suppose that \(\rho_n\to\rho>0\), where \(\rho\) is sufficiently +small. Around \(x_{i,n}\), cut out the interiors in the array +division with hole shape \(\rho_nP\), using the rounding conventions +of Section~\ref{sec:sectors}. Let \(F_{\rm ext}^s\) be the sector +ratio in the remaining dual graph, with periods +\(2\pi\sigma_i\) about these holes. Then +\begin{equation} + \limsup_n + \left| + \log F_{\rm ext}^s+ + \frac{2\pi k\log(1/\rho)+4\pi^2H_D(\mathbf z,\boldsymbol\sigma)} + {2a} + \right| + \leq C(\varepsilon+\rho). + \tag{13}\label{eq:source-13} +\end{equation} +The constant \(C\) depends on the separated limiting configuration +but not on sufficiently small \(\rho\) or \(\varepsilon\). +The same assertion holds after bounded-step mesh rounding. +\end{lemma} + +\begin{proof} +Bracket the moving holes by fixed, positively separated rectilinear +holes with dyadic coordinates. To obtain strict eventual inclusions, +start with circles of radii \((1-3\varepsilon)\rho\) and +\((1+3\varepsilon)\rho\) about the limiting points, and approximate +them and their centers by still smaller dyadic perturbations. +The perturbations can be chosen after fixing \(\rho\) and +\(\varepsilon\). For all sufficiently large \(n\), the rounded +moving holes lie between these fixed approximations and contain +their respective sources. The outer boundary remains the same square. + +Use one connection before making any of these cuts. Its restrictions +to the three exteriors have the same winding data. The cut comparison +in Equation~\eqref{eq:source-2} therefore brackets their sector ratios. +Theorem~\ref{sectors:positive-radius} applies to each fixed polygonal +exterior. Its continuum minimum energy is in turn bracketed by the +energies for the corresponding circular holes: restriction of an +admissible affine gradient preserves all the remaining periods. +Changing each radius by a factor \(1+O(\varepsilon)\) changes the +logarithmic term of Equation~\eqref{eq:source-12} by +\(O(\varepsilon)\). That equation now proves the claim. + +Every domain used in this argument has positive passages and +separated hole boundaries. Coherent rounding consequently preserves +connectivity and the cycles defining the periods. In particular the +argument uses only fixed-domain limits; it does not require uniform +convergence under arbitrary motion of the lattice marks. +\end{proof} + +\subsection{Identical nests and their exterior attachment} + +Fix the annular parameters \(\varepsilon,\lambda,R_0\) as in +Section~\ref{sec:kernels}, and write +\[ + R_j=R_0\lambda^j,\qquad + N=N(n)=\max\{j:R_j\leq\eta n\}, +\] +where \(\eta>0\) is small. The \(R_j\) are now lattice radii. +Place identical translated nests, from their filled cores through +their \(N\)-th annuli, around all \(x_{i,n}\). +For sufficiently small \(\eta\), these nests have disjoint closures, +are contained in the spin square, and leave a connected exterior. +Use the same parameters and the same \(N\) for a single nest centered +at \(0\), which will represent the observable \(a_n\). +Since \(nx_{i,n}\in\Z^2\), the microscopic cores and all annuli are +exact lattice translates. The restricted connections differ only +by integral gauge transformations; reversing a unit period also +leaves a bare nest ratio unchanged. + +We may pass further along any subsequence so that +\[ + R_N/n\longrightarrow t\in[\eta/\lambda,\eta]. +\] +For fixed \(\eta,\lambda\), this is a positive macroscopic radius. +The following estimate concerns only the joint kernel which attaches +the last annuli to the exterior. Its notation is that of +Lemma~\ref{kernels:bounds}: \(d_{\rm ext}\) is the centered kernel +at zero relative guard data, and \(m_{N,i}^s\) is the cut law of +the last guard in the \(i\)-th annulus. + +\begin{lemma}[Exterior attachment]\label{correlations:attachment} +For every sufficiently small \(\xi>0\), the parameters may be chosen +so that Equation~\eqref{eq:source-11} holds throughout the regular +annuli and +\begin{equation} + \limsup_n\int + \left(1-\frac{T_{\rm ext}^s}{d_{\rm ext}}\right) + \prod_{i=1}^k dm_{N,i}^s + \leq\xi . + \tag{14}\label{eq:source-14} +\end{equation} +This holds both for the shifted and centered calculations and for +the single center nest. The parameters depend on the fixed +separated configuration and on \(\xi\), but not on \(n\). +The errors in Equation~\eqref{eq:source-13} at the radii in this +construction may simultaneously be made at most \(\xi\). +\end{lemma} + +\begin{proof} +For the centered calculation, Equation~\eqref{eq:source-9} bounds +the attachment loss by the sum of the flat-pin interactions in +the incident last annuli. These interactions can be made as small +as desired by the choices in Section~\ref{sec:annuli}. +The bound depends only on these annuli, not on the exterior. + +For the shifted calculation use the exact identity +in Equation~\eqref{eq:source-10}, now for the last annuli together +with their common exterior. Restoring them produces an exterior +whose smaller holes have radius parameter tending to \(t/\lambda\). +Before restoration, the factors are the \(k\) last-annulus sector +ratios and the exterior ratio with holes of radius parameter \(t\). +By Theorem~\ref{sectors:positive-radius}, each annular energy is +\(2\pi\log\lambda+O(\varepsilon)\). By +Equation~\eqref{eq:source-13}, the restored and cut exterior energies +have the same term \(4\pi^2H_D(\mathbf z,\boldsymbol\sigma)\). +The difference between their radius terms is exactly +\(k\,2\pi\log\lambda\). Consequently the logarithm of the +restored sector ratio divided by the product of the cut ratios +has absolute limiting error at most \(C(\varepsilon+\eta)\). +The circular energy brackets make this error uniform over +\(t\in[\eta/\lambda,\eta]\) and over sufficiently large fixed +\(\lambda\). + +Choose \(\varepsilon\) and \(\eta\) small, then choose \(\lambda\) +large enough for the flat-pin interactions, and finally choose +\(R_0\) large enough for the fixed annular geometry. These choices +retain the regular-interface estimate +Equation~\eqref{eq:source-11}. The preceding centered estimate, +the restored-to-cut identity, and the upper bound +\(T_{\rm ext}^s/d_{\rm ext}\leq1\) now give +Equation~\eqref{eq:source-14}, after making the preliminary errors +small enough in terms of \(\xi\). Apply the same choices to the +one-source configuration at \(0\). + +The argument was given after extraction of a limit \(t\). +Every subsequence has such a further subsequence, and the error +bounds are uniform over its possible values. It therefore proves +the stated limiting bounds on the full sequence. All height and +pin limits used here precede variation of the fixed geometric +parameters; no rate uniform in a shrinking physical radius has +been assumed. +\end{proof} + +\begin{proposition}[Center-normalized correlations] +\label{correlations:separated} +For the separated moving marks fixed at the beginning of this section, +\begin{equation} + \lim_n\log\left( + \frac{\E_n^0\exp(i\sum_i\sigma_i\theta_{x_{i,n}})}{a_n^k} + \right) + = + -\frac{H_D(\mathbf z,\boldsymbol\sigma)-kR_D(0,0)} + {2K_*}. + \tag{15}\label{eq:source-15} +\end{equation} +\end{proposition} + +\begin{proof} +Let \(F_{\rm nest}(R_N)\) denote the bare sector ratio of one whole +free nest with a unit source, including its core. It is the same +for all the translated sources, for either sign, and for the +center reference. By Lemma~\ref{kernels:multiplication}, the +partition function of the whole graph is the product of its cut +block partition functions times its gluing-kernel expectation. +After the kernels internal to each nest have been integrated, its +last-guard law has density at most \(2\) relative to \(m_N^s\), +for all sufficiently large \(N\), by +Corollary~\ref{kernels:last-guard}. + +Write \(e_s\) for the expectation of +\(T_{\rm ext}^s/d_{\rm ext}\) under the product of these propagated +last-guard laws, and \(e_0\) for the corresponding centered +expectation. Equation~\eqref{eq:source-14} and the product density +bound give +\[ + 1-2^k\xi-o(1)\leq e_s\leq1 +\] +in both sectors. The centered factor \(d_{\rm ext}\) cancels +between the numerator and denominator, leaving the exact identity +\[ + \E_n^0\exp\left(i\sum_i\sigma_i\theta_{x_{i,n}}\right) + =F_{\rm nest}(R_N)^k F_{\rm ext}^s\,\frac{e_s}{e_0}. +\] +Taking logarithms, with \(\xi\) small at this fixed \(k\), gives +\[ + \log\E_n^0\exp\left(i\sum_i\sigma_i\theta_{x_{i,n}}\right) + =k\log F_{\rm nest}(R_N)+\log F_{\rm ext}^s + +O_k(\xi)+o(1). +\] +The same calculation with the single center source expresses +\(\log a_n\) using this identical nest ratio. Subtracting \(k\) +times that expression cancels the microscopic term exactly. +In Equation~\eqref{eq:source-13}, the common-radius logarithms +also cancel. The remaining coefficient is +\[ + \frac{4\pi^2}{2a}=\frac1{2K_*}, + \qquad a=8\pi,\quad K_*=\frac2\pi. +\] +This proves Equation~\eqref{eq:source-15} up to an error that can +be made arbitrarily small. + +For completeness, the choices of nests may depend on that accuracy. +This is harmless: the quantity on the left is always the same exact +observable. If its asserted convergence failed, a subsequence with +a fixed positive error would, at a sufficiently high chosen accuracy, +have a further subsequence with \(R_N/n\to t\) and a smaller error +by the preceding calculation, a contradiction. +\end{proof} + +\subsection{The deterministic diagonal correction} + +\begin{lemma}[Green diagonal differences] +\label{correlations:green-diagonal} +If \(x_n\in D_n^\circ\) and \(x_n\to z\in D\), then +\begin{equation} + G_n(x_n,x_n)-G_n(0,0) + \longrightarrow R_D(z,z)-R_D(0,0). + \tag{16}\label{eq:source-16} +\end{equation} +These differences are bounded uniformly when \(x_n\) ranges over +a fixed compact subset of \(D\). +\end{lemma} + +\begin{proof} +The killed Green estimate in \cite[Lemma~4.2]{B} gives local uniform +convergence of \(G_n\) to \(G_D\) off the diagonal, with exactly the +unscaled Laplacian convention used here. We explain how to recover +the difference of the diagonals without specifying their common +lattice constant. + +Choose a small fixed box \(Q\) of displacement vectors about \(0\) +such that \(x_n+Q\) and \(Q\) remain in \(D\) for all large \(n\). +For lattice displacement vectors \(v\), set +\[ + h_n(v)=G_n(x_n,x_n+v)-G_n(0,v). +\] +Both kernels have the same unit source at \(v=0\). +Their difference is therefore discrete harmonic throughout \(Q\), +including at \(v=0\). On a coherently rounded boundary of \(Q\), +the off-diagonal convergence identifies its limit as +\[ + G_D(z,z+v)-G_D(0,v) + =R_D(z,z+v)-R_D(0,v). +\] +The discrete maximum principle bounds \(h_n(0)\) between the +minimum and maximum of these boundary values, up to an error +tending to zero. Shrinking \(Q\) and using continuity gives +Equation~\eqref{eq:source-16}. +If uniform boundedness on an interior compact failed, a violating +sequence would have a convergent subsequence of its locations. +The convergence just proved would contradict that violation. +The finitely many remaining lattice sizes cause no difficulty. +\end{proof} + +For \(\sigma\in\{-1,1\}\), define the normalized point fields +\[ + W_n^\sigma(x) + =a_n^{-1} + \exp\left\{\frac{G_n(x,x)-G_n(0,0)}{2K_*}\right\} + e^{i\sigma\theta_x}. +\] +Thus \(S_n(f)=n^{-2}\sum_x f(x)W_n^+(x)\), and changing +the sign conjugates the point field without conjugating a test +function. + +\begin{corollary}[Normalized separated correlations] +\label{correlations:normalized} +For every fixed separated moving configuration as above, +\begin{equation} + \lim_n\E_n^0\prod_{i=1}^kW_n^{\sigma_i}(x_{i,n}) + =\exp\left\{-\frac1{K_*} + \sum_{i0\), independent of \(n,N,j\), such that, +for all sufficiently large \(n\), +\begin{equation} + a_n\geq c\,F_{\rm nest}(R_{N(n)}), + \tag{18}\label{eq:source-18} +\end{equation} +and, for every \(N\geq j\geq0\), +\begin{equation} + F_{\rm nest}(R_N) + \geq c'\lambda^{-(N-j)/2}F_{\rm nest}(R_j). + \tag{19}\label{eq:source-19} +\end{equation} +Here \(j=0\) means the filled core alone. +\end{lemma} + +\begin{proof} +For the single center source, the normalized exterior attachment +has a positive lower bound in the shifted sector and is at most +one in the centered sector. Its exterior sector ratio is also +bounded away from zero by Equation~\eqref{eq:source-13}: +every subsequential hole radius lies in the fixed interval +\([\eta/\lambda,\eta]\). The exact gluing identity therefore +gives Equation~\eqref{eq:source-18}. The compact subsequence +argument makes the lower bound uniform for all sufficiently +large \(n\). + +Next add one regular annulus to a nest. Its sector ratio +multiplies the nest ratio, together with the ratio of the +shifted and centered attachment expectations. Normalize these +two expectations by their common centered zero-data value. +The centered expectation is at most one. Once the nest is +sufficiently long, its last-data density is at most two, so +Equation~\eqref{eq:source-11} makes the shifted expectation at +least \(1-2\xi\). + +By Theorem~\ref{sectors:positive-radius} and the circular annular +energy bracket, the annular sector factor at large lattice scales +is bounded below by +\[ + \lambda^{-1/8}\exp\{-O(\varepsilon)-o(1)\}, + \qquad \frac{\pi}{a}=\frac18. +\] +Include in the fixed parameter choices that \(\lambda\) and then +the lattice threshold are sufficiently large. Each sufficiently +late extension factor is then at least \(\lambda^{-1/2}\). +There are only finitely many earlier extension factors, all +strictly positive. Their losses relative to this lower bound +can be absorbed into a single \(c'>0\), valid for every interval +of extension indices. Multiplication proves +Equation~\eqref{eq:source-19}. +\end{proof} + +\begin{proposition}[Two-point collision bound] +\label{correlations:diagonal} +For every compact \(E\Subset D\), there is \(C_E<\infty\) such that, +for all \(n\geq1\), all \(x,y\in E\cap D_n^\circ\), including \(x=y\), +and all \(\sigma,\tau\in\{-1,1\}\), +\begin{equation} + 0\leq\E_n^0\bigl[W_n^\sigma(x)W_n^\tau(y)\bigr] + \leq C_E\bigl(|x-y|+1/n\bigr)^{-1}. + \tag{20}\label{eq:source-20} +\end{equation} +Expectations of products of any number of the point fields +\(W_n^\sigma\), with arbitrary signs and repeated locations, are +also nonnegative. +\end{proposition} + +\begin{proof} +Write \(r=n|x-y|\). Choose \(c_E>0\) so small that nests of +radius \(R\leq c_Er\), centered at \(x\) and \(y\) in lattice +units, are disjoint and contained in the spin square. Decrease +\(c_E\), if necessary, so that \(c_Er\leq\eta n\) for all +locations under consideration. + +If there is a largest \(j\geq0\) with \(R_j\leq c_Er\), cut +out these two nests. The restriction of the magnetic sector +to each nest has its single unit source. The cut comparison +and the upper bound one for the remaining sector ratio give +\[ + \E_n^0e^{i\sigma\theta_x+i\tau\theta_y} + \leq F_{\rm nest}(R_j)^2. +\] +Equations~\eqref{eq:source-18} and \eqref{eq:source-19} yield +\[ + \frac{\E_n^0e^{i\sigma\theta_x+i\tau\theta_y}}{a_n^2} + \leq C\lambda^{N-j}. +\] +The maximality of \(j\) gives \(R_j>c_Er/\lambda\), while +\(R_N\leq\eta n\). Since \(R_N/R_j=\lambda^{N-j}\), +this is at most \(C_E n/(1+r)\), after increasing the constant. + +If no such \(j\) exists, then \(r0$ and +multiplicity. Since uniform rotation gives $\E X^2=v/2$, the +inequality $1+u\leq e^u$ yields +\begin{equation}\label{fields:ly-laplace} + \E e^{tX}\leq e^{t^2v/4},\qquad t\in\R. +\end{equation} +The variable $Y=\operatorname{Im}(e^{iU}W)$ has the same law as $X$. +If $v>0$, Chernoff's Inequality applied to both signs of $X$ and $Y$ +therefore gives +\[ + \Pbb\{|W|>r\} + \leq \Pbb\{|X|>r/\sqrt2\}+\Pbb\{|Y|>r/\sqrt2\} + \leq4e^{-r^2/(2v)},\qquad r>0. +\] +Integrating this bound against $p r^{p-1}\,dr$ gives +$\E|W|^p\leq2p(2v)^{p/2}\Gamma(p/2)$, which implies +Equation~\eqref{fields:ly-bound}. When $v=0$, the assertion is +immediate. The auxiliary rotation has left $|W|$ unchanged. + +For Equation~\eqref{fields:positive-moments}, orient the edges +arbitrarily and expand each interaction in its absolutely convergent +Fourier series. If an edge has coupling $b\geq0$, its coefficients are +\[ + I_k(b)=\sum_{r=0}^\infty + \frac{(b/2)^{2r+|k|}}{r!(r+|k|)!}\geq0, + \qquad k\in\Z. +\] +At $b=0$ these are interpreted as $I_0(0)=1$ and $I_k(0)=0$ +for $k\ne0$. Integrating each interior angle imposes the divergence +constraint prescribed by the inserted charges on the integer edge +currents. Boundary angles are zero, so they contribute a factor one +and impose no divergence constraint. Each surviving summand is +nonnegative, and the partition function in the denominator is +positive. Repeated vertices simply add their integer charges. +\end{proof} + +\subsection{The continuum circle field} + +We next identify the continuum moments which arise from +Equation~\eqref{eq:source-17}. We state the construction for the +whole range in which the collision bound below is integrable. +The circle-averaging argument follows \cite[Lemma~5.11]{B}; +its use of covariance positivity is related to the Onsager estimate +in \cite[Proposition~3.9]{JSW2020}. We include the bound for unequal +radii, which is needed to compare two regularizations. + +\begin{lemma}[Circle Wick field and its mixed moments] +\label{fields:circle-chaos} +Let $K>1/(4\pi)$ and let $\Phi_D$ be the zero-Dirichlet Gaussian free +field with covariance $G_D=(-\Delta_D)^{-1}$. Where the circle of +radius $\varepsilon$ about $x$ lies in $D$, define +\[ + V_\varepsilon(x)= + \exp\left\{\frac{i\Phi_{D,\varepsilon}(x)}{\sqrt K} + +\frac{\E\Phi_{D,\varepsilon}(x)^2}{2K}\right\}. +\] +There exists a complex random distribution +$V_{K,D}\in H^{-2}_{\mathrm{loc}}(D)$ such that, for every +$f\in C_c^\infty(D;\C)$, +$V_\varepsilon(f)\to V_{K,D}(f)$ in every finite $L^p$. +For every smooth compactly supported cutoff $\chi$, +$\chi V_\varepsilon\to\chi V_{K,D}$ in +$L^2(\Omega;H^{-2})$. +For $m\geq1$, arbitrary signs $\sigma_i\in\{-1,1\}$, and +$f_i\in C_c^\infty(D;\C)$, +\begin{equation}\label{fields:continuum-moments} + \E\prod_{i=1}^m V_{K,D}^{\sigma_i}(f_i) + =\int_{D^m}\prod_{i=1}^m f_i(x_i) + \exp\left\{-\frac1K\sum_{i0$. +Each Fourier coefficient is Cauchy in $L^2$, and +$\sum_{\nu\in\Z^2}(1+|\nu|^2)^{-2}<\infty$. +Dominated convergence in this series proves that +$\chi V_\varepsilon$ is Cauchy in $L^2(\Omega;H^{-2})$ on +the torus. Choose a countable exhaustion by smooth cutoffs +which equal one near the supports of their predecessors. +Their limits are compatible, since multiplication by a smooth +function is continuous on $H^{-2}$. They define an almost surely +$H^{-2}_{\mathrm{loc}}$ distribution $V_{K,D}$, and testing +recovers the scalar limits already constructed. Localization +identifies these torus Sobolev spaces with the Euclidean local +spaces on $D$. This proves all the asserted convergence. +In particular the normalization uses the complete Dirichlet +variance, including its finite spatial part. +\end{proof} + +At $K=K_*=2/\pi$ the parameter $q$ in the proof is $1/4$, +and Lemma~\ref{fields:circle-chaos} constructs precisely the +field $V_*$ in Theorem~\ref{thm:main}. We next prove that its +mixed moments are the limits of the lattice moments. + +\subsection{Uniform bounds and removal of collisions} + +\begin{lemma}[Bounds for smeared lattice fields] +\label{fields:moment-bounds} +For every compact $E\Subset D$, every bounded complex function $f$ +supported in $E$, and every $p\geq2$, +\begin{equation}\label{fields:test-bounds} + \E_n^0|S_n^\sigma(f)|^2\leq C_E\|f\|_\infty^2, + \qquad + \|S_n^\sigma(f)\|_{L^p}\leq C_E\sqrt p\,\|f\|_\infty, +\end{equation} +uniformly in $n$ and $\sigma\in\{-1,1\}$. +If $B\subset E$ is a square of side $d\leq1$, then +\begin{equation}\label{fields:box-bounds} + \E_n^0|S_n^\sigma(1_B)|^2\leq C_E(d+n^{-1})^3, + \qquad + \|S_n^\sigma(1_B)\|_{L^p} + \leq C_E\sqrt p\,(d+n^{-1})^{3/2}. +\end{equation} +\end{lemma} + +\begin{proof} +Write $h=n^{-1}$. For a fixed $x\in B\cap D_n^\circ$, +the shell $jh\leq|x-y|<(j+1)h$ contains at most $C(j+1)$ +lattice points. Equation~\eqref{eq:source-20} thus implies +\[ + h^2\sum_{y\in B\cap D_n^\circ}(|x-y|+h)^{-1} + \leq Ch\sum_{0\leq j\leq C(d/h+1)}1 + \leq C(d+h). +\] +There are at most $C(d/h+1)^2$ choices of $x$, so a second +summation, including its factor $h^2$, proves the first bound +in Equation~\eqref{fields:box-bounds}. Summation over a fixed +compact set gives a uniform bound for its indicator. Positivity +of the opposite-sign two-point expectation then gives, for +arbitrary complex $f$ supported there, +\[ + \E_n^0|S_n^+(f)|^2 + \leq h^4\sum_{x,y\in E\cap D_n^\circ} + |f(x)|\,|f(y)|\, + \E_n^0[W_n^+(x)W_n^-(y)] + \leq C_E\|f\|_\infty^2. +\] + +For a nonnegative weight $w$, the coefficients of the spins in +$S_n^+(w)$ are +\[ + \lambda_x=n^{-2}w(x)a_n^{-1} + \exp\left\{\frac{G_n(x,x)-G_n(0,0)}{2K_*}\right\}\geq0. +\] +Lemma~\ref{fields:lee-yang} applies to these exact coefficients +for each $n$. It proves the second box bound from the first. +For a complex weight, decompose +\[ + f=(\operatorname{Re}f)_+-(\operatorname{Re}f)_- + +i(\operatorname{Im}f)_+-i(\operatorname{Im}f)_-. +\] +Each nonnegative summand is bounded by $\|f\|_\infty$ and +supported in $E$. Minkowski's Inequality and the Lee--Yang +bound prove the second estimate in +Equation~\eqref{fields:test-bounds}. Conjugation proves the +same bounds for the minus field. +\end{proof} + +\begin{proposition}[Convergence of every mixed moment] +\label{fields:moments} +For every $m\geq1$, signs $\sigma_i\in\{-1,1\}$, and +$f_i\in C_c^\infty(D;\C)$, +\begin{equation}\label{fields:lattice-moment-limit} + \lim_{n\to\infty}\E_n^0\prod_{i=1}^m S_n^{\sigma_i}(f_i) + =\E\prod_{i=1}^m V_*^{\sigma_i}(f_i). +\end{equation} +The right side is the integral in +Equation~\eqref{fields:continuum-moments} with $K=K_*$. +\end{proposition} + +\begin{proof} +Let $E\Subset D$ contain all supports. The case $m=1$ +follows from Equation~\eqref{eq:source-17}, uniformly on $E$, +and Riemann summation. Assume $m\geq2$ and expand the moment +as a sum over $(x_1,\ldots,x_m)\in(D_n^\circ)^m$. +All signed correlations in this sum are nonnegative by +Equation~\eqref{fields:positive-moments} and the positivity +of the deterministic normalizers. + +Fix two indices $i\ne j$ and a small $d>0$. Enlarging the squares +of a grid of side $d$ by a fixed factor gives a family +$\mathcal B_d$ of at most $C_Ed^{-2}$ squares, each of side +at most $C d$, such that every pair $x_i,x_j\in E$ with +$|x_i-x_j|0$, choose $R_j<\infty$ so that the sum of the +Markov bounds in Equation~\eqref{fields:sobolev-bound}, +with thresholds $R_j^2$, is less than $\eta$. +With probability at least $1-\eta$, all inequalities +$\|\chi_jS_n\|_{H^{-2}}\leq R_j$ hold simultaneously. +The set of distributions satisfying these inequalities has +relatively compact closure in $H^{-3}_{\mathrm{loc}}(D)$: +compact embedding gives a convergent subsequence for each +localized field, and a diagonal choice gives convergence for +all $j$. The identities +$\chi_j(\chi_{j+1}S_n)=\chi_jS_n$ pass to the limits because +smooth multiplication is continuous on $H^{-3}$. Thus the +limits define one distribution on $D$. This proves tightness. + +For completeness, the topology just used is the Euclidean +local topology in the statement of Theorem~\ref{thm:main}. +On distributions supported in any fixed compact subset of $D$, +the torus Sobolev norms and the corresponding Euclidean +Sobolev norms are equivalent by localization inside its +fundamental square. Moreover every compactly supported +cutoff is supported where some $\chi_j$ equals one. +The countable family of these localized norms therefore +generates exactly $H^{-3}_{\mathrm{loc}}(D)$. + +Every subsequential limit in this space has the test-function +laws in Proposition~\ref{fields:law}, since evaluation on a +smooth compactly supported test is continuous. +These laws determine the distribution-valued law. For +example, the countably many tests $\chi_j e_\nu$ determine +every localized Fourier coefficient; their real and imaginary +parts generate the Borel sigma-field of the compatible +subspace of the countable product of the separable spaces +$H^{-3}$. Lemma~\ref{fields:circle-chaos} places $V_*$ +in this space, so all subsequential limits have its law. +Tightness and this uniqueness prove full-sequence convergence. +\end{proof} + +Propositions~\ref{fields:moments}, \ref{fields:law}, +and~\ref{fields:tightness} complete the proof of +Theorem~\ref{thm:main}. Every lattice field in this argument +uses the same exact center magnetization $a_n$ and the same +Green-function factor fixed in its definition. diff --git a/preprints/The-Critical-Spin-Field-of-the-Planar-XY-Model-October-5-2026/paper.pdf b/preprints/The-Critical-Spin-Field-of-the-Planar-XY-Model-October-5-2026/paper.pdf new file mode 100644 index 0000000000000000000000000000000000000000..798cbeb22f4e0d84e0c0b13573d6f12144f2d168 --- /dev/null +++ b/preprints/The-Critical-Spin-Field-of-the-Planar-XY-Model-October-5-2026/paper.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:486481fbcc54fe62c8d14e4f4a65869facc2a7f8a4c32647649468f91825ac47 +size 577672 diff --git a/preprints/The-Deligne-Drinfeld-conjecture-September-23-2026/README.md b/preprints/The-Deligne-Drinfeld-conjecture-September-23-2026/README.md new file mode 100644 index 0000000000000000000000000000000000000000..9ac5ce274571540bad55372daf4d1c35ce34d079 --- /dev/null +++ b/preprints/The-Deligne-Drinfeld-conjecture-September-23-2026/README.md @@ -0,0 +1,18 @@ +# [The Deligne-Drinfeld conjecture](paper.pdf) + +**Author:** OpenAI + +**Date:** September 23, 2026 + +## Citation + +```bibtex +@misc{OAI:The-Deligne-Drinfeld-conjecture-September-23-2026, + author = {{OpenAI}}, + title = {{The Deligne--Drinfeld conjecture}}, + howpublished = {OpenAI Math Release preprint + \href{https://github.com/openai/math/blob/main/preprints/The-Deligne-Drinfeld-conjecture-September-23-2026/paper.pdf}{OAI:The-Deligne-Drinfeld-conjecture-September-23-2026}}, + year = {2026} +} +``` + diff --git a/preprints/The-Deligne-Drinfeld-conjecture-September-23-2026/build/figures/deletion-block.tex b/preprints/The-Deligne-Drinfeld-conjecture-September-23-2026/build/figures/deletion-block.tex new file mode 100644 index 0000000000000000000000000000000000000000..60c66631c98678d7ae1c30a72af47f280c5d65e3 --- /dev/null +++ b/preprints/The-Deligne-Drinfeld-conjecture-September-23-2026/build/figures/deletion-block.tex @@ -0,0 +1,60 @@ +\begin{figure}[tbp] + \centering + \begin{tikzpicture}[ + x=1cm,y=1cm, + letter/.style={minimum width=.48cm,minimum height=.48cm,inner sep=1pt}, + boundary/.style={letter,draw,line width=.6pt,rounded corners=1pt}, + block/.style={draw=black!55,fill=black!4,rounded corners=2pt, + minimum width=3.25cm,minimum height=.68cm,inner sep=3pt}, + move/.style={-{Stealth[length=2mm]},line width=.65pt}, + note/.style={font=\small,align=center,inner sep=2pt} + ] + \node[note,anchor=west] at (-5.7,1.15) + {A nonempty block and its boundaries}; + \node[boundary] (left) at (-4.7,0) {$e$}; + \node[block] (word) at (-2.45,0) {$p\;0\;\cdots\;p\;0$}; + \node[boundary] (right) at (-.2,0) {$e$}; + \node[letter] (end) at (1.0,0) {$w$}; + \draw (left) -- (word) -- (right) -- (end); + \draw[decorate,decoration={brace,amplitude=4pt,mirror}] + (-4.0,-.48) -- (-.9,-.48) + node[note,midway,below=5pt] {letters in $\{p,0\}$}; + \node[note,align=left,anchor=west] at (1.75,0) + {If the right $e$ is terminal,\\it may be removed.}; + + \draw[move] (-2.45,-1.2) -- (-2.45,-1.85); + \node[boundary] (leftnext) at (-4.7,-2.35) {$e$}; + \node[block] (wordnext) at (-2.45,-2.35) {$p\;0\;\cdots\;p\;0$}; + \node[boundary] (rightnext) at (-.2,-2.35) {$w$}; + \draw (leftnext) -- (wordnext) -- (rightnext); + \node[note,align=left,anchor=west] at (1.75,-2.35) + {The left $e$ is still not terminal:\\it cannot be removed first.}; + + \draw[move] (-2.45,-2.95) -- (-2.45,-3.6) + node[note,midway,right=4pt] {any further deletions}; + \node[note,anchor=west] at (-5.7,-4.0) + {The final surviving block letter, with $q\in\{e,w\}$}; + + \node[boundary] (ep) at (-4.7,-4.8) {$e$}; + \node[letter] (p) at (-3.5,-4.8) {$p$}; + \node[boundary] (qp) at (-2.3,-4.8) {$q$}; + \draw (ep) -- node[above,font=\small] {$1$} (p) + -- node[above,font=\small] {$1$} (qp); + \node[note] at (-3.5,-5.5) {$C$-weight: $1+1=0$}; + + \node[boundary] (ez) at (.6,-4.8) {$e$}; + \node[letter] (zero) at (1.8,-4.8) {$0$}; + \node[boundary] (qz) at (3.0,-4.8) {$q$}; + \draw (ez) -- node[above,font=\small] {$0$} (zero) + -- node[above,font=\small] {$0$} (qz); + \node[note] at (1.8,-5.5) {$B$-weight: $0+0=0$}; + \end{tikzpicture} + \caption{The obstruction in the second summand of the leading pentagon, + after the minimal $A$-count forces every $e$ to be removed by an + $A$-operation. Such an operation removes only a terminal $e$. + While the displayed $p,0$ block is nonempty, its left boundary remains + $e$; removing a terminal right boundary changes $e$ to $w$. + With either right boundary, the final surviving $p$ or $0$ has deletion weight + zero in $\mathbb F_2$. Thus no nonzero deletion sequence empties the block.} + \label{fig:deletion-block} +\end{figure} diff --git a/preprints/The-Deligne-Drinfeld-conjecture-September-23-2026/build/figures/reassociation.tex b/preprints/The-Deligne-Drinfeld-conjecture-September-23-2026/build/figures/reassociation.tex new file mode 100644 index 0000000000000000000000000000000000000000..cff05c907512c41ef307fa0b4f1769a5adc56f15 --- /dev/null +++ b/preprints/The-Deligne-Drinfeld-conjecture-September-23-2026/build/figures/reassociation.tex @@ -0,0 +1,92 @@ +\begin{figure}[!t] + \centering + \begin{tikzpicture}[ + x=1cm,y=1cm, + vertex/.style={minimum width=2.7cm,minimum height=1.9cm,inner sep=0pt}, + tree/.style={line width=.65pt,line cap=round,line join=round}, + reassociate/.style={-{Stealth[length=2mm]},line width=.7pt}, + edge value/.style={fill=white,inner sep=3pt,font=\small}, + leaf/.style={font=\scriptsize,inner sep=1pt}, + tree caption/.style={font=\small,inner sep=1pt} + ] + \node[vertex] (A) at (-4.2,0) {}; + \node[vertex] (D) at (0,2.4) {}; + \node[vertex] (E) at (4.2,0) {}; + \node[vertex] (B) at (-2.7,-3) {}; + \node[vertex] (C) at (2.7,-3) {}; + + \draw[reassociate] (A) -- node[edge value,above left] {$p_1$} (D); + \draw[reassociate] (D) -- node[edge value,above right] {$p_2$} (E); + \draw[reassociate] (A) -- node[edge value,left] {$p_3$} (B); + \draw[reassociate] (B) -- node[edge value,above] {$p_4$} (C); + \draw[reassociate] (C) -- node[edge value,right] {$p_5$} (E); + + \begin{scope}[shift={(A)}] + \draw[tree] (0,.75) -- (-.3,.4) -- (-.6,.05) -- (-.9,-.3) + (-.6,.05) -- (-.3,-.3) (-.3,.4) -- (.3,-.3) + (0,.75) -- (.9,-.3); + \foreach \x/\i in {-.9/1,-.3/2,.3/3,.9/4} + \node[leaf,below] at (\x,-.3) {$\i$}; + \node[tree caption] at (0,-.78) {$\mathsf{A}:\ (((12)3)4)$}; + \end{scope} + \begin{scope}[shift={(D)}] + \draw[tree] (0,.75) -- (-.6,.2) -- (-.9,-.3) + (-.6,.2) -- (-.3,-.3) + (0,.75) -- (.6,.2) -- (.3,-.3) + (.6,.2) -- (.9,-.3); + \foreach \x/\i in {-.9/1,-.3/2,.3/3,.9/4} + \node[leaf,below] at (\x,-.3) {$\i$}; + \node[tree caption] at (0,-.78) {$\mathsf{D}:\ ((12)(34))$}; + \end{scope} + \begin{scope}[shift={(E)}] + \draw[tree] (0,.75) -- (-.9,-.3) + (0,.75) -- (.3,.4) -- (-.3,-.3) + (.3,.4) -- (.6,.05) -- (.3,-.3) + (.6,.05) -- (.9,-.3); + \foreach \x/\i in {-.9/1,-.3/2,.3/3,.9/4} + \node[leaf,below] at (\x,-.3) {$\i$}; + \node[tree caption] at (0,-.78) {$\mathsf{E}:\ (1(2(34)))$}; + \end{scope} + \begin{scope}[shift={(B)}] + \draw[tree] (0,.75) -- (-.3,.4) -- (-.9,-.3) + (-.3,.4) -- (0,.05) -- (-.3,-.3) + (0,.05) -- (.3,-.3) (0,.75) -- (.9,-.3); + \foreach \x/\i in {-.9/1,-.3/2,.3/3,.9/4} + \node[leaf,below] at (\x,-.3) {$\i$}; + \node[tree caption] at (0,-.78) {$\mathsf{B}:\ ((1(23))4)$}; + \end{scope} + \begin{scope}[shift={(C)}] + \draw[tree] (0,.75) -- (-.9,-.3) + (0,.75) -- (.3,.4) -- (0,.05) -- (-.3,-.3) + (0,.05) -- (.3,-.3) (.3,.4) -- (.9,-.3); + \foreach \x/\i in {-.9/1,-.3/2,.3/3,.9/4} + \node[leaf,below] at (\x,-.3) {$\i$}; + \node[tree caption] at (0,-.78) {$\mathsf{C}:\ (1((23)4))$}; + \end{scope} + \node[align=center,font=\small] at (0,-.55) + {$p_1+p_2=p_3+p_4+p_5$}; + \end{tikzpicture} + + \smallskip + {\small + \begin{tabular}{@{}cl@{\qquad}cl@{}} + \toprule + Label & Infinitesimal value & Label & Infinitesimal value \\ + \midrule + $p_1$ & $\psi(t_{13}+t_{23},t_{34})$ + & $p_3$ & $\psi(t_{12},t_{23})$ \\ + $p_2$ & $\psi(t_{12},t_{23}+t_{24})$ + & $p_4$ & $\psi(t_{12}+t_{13},t_{24}+t_{34})$ \\ + & & $p_5$ & $\psi(t_{23},t_{34})$ \\ + \bottomrule + \end{tabular}} + \caption{The five parenthesizations of four ordered leaves. Each arrow + reassociates $((IJ)K)$ to $(I(JK))$, with the appropriate fixed subtrees + inserted. The labels $p_1,\ldots,p_5$ denote the infinitesimal values + in the table, not the arrows themselves. + The two paths $\mathsf{A}\to\mathsf{D}\to\mathsf{E}$ and + $\mathsf{A}\to\mathsf{B}\to\mathsf{C}\to\mathsf{E}$ have the same unit + arrow as composite. Their infinitesimal values add, yielding the pentagon + identity.} + \label{fig:reassociation} +\end{figure} diff --git a/preprints/The-Deligne-Drinfeld-conjecture-September-23-2026/build/paper.tex b/preprints/The-Deligne-Drinfeld-conjecture-September-23-2026/build/paper.tex new file mode 100644 index 0000000000000000000000000000000000000000..f17500a86d4cf7e344e8f1c2c193223ede60c56a --- /dev/null +++ b/preprints/The-Deligne-Drinfeld-conjecture-September-23-2026/build/paper.tex @@ -0,0 +1,21 @@ +\documentclass[11pt]{article} +\input{preamble.tex} +\title{The Deligne--Drinfeld conjecture} +\author{OpenAI} +\date{September 23, 2026} +\begin{document} +\maketitle +\begin{abstract} +We prove the Deligne--Drinfeld conjecture: the rational Grothendieck--Teichm\"uller Lie algebra is freely generated by one element in every odd weight at least three, with the corresponding isomorphism after weight completion. +\end{abstract} +\tableofcontents +\input{sections/01-introduction.tex} +\input{sections/02-algebra.tex} +\input{sections/03-degeneration.tex} +\input{sections/04-deletion.tex} +\input{sections/05-image.tex} +\input{sections/06-category.tex} +\input{sections/07-holonomy.tex} +\input{sections/08-induction.tex} +\input{references.tex} +\end{document} diff --git a/preprints/The-Deligne-Drinfeld-conjecture-September-23-2026/build/preamble.tex b/preprints/The-Deligne-Drinfeld-conjecture-September-23-2026/build/preamble.tex new file mode 100644 index 0000000000000000000000000000000000000000..b2814d51d0dc232a8cdc1920f0824d90c92b6f61 --- /dev/null +++ b/preprints/The-Deligne-Drinfeld-conjecture-September-23-2026/build/preamble.tex @@ -0,0 +1,41 @@ +\usepackage[T1]{fontenc} +\usepackage[utf8]{inputenc} +\usepackage{lmodern} +\usepackage[margin=1.03in]{geometry} +\usepackage{amsmath,amssymb,amsthm,mathtools,mathrsfs} +\usepackage{enumitem,booktabs,array,longtable,graphicx} +\usepackage{tikz} +\usetikzlibrary{arrows.meta,positioning,calc,decorations.pathreplacing} +\usepackage{microtype} +\usepackage[hidelinks]{hyperref} +\usepackage{bookmark} +\urlstyle{same} +\hypersetup{ + pdftitle={The Deligne-Drinfeld conjecture}, + pdfauthor={OpenAI}, + pdfsubject={The Deligne-Drinfeld conjecture}, + pdfkeywords={Grothendieck-Teichmuller Lie algebra, Deligne-Drinfeld conjecture, Ihara bracket} +} +\newcommand{\Q}{\mathbb Q} +\newcommand{\Z}{\mathbb Z} +\newcommand{\F}{{\mathbb F_2}} +\newcommand{\CC}{\mathbb C} +\newcommand{\Lie}{\operatorname{Lie}} +\newcommand{\grt}{\mathfrak{grt}_1} +\DeclareMathOperator{\ad}{ad} +\DeclareMathOperator{\wt}{wt} +\DeclareMathOperator{\depth}{depth} +\DeclareMathOperator{\gr}{gr} +\newcommand{\id}{\mathrm{id}} +\newtheorem{theorem}{Theorem}[section] +\newtheorem{proposition}[theorem]{Proposition} +\newtheorem{lemma}[theorem]{Lemma} +\newtheorem{corollary}[theorem]{Corollary} +\theoremstyle{definition} +\newtheorem{definition}[theorem]{Definition} +\theoremstyle{remark} +\newtheorem{remark}[theorem]{Remark} +\numberwithin{equation}{section} +\setcounter{tocdepth}{1} +\setlist{topsep=4pt,itemsep=2pt,parsep=0pt} +\setlength{\emergencystretch}{1em} diff --git a/preprints/The-Deligne-Drinfeld-conjecture-September-23-2026/build/references.tex b/preprints/The-Deligne-Drinfeld-conjecture-September-23-2026/build/references.tex new file mode 100644 index 0000000000000000000000000000000000000000..4f2fdc5cf33232f73272b3944a8b95687d782f4e --- /dev/null +++ b/preprints/The-Deligne-Drinfeld-conjecture-September-23-2026/build/references.tex @@ -0,0 +1,127 @@ +\begin{thebibliography}{99} + +\bibitem{Arnold} +V.~I.~Arnol'd, +\emph{The cohomology ring of the colored braid group}, +Mathematical Notes \textbf{5} (1969), no.~2, 138--140; +Russian original: Matematicheskie Zametki \textbf{5} (1969), no.~2, 227--231. +\href{https://doi.org/10.1007/BF01098313}{doi:10.1007/BF01098313}. + +\bibitem{BarNatan} +D.~Bar-Natan, +\emph{On associators and the Grothendieck--Teichm\"uller group I}, +Selecta Mathematica (N.S.) \textbf{4} (1998), no.~2, 183--212. +Preprint version cited: +\href{https://arxiv.org/abs/q-alg/9606021v4}{arXiv:q-alg/9606021v4}. + +\bibitem{Brown} +F.~Brown, +\emph{Mixed Tate motives over \(\mathbb{Z}\)}, +Annals of Mathematics (2) \textbf{175} (2012), no.~2, 949--976. +\href{https://doi.org/10.4007/annals.2012.175.2.10}{doi:10.4007/annals.2012.175.2.10}. + +\bibitem{BrownDepth} +F.~Brown, +\emph{Depth-graded motivic multiple zeta values}, +Compositio Mathematica \textbf{157} (2021), no.~3, 529--572. +\href{https://doi.org/10.1112/S0010437X20007654}{doi:10.1112/S0010437X20007654}. +Preprint version cited: +\href{https://arxiv.org/abs/1301.3053v2}{arXiv:1301.3053v2}. + +\bibitem{Deligne} +P.~Deligne, +\emph{Le groupe fondamental de la droite projective moins trois points}, +in \emph{Galois Groups over \(\mathbb Q\)}, +Mathematical Sciences Research Institute Publications, vol.~16, +Springer, New York, 1989, pp.~79--297. +\href{https://publications.ias.edu/sites/default/files/61_LeGroupeFondamentalDroite.pdf}{Author-hosted text}. + +\bibitem{Drinfeld} +V.~G.~Drinfel'd, +\emph{On quasitriangular quasi-Hopf algebras and a group closely +connected with \(\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\)}, +Leningrad Mathematical Journal \textbf{2} (1991), no.~4, 829--860; +Russian original: Algebra i Analiz \textbf{2} (1990), no.~4, 149--181. +\href{https://www.mathnet.ru/eng/aa199}{Journal archive}. + +\bibitem{Enriquez} +B.~Enriquez, +\emph{Quasi-reflection algebras and cyclotomic associators}, +Selecta Mathematica (N.S.) \textbf{13} (2007), no.~3, 391--463. +\href{https://doi.org/10.1007/s00029-007-0048-2}{doi:10.1007/s00029-007-0048-2}. +Preprint version cited: +\href{https://arxiv.org/abs/math/0408035v2}{arXiv:math/0408035v2}. + +\bibitem{Goncharov} +A.~B.~Goncharov, +\emph{Multiple polylogarithms and mixed Tate motives}, +preprint, 2001. +\newline +\href{https://arxiv.org/abs/math/0103059v4}{arXiv:math/0103059v4}. + +\bibitem{Hirose} +M.~Hirose, +\emph{The cyclotomic Grothendieck--Teichm\"uller group and the motivic +Galois group}, +preprint, 2023. +\href{https://arxiv.org/abs/2301.04064v1}{arXiv:2301.04064v1}. + +\bibitem{HiroseSato} +M.~Hirose and N.~Sato, +\emph{Algebraic differential formulas for the shuffle, stuffle and duality +relations of iterated integrals}, +Journal of Algebra \textbf{556} (2020), 363--384. +\href{https://doi.org/10.1016/j.jalgebra.2020.01.032}{\nolinkurl{doi:10.1016/j.jalgebra.2020.01.032}}. +Preprint version cited: +\href{https://arxiv.org/abs/1801.03165v3}{arXiv:1801.03165v3}. + +\bibitem{Kohno} +T.~Kohno, +\emph{Monodromy representations of braid groups and Yang--Baxter equations}, +Annales de l'Institut Fourier \textbf{37} (1987), no.~4, 139--160. +\href{https://doi.org/10.5802/aif.1114}{doi:10.5802/aif.1114}. + +\bibitem{MelanconReutenauer} +G.~Melan\c{c}on and C.~Reutenauer, +\emph{Lyndon words, free algebras and shuffles}, +Canadian Journal of Mathematics \textbf{41} (1989), no.~4, 577--591. +\href{https://doi.org/10.4153/CJM-1989-025-2}{doi:10.4153/CJM-1989-025-2}. + +\bibitem{NaefWillwacher} +F.~Naef and T.~Willwacher, +\emph{Numerical computation of linearized KV and the Deligne--Drinfeld and +Broadhurst--Kreimer conjectures}, +preprint, 2025. +\href{https://arxiv.org/abs/2508.08081v1}{arXiv:2508.08081v1}. + +\bibitem{Panzer} +E.~Panzer, +\emph{Feynman integrals and hyperlogarithms}, +Ph.D. thesis, Humboldt-Universit\"at zu Berlin, 2015. +Updated version cited: +\href{https://arxiv.org/abs/1506.07243v1}{arXiv:1506.07243v1}. + +\bibitem{Reutenauer} +C.~Reutenauer, +\emph{Free Lie Algebras}, +London Mathematical Society Monographs, New Series, vol.~7, +Clarendon Press, Oxford University Press, 1993. +\newline +\href{https://doi.org/10.1093/oso/9780198536796.001.0001}{\nolinkurl{doi:10.1093/oso/9780198536796.001.0001}}. + +\bibitem{Willwacher} +T.~Willwacher, +\emph{M.~Kontsevich's graph complex and the Grothendieck--Teichm\"uller +Lie algebra}, +Inventiones Mathematicae \textbf{200} (2015), 671--760. +\href{https://doi.org/10.1007/s00222-014-0528-x}{doi:10.1007/s00222-014-0528-x}. +Preprint version cited: +\href{https://arxiv.org/abs/1009.1654v4}{arXiv:1009.1654v4}. + +\bibitem{WillwacherFree} +T.~Willwacher, +\emph{On free generators in the Grothendieck--Teichm\"uller Lie Algebra}, +preprint, 2026. +\href{https://arxiv.org/abs/2607.18793v2}{arXiv:2607.18793v2}. + +\end{thebibliography} diff --git a/preprints/The-Deligne-Drinfeld-conjecture-September-23-2026/build/sections/01-introduction.tex b/preprints/The-Deligne-Drinfeld-conjecture-September-23-2026/build/sections/01-introduction.tex new file mode 100644 index 0000000000000000000000000000000000000000..11c104e7deedd05826e466b6f7f4eea1a69a01b3 --- /dev/null +++ b/preprints/The-Deligne-Drinfeld-conjecture-September-23-2026/build/sections/01-introduction.tex @@ -0,0 +1,204 @@ +\section{Introduction}\label{sec:introduction} + +The Grothendieck--Teichm\"uller Lie algebra records the infinitesimal +symmetries of the associativity and braiding constraints for +parenthesized braids. Its defining equations are short, but they place +conditions in every weight on a free Lie algebra in two letters. +The Deligne--Drinfeld conjecture predicts that these conditions leave +exactly a free Lie algebra with one generator in each odd weight +starting at three. We prove this conjecture, including generation of +the entire solution space. + +\subsection{The statement} +Let \(L=\Lie_{\Q}\langle x,y\rangle\), graded by the total number of +letters. We call this degree the \emph{weight}. For a finite set of +labels, the infinitesimal pure braid Lie algebra \(\mathfrak t_n\) +has generators \(t_{ij}=t_{ji}\), \(i\ne j\), and relations +\begin{equation}\label{eq:braid-relations} + [t_{ij},t_{k\ell}]=0\quad(i,j,k,\ell\text{ distinct}),\qquad + [t_{ij},t_{ik}+t_{jk}]=0\quad(i,j,k\text{ distinct}). +\end{equation} +Every generator has weight one. +Let \(W\subset L\) be the graded vector space of polynomials satisfying +\begin{align} + \psi(x,y)+\psi(y,x)&=0,\label{eq:antisymmetry}\\ + \psi(x,y)+\psi(y,-x-y)+\psi(-x-y,x)&=0.\label{eq:three-term} +\end{align} +\begin{equation}\label{eq:pentagon} +\begin{aligned} + &\psi(t_{12},t_{23}+t_{24})+\psi(t_{13}+t_{23},t_{34})\\ + &\qquad= + \psi(t_{23},t_{34})+\psi(t_{12}+t_{13},t_{24}+t_{34}) + +\psi(t_{12},t_{23}). +\end{aligned} +\end{equation} +The last equation is imposed in \(\mathfrak t_4\). +These are the usual defining equations of \(\grt\); +see \cite[Section~6.1]{Willwacher}. We work first with their polynomial +solution space \(W=\bigoplus_n W_n\), where \(W_n\) is its weight-\(n\) +piece. The usual completed algebra is \(\widehat W=\prod_n W_n\). + +For \(\psi\in L\), define the derivation +\begin{equation}\label{eq:ihara-derivation} + D_\psi(x)=0,\qquad D_\psi(y)=[y,\psi]. +\end{equation} +The Ihara bracket, with the convention used throughout this paper, is +\begin{equation}\label{eq:ihara} + \{\psi,\phi\}=D_\psi(\phi)-D_\phi(\psi)+[\psi,\phi]. +\end{equation} + +\begin{theorem}\label{thm:main} +There are homogeneous elements +\(\sigma_{2k+1}\in W_{2k+1}\), one for each \(k\ge1\), for which +\[ + \Lie_{\Q}\langle e_3,e_5,e_7,\ldots\rangle + \longrightarrow (W,\{\, ,\,\}),\qquad + e_{2k+1}\longmapsto\sigma_{2k+1}, +\] +is an isomorphism of graded Lie algebras. +In particular, \(W\) is closed under the bracket +\eqref{eq:ihara}. The induced map on completions by weight is a +continuous graded isomorphism. +\end{theorem} + +The generators in Theorem~\ref{thm:main} are not canonical. +The assertion combines two requirements: there are no additional Lie +relations among them, and they span every solution of +\eqref{eq:antisymmetry}--\eqref{eq:pentagon}. + +\subsection{Context and the main estimate} +The arithmetic study of the fundamental group of the projective line +minus three points provides a central setting for these symmetry +questions; Deligne developed its motivic and tangential-basepoint +framework in \cite[Introduction and Section~15]{Deligne}. +Drinfeld's associators express compatibility between reassociation and +braiding, and their symmetries led to the +Grothendieck--Teichm\"uller group and its Lie algebra +\cite{Drinfeld}. The equations above are the corresponding +infinitesimal compatibility conditions. + +The conjecture has three parts: existence of the proposed odd-weight +elements, freeness of the Lie algebra they generate, and exhaustion of +the equation space. Drinfeld's Proposition~6.3 credits Ihara with the +odd-weight existence result and gives an associator proof. +Brown's work on mixed Tate motives over \(\mathbb Z\) proves the +faithfulness of the motivic action on the fundamental group of the +three-punctured line \cite{Brown}. The graded Lie algebra of the +prounipotent motivic Galois group is free on generators corresponding to weights +\(3,5,7,\ldots\); its inclusion in the associator symmetry algebra +therefore supplies a free Lie subalgebra of \(W\) +\cite[Section~1.4.2]{BrownDepth}. +Willwacher subsequently proved freeness for any homogeneous family of +solutions, one in each odd weight \(2k+1\ge3\), whose corresponding +element has nonzero coefficient of +\(x^{2k}y\) \cite[Theorem~1.2]{WillwacherFree}. +This coefficient lies in the \emph{depth-one} component, consisting of +words with exactly one occurrence of \(y\). + +Freeness leaves open whether there are additional solutions of the +defining equations; see \cite[Section~9]{Willwacher}. +Naef and Willwacher's computations of the linearized +Kashiwara--Vergne algebra, together with the known inclusions, +establish the required equality through weight~29 +\cite[Theorem~5 and Corollary~7]{NaefWillwacher}. +Our main estimate is the all-weight upper bound +\[ + \dim_{\Q}W_n\leq + \dim_{\Q}\Lie_{\Q}\langle e_3,e_5,e_7,\ldots\rangle_n, + \qquad \wt(e_j)=j, +\] +proved in Corollary~\ref{cor:dimension-bound} for the full rational +equation space. The known free subalgebra gives the reverse inequality. +The proof here also obtains that lower bound internally: +Sections~\ref{sec:category}--\ref{sec:induction} construct a rational +subspace of solutions closed under the Ihara bracket, supply its +odd-weight depth-one values, and choose integral generators attaining +the bound. These choices connect the rational construction to the +characteristic-two argument. + +The completed assertion has a graph-theoretic consequence. Over +\(\Q\), Willwacher's isomorphism +\cite[Theorem~1.1]{Willwacher} identifies the zeroth cohomology of +Kontsevich's graph complex with \(\widehat W\). Thus +\[ + H^0(\mathrm{GC}_2)\cong + \widehat{\Lie_{\Q}\langle e_3,e_5,e_7,\ldots\rangle}. +\] +The graph complex and its completion are taken in the conventions of +that source. The superscript zero is a cohomological degree, whereas +the generator indices specify weights; no other cohomological degree +is computed here. + +\subsection{Proof strategy} + +The main step is a degeneration of the pentagon in characteristic two. +Inside the free associative algebra, put +\[ + A=x^2,\qquad C=[x,y],\qquad B=y, +\] +and filter by the number of occurrences of \(B\). +The pentagon for a leading part becomes a relation in the enveloping +algebra of the level-two cyclotomic hyperplane arrangement +\cite[Section~1.4]{Enriquez}, modulo terms of lower count. +The comparison uses a common ordering procedure: corrections +raise the count in the braid algebra and lower it in the arrangement +algebra. Keeping only the common count-preserving terms transfers the +relation without any independence assumption on the target algebra. + +Integral deletion operators on words then detect the leading part +after setting \(A=0\). Their analytic antecedents are the +total-differential formulas for iterated integrals and hyperlogarithms +of Goncharov \cite[Theorem~2.1]{Goncharov} and +Panzer \cite[Lemma~3.3.30]{Panzer}. +The operators used here specialize the integral +algebraic differential operators of Hirose and Sato +\cite[Section~2]{HiroseSato}; their use with cyclotomic braid algebras +and ratios of sums and differences follows Hirose's construction +\cite[Sections~4.3 and~6]{Hirose}. Here the arrangement relations are proved over the +integers before reduction, and the characteristic-two comparison is +proved directly. A shift symmetry +\(B\mapsto B+s(C)\), followed by a Vandermonde argument, confines the +image to the ordinary free Lie algebra on +\[ + \ad_C^kB,\qquad k\ge1. +\] +These letters have exactly the weights predicted by the conjecture. +The ordering comparison and the integral deletion test provide +separate tools for extracting information from filtered braid +relations. + +To attain the bound within the proof, we construct rational solutions with a nonzero +coefficient of \(x^{n-1}y\) in every odd weight \(n\ge3\). +Following Drinfeld's associator proof +\cite[Proposition~6.3]{Drinfeld}, we compare regularized braid holonomy +with the rule obtained by conjugating paths and negating the chord +generators. We implement +this comparison in the parenthesized-chord formalism of Bar-Natan +\cite[Sections~2--3]{BarNatan}, with the required operations and +limits proved explicitly. The logarithm of the comparison, +evaluated on a reassociation arrow, has the required nonzero +coefficient. This uses the nonvanishing of a convergent integral, +and requires no arithmetic independence assertion about its values. + +Finally, we work with saturated lattices over \(\Z_{(2)}\). +The characteristic-two bound applies to reductions of rational +solutions. Independent Hall words in previously chosen generators, +together with a new depth-one value in odd weight, attain the bound +at each step. +Equality forces the next integral generator to occupy the required +leading filtration piece. This proves generation and freeness +simultaneously. + +\subsection{Organization and conventions} +Section~\ref{sec:algebra} establishes the integral algebra and a +special-derivation identity. Sections~\ref{sec:degeneration}--\ref{sec:image} +prove the dimension bound. Sections~\ref{sec:category} +and~\ref{sec:holonomy} construct the odd-weight values. +Section~\ref{sec:induction} completes the induction. + +All Lie algebras in characteristic two are \emph{ordinary} Lie +algebras, with \([u,u]=0\). Their enveloping algebras are ordinary +associative enveloping algebras. We never identify all primitive +elements in characteristic two with ordinary Lie elements. +All completions are by weight, and all infinite constructions are +interpreted first in each finite weight quotient. diff --git a/preprints/The-Deligne-Drinfeld-conjecture-September-23-2026/build/sections/02-algebra.tex b/preprints/The-Deligne-Drinfeld-conjecture-September-23-2026/build/sections/02-algebra.tex new file mode 100644 index 0000000000000000000000000000000000000000..5ededdcd6f8cf5157e331a443573ca15ef805acb --- /dev/null +++ b/preprints/The-Deligne-Drinfeld-conjecture-September-23-2026/build/sections/02-algebra.tex @@ -0,0 +1,422 @@ +\section{Integral Lie algebras and the special identity} +\label{sec:algebra} + +Throughout, a Lie algebra is an ordinary Lie algebra: its bracket is +alternating, including in characteristic two. Put +\(R=\Z_{(2)}=\{a/b\in\Q:b\text{ is odd}\}\), with residue field +\(\F=R/2R\). We will pass from the rational solution space to an +embedded reduction of the same dimension. For this passage we need +torsion-free integral targets for the defining equations, together with +an additional identity proved over \(\Q\) before reduction. + +\begin{lemma}\label{lem:low-weights} +The solution spaces \(W_1\) and \(W_2\) are zero. +\end{lemma} +\begin{proof} +For \(\psi(x,y)=\alpha x+\beta y\), the left side minus the right side of +the pentagon is \(-\alpha t_{12}-\beta t_{34}\). The degree-one braid +generators are linearly independent, so \(\alpha=\beta=0\). +The degree-two free Lie algebra is spanned by \([x,y]\), and the +three-term expression for this polynomial is \(3[x,y]\). Thus it too +has no nonzero rational solution. +\end{proof} + +\begin{samepage} +\begin{lemma}[Ordinary Lie lattices]\label{lem:lie-lattice} +Let \(X\) be a finite alphabet, and let \(K(X)\) be the \(R\)-span of +all bracket monomials in the free associative algebra +\(R\langle X\rangle\). It is the free ordinary Lie algebra over \(R\). +Each multihomogeneous piece is finite free, and its rationalization and +reduction are, respectively, +\[ + K(X)\otimes_R\Q=\Lie_{\Q}\langle X\rangle, + \qquad + K(X)/2K(X)=\Lie_{\F}\langle X\rangle + \ \subset\ \F\langle X\rangle. +\] +In particular, this reduction is injective in associative words, and +free ordinary Lie dimensions in every multidegree agree over \(\Q\) +and \(\F\). The same statements hold for an alphabet with positive +integer weights and finitely many letters in each bounded weight. +\end{lemma} +\end{samepage} +\begin{proof} +We recall the integral free-Lie and PBW facts being used. The +Hall--Lyndon construction gives a basis of the free Lie algebra over +\(\Z\): iterated use of alternation and Jacobi expresses bracket +monomials in the standard Lyndon brackets, and the associative +expansion of each such bracket has its Lyndon word as leading word, +with coefficient one. Distinct leading words give independence over +every coefficient ring. Thus base change to \(R\) identifies the +abstract free Lie algebra with precisely the displayed bracket span. +PBW applies to this free \(R\)-module; in particular its universal +enveloping algebra is \(R\langle X\rangle\). These are the usual +integral versions of the free-Lie basis and PBW theorems +\cite[Theorems~0.2, 0.5, 4.9 and~5.1]{Reutenauer}; +the integral Lyndon and PBW bases are also recalled in +\cite[Section~2]{MelanconReutenauer}. + +The same Lyndon brackets form bases after base change to $\Q$ and +$\F$, with unchanged multidegrees. This proves the asserted +injectivity and equality of dimensions. For a weighted alphabet, +restriction to any bounded weight leaves finitely many letters, +so the same basis argument applies. +\end{proof} + +In particular, ordinary Lie membership is a condition stronger than +being primitive in a characteristic-two enveloping algebra. For +example, \(x^2\) is primitive in \(\F\langle x\rangle\), whereas the +ordinary Lie algebra generated by \(x\) is just \(\F x\). We will +always retain ordinary Lie membership when using associative words. + +For a coefficient ring \(k\in\{R,\Q,\F\}\), write +\(\mathfrak t_q(k)\) for the infinitesimal braid Lie algebra on +\(q\) strands, with the presentation in \eqref{eq:braid-relations}. + +\begin{lemma}[The free fiber]\label{lem:free-fiber} +The map \(\mathfrak t_q(k)\longrightarrow\mathfrak t_{q-1}(k)\) +which forgets the last strand is split, and its kernel is the free +ordinary Lie algebra on \(s_i=t_{iq}\), \(1\leq i2\). In the free Lie algebra on +\(a,b\), with \(c=-a-b\), one has +\begin{equation}\label{eq:special} + [a,\psi(b,a)]+[c,\psi(b,c)]=0. +\end{equation} +In addition, for an extra free letter \(T\), +\begin{equation}\label{eq:special-linearization} + \partial_2\psi(a,-a)T=0, +\end{equation} +where \(\partial_2\) means the coefficient of a central parameter +\(t\) in \(\psi(a,-a+tT)\). +\end{lemma} +\begin{proof} +All calculations in this proof take place over \(\Q\). We first +rewrite the pentagon in a free kernel, then linearize at \(a+b+c=0\). +Equality of mixed derivatives will force the possible linearization +coefficient to vanish. + +\emph{The free-fiber identity.} +Forget strand \(2\) in \(\mathfrak t_4(\Q)\), and put +\[ + a=t_{12},\qquad b=t_{23},\qquad c=t_{24},\qquad + u=t_{13},\qquad v=t_{34}. +\] +By Lemma~\ref{lem:free-fiber}, the kernel is the actual free Lie +algebra on \(a,b,c\). Define +\begin{align*} + F_a&=\psi(u+b,v)-\psi(u,v),\\ + F_b&=\psi(u+a,v+c)-\psi(u,v),\\ + F_c&=\psi(u,v+b)-\psi(u,v). +\end{align*} +Forgetting the fiber sends each displayed difference to zero, so +\(F_a,F_b,F_c\) are elements of this free kernel. They are +homogeneous of degree \(n\). + +Each of the three expressions before subtracting \(\psi(u,v)\) +commutes with its corresponding letter. For example, +\([a,u+b]=[a,v]=0\) and +\([b,u+a]=[b,v+c]=0\); the third case is the same braid relation at +\(c\). The base action fixes \(S=a+b+c\). Summing the resulting +commutators therefore gives +\[ + [a,F_a]+[b,F_b]+[c,F_c]=0. +\] +Set \(\Delta_a=F_b-F_a\) and \(\Delta_c=F_b-F_c\). We obtain the +identity in the free fiber +\begin{equation}\label{eq:special-fiber} + [S,F_b]=[a,\Delta_a]+[c,\Delta_c]. +\end{equation} + +The pentagon gives +\(\Delta_a=\psi(a,b+c)-\psi(a,b)-\psi(b,v)\). +The element \(v+b+c\) is central in the subalgebra on \(v,b,c\). +A Lie polynomial of degree greater than one is unchanged when a +central element is added to an argument: every term in the difference +contains that element in a bracket. Thus \(\psi(b,v)=\psi(b,-b-c)\). +Exchanging strands \(1\) and \(4\) in the same calculation and using +antisymmetry gives the second formula below: +\begin{align} + \Delta_a&=\psi(a,b+c)-\psi(a,b)-\psi(b,-b-c), + \label{eq:special-delta-a}\\ + \Delta_c&=-\psi(c,a+b)+\psi(c,b)+\psi(b,-b-a). + \label{eq:special-delta-c} +\end{align} +Both differences vanish when \(S=0\): the two remaining terms cancel +by antisymmetry and \(\psi(z,-z)=0\) in degree greater than one. + +The ideal generated by \(S\) is stable under the base action, so the +quotient fiber is \(\Lie_{\Q}\langle a,b\rangle\), with +\(c=-a-b\). In this quotient the actions of \(u,v\) are +\(\ad_c,\ad_a\), respectively. For example, +\[ + [u,a]=[a,b]=[c,a],\quad [u,b]=[b,a]=[c,b],\quad [u,c]=0, +\] +and +\([v,a]=0\), \([v,b]=[b,c]=[a,b]\), +\([v,c]=[c,b]=[a,c]\). It follows that \(\psi(u,v)\) acts on +the quotient fiber by \(\ad_{\psi(c,a)}\). Since the expression +defining \(F_b+\psi(u,v)\) commutes with \(b\), we have +\[ + [b,F_b+\psi(c,a)]=0 +\] +in this free Lie quotient. + +The centralizer of \(b\) in the ordinary free Lie algebra on \(a,b\) +is \(\Q b\). To see this directly, embed it in associative words. +For a homogeneous polynomial \(P=\sum_w p_w w\) commuting with +\(b\), comparison in \(bP=Pb\) gives +\(p_{av}=0\) and +\(p_{b^j av}=p_{b^{j-1}avb}\) for every word \(v\) and \(j>0\). +Iterating shows that every word containing \(a\) has coefficient +zero. Hence \(P\) is a polynomial in \(b\). Setting \(a=0\) now shows that a +Lie element of this form belongs to the one-dimensional ordinary +Lie algebra on \(b\). The homogeneous element at hand has degree +\(n>1\), and is therefore zero. We have proved an equality of +quotient fiber elements, not just of their induced derivations: +\begin{equation}\label{eq:special-fb} + F_b\big|_{S=0}=-\psi(c,a). +\end{equation} + +\emph{The first variation at \(S=0\).} +We have determined the quotient value of \(F_b\). We now differentiate +\eqref{eq:special-fiber} transversely to this quotient and compare the +result with the derivative of the expression in \eqref{eq:special}. +All subsequent derivatives mean coefficient +extraction after adjoining free direction letters and commuting +scalar parameters. Thus +\[ + \partial_1\psi(p,q)T=[t]\psi(p+tT,q),\qquad + \partial_2\psi(p,q)T=[t]\psi(p,q+tT). +\] +The degree-\(n\) Lie polynomials on \(a,T\) containing exactly one +\(T\) and \(n-1\) copies of \(a\) form the one-dimensional space +spanned by \(\ad_a^{n-1}T\): in a nonzero bracket monomial, each +branch not containing \(T\) must be a single \(a\). Consequently +there is a scalar \(k\in\Q\), depending only on \(\psi\), such that +\begin{equation}\label{eq:special-k} + \partial_2\psi(a,-a)T=k\ad_a^{n-1}T. +\end{equation} +Differentiating the identity \(\psi(a,-a)=0\) while varying both +arguments gives +\begin{equation}\label{eq:special-diagonal} + \partial_1\psi(a,-a)T=\partial_2\psi(a,-a)T. +\end{equation} + +Take \(c=S-a-b\), keep \(a,b\) fixed, and extract the coefficient +linear in \(S=tT\). In these same coordinates the two differences +are +\begin{align*} + \Delta_a&=\psi(a,S-a)-\psi(a,b)-\psi(b,a-S),\\ + \Delta_c&=-\psi(S-a-b,a+b)+\psi(S-a-b,b)+\psi(b,-b-a). +\end{align*} +Writing \(c=-a-b\) after differentiation, their derivatives at zero +are exactly +\begin{align*} + d_S\Delta_a(T) + &=k\ad_a^{n-1}T+\partial_2\psi(b,a)T,\\ + d_S\Delta_c(T) + &=-\partial_1\psi(c,-c)T+\partial_1\psi(c,b)T\\ + &=-k\ad_c^{n-1}T-\partial_2\psi(b,c)T. +\end{align*} +The last equality uses \eqref{eq:special-diagonal} and +antisymmetry. In differentiating \eqref{eq:special-fiber}, the left +side becomes \([T,F_b|_{S=0}]\): its other product-rule term has +outside letter \(S=0\). On the right the outside \(a\) is fixed, +and the derivative of the outside \(c\) contributes +\([T,\Delta_c|_{S=0}]=0\). Thus no outside-letter term remains, +and \eqref{eq:special-fb} gives +\begin{equation}\label{eq:special-transverse} + \begin{split} + -[T,\psi(c,a)]={}&k(\ad_a^n-\ad_c^n)T\\ + &+[a,\partial_2\psi(b,a)T] + -[c,\partial_2\psi(b,c)T]. + \end{split} +\end{equation} + +Now define +\(G(a,b)=[a,\psi(b,a)]+[c,\psi(b,c)]\), where \(c=-a-b\). +Hold \(b\) fixed and vary \(a\) in direction \(T\), so that \(c\) +varies in direction \(-T\). The full product rule is +\begin{align*} + d_aG(T)={}&[T,\psi(b,a)-\psi(b,c)]\\ + &+[a,\partial_2\psi(b,a)T] + -[c,\partial_2\psi(b,c)T]. +\end{align*} +The three-term identity and antisymmetry give +\(\psi(b,a)-\psi(b,c)=\psi(c,a)\). Comparing with +\eqref{eq:special-transverse}, we conclude that +\begin{equation}\label{eq:special-dg} + d_aG(T)=-k(\ad_a^n-\ad_c^n)T. +\end{equation} + +\emph{Vanishing of the scalar obstruction.} +Equation~\eqref{eq:special-dg} expresses the derivative of \(G\) in +terms of the single scalar \(k\). To show that this scalar is zero, +introduce two independent constant directions \(U,V\). The +coefficients of their two commuting scalar parameters in \(G\) +are unchanged when the differentiations are interchanged. After +forming these mixed derivatives, specialize +\[ + a=a_0,\qquad b=-a_0,\qquad c=0,\qquad U=a_0,\qquad V=T, +\] +where \(a_0,T\) are free letters. The derivative of \(\ad_c^n\) +vanishes there, since every summand contains \(n-1>0\) copies of +\(\ad_c\). The two derivatives of the other term are +\begin{align*} + d_U(\ad_a^nV)\big|&=n\ad_{a_0}^nT,\\ + d_V(\ad_a^nU)\big| + &=\sum_{j=0}^{n-1} + \ad_{a_0}^{j}\ad_T\ad_{a_0}^{n-1-j}a_0 + =-\ad_{a_0}^{n}T. +\end{align*} +Only \(j=n-1\) survives in the second sum. Equality of the mixed +derivatives in \eqref{eq:special-dg} implies +\[ + (n+1)k\ad_{a_0}^{n}T=0. +\] +The word \(a_0^nT\) has coefficient one in +\(\ad_{a_0}^{n}T\). Since the calculation is over \(\Q\), it +follows that \(k=0\). This proves +\eqref{eq:special-linearization}. + +Equation~\eqref{eq:special-dg} now says that every directional +derivative of \(G\) in \(a\) vanishes. Equivalently, setting the +direction equal to \(a\) multiplies its component with \(j\) +occurrences of \(a\) by \(j\). In characteristic zero every such +component with \(j>0\) is zero. Thus \(G\) is independent of +\(a\), and evaluation at \(a=0,c=-b\) gives \(G=0\). This is +\eqref{eq:special}. +\end{proof} + +\begin{proposition}[Saturated reduction]\label{prop:saturated-reduction} +Let +\[ + K_n=\Lie_R\langle x,y\rangle_n, + \qquad L_n=W_n\cap K_n. +\] +Then \(L_n\) is a saturated finite free \(R\)-submodule of \(K_n\), +and its reduction embeds in \(K_n/2K_n\). Writing its image as +\(\overline L_n\), one has +\begin{equation}\label{eq:saturated-dimension} + \dim_{\F}\overline L_n=\dim_{\Q}W_n. +\end{equation} +More generally, for any rational subspace \(V_n\subset W_n\), the +lattice \(M_n=V_n\cap K_n\) has an embedded reduction +\(\overline M_n\subset\overline L_n\) of dimension +\(\dim_{\Q}V_n\). + +Every element of \(\overline L_n\) satisfies the reductions of the +three defining identities. If \(n>2\), it also satisfies +\eqref{eq:special} and \eqref{eq:special-linearization} after +reduction, together with all formal coefficient consequences of +these identities. In particular, putting \(z=x+y\), +\begin{equation}\label{eq:special-mod-two} + [x,\overline\psi(x,y)+\overline\psi(z,y)] + +[y,\overline\psi(z,y)]=0 + \qquad(\overline\psi\in\overline L_n). +\end{equation} +\end{proposition} +\begin{proof} +An intersection of a rational subspace with a finite free +\(R\)-module is saturated: if \(v\in K_n\) and \(2v\in L_n\), +then \(v\in W_n\), hence \(v\in L_n\). Thus +\(L_n\cap2K_n=2L_n\), which gives the asserted injection after +reduction. It is a finite free module because \(R\) is a principal +ideal domain. Clearing denominators in a rational basis of +\(W_n\) shows that \(\Q L_n=W_n\), so its rank and its reduction +dimension are \(\dim_{\Q}W_n\). Exactly the same argument applies +to \(M_n\); embedding both reductions in \(K_n/2K_n\) proves their +stated inclusion. + +For a representative \(\psi\in L_n\), each defining identity is +an expression with coefficients in \(R\). Its target is either a +free Lie lattice or the braid Lie algebra over \(R\). These targets +embed in their rationalizations by Lemmas~\ref{lem:lie-lattice} and +\ref{lem:free-fiber}. An expression which vanishes over \(\Q\) +therefore already vanishes over \(R\), and can be reduced modulo +two. The identical argument applies to the special identity and +its linearization, which were established over \(\Q\) in +Lemma~\ref{lem:special}. Adjoining free letters and commuting +parameters preserves the word bases; extracting a parameter +coefficient involves no division and commutes with reduction. +This justifies the assertion about formal coefficient consequences. + +Finally, reduce \eqref{eq:special}, take \(a=x,b=y,c=x+y\), and +use the reduced antisymmetry to replace \(\overline\psi(y,x)\) +and \(\overline\psi(y,x+y)\) by +\(\overline\psi(x,y)\) and \(\overline\psi(x+y,y)\). +Expanding the outside bracket with \(x+y\) gives +\eqref{eq:special-mod-two}. +\end{proof} + +In the characteristic-two arguments that follow, the permitted +polynomials are these reductions of rational solutions. No +identification of \(\overline L_n\) with the full solution space of +equations written over \(\F\) is asserted or needed. In particular, +the characteristic-zero mixed-derivative argument proving +\((n+1)k=0\) has already been completed before reduction. diff --git a/preprints/The-Deligne-Drinfeld-conjecture-September-23-2026/build/sections/03-degeneration.tex b/preprints/The-Deligne-Drinfeld-conjecture-September-23-2026/build/sections/03-degeneration.tex new file mode 100644 index 0000000000000000000000000000000000000000..e5239c9f327b200928c605d615851c04a6c86805 --- /dev/null +++ b/preprints/The-Deligne-Drinfeld-conjecture-September-23-2026/build/sections/03-degeneration.tex @@ -0,0 +1,459 @@ +\section{A filtered degeneration of the pentagon} +\label{sec:degeneration} + +We now work over $\F$. The elements to which we apply the +argument are the reductions $\bar\psi\in\overline L_n$ of +Proposition~\ref{prop:saturated-reduction}, with $n>2$. In particular, +they are ordinary Lie polynomials and satisfy the reduced pentagon. +We first introduce a filtration in which a leading part of that pentagon +can be tested in a different Lie algebra. + +\begin{samepage} +\begin{lemma}\label{lem:abc-free} +The assignments +\[ + A\longmapsto x^2,\qquad C\longmapsto[x,y],\qquad B\longmapsto y +\] +define an injective homomorphism +$\F\langle A,C,B\rangle\longrightarrow\F\langle x,y\rangle$. +Its restriction embeds the ordinary free Lie algebra +$\Lie_{\F}\langle A,C,B\rangle$ in the latter associative algebra. +Every homogeneous element of $\Lie_{\F}\langle x,y\rangle$ of weight +greater than one belongs to this embedded Lie algebra. +\end{lemma} +\end{samepage} + +\begin{proof} +Order words of each fixed length lexicographically with $x>y$. +The leading words of $x^2,[x,y],y$ are respectively $xx,xy,y$, each with +coefficient one. These form a prefix code: none is a proper initial +segment of another. Thus their concatenations have unique decodings, +so distinct words in $A,C,B$ have distinct leading words. Split a +putative relation by its total $x,y$ length and take the largest +leading word in each part; its coefficient proves associative +injectivity. Ordinary free Lie algebras embed in their free associative +enveloping algebras, so the Lie assertion follows as well. + +We use the ordinary free-Lie elimination construction +\cite[Theorem~0.6]{Reutenauer}, whose needed instance can be seen directly. +Set $e_j=\ad_x^j y$ for $j\geq0$. The Lie subalgebra generated by the +$e_j$ is stable under $\ad_x$, because $\ad_x(e_j)=e_{j+1}$ and +$\ad_x$ is a derivation. Its sum with the line $\F x$ is therefore a +Lie algebra containing $x,y$, hence the whole free Lie algebra. The +subalgebra is thus an ideal with one-dimensional quotient spanned by +the class of $x$. Every homogeneous Lie polynomial of weight greater +than one consequently lies in this subalgebra. It is itself free on +the $e_j$: their leading words $x^j y$ form a prefix code, which proves +injectivity of the corresponding free associative algebra and hence +of its ordinary free Lie algebra. + +In characteristic two, the associative identity +$\ad_x^2(P)=[x^2,P]$ gives +\begin{equation}\label{eq:elimination-abc} + e_{2i}=\ad_A^i B,\qquad e_{2i+1}=\ad_A^i C\qquad(i\geq0). +\end{equation} +Both expressions belong to the ordinary Lie algebra on the three +abstract letters $A,C,B$, which proves the final assertion. +\end{proof} + +The statement does not assert that $x^2$ is an ordinary Lie polynomial +in $x,y$. Rather, it embeds an ordinary Lie algebra on three new +letters into an associative algebra; the original Lie polynomials of +weight greater than one happen to lie in its image. + +Give $A,C$ weight two and $B$ weight one, and let $F^r$ be the span of +ordinary Lie monomials containing at least $r$ occurrences of $B$. +The multigrading of the free Lie algebra makes this a decreasing +filtration. Through Lemma~\ref{lem:abc-free}, we regard +$\overline L_n$ as a subspace of +$\Lie_{\F}\langle A,C,B\rangle$ and give it the induced filtration +$F^r\overline L_n=\overline L_n\cap F^r$; write +$\operatorname{gr}^r_F\overline L_n= +F^r\overline L_n/F^{r+1}\overline L_n$. +Write $\Psi(A,C,B)$ for the unique polynomial representing +$\bar\psi$ under Lemma~\ref{lem:abc-free}. If $\Psi\in F^r$, let +$\chi(A,C,B)$ be its component containing exactly $r$ letters $B$. +Whenever $\chi\ne0$, there is an integer $m\geq0$ such that +\begin{equation}\label{eq:leading-count-weight} + n=2m+r, +\end{equation} +and every associative word of $\chi$ has $m$ letters from $\{A,C\}$ +and total length $m+r$. The case $m=0$ is zero: an ordinary Lie +polynomial in $B$ alone has only its degree-one component, whereas +$n>2$. Thus a nonzero leading part always has $m>0$, and in particular +$n>r$. We shall later study the ordinary Lie polynomial +$f=\chi(0,C,B)$. This section establishes the relation satisfied by +$\chi$ before that projection. + +We need two algebras with the same named symbols. The source is the +ordinary enveloping algebra of the infinitesimal braid Lie algebra +on strands $0,1,\ldots,d$, over $\F$. In it put +\begin{equation}\label{eq:source-symbols} +\begin{gathered} + X_i=t_{0i},\qquad u_{ij}=t_{ij},\qquad + a_i=X_i^2,\\ + v_{ij}=[X_i,u_{ij}]=[X_j,u_{ij}]=[X_i,X_j]. +\end{gathered} +\end{equation} +The equalities defining $v_{ij}$ follow from the three-strand braid +relations and characteristic two. Here the $a_i$ are associative +squares. + +For the target, let $\mathcal A_d$ be the arrangement in $\Q^d$ with +hyperplanes +\[ + \begin{gathered} + z_i=0\quad(1\leq i\leq d),\\ + z_i-z_j=0,\qquad z_i+z_j=0\quad(1\leq i2$, and suppose its expression +$\Psi(A,C,B)$ lies in $F^r$. Let $\chi$ be its component of count $r$, +and write $n=2m+r$, with $m>0$. In $U(\mathfrak h)$ let +$\mathcal U_{\ell,r$. All remaining replacements preserve the original $B$-count. +Thus the full source pentagon has the form $P+E=0$, where every word +of $E$ has count greater than $r$. + +The distinction between original weight and symbolic length is +essential here. Assign source weight two to each $a,v,H,V$ and +weight one to each $u,U$. A word of length $\ell$ and count $q$ has +original weight +\begin{equation}\label{eq:weight-length-count} + n=2\ell-q. +\end{equation} +All source action formulas preserve this weight. Thus a correction +that raises count by two raises length by one, and every ordered +word of weight $n$ and count $r$ has length +$(n+r)/2=m+r$. In particular $P$ has that length, while the +higher-count expressions are allowed to have greater length. + +Apply the deterministic source ordering to $P+E=0$. Since count +never decreases, $E$ contributes nothing at count $r$. Independence +of the ordered restricted source products, proved above from the +free fibers, implies that every ordered coefficient of count $r$ +in $P$ is zero separately. This conclusion uses independence +across all the restricted words, so expressions of other lengths +or counts cannot cancel a coefficient under consideration. + +Now interpret only the polynomial $P$, of length $m+r$, in the +target. Apply the same deterministic ordering, discarding a branch +as soon as its count falls below $r$. Such a branch can never return +to count $r$. The retained branches use the identical quadratic +formulas that retained count $r$ in the source. Their ordered +coefficient vector is therefore the zero vector just obtained. +Every target replacement preserves length, so every discarded +branch has length $m+r$ and count less than $r$. Their sum belongs +to $\mathcal U_{m+r,2$. +Every word of $\chi$ has $m$ letters from $\{A,C\}$ and $r$ letters +$B$. Proposition~\ref{prop:leading-pentagon}, tested on a word with +$m$ nonzero internal letters and $r$ zeros, is an exact scalar identity: +the lower-count remainder is killed by Lemma~\ref{lem:edge-table}. +We use this exact identity to eliminate every component involving +both $A$ and $C$ from a possible kernel of the projection $A=0$. + +\begin{lemma}\label{lem:deletion-kernel} +If $\chi(0,C,B)=0$, then $\chi$ belongs to the ordinary free Lie +algebra on $A,B$; in particular it has the form $P(A,B)$ with exactly +$m$ letters $A$ and $r$ letters $B$. +\end{lemma} + +\begin{proof} +The associative multihomogeneous components of an ordinary Lie +polynomial are again ordinary Lie polynomials. Since the component +with no $A$ vanishes, suppose that $h>0$ is the least $A$-count of a +nonzero component $\chi_h$. We will rule out $h0$. The third summand vanishes because $h>0$ and its $A$-operator +is zero. + +Consider the second summand. Suppose an $e$ is followed by a nonempty +maximal block consisting of $p$'s and zeros, ending at the next $e$ or +at $w$. Its left $e$ cannot disappear while any of this block remains: +all $e$'s must be removed by $A$-slots, which remove only a terminal +$e$. If the right boundary is an $e$ and that $e$ disappears before +the block does, it must already be terminal, so the new right boundary +is $w$. Thus a surviving nonempty part of the block always has +boundaries $e,e$ or $e,w$. The final surviving letter of this block +cannot be removed. If that letter is $p$, its $C$-weight is $1+1=0$; +if it is $0$, its $B$-weight +is $0+0=0$. Consequently the second summand has no complete deletion +sequence whenever some $e$ occurs before a non-$e$. +Figure~\ref{fig:deletion-block} records the two possible boundaries and +the last-letter obstruction. + +\input{figures/deletion-block.tex} + +The scalar leading pentagon now gives $[s']\chi_h=0$ for every such +$s$. The only remaining words have the form $v e^h$, with $v$ a word +of $p$'s and zeros. Because $m-h>0$, the word $v$ is nonempty. Let +$\operatorname{rev}$ denote reversal of associative words. For every +ordinary homogeneous Lie polynomial $q$ of associative length $d$, +\[ + \operatorname{rev}(q)=(-1)^{d-1}q. +\] +Indeed reversal sends $[q_1,q_2]$ to +$-[\operatorname{rev}(q_1),\operatorname{rev}(q_2)]$, proving the +formula by induction on Lie monomials. In characteristic two it fixes +$\chi_h$. The coefficient of $v e^h$ therefore equals the coefficient +of $e^h\operatorname{rev}(v)$, which has already been shown to vanish. +Every coefficient of $\chi_h$ is zero, a contradiction. + +Thus a least nonzero $A$-count can only be $m$. All $A$-counts are at +most $m$, so only the component using $A,B$ remains. It is an ordinary +Lie polynomial as asserted. +\end{proof} + +\subsection{Excluding the residual kernel} +Deletion has reduced the possible kernel to polynomials $P(A,B)$. +To exclude them, we now use the special identity rather than further +deletion tests. Its lowest-count part compares coefficients obtained +by exchanging an $A$ and a $B$; antisymmetry will handle the remaining +case with only one $A$. + +\begin{proposition}\label{prop:projection-injective} +For $n>2$ and every $r\geq0$, the leading projection +\[ + \operatorname{gr}^{r}\overline L_n\longrightarrow + \Lie_{\F}\langle C,B\rangle,\qquad + \chi\longmapsto\chi(0,C,B) +\] +is injective. Its source is zero unless $n=2m+r$ for an integer +$m\geq0$; in that case its image has $C$-count $m$ and $B$-count $r$. +\end{proposition} + +\begin{proof} +The map is well defined because changing a representative by an +element of $F^{r+1}$ does not change its count-$r$ component. The +parity assertion and the multidegree of its image follow from +$\wt(A)=\wt(C)=2$ and $\wt(B)=1$. If $m=0$, the leading part is an ordinary Lie polynomial +in $B$ alone and is zero in the weights at issue. We may therefore +assume $m>0$. + +Suppose that $\chi(0,C,B)=0$. By Lemma~\ref{lem:deletion-kernel}, +$\chi=P(A,B)$. If $r=0$, this is an ordinary Lie polynomial in $A$ +alone, and hence zero except possibly in weight two. Thus a nonzero +kernel would require $r\geq1$. + +Apply the reduction of Lemma~\ref{lem:special} and antisymmetry with +$z=x+y$. They give +\begin{equation}\label{eq:deletion-special} + [x,\bar\psi(x,y)+\bar\psi(z,y)]+[y,\bar\psi(z,y)]=0. +\end{equation} +Indeed the special identity with $a=x$, $b=y$, $c=z$ becomes +$[x,\bar\psi(y,x)]+[z,\bar\psi(y,z)]=0$, and antisymmetry in +characteristic two interchanges each pair of arguments. + +Under $x\mapsto x+y$, the three associative elements become +\[ + A\mapsto A+C+B^2,\qquad C\mapsto C,\qquad B\mapsto B. +\] +Moreover, on their free associative algebra, +\[ + [x,A]=0,\qquad [x,B]=C,\qquad [x,C]=[A,B]. +\] +The part of $\ad_x$ that lowers $B$-count by one is consequently the +derivation +\[ + D(A)=D(C)=0,\qquad D(B)=C; +\] +its other part raises $B$-count by one. The substitution +$A\mapsto A+C+B^2$ never lowers that count. Therefore components of +$\bar\psi$ with count greater than $r$ cannot contribute to count +$r-1$ in \eqref{eq:deletion-special}, and its outside bracket with $y$ +has count at least $r+1$. Within the leading part, using any $B^2$ +in the substitution also raises the count by two. The count-$r-1$ +component of \eqref{eq:deletion-special} is exactly +\begin{equation}\label{eq:residual-kernel-identity} + D\bigl(P(A+C,B)+P(A,B)\bigr)=0. +\end{equation} + +Write $c(v)$ for the coefficient of a word $v$ with $m$ letters $A$ +and $r$ letters $B$ in $P$. Fix any target word with $m-1$ letters +$A$, $r-1$ letters $B$, and two letters $C$. Its coefficient on the +left of \eqref{eq:residual-kernel-identity} is the sum of exactly two +coefficients of $P$: fill the two marked $C$-positions with $(A,B)$ +or with $(B,A)$. In each case the original $A$ is replaced by $C$ +in the substitution and the original $B$ by $C$ under $D$; these are +the only possibilities. The term $D(P(A,B))$ has only one $C$ and +contributes nothing here. Thus the coefficients of $P$ are invariant +under exchanging any chosen $A$-position and $B$-position. These +exchanges connect all words with the specified multiplicities, so +\begin{equation}\label{eq:constant-residual-coefficients} + c(v)=\lambda\quad\text{for every such word }v +\end{equation} +for one $\lambda\in\F$. + +If $m\geq2$, fix a target word with $m-2$ letters $A$, $r-1$ letters +$B$, and three letters $C$. Exactly one of its three $C$-positions +must have been the $B$ differentiated by $D$; the other two must have +been $A$'s replaced in the substitution. There are exactly three +choices. Its coefficient is therefore $3\lambda=\lambda$, which +\eqref{eq:residual-kernel-identity} forces to vanish. Hence $P=0$. + +It remains to consider $m=1$. The polynomial $P$ has original +$(x,y)$-bidegree $(2,r)$. An $A,C,B$ word with counts $(a,c,b)$ has +original bidegree $(2a+c,c+b)$. Solving for bidegree $(2,r)$ gives +only +\[ + (a,c,b)=(1,0,r)\quad\text{or}\quad(0,2,r-2). +\] +The latter possibility is excluded by the assumption +$\bar\psi\in F^r$, and the former is precisely $P$. Thus if $P\ne0$, +the entire bidegree-$(2,r)$ component of $\bar\psi$ is nonzero. +Symmetry under $x\leftrightarrow y$ forces a nonzero component of +bidegree $(r,2)$. Its $B$-count is at most its $y$-count, namely two, +so $F^r$ forces $r\leq2$. + +For $r=2$, the ordinary free Lie component with one $A$ and two $B$'s +is spanned by +\[ + [B,[A,B]]=ABB+BBA. +\] +To see the spanning assertion directly, every three-letter bracketing +with two identical $B$'s either has the inner bracket $[B,B]=0$, or +is $[B,[A,B]]$ up to sign and skew-symmetry. The coefficients on +$ABB,BAB,BBA$ are therefore $(\mu,0,\mu)$, which are all equal as +required by \eqref{eq:constant-residual-coefficients} only when +$\mu=0$. + +For $r=1$, the forced swapped bidegree is $(1,2)$. The count equations +$2a+c=1$, $c+b=2$ have the unique nonnegative solution +$(a,c,b)=(0,1,1)$. Thus this nonzero component is a multiple of +$[C,B]$, lies in count one, and survives the projection $A=0$. +No higher-filtration component has this bidegree, so it cannot cancel +this projection. This contradicts $\chi(0,C,B)=0$. Both exceptional +cases are excluded, proving injectivity. +\end{proof} diff --git a/preprints/The-Deligne-Drinfeld-conjecture-September-23-2026/build/sections/05-image.tex b/preprints/The-Deligne-Drinfeld-conjecture-September-23-2026/build/sections/05-image.tex new file mode 100644 index 0000000000000000000000000000000000000000..d1e51501ff0e9831ea5df2516ea63eda9165be62 --- /dev/null +++ b/preprints/The-Deligne-Drinfeld-conjecture-September-23-2026/build/sections/05-image.tex @@ -0,0 +1,522 @@ +\section{The image of the leading projection} +\label{sec:image} + +Throughout this section the coefficient field is $\F$. +Recall that $\overline L_n$ denotes the reduction of the saturated +$\Z_{(2)}$-lattice of rational solutions of weight $n$, and that +$F^r$ consists of polynomials containing at least $r$ occurrences of $B$ +in the free associative algebra on +\[ + A=x^2,\qquad C=[x,y],\qquad B=y. +\] +We use the ordinary free Lie algebra on $A,C,B$ inside this associative +algebra, as justified in Lemma~\ref{lem:abc-free}. +For $\psi\in F^r\overline L_n$, let $\chi(A,C,B)$ be its component +with exactly $r$ occurrences of $B$, and put +\[ + f(C,B)=\chi(0,C,B). +\] +If this component is nonzero, then $n=2m+r$, where $m$ is the total +number of $A$'s and $C$'s in $\chi$. +Proposition~\ref{prop:projection-injective} says that this projection +is injective on the corresponding associated graded space. +We shall identify a free Lie algebra containing its image. + +The argument begins with a restriction on products of iterated adjoints, +obtained from symmetry in $x,y$ in Lemma~\ref{lem:even-indices}. +This restriction gives invariance under the polynomial shifts +$B\mapsto B+s(C)$, and a Vandermonde argument then places $f$ in the +ordinary free Lie algebra generated by $\ad_C^kB$, $k\geq1$. +This proves the all-weight dimension bound. We then establish a +separate compatibility: projected leading Ihara brackets become +ordinary brackets. It will provide the independent Hall words in +Section~\ref{sec:induction}. + +\subsection{Symmetry and polynomial shifts} + +\begin{lemma}[The even-index bound] +\label{lem:even-indices} +Suppose that $n>2$, $\psi\in F^r\overline L_n$, and $n=2m+r$. +The component of $\psi$ of original $(x,y)$-bidegree $(m,m+r)$ +is an associative linear combination of products +\begin{equation} + \label{eq:swapped-products} + (\ad_y^{j_1}x)\cdots(\ad_y^{j_m}x) +\end{equation} +in which at least $r$ of the indices $j_1,\ldots,j_m$ are even, +with zero counted as even. +Its image under $A=0$ is exactly $f(C,B)$. +In particular, if $r>m$, then $f=0$. +\end{lemma} + +\begin{proof} +By the elimination argument in Lemma~\ref{lem:abc-free}, every +homogeneous Lie polynomial in $x,y$ of weight greater than one is a +Lie polynomial in $U_j=\ad_x^j y$, $j\geq0$. +In characteristic two these elimination letters are +\[ + U_{2i}=E_i=\ad_A^i B, + \qquad + U_{2i+1}=O_i=\ad_A^i C. +\] +Associative words in the $E_i,O_i$ are linearly independent. +Indeed, order the letters with $A$ largest and use degree followed +by lexicographic order. The leading words of these letters are +$A^iB,A^iC$, respectively, with coefficient one. +They form a prefix code: one reads a string of $A$'s up to its +next $B$ or $C$, thereby determining each successive codeword. +Distinct products consequently have distinct leading words. + +Each $E_i$ is homogeneous of $B$-count one, whereas each $O_i$ +has $B$-count zero. Thus the expansion of $\psi$ in independent +associative words in this alphabet is graded by the number of +$E$-factors. The hypothesis $\psi\in F^r$ forces the coefficients +of all words with fewer than $r$ such factors to vanish separately. +This establishes a termwise restriction before any substitution +or quotient is taken. + +Select the component of original bidegree $(m+r,m)$. +Every $U_j$ has exactly one $y$, so every product in this component +has exactly $m$ factors. The reduced antisymmetry equation gives +$\psi(x,y)=\psi(y,x)$. Swapping $x$ and $y$ therefore gives +the desired expression~\eqref{eq:swapped-products} for bidegree +$(m,m+r)$, with the same lower bound on the number of even indices. +If $r>m$, no such product exists. + +For completeness, an $A,C,B$-word with respective counts $(a,c,b)$ +has original bidegree $(2a+c,c+b)$. In bidegree $(m,m+r)$ this means +\begin{equation} + \label{eq:bidegree-counts} + c=m-2a,\qquad b=r+2a. +\end{equation} +The terms with $a>0$ have higher $B$-count and vanish under $A=0$; +the terms with $a=0$ have $c=m,b=r$ and are precisely $f$. +Thus the even-index bound applies to the whole bidegree component, +including every term subsequently removed by the projection. +\end{proof} + +\begin{lemma}[Polynomial shift invariance] +\label{lem:shift-invariance} +For every $f$ obtained as above and every polynomial $s(C)$, +\begin{equation} + \label{eq:polynomial-shift} + f(C,B+s(C))=f(C,B). +\end{equation} +The identity holds after adjoining any finite collection of +commuting scalar indeterminates to the coefficients; in particular, +it is a polynomial identity in the coefficients of $s$. +\end{lemma} + +\begin{proof} +Let $R$ be any commutative $\F$-algebra, and work in +$S=R\langle C,B\rangle$. Define an $R$-linear derivation $D$ by +\[ + DB=C,\qquad DC=0. +\] +In characteristic two the square of a derivation is again a +derivation: the two middle terms in the twice-applied product rule +cancel. Since $D^2$ vanishes on $B,C$, it vanishes on all of $S$. +Consequently the Ore extension, followed by the indicated quotient, +\[ + \mathcal E=S[X;D]/(X^2),\qquad Xq=qX+Dq\quad(q\in S), +\] +has underlying left $S$-module $S\oplus SX$. +Indeed $X^2q=qX^2+D^2q=qX^2$, so $X^2$ is central in the Ore +extension and its quotient introduces no relation in $S$. +In particular, $S$ embeds in $\mathcal E$. +The substitutions $x\mapsto X$, $y\mapsto B$ give +\[ + x^2\mapsto0,\qquad [x,y]\mapsto C. +\] +Equivalently, $\mathcal E$ acts on $S$ with $X$ acting by $D$ +and elements of $S$ acting by left multiplication. +For an element written $q+q'X$, its value on $1$ is its constant +term $q$. + +Fix $s\in R[C]$, and put $F=B+s(C)$. +The substitution $B\mapsto F$, $C\mapsto C$ commutes with $D$, +since $D(s)=0$. It therefore extends to an automorphism of +$\mathcal E$ fixing $X$. +Let $h(x,y)$ be the whole bidegree-$(m,m+r)$ component in +Lemma~\ref{lem:even-indices}. Evaluation at $x=X,y=B$ kills the +terms containing $A$ in its $A,C,B$ expression, so that +$h(X,B)=f(C,B)$ in the embedded subalgebra $S$ of $\mathcal E$. +Applying the shift automorphism just constructed gives +$h(X,F)=f(C,F)$. We may therefore evaluate the even-index product +expansion of $h$ term by term in $\mathcal E$. Its sum belongs to +$S$, so its value on $1$ is exactly $f(C,F)$. +The factors of positive index become elements of $S$: +\[ + \ad_F^jX=P_j:=\ad_F^{j-1}C\quad(j\geq1), +\] +whereas the index-zero factor is $X$. +We shall bound the $B$-degree of the constant term of each product +separately. + +Write +\[ + L=\ad_B,\qquad Q=\ad_{s(C)},\qquad K=\ad_C, + \qquad P_j=(L+Q)^{j-1}C. +\] +Here $L$ raises $B$-degree by one and $Q$ preserves it. +The relations needed below are +\[ + Q(C)=0,\qquad QK=KQ,\qquad DL=LD+K,\qquad DQ=QD. +\] +For the two smallest positive indices one has +\begin{equation} + \label{eq:small-indices} + P_1=C,\quad DP_1=0,\qquad + P_2=[B,C],\quad DP_2=0. +\end{equation} +For $j\geq2$, every nonzero operator word in $(L+Q)^{j-1}C$ +has rightmost operator $L$, because $Q(C)=0$. +Thus $P_j$ has no $B$-degree-zero component, and its degree-one +component is +\[ + (P_j)_{[1]}=Q^{j-2}LC. +\] +This is killed by $D$, since $DLC=KC=[C,C]=0$. +As $D$ lowers $B$-degree by one, $DP_j$ has no degree-zero +component for any $j\geq1$. + +For $j\geq3$, the degree-two component is explicitly +\[ + (P_j)_{[2]}= + \sum_{a+b=j-3}Q^aLQ^bLC. +\] +Upon applying $D$, differentiating the rightmost $L$ gives zero, +and differentiating the other $L$ gives $Q^aKQ^bLC$. +The commutation of $Q$ and $K$ therefore gives +\begin{equation} + \label{eq:differentiated-linear-part} + (DP_j)_{[1]}=(j-2)Q^{j-3}KLC. +\end{equation} +For even $j$ this is zero in $R$. +Together with~\eqref{eq:small-indices}, these calculations prove +the following lower bounds; the zero polynomial satisfies every +listed bound: +\[ +\begin{array}{c|cc} + & P_j & DP_j\\ \hline + j\geq1\text{ odd} & 0 & 1\\ + j\geq2\text{ even} & 1 & 2 +\end{array} +\qquad\text{(minimum $B$-degree).} +\] +The bound for odd $P_j$ is deliberately weak, so that it includes +$P_1=C$. +All these calculations allow constant terms in $s$ and arbitrary +scalar coefficients in $R$. + +Now consider one product with $q$ index-zero factors and $e$ positive +even-index factors. Move each $X$ to the right using +$XP=PX+DP$, and apply the resulting expression to $1$. +Every surviving term must use each of the original $q$ copies of +$X$ as a differentiation, since a terminal $X$ kills $1$. +Repeated use of the product rule assigns these differentiations +to positive-index factors to their right. An allocation assigning +two differentiations to the same factor vanishes, because $D^2=0$. +Hence every surviving term is a product in which exactly $q$ +distinct positive-index factors have been differentiated once. +An undifferentiated positive even-index factor contributes a lower +bound of one, and an odd-index factor contributes a lower bound +of zero. Differentiating either type increases the respective +lower bound by one. The resulting $B$-degree is therefore at least +\[ + e+q, +\] +which is precisely the number of all even indices, including zero. +If no such allocation is possible the constant term is zero, which +also satisfies the bound. Cancellations among allocations cannot +create lower-degree terms. + +Lemma~\ref{lem:even-indices} now shows that $f(C,B+s(C))$ has no +$B$-degree below $r$. On the other hand, $f$ is homogeneous of +$B$-degree $r$, and replacing $B$ by $B+s(C)$ cannot increase that +degree. Its degree-$r$ component is exactly $f(C,B)$, obtained by +choosing $B$ at every occurrence. This proves +\eqref{eq:polynomial-shift} over $R$, and in particular over any +polynomial ring in scalar indeterminates. +\end{proof} + +\subsection{The ordinary Lie image and the dimension bound} +\begin{proposition}[The image bound] +\label{prop:image-bound} +Set $g_k=\ad_C^kB$ for $k\geq1$. +These elements freely generate an ordinary free Lie algebra inside +$\Lie_{\F}\langle C,B\rangle$. +For $n>2$, the image of the leading projection of +$\operatorname{gr}^r_F\overline L_n$ belongs to its weight-$n$, +length-$r$ component, where $\wt(g_k)=2k+1$ and each $g_k$ has +length one. +The associated graded piece with $r=0$ is zero. +\end{proposition} + +\begin{proof} +First suppose $r\geq1$. +Encode the associative words of $B$-degree $r$ by the vector-space +isomorphism +\[ + \mathcal P_r: + C^{i_0}BC^{i_1}\cdots BC^{i_r} + \longmapsto s_0^{i_0}s_1^{i_1}\cdots s_r^{i_r} + \quad\text{in }\F[s_0,\ldots,s_r]. +\] +Put $P=\mathcal P_r(f)$. +For $1\leq i\leq r$, let $R_iP$ be its restriction obtained by +identifying $s_{i-1}$ and $s_i$, written in the same polynomial +ring $\F[t_0,\ldots,t_{r-1}]$ for every $i$: +\[ + R_iP=P(t_0,\ldots,t_{i-2},t_{i-1},t_{i-1}, + t_i,\ldots,t_{r-1}). +\] +Empty initial or final strings in this formula are omitted. +In the coefficient of $\lambda$ in $f(C,B+\lambda C^j)$, replacing +the $i$th $B$ merges its two neighboring powers of $C$ and adds +$j$ to their exponent. Its encoding with $r-1$ remaining $B$'s +is consequently $t_{i-1}^{j}R_iP$. +Lemma~\ref{lem:shift-invariance}, applied with the independent +scalar $\lambda$, gives +\begin{equation} + \label{eq:vandermonde-system} + \sum_{i=1}^r t_{i-1}^{j}R_iP=0\qquad(j\geq0). +\end{equation} +Taking $j=0,\ldots,r-1$ produces a Vandermonde matrix whose +determinant is +\[ + \prod_{0\leq a2$. +Proposition~\ref{prop:projection-injective} then gives +$\operatorname{gr}^0_F\overline L_n=0$. +The possible boundary $m=0$ also contributes nothing: its leading +component would be an ordinary Lie polynomial in $B$ alone, +whose only possible positive weight is one. +\end{proof} + +Define the ordinary free Lie algebra +\[ + \mathfrak f_{\mathrm{odd}} + =\Lie_{\F}\langle z_k:k\geq1\rangle, + \qquad \wt(z_k)=2k+1, +\] +and give each $z_k$ length one. +Write $\mathfrak f_{\mathrm{odd},n,r}$ for its component of +weight $n$ and length $r$, and set +\begin{equation} + \label{eq:free-dimensions} + d_{n,r}=\dim_{\F}\mathfrak f_{\mathrm{odd},n,r}, + \qquad d_n=\sum_{r\geq0}d_{n,r}. +\end{equation} +These are also the corresponding dimensions over $\Q$, by +the ordinary free Lie lattice of Lemma~\ref{lem:lie-lattice}. +The sum is finite: a length-$r$ word has weight at least $3r$, +and only finitely many generators can occur in any fixed weight. + +\begin{corollary}[The all-weight upper bound] +\label{cor:dimension-bound} +For every $n$ and $r\geq0$ there is an injection +\[ + \operatorname{gr}^r_F\overline L_n + \lhook\joinrel\longrightarrow + \mathfrak f_{\mathrm{odd},n,r}, +\] +obtained by the leading projection and the identification +$g_k\leftrightarrow z_k$. In particular, +\begin{equation} + \label{eq:dimension-upper-bound} + \dim_{\Q}W_n + =\dim_{\F}\overline L_n + \leq\sum_{r\geq0}d_{n,r}=d_n. +\end{equation} +\end{corollary} + +\begin{proof} +For $n>2$, combine Proposition~\ref{prop:projection-injective} +with Proposition~\ref{prop:image-bound}. +If $n-r$ is odd or negative the source piece is zero by the +weights of $A,C,B$, and the target piece is likewise zero since +the weight of a length-$r$ word in odd-weight letters has parity +$r$. The cases $r=0$ and $m=0$ were disposed of in +Proposition~\ref{prop:image-bound}. +Weights one and two have zero solution space by +Lemma~\ref{lem:low-weights}, and have no generators or Lie words +in $\mathfrak f_{\mathrm{odd}}$. +Finally, $F^{n+1}\overline L_n=0$, so this is a finite filtration +in weight $n$ and the dimensions of its associated graded pieces +sum to $\dim_{\F}\overline L_n$. +The equality of this dimension with $\dim_{\Q}W_n$ is +Proposition~\ref{prop:saturated-reduction}. +\end{proof} + +\subsection{Leading Ihara brackets} +The upper bound concerns the full equation space. To attain it in +Section~\ref{sec:induction}, we will construct elements whose projected +leading terms are the $g_k$ and compare their iterated Ihara brackets +with ordinary Lie words. The following compatibility is computed in +the ambient Lie algebra and does not require closure of $W$. + +\begin{proposition}[The leading Ihara bracket] +\label{prop:leading-ihara} +Let $\psi\in F^r\overline L_n$ and +$\phi\in F^s\overline L_{n'}$, where $n,n'>2$ and $r,s\geq1$. +Write $f,g$ for their leading projections at counts $r,s$. +Their Ihara bracket, computed in the ambient Lie algebra in +$x,y$, belongs to $F^{r+s}$ in the $A,C,B$ presentation, and +the projection of its component of count $r+s$ is +\begin{equation} + \label{eq:leading-ihara} + [f,g]. +\end{equation} +This assertion does not require prior closure of the full solution +space under the Ihara bracket. +\end{proposition} + +\begin{proof} +We first prove the filtration assertion before applying $A=0$. +The derivation $\delta=\ad_x$ acts on the free $A,C,B$ algebra by +\[ + \delta A=0,\qquad \delta B=C,\qquad \delta C=[A,B]. +\] +Write $\delta=\delta_-+\delta_+$, where +\[ + \delta_-B=C,\quad \delta_-A=\delta_-C=0, + \qquad + \delta_+C=[A,B],\quad \delta_+A=\delta_+B=0. +\] +These two derivations respectively lower and raise the $B$-count +by one. In particular, $\delta(F^r)\subset F^{r-1}$. +Directly from the definition of the Ihara derivation, +\begin{equation} + \label{eq:ihara-abc-action} + D_\psi A=0,\qquad + D_\psi B=[B,\psi],\qquad + D_\psi C=[C,\psi]+[B,\delta\psi]. +\end{equation} +The last formula is the derivation identity applied to $[x,B]$. +Thus $D_\psi B\in F^{r+1}$ and $D_\psi C\in F^r$: +the outside $B$ in $[B,\delta\psi]$ restores the one count that +$\delta_-$ can remove. Replacing a $B$ in a word of count $s$ +adds at least $r$ to its count, and replacing a $C$ also adds at +least $r$; the derivative of an $A$ vanishes. +It follows that +\[ + D_\psi(F^s)\subset F^{r+s}. +\] +Interchanging $\psi,\phi$ and including their ordinary bracket +proves the asserted containment of the Ihara bracket. + +Let $\chi,\eta$ be the components of $\psi,\phi$ of counts +$r,s$. In~\eqref{eq:ihara-abc-action}, only $\chi$ can contribute +to the part of the derivation that raises count by exactly $r$. +Its action is +\[ + A\longmapsto0,\qquad + B\longmapsto[B,\chi],\qquad + C\longmapsto[C,\chi]+[B,\delta_-\chi]. +\] +Indeed, a higher-count component of $\psi$ still has higher count +after applying $[B,\delta_-(-)]$, while +$[B,\delta_+\chi]$ raises the count by two more. +This derivation preserves the ideal generated by $A$, so it +descends under $A=0$. +The projection commutes with $\delta_-$, whose induced action +on $\F\langle C,B\rangle$ is the derivation $D$ from +Lemma~\ref{lem:shift-invariance}. +The induced leading action is consequently +\[ + B\longmapsto[B,f],\qquad + C\longmapsto[C,f]+[B,Df]. +\] +By Proposition~\ref{prop:image-bound}, $f$ is an ordinary Lie +polynomial in $g_k$, $k\geq1$; each of these is killed by $D$, +because +\[ + Dg_k=\ad_C^k(C)=0\quad(k\geq1). +\] +Thus the extra term $[B,Df]$ vanishes, and the induced derivation +on $\F\langle C,B\rangle$ is $q\mapsto[q,f]$. +One can also read $Df=0$ directly as the coefficient of $\lambda$ +in~\eqref{eq:polynomial-shift} with $s(C)=\lambda C$. + +The two Ihara derivations therefore project to $[g,f]$ and +$[f,g]$, respectively, and the ordinary bracket projects to +$[f,g]$. Their specified combination is +\[ + [g,f]-[f,g]+[f,g]=[g,f]=[f,g] +\] +over $\F$. +All higher components of $\psi$ or $\phi$ have already been +excluded by the filtration calculation, so this proves +\eqref{eq:leading-ihara}. +\end{proof} diff --git a/preprints/The-Deligne-Drinfeld-conjecture-September-23-2026/build/sections/06-category.tex b/preprints/The-Deligne-Drinfeld-conjecture-September-23-2026/build/sections/06-category.tex new file mode 100644 index 0000000000000000000000000000000000000000..dbdb648d08f710567e8f0fc4da1a009909a1e562 --- /dev/null +++ b/preprints/The-Deligne-Drinfeld-conjecture-September-23-2026/build/sections/06-category.tex @@ -0,0 +1,238 @@ +\section{A rational Lie algebra of categorical values}\label{sec:category} + +The upper bound on the full equation space is now established. To +attain it, we construct a rational subspace that is closed under the +Ihara bracket and contains an element with nonzero coefficient of +$x^{n-1}y$ in every odd weight $n\geq3$. This section proves the +closure property; the next constructs the nonzero values. +Only configurations with at most four labels will be needed. +We use the parenthesized-chord formalism of Bar-Natan +\cite[Sections~2--3]{BarNatan}, restricted to the finite arities and +operations specified below. +All completions in this section are by weight, and all tensor products of +completed spaces are completed by total weight. + +We use the standard label sets $[k]=\{1,\ldots,k\}$ for +$1\leq k\leq4$; other label sets denote their relabeled copies. +For such a finite label set $I$, let +$\mathcal T(I)$ be the set of planar binary trees whose leaves are +bijectively labeled by $I$. Thus the objects record both parentheses and +a linear ordering. Put $\mathfrak t_I=0$ for $|I|=1$, and otherwise use +the infinitesimal braid algebra on $I$. Over a field $K$ of +characteristic zero define a $K$-linear category $\mathcal C(I;K)$ by +\[ + \operatorname{Hom}(p,q)=\widehat U(\mathfrak t_I\otimes K) + \qquad(p,q\in\mathcal T(I)). +\] +Composition is multiplication, with the last arrow on the left. Write +$1_{pq}$ for the arrow with coefficient $1$, so that +$1_{qr}1_{pq}=1_{pr}$. Each arrow space carries the coproduct determined +by +\[ + \Delta(t_{ij})=t_{ij}\otimes1+1\otimes t_{ij}, + \qquad \Delta(1_{pq})=1_{pq}\otimes1_{pq}. +\] +In particular, the unit coefficient arrow between different objects is +distinguished from an identity endomorphism. + +Here are all the additional operations that we impose. Relabeling is +allowed along every bijection of label sets. For an outer tree on $I$ +and fixed trees on disjoint nonempty sets $B_i$, graft the latter at its +leaves. On arrows this operation is the algebra homomorphism +\begin{equation}\label{eq:category-block-sum} + t_{ij}\longmapsto t_{B_iB_j}:= + \sum_{a\in B_i,\ b\in B_j}t_{ab}. +\end{equation} +There is also insertion of a varying tree in one fixed leaf of a fixed +outer tree: on arrows this retains each inner chord $t_{ab}$ with its +labels. Both operations are used whenever the resulting number of +leaves is at most four; single-leaf trees permit identity insertions. +These are functors preserving coproducts. Indeed, the disjoint-chord +relations and the three-label relations imply the corresponding +relations between the block sums in +\eqref{eq:category-block-sum}, by summing first over labels in each +block. Inner chords commute with outer block sums: for +$a,b\in B_i$ and $c\notin B_i$ the only possibly nonzero terms are +$[t_{ab},t_{ac}+t_{bc}]=0$. These observations also show that successive +insertions agree with grafting the same trees in one step. + +A \emph{compatible derivation of weight $n$} is a family $\delta$ of +$K$-linear maps on these arrow spaces, raising weight by $n$, such that +\[ + \delta(ba)=\delta(b)a+b\delta(a),\qquad + \Delta\delta=(\delta\otimes\id+\id\otimes\delta)\Delta. +\] +It must commute with all the operations just specified and vanish on +all arrow spaces in arity two. The arity-one derivation is necessarily +zero. Let $\mathcal D_n(K)$ denote this space. Commutators of such +families are again compatible derivations, of the sum of their weights. + +Set +\[ + a=((12)3),\qquad b=(1(23)),\qquad + \alpha=1_{ab},\qquad x=t_{12},\quad y=t_{23}. +\] +Evaluation on $\alpha$ will be understood as its coefficient in +$\widehat U(\mathfrak t_3\otimes K)$. + +\begin{proposition}\label{prop:categorical-values} +For $n\geq2$, evaluation $\delta\mapsto\delta(\alpha)$ takes +$\mathcal D_n(\Q)$ into $W_n$. Its images +\[ + V_n=\{\delta(\alpha):\delta\in\mathcal D_n(\Q)\},\qquad + V=\bigoplus_{n\geq2}V_n, +\] +form a rational Lie subalgebra under the Ihara bracket \eqref{eq:ihara}. +\end{proposition} + +\begin{proof} +Write $\psi=\delta(\alpha)$. The coderivation rule at the group-like +arrow $\alpha$ says +\[ + \Delta\psi=\psi\otimes1+1\otimes\psi. +\] +Primitives in a characteristic-zero enveloping algebra are its Lie +algebra: by the PBW filtration, a primitive of filtration degree +$d>1$ would give a primitive homogeneous polynomial of degree $d$ in a +symmetric algebra, whereas its coproduct has a nonzero component of +bidegree $(1,d-1)$. Induction on filtration degree proves the assertion. +Now +\begin{equation}\label{eq:category-three-split} + \mathfrak t_3=\Lie_K\langle x,y\rangle\oplus K T, + \qquad T=t_{12}+t_{13}+t_{23}, +\end{equation} +with $T$ central: replacing $t_{13}$ by $T-x-y$ transforms exactly the +three defining relations into the centrality of $T$. Thus a primitive +of weight $n\geq2$ is a Lie polynomial $\psi(x,y)$. + +Reversing the two children at any fork is insertion, followed if +necessary by an inner insertion, of a unit arrow in arity two. Its +$\delta$-value is zero. The values of unit coefficient arrows add under +composition, and inverse unit arrows have opposite values. Relabel +$\alpha$ by $1\leftrightarrow3$. Its source $((32)1)$ is carried to $b$ +by fork reversals, and its target $(3(21))$ is carried to $a$ in the same +way. Its value is therefore both $\psi(y,x)$ and $-\psi(x,y)$, giving +\begin{equation}\label{eq:category-antisymmetry} + \psi(x,y)+\psi(y,x)=0. +\end{equation} +For the cyclic relation, use the successive trees +$((12)3),((23)1),((31)2)$. Reassociation on the ordered triples +$(1,2,3),(2,3,1),(3,1,2)$, each followed by a root-fork reversal, +forms a cycle of unit arrows. Consequently +\[ + \psi(t_{12},t_{23})+\psi(t_{23},t_{31})+ + \psi(t_{31},t_{12})=0. +\] +An occurrence of the central $T$ in a Lie monomial of degree greater +than one contributes zero. Substitution of $t_{31}=T-x-y$ gives +\begin{equation}\label{eq:category-cyclic} + \psi(x,y)+\psi(y,-x-y)+\psi(-x-y,x)=0. +\end{equation} + +Figure~\ref{fig:reassociation} shows the five four-leaf trees and +the two paths of reassociation arrows: +\[ +\begin{gathered} + A=(((12)3)4),\quad B=((1(23))4),\quad C=(1((23)4)),\\ + D=((12)(34)),\qquad E=(1(2(34))),\\ + A\longrightarrow D\longrightarrow E, + \qquad A\longrightarrow B\longrightarrow C\longrightarrow E. +\end{gathered} +\] +The two edge values on the first path are +$\psi(t_{13}+t_{23},t_{34})$ and +$\psi(t_{12},t_{23}+t_{24})$. Those on the second are +$\psi(t_{12},t_{23})$, +$\psi(t_{12}+t_{13},t_{24}+t_{34})$, and +$\psi(t_{23},t_{34})$. +Both composites are $1_{AE}$, hence +\begin{equation}\label{eq:category-pentagon} +\begin{split} + \psi(t_{12},t_{23}+t_{24})+ + \psi(t_{13}+t_{23},t_{34}) + ={}&\psi(t_{23},t_{34})\\ + &+\psi(t_{12}+t_{13},t_{24}+t_{34}) + +\psi(t_{12},t_{23}). +\end{split} +\end{equation} +This proves membership in $W_n$ using only the stated operations. + +\input{figures/reassociation.tex} + +It remains to check the bracket, including its sign. Let $d_a,d_b$ +be the loop derivations at $a,b$. Inner insertion gives $d_a(x)=0$ +and $d_b(y)=0$. Differentiating the equality of arrows +$\alpha y_a=y_b\alpha$ gives +\[ + \psi y+d_a(y)=y\psi, + \qquad d_a(y)=[y,\psi]. +\] +Outer arity-two insertion also kills $t_{13}+t_{23}$ at $a$, so +$d_a(T)=0$. In the decomposition +\eqref{eq:category-three-split} its restriction is exactly $D_\psi$. +An arrow from $a$ to $b$ with coefficient $h$ equals $\alpha h_a$; +therefore +\begin{equation}\label{eq:category-arrow-derivative} + \delta_{ab}(h)=\psi h+D_\psi(h) + \qquad(h\in K\langle\!\langle x,y\rangle\!\rangle). +\end{equation} +If another derivation $\varepsilon$ has value $\phi$, their commutator +has value +\[ + [\delta,\varepsilon](\alpha) + =D_\psi(\phi)-D_\phi(\psi)+\psi\phi-\phi\psi + =\{\psi,\phi\}. +\] +Evaluation is thus a Lie homomorphism onto its image. No injectivity +of evaluation is required. +\end{proof} + +\begin{lemma}[Rational finite data]\label{lem:rational-descent} +For every $n>0$ and every characteristic-zero extension $K/\Q$, +\[ + \mathcal D_n(K)=\mathcal D_n(\Q)\otimes_{\Q}K. +\] +For $n\geq2$, the image of evaluation is $V_n\otimes_{\Q}K$. +In particular, if the coefficient of a fixed associative word is +nonzero on a complex categorical value of weight $n$, it is nonzero +on some rational categorical value of that weight. +\end{lemma} + +\begin{proof} +At each arity choose a base object $o$. For a composition derivation +put $h_p=\delta(1_{op})$, with $h_o=0$, and let $d$ be its derivation on +loops at $o$. Since every arrow has the factorization +$1_{oq}\,u_o\,1_{po}$, the product and inverse rules force +\begin{equation}\label{eq:category-finite-data} + \delta_{pq}(u)=h_qu+d(u)-uh_p. +\end{equation} +Conversely, an algebra derivation $d$ and elements $h_p$ define a +composition derivation by this formula. In weight $n$ the data are +\[ + h_p\in U(\mathfrak t_I)_n, + \qquad d(t_{ij})\in U(\mathfrak t_I)_{n+1}. +\] +There are finitely many objects and generators, and every displayed +space is finite dimensional over $\Q$. + +The conditions that $d$ preserve the defining braid relations are +rational linear equations in these data. The coderivation conditions +are equivalent to primitivity of all $h_p$ and $d(t_{ij})$: necessity +follows on the reference unit arrows and generator loops, and +sufficiency follows by the product rule and +\eqref{eq:category-finite-data}. These are again finitely many rational +linear equations, since coproducts in the indicated weights take +values in finite sums of finite-dimensional spaces. + +Finally, there are only finitely many label permutations, object +graftings, and inner placements through arity four. For each associated +functor $\rho$, the condition $\delta\rho=\rho\delta$ need only be +checked on the reference unit arrows and generator loops: all other +arrows follow by composition, the product rule, and continuity. +These checks, and the requirement of zero arity-two action, are +rational linear equations. Thus $\mathcal D_n$ is the kernel of a +linear map between finite-dimensional rational spaces. Kernels and +images of such maps commute with scalar extension. Evaluation and +coefficient extraction are rational linear maps, proving the last +assertion as well. +\end{proof} diff --git a/preprints/The-Deligne-Drinfeld-conjecture-September-23-2026/build/sections/07-holonomy.tex b/preprints/The-Deligne-Drinfeld-conjecture-September-23-2026/build/sections/07-holonomy.tex new file mode 100644 index 0000000000000000000000000000000000000000..f330322b21b980bbac6c521d746cedde2324c253 --- /dev/null +++ b/preprints/The-Deligne-Drinfeld-conjecture-September-23-2026/build/sections/07-holonomy.tex @@ -0,0 +1,553 @@ +\section{Holonomy and odd depth-one values}\label{sec:holonomy} + +By Lemma~\ref{lem:rational-descent}, a complex categorical value with +nonzero coefficient of $x^{n-1}y$ gives a rational value with the same +nonvanishing property. We construct these complex values directly +from a flat connection. +This follows the associator approach to odd-weight elements in +\cite[Proposition~6.3]{Drinfeld}; we include the regularization and +compatibility arguments needed here. The section first constructs +regularized transport (Proposition~\ref{prop:regularized-holonomy}), +then compares the two transport rules through their operator logarithm, +and finally extracts the nonzero coefficient in +Proposition~\ref{prop:odd-depth-one}. + +\subsection{Tree endpoints and regularized transport} +Write $\lambda=(2\pi i)^{-1}$ and +\[ + X_I=\{(z_i)_{i\in I}\in\CC^I:z_i\ne z_j\text{ for }i\ne j\}. +\] +On $X_I$ consider the universal logarithmic braid connection +\cite[Section~1.3]{Kohno} +\begin{equation}\label{eq:holonomy-connection} + \Omega=\lambda\sum_{i0$. +Thus the coordinate $(z_2-z_1)/(z_3-z_1)$ approaches $0$ and $1$ +from inside the real interval, as will be used for reassociation. +Let +\[ + K_d=\sum_{\substack{i\text{ in the left child of }d\\ + j\text{ in the right child of }d}}t_{ij}, + \qquad + E_p(\epsilon)= + \exp\left(\lambda\sum_dK_d\log s_d\right) +\] +for the object $p$. All $K_d$ of a fixed tree commute. Disjoint nodes +are immediate from the disjoint-chord relation. For nested nodes, +any chord within the smaller cluster commutes with the total of the +chords from that cluster to a fixed outside point, since the two +incident terms give $[t_{ij},t_{ik}+t_{jk}]=0$; summing proves the +claim. + +For these chosen collar families, a path between limiting endpoints +means a homotopy class represented by an initial outward collar, a +fixed piecewise smooth path between ordinary configurations, and a +terminal inward collar. Denote their groupoid by $\mathcal P(I)$. +Truncations at different $\epsilon$ are identified by collar segments. +It is an ordinary fundamental groupoid after identifying each limiting +endpoint with a chosen point on its collar. Different admissible +exponent systems will be compared below, rather than identified in +this definition. + +For $\gamma:p\longrightarrow q$, truncate the collars at $\epsilon$ +and write $G(\gamma_\epsilon)$ for ordinary transport. The proposed +regularized rule is +\begin{equation}\label{eq:holonomy-regularized} + H_+(\gamma)=\lim_{\epsilon\to0^+} + E_q(\epsilon)^{-1}G(\gamma_\epsilon)E_p(\epsilon). +\end{equation} +We also consider the rule $H_-$ obtained by conjugating the path and +replacing every chord $t_{ij}$ by $-t_{ij}$; its endpoint frame is +$E_p^{-1}$. Existence of these limits is part of the next proposition. + +The path representatives for the two insertion operations are equally +concrete. For outer insertion, replace an outer coordinate $z_i(s)$ +by the cluster $z_i(s)+\epsilon^M w_{ia}$, where the $w_{ia}$ are fixed +along the outer path at each $\epsilon$ and give the coordinates of +the inserted tree. For inner insertion, fix the outer +anchors and let one such cluster follow a scaled inner path +$\epsilon^M w_{ia}(s)$. Use nested smaller scales within each tree, +and choose the inserted scale sufficiently small relative to every +outer separation. These recipes are used only when the resulting +number of labels is at most four. The proof below establishes that +they define coherent operations on continued path classes, independent +of the sufficiently separated scale choices. + +\begin{proposition}[Regularized transport]\label{prop:regularized-holonomy} +For the tree endpoints and frames just defined, +\eqref{eq:holonomy-regularized} has a limit in every weight. +The limit is independent of the admissible scale exponents under +real interpolation of the endpoint collars. The insertion recipes +define coherent operations on the continued path classes, and the +limit respects relabeling, outer block insertion, and inner insertion. +The same assertions hold for $H_-$. The two transport rules agree +on all arity-two paths, including paths reversing the two real points. +\end{proposition} + +\begin{proof} +\emph{Convergence at each endpoint.} +If $d$ is the lowest common ancestor of $i,j$, formula +\eqref{eq:holonomy-tree-coordinate} gives +\begin{equation}\label{eq:holonomy-collar-difference} + z_i-z_j=\sigma_{ij}\epsilon^{a_d}(1+r_{ij}(\epsilon)), + \qquad \sigma_{ij}\in\{1,-1\}, +\end{equation} +where $r_{ij}$ is a finite signed sum of powers +$\epsilon^{a_e-a_d}$ with positive exponents. Choose a positive lower +bound $\eta$ for all these exponent gaps. Then +\[ + r_{ij}=O(\epsilon^\eta),\qquad + r_{ij}'=O(\epsilon^{\eta-1}),\qquad + \Omega=\lambda\sum_d K_d\,d\log s_d+R(\epsilon)\,d\epsilon, +\] +where every coefficient of $R$ is $O(\epsilon^{\eta-1})$. +The sign $\sigma_{ij}$ contributes no logarithmic differential. + +Here and below estimates are at a fixed weight cutoff $N$, using any +norm on the finite-dimensional quotient by weights greater than $N$. +Changing from $g$ to $E_p^{-1}g$ removes the singular term exactly, +because the $K_d$ commute. The remaining form is +$E_p^{-1}R E_p\,d\epsilon$. In weight at most $N$, conjugation adds +only polynomial powers of $\log\epsilon$, so its norm is bounded by +\begin{equation}\label{eq:holonomy-collar-bound} + C_N\epsilon^{\eta-1}(1+|\log\epsilon|)^N\,d\epsilon. +\end{equation} +Its integral between $0$ and $\epsilon$ is +$O_N(\epsilon^\eta(1+|\log\epsilon|)^N)$. +The finite iterated-integral formula, or the differential equation in +the truncated algebra, therefore proves convergence and gives the +same bound for the difference between collar transport in these +frames and the identity. + +The collar bound proves the limit in +\eqref{eq:holonomy-regularized}; concatenation, homotopy invariance +and group-likeness pass to the limit. Different choices of middle +representatives of a continued homotopy class give the same answer by +flatness. + +\emph{Independence of endpoint scales.} +To check exponent independence, interpolate two systems of exponents +linearly with a parameter $u\in[0,1]$. The inequalities between parent +and child exponents persist, and their finitely many gaps have a +uniform positive lower bound $\eta$. On this interpolation at fixed +$\epsilon$, +\[ + |r_{ij}|\leq C\epsilon^\eta, + \qquad |\partial_u r_{ij}| + \leq C\epsilon^\eta|\log\epsilon|. +\] +Again the singular part is precisely $E^{-1}dE$ before changing frame. +The remaining form on the interpolation has integral norm at most +$C_N\epsilon^\eta(1+|\log\epsilon|)^{N+1}$ after changing frame. +Its transport tends to the identity. Thus identifying endpoints by +these real interpolation paths does not change +\eqref{eq:holonomy-regularized}. + +\emph{The two insertion identities.} +We have constructed transport on $\mathcal P(I)$, independently of +the endpoint exponents. It remains to verify the insertion operations +required of a functor to the chord categories in +Section~\ref{sec:category}. We first compare the connection forms for +each kind of insertion, then check the operations on path classes. +On a truncated +outer path of the above form, there are constants $c,K>0$ such that +all distinct outer coordinates are separated by at least +$c\epsilon^K$. Their total variations are bounded independently of +$\epsilon$, and the integral norms of their logarithmic differences +are $O(1+|\log\epsilon|)$. These statements follow from +\eqref{eq:holonomy-collar-difference} on collars and compactness on +the middle path. + +Insert fixed configurations at offsets of size $O(\epsilon^M)$, +with their own smaller nested scales, and choose $M>2K$, also larger +than all outer scale exponents. Write $D=z_i-z_j$ for an outer +difference and $e$ for the constant difference of the two inserted +offsets. Then +\begin{equation}\label{eq:holonomy-outer-error} + d\log(D+e)-d\log D + =-\frac{e\,dD}{D(D+e)}, + \qquad + \int\left|d\log(D+e)-d\log D\right| + \leq C\epsilon^{M-2K}. +\end{equation} +Internal differences are constant along the outer path. Replacing +all cross-cluster forms by the outer forms gives precisely the +connection obtained from \eqref{eq:holonomy-connection} by +$t_{ij}\mapsto t_{B_iB_j}$. A difference of iterated integrals of +length at most $N$ is expanded by changing one factor at a time. +The error bound \eqref{eq:holonomy-outer-error} and the logarithmic +bounds for the other factors give an error bounded by a positive +power of $\epsilon$ times a power of $1+|\log\epsilon|$. +Multiplication by the two endpoint frames preserves convergence to +zero: their coefficients of weight at most $N$ are polynomials in +$\log\epsilon$. A single bound sufficient for all these errors is +\begin{equation}\label{eq:holonomy-insertion-bound} + C_N\epsilon^\gamma(1+|\log\epsilon|)^{3N},\qquad \gamma>0. +\end{equation} + +The full endpoint frame factors exactly as the outer frame under the +block substitution, times the internal frames of the fixed inserted +trees. The latter frames are the same at both endpoints, and every +inner chord commutes with every outer block sum, as checked in +Section~\ref{sec:category}. They therefore cancel from normalized +transport. This proves compatibility with outer insertion. + +For inner insertion, keep the outer configuration fixed and insert a +varying inner path $\xi$ scaled uniformly by $\epsilon^M$. Choose +$M>K$, where $c\epsilon^K$ bounds the distance between its anchor and +outside points from below. Inner coordinate variations are bounded +as above. Each cross-cluster differential now has integral norm at +most $C\epsilon^{M-K}$; the inner logarithmic differentials are +unchanged by uniform scaling. The same iterated-integral argument +gives \eqref{eq:holonomy-insertion-bound}. There is one additional +frame factor. If $B$ is the inner label set, it is +\begin{equation}\label{eq:holonomy-central-scale} + \exp\bigl(\lambda M\log\epsilon\,T_B\bigr), + \qquad T_B=\sum_{\{i,j\}\subset B}t_{ij} + =\sum_{d\text{ inner}}K_d. +\end{equation} +Indeed scaling multiplies every inner scale by $\epsilon^M$. +The total chord $T_B$ is central in $\mathfrak t_B$, by summing its +three-label relations. All outer frame factors commute with inner +chords as well. These common factors at the two endpoints cancel, +leaving the unscaled inner transport embedded with its original +labels. The argument includes insertion into a single leaf. +For sufficiently small $\epsilon$ all the cabled paths remain in the +configuration space and vary continuously with $\epsilon$; after +collar identification they consequently define a single continued +homotopy class. A homotopy of representatives has a compact middle +part, so the inserted scales can be chosen sufficiently small uniformly +on it; applying the same insertion throughout gives a homotopy of the +cabled paths. Increasing the scale exponent also gives a homotopy +through collision-free paths. Thus the path operations are well +defined on the continued classes and compatible with concatenation, +and the limiting identities are identities for these operations. + +\emph{Coherence of the path operations.} +For truncated outer and inner paths, with parameters $s,t\in[0,1]$ +and the uniform bounds above, use the homotopy square +\[ + F(s,t)_{ia}=z_i(s)+\epsilon^M w_{ia}(t). +\] +If $|z_i(s)-z_j(s)|\ge c\epsilon^K$ and +$|w_{ia}(t)|\le C$, then distinct clusters remain separated by +$c\epsilon^K-2C\epsilon^M>c\epsilon^K/2$ for sufficiently small +$\epsilon$ and $M>2K$; within a cluster, differences are +$\epsilon^M(w_{ia}(t)-w_{ib}(t))\ne0$. Thus the two boundary +routes give the same path class. Nested insertions agree after +flattening, since +$z+\epsilon^M(w+\epsilon^Nv)=z+\epsilon^Mw+\epsilon^{M+N}v$. +The exponent-interpolation estimate identifies other admissible +scale choices with these, with normalized transport tending to the +identity. The commuting frame factors above therefore identify +these operations exactly. For a one-leaf outer tree, restoring +root scale one uses a positive real dilation by $\rho$; its transport +is $\exp(\lambda\log\rho\,T_B)$ and cancels from normalized transport +at the two endpoints by centrality of $T_B$. + +\emph{The second transport rule and arity two.} +For $H_-$, apply the same construction to $\bar\gamma$ with +$t_{ij}$ replaced by $-t_{ij}$ and endpoint frame $E_p^{-1}$. +All estimates and cancellation identities just proved remain valid; +relabeling is immediate for both rules. In arity two the root scale +is $1$, and the difference of the two endpoint coordinates is $1$ or +$-1$. Every path has logarithmic-difference integral $k\pi i$ for +some integer $k$, of the parity determined by the endpoint orders. +Its first holonomy is $\exp(kt_{12}/2)$. Conjugating the path negates +$k$, and negating the chord negates it once again, so the second +holonomy is the same. This checks both full turns and half-turns. +\end{proof} + +\subsection{Comparison of the completed path categories} +The two transport rules now respect exactly the operations imposed in +Section~\ref{sec:category}. To compare them by an automorphism of the +chord category, we first show that each becomes an isomorphism after +completion. +Linearize $\mathcal P(I)$ over $\CC$. +At an object, filter its loop group algebra by powers +of the augmentation ideal; transport this filtration to every arrow +space by any chosen reference path. Changing the reference path +multiplies by a group element and preserves each filtration step. +Write $\widehat{\CC\mathcal P(I)}$ for the resulting completion, with +coproduct $\Delta\gamma=\gamma\otimes\gamma$. + +The following associated-graded comparison is the standard pure-braid +holonomy argument; compare \cite[Proposition~3.6]{BarNatan}. +We give the meridian and completion steps for the path categories just +constructed. + +\begin{lemma}\label{lem:holonomy-isomorphism} +Each of $H_+$ and $H_-$ extends to an isomorphism of the completed +linear path categories with $\mathcal C(I;\CC)$. Both preserve +coproducts and all specified operations, and their associated graded +maps are identical. +\end{lemma} + +\begin{proof} +Fix an object, let $P=\pi_1(X_I)$ and let $I_P$ be the augmentation +ideal in $\CC[P]$. Pair-collision meridians normally generate $P$. +To see this directly, fill a loop by a disk in $\CC^I$ and perturb its +interior to meet the collision hyperplanes transversely in finitely +many smooth points, avoiding their complex-codimension-two +intersections. Removing small disks about these points expresses the +loop as a product of conjugated positive or negative meridians. +Meridians around the same hyperplane are conjugate, since its smooth +part outside the other hyperplanes is path connected. It follows +that their classes generate $I_P/I_P^2$. They are also independent: +the winding numbers of the functions $z_i-z_j$ take value $1$ on the +corresponding positive meridian and $0$ on all the others. + +Every augmentation associated graded algebra is generated by its +degree-one part, since $I_P^r/I_P^{r+1}$ is spanned by products of $r$ +classes from $I_P/I_P^2$. These meridian classes satisfy the +infinitesimal braid relations. Near a disjoint pair of collisions +the two local meridians commute. Near a triple collision, consider +the orbit loop rotating the small three-point cluster simultaneously +through $2\pi$. It commutes with every local meridian: a circle +action carries any such loop through a homotopy whose two boundary +composites are the two orders of these loops. Its degree-one class +is the sum of the three pair-meridian classes, because each of the +three differences winds once and all other differences wind zero +times. The degree-two commutator therefore says +$[t_{ij},t_{ik}+t_{jk}]=0$ (the commutator with $t_{ij}$ itself is +zero). Transporting these local loops to the basepoint only changes +them by conjugation, which changes their classes by terms in $I_P^2$ +and does not affect their degree-two commutators. Hence there is a +surjective graded algebra map +\begin{equation}\label{eq:holonomy-meridian-surjection} + f:U(\mathfrak t_I\otimes\CC) + \longrightarrow\operatorname{gr}_{I_P}\CC[P]. +\end{equation} + +Residues show that $H_+$ of a positive pair meridian is +$1+t_{ij}+\text{terms of weight at least two}$: the integral of the +corresponding logarithmic form is $2\pi i$. For $H_-$ both the turn +and the chord change sign, giving the same result. Endpoint +normalization conjugates loop transport by a series with constant +term $1$ and hence does not change its degree-one term. Thus both +holonomies are filtered and each induces a graded map +\[ + g:\operatorname{gr}_{I_P}\CC[P] + \longrightarrow U(\mathfrak t_I\otimes\CC) + \quad\text{with}\quad g f(t_{ij})=t_{ij}. +\] +Since the chords generate the enveloping algebra, $gf=\id$. +Surjectivity of $f$ now implies that $f$ and $g$ are inverse +isomorphisms. In particular both associated graded holonomies are +the same inverse of \eqref{eq:holonomy-meridian-surjection}. + +An isomorphism on associated graded spaces lifts inductively to an +isomorphism on every finite filtration quotient: lift the constant +term, then correct the error successively in each weight; injectivity +follows by the first nonzero weight. Passing to inverse limits proves +the completed assertion on loop algebras. A reference path identifies +each other arrow space with a loop-algebra torsor, so the same assertion +holds on the whole category. Coproduct preservation follows from +group-likeness, and preservation of insertions and relabeling follows +from Proposition~\ref{prop:regularized-holonomy}, first on paths and +then on their linear completions by continuity. +\end{proof} + +We can now compare the two isomorphisms on the same category: +\[ + S=H_-H_+^{-1}:\mathcal C(I;\CC)\longrightarrow\mathcal C(I;\CC). +\] +It fixes the objects, preserves the coproducts and all operations, +and is the identity on arity two. Its associated graded is the +identity, so $S-\id$ raises weight on every arrow space. The operator +\begin{equation}\label{eq:holonomy-operator-log} + \delta=\log S= + \sum_{j\geq1}\frac{(-1)^{j+1}}j(S-\id)^j +\end{equation} +is well defined weight by weight and is a compatible derivation and +coderivation. Here is a direct justification of both rules. Modulo +weights greater than $N$, the binomial polynomial +\[ + S^s=\sum_{j=0}^{N}\binom{s}{j}(S-\id)^j +\] +agrees with the usual power for every nonnegative integer $s$. +The assertions that it preserve composition, coproduct, and every +insertion are polynomial identities in $s$. They hold at all such +integers and hence identically. For the coproduct use the tensor +product truncated by total weight $N$. Differentiating at $s=0$ +gives exactly the derivation and coderivation rules for +\eqref{eq:holonomy-operator-log}, as well as compatibility with the +operations. Taking homogeneous components gives +$\delta_n\in\mathcal D_n(\CC)$ for all $n>0$. + +\subsection{A nonzero depth-one value} +The compatible derivations are now available. We evaluate their +homogeneous components on the reassociation unit arrow and compare +this operator-logarithm value with the holonomy of a real path. + +\begin{proposition}\label{prop:odd-depth-one} +For every odd $n\geq3$ there exists $\psi_n\in V_n$ for which the +coefficient of $x^{n-1}y$ is nonzero. Equivalently, its ordinary +$y$-depth-one part is a nonzero multiple of $\ad_x^{n-1}y$. +\end{proposition} + +\begin{proof} +Use the objects $a,b$ and unit arrow $\alpha$ from +Section~\ref{sec:category}. Let $\beta$ be the real order-preserving +path of reassociation from $a$ to $b$, and put +\[ + \Phi=H_+(\beta),\qquad U=S(\alpha),\qquad + \theta=S|_{\operatorname{End}(a)}. +\] +We identify $\Phi,U$ with arrow coefficients. Since an arrow with +coefficient $h$ is $\alpha h_a$, one has exactly +\begin{equation}\label{eq:holonomy-arrow-action} + S_{ab}(h)=U\theta(h). +\end{equation} +The path $\beta$ is real, so its conjugate is itself and +$H_-(\beta)=\Phi(-t_{ij})$. Hence +\begin{equation}\label{eq:holonomy-exact-comparison} + U\theta(\Phi)=\Phi(-t_{ij}). +\end{equation} +Source $x$ and target $y$ are fixed by the arity-two insertion +operations. Applying $S$ to $\alpha y_a=y_b\alpha$ therefore yields +\begin{equation}\label{eq:holonomy-source-action} + \theta(x)=x,\qquad \theta(y)=U^{-1}yU. +\end{equation} +The source total chord $T$ is fixed as well. + +The linear term of $\Phi$ is zero. Along a real order-preserving +path the sign of each difference is constant, so its integral is the +logarithm of the ratio of endpoint absolute differences. Normalizing +by the frames subtracts the logarithms of their lowest-common-ancestor +scales. By \eqref{eq:holonomy-collar-difference}, each remaining +ratio tends to $1$. This proves the claim for every chord separately. +As $\theta$ preserves filtration, equation +\eqref{eq:holonomy-exact-comparison} implies that $U$ too has zero +linear term. Both series are group-like. Their logarithms are +primitive and have weight at least two, so +\eqref{eq:category-three-split} places these logarithms in the +completed free Lie algebra on $x,y$. Every such Lie monomial uses +both letters, since the free Lie algebra on one letter has only +weight one. Consequently $\Phi,U$ depend only on $x,y$ and +\begin{equation}\label{eq:holonomy-exact-xy} + U\theta(\Phi)=\Phi(-x,-y),\qquad + \Phi-1,\ U-1\in J, +\end{equation} +where $J$ is the closed two-sided ideal of series having at least one +$y$ in $\CC\langle\!\langle x,y\rangle\!\rangle$. + +Conjugation in \eqref{eq:holonomy-source-action} changes $y$ only by +terms in $J^2$, because $U-1\in J$. Substitution therefore gives +\[ + (\theta-\id)(J^r)\subseteq J^{r+1}\quad(r\geq1), + \qquad\theta\equiv\id\pmod{J^2} +\] +on the full series algebra (it fixes the pure-$x$ terms). +Reducing \eqref{eq:holonomy-exact-xy} modulo $J^2$ gives +\begin{equation}\label{eq:holonomy-unit-difference} + U-1=\Phi(-x,-y)-\Phi(x,y)\pmod{J^2}. +\end{equation} +This is a statement about the image of the unit arrow. To pass to its +infinitesimal value, put $N=S_{ab}-\id$ on this arrow space. By +\eqref{eq:holonomy-arrow-action}, for $h\in J$, +\[ + N(h)=(U-1)\theta(h)+(\theta(h)-h)\in J^2. +\] +More generally $N(J^r)\subseteq J^{r+1}$ for $r\geq1$. +Since $N(1)=U-1\in J$, all $N^j(1)$ with $j\geq2$ belong to $J^2$. +The operator logarithm consequently gives +\begin{equation}\label{eq:holonomy-log-depth-one} + \delta(\alpha)=U-1 + =\Phi(-x,-y)-\Phi(x,y)\pmod{J^2}. +\end{equation} + +It remains to compute a single convergent coefficient of $\Phi$. +Set the central $T$ to zero and use +$s=(z_2-z_1)/(z_3-z_1)$. Writing the three differences as a common +factor times $s,s-1,1$ shows that the common logarithmic differential +has coefficient $T$, while the remaining connection is +\begin{equation}\label{eq:holonomy-interval-connection} + \lambda\left(x\frac{ds}{s}+y\frac{ds}{s-1}\right), + \qquad 00\), depending +on \(C\) and \(K\), such that every mixed point \(s\in C(\Qbar)\) +satisfies +\[ + \#\bigl(\Gal(\Qbar/K)\cdot s\bigr) + \ge c\,\cplx(s)^{1/42}. +\] +\end{theorem} + +The proof below uses Siegel's potentially ineffective class-number +bound. Remark~\ref{rem:effective-cm} gives an effective alternative for +that input alone; we make no effectivity claim for the constants in +Theorem~\ref{thm:orbits}. The exponent records the positive power needed +by the counting theorem; its optimization is not needed here. +Section~\ref{sec:tools} fixes the height and period normalizations and +collects the CM estimates. Section~\ref{sec:gluing} proves the canonical +gluing description and its reciprocal degeneration formula. +Section~\ref{sec:height} turns that formula into the height bound on +the fixed curve, treating an empty boundary separately. +Section~\ref{sec:threefold} constructs and balances the essential +period. Section~\ref{sec:orbits} combines the estimates to prove +Theorem~\ref{thm:orbits}, and Section~\ref{sec:finiteness} applies +the corrected Daw--Orr implication and Andr\'e--Oort. + +\section{Heights, periods, and CM elliptic curves}\label{sec:tools} + +\subsection{Conventions} + +Endomorphism rings and homomorphism groups are geometric unless a +ground field is indicated. All heights are absolute logarithmic heights. We write \(h(j_P)\) +for the usual height of the \(j\)-invariant of an elliptic curve \(P\), +and \(\hF(B)\) for stable Faltings height. Set +\(\hFp(B)=\max\{1,\hF(B)\}\). +For a line bundle \(L\) on a \(g\)-dimensional abelian variety, +\(\deg_L B=c_1(L)^g[B]\). Thus a principal polarization has degree \(g!\) +in this convention. The associated period norm is the norm of its +positive Hermitian form. + +For an elliptic curve with its principal polarization, choose a period +basis \(e,e\tau\) with \(\tau\) in the standard closed fundamental region. +Writing \(y=\Im\tau\), one has +\begin{equation}\label{eq:elliptic-norms} + y\ge\sqrt3/2,\qquad + \|e\|^2=1/y,\qquad + \|e\tau\|^2=|\tau|^2/y\ll y. +\end{equation} +The Fourier expansion of \(j\), together with compactness away from +the cusp, gives +\begin{equation}\label{eq:j-y} + \abs{\log^+|j(\tau)|-2\pi y}\ll1. +\end{equation} +Constants in these two formulas are absolute. +Other implicit constants may depend on a fixed curve, family, and +ground field, but never on a varying point or its field of definition. + +For a number field \(k\), absolute values at finite places extend the +standard absolute values on \(\Q\); they are not renormalized after a +field extension. Heights are sums with weights +\([k_v:\Q_v]/[k:\Q]\). Equivalently, the archimedean contribution is +the average over all complex embeddings, with conjugate embeddings +counted separately. + +\subsection{Two established height and period estimates} + +We use product additivity and the isogeny inequality for stable +Faltings height: +\begin{equation}\label{eq:faltings-isogeny} + \hF(B_1\times B_2)=\hF(B_1)+\hF(B_2),\qquad + |\hF(B_1)-\hF(B_2)|\le\tfrac12\log\deg\phi +\end{equation} +for an isogeny \(\phi:B_1\to B_2\); see \cite[\S2.3]{GR}. +For elliptic curves, \(\hF\) is bounded below and +\begin{equation}\label{eq:elliptic-height} + 1+h(j_P)\asymp 1+\hFp(P). +\end{equation} +These are stable-height statements, so passing to a semistable +extension does not change the heights. For the comparison with +\(j\)-height, see Silverman \cite[Proposition~2.1]{Silverman} +and the normalization recorded in \cite[Proposition~3.1]{Lobrich}. + +\begin{lemma}\label{lem:family-height} +Let \(\mathcal X\to S\) be a fixed principally polarized abelian scheme +over a smooth curve defined over a number field, and let \(\bar S\) +be its smooth projective completion. If \(H\) is an ample divisor on +\(\bar S\), and \(h_H\) is shifted to be nonnegative, then +\[ + \hFp(\mathcal X_t)\ll 1+h_H(t) + \qquad(t\in S(\Qbar)). +\] +\end{lemma} + +\begin{proof} +After a fixed finite cover and restriction to a dense open, choose +a symmetric ample line bundle representing the polarization and +theta data as in \cite[\S2.3]{Pazuki}, for one fixed even integer +\(r\ge2\). This is possible by choosing the data over a finite +extension of the function field and spreading out. The theta-null map is an algebraic map to projective space. +Pazuki's theta/Faltings comparison +\cite[Theorem~1.1 and Corollary~1.3]{Pazuki} gives +\(\hFp\ll1+h_\Theta\), with constants depending only on the dimension +and the fixed theta level. + +The theta-null map extends to the smooth projective covering curve. +If \(p\) is the covering map and \(T\) is the pullback of the +hyperplane divisor, choose an integer \(a\) such that +\(ap^*H-T\) is ample. Its height is bounded below, so the height +machine gives \(h_\Theta\le a h_H\circ p+O(1)\). +This proves the asserted bound after lifting \(t\); stable Faltings +height is unchanged by the lift. The finite set omitted when choosing the data can be included +by enlarging the constant. +\end{proof} + +Call a nonzero period \(\omega\) of a complex abelian variety +\emph{essential} if it belongs to the Lie algebra of no proper +abelian subvariety. For a polarized abelian variety \((B,L)\) of +dimension \(g\) over a number field \(k\), and an essential period +\(\omega\) at one complex +embedding of \(k\), \cite[Theorem~1.2]{GR} gives +\begin{equation}\label{eq:period-theorem} + \frac{(\deg_L B)^{1/g}}{\|\omega\|_L^2} + \ll_g [k:\Q]\max\{1,\hF(B),\log\deg_L B\}. +\end{equation} +To explain the field-degree factor, let +\(\delta_0(B_\sigma,L_\sigma)\) be the least norm of an essential +period at the embedding \(\sigma\), with value \(+\infty\) if +there is none. The essential-minimum form of their theorem bounds +\[ + \frac1{[k:\Q]}\sum_{\sigma:k\hookrightarrow\C} + \frac{(\deg_LB)^{1/g}}{\delta_0(B_\sigma,L_\sigma)^2} + \ll_g \max\{1,\hF(B),\log\deg_LB\}. +\] +At the chosen embedding, +\(\delta_0(B_\sigma,L_\sigma)\le\|\omega\|_L\). +Keeping this one nonnegative summand proves +\eqref{eq:period-theorem}. The estimate allows arbitrary +polarization degree. + +\subsection{CM estimates}\label{subsec:cm} + +\begin{lemma}\label{lem:cm} +Let \(k\) be a number field and \(F/k\) a CM elliptic curve. Set +\(\mathcal O=\End_{\Qbar}(F)\), \(D=|\disc\mathcal O|\), and \(d=[k:\Q]\). +Then +\begin{equation}\label{eq:cm-bounds} + D\ll d^4,\qquad h(j_F)\ll1+\sqrt D. +\end{equation} +At any complex embedding, choose a reduced period basis \(f,f\tau_F\) +and put \(y_F=\Im\tau_F\). Then +\begin{equation}\label{eq:b-index} + b=[\Lambda_F:\mathcal O f] + =\frac{\sqrt D}{2y_F}\in\Z_{>0},\qquad + y_F\le\sqrt D/2,\qquad b\ll\sqrt D. +\end{equation} +\end{lemma} + +\begin{proof} +The degree of \(j_F\) over \(\Q\) is the class number \(h(\mathcal O)\), +and \(j_F\) is an algebraic integer \cite[Theorem~11.1 and Proposition~13.2]{Cox}. +Write \(D=f_0^2D_0\), where \(D_0\) is the absolute discriminant of +the maximal order. Siegel's class-number bound gives +\(h(\mathcal O_{D_0})\gg D_0^{1/4}\) \cite{Siegel}. +The conductor formula \cite[Theorem~7.24]{Cox} multiplies this by +\[ + \frac{f_0}{[\mathcal O_{D_0}^{\times}:\mathcal O^\times]} + \prod_{p\mid f_0}\left(1-\frac{\chi_{-D_0}(p)}p\right), +\] +which is at least \(\varphi(f_0)/3\gg\sqrt{f_0}\). +Here \(\chi_{-D_0}\) is the quadratic character of the imaginary +quadratic field of discriminant \(-D_0\), defined by the Kronecker +symbol, and \(\varphi\) is Euler's totient function. The elementary bound +\(\varphi(f_0)\ge\sqrt{f_0/2}\) suffices. +Thus \(d\ge h(\mathcal O)\gg D^{1/4}\), including every conductor. +The constant may be ineffective. + +The covolume of \(\mathcal O\) in \(\C\) is \(\sqrt D/2\), whereas +that of \(\Lambda_F/f=\Z+\tau_F\Z\) is \(y_F\). Since +\(\mathcal O f\subset\Lambda_F\), their ratio gives +\eqref{eq:b-index}. The lower bound for \(y_F\) in +\eqref{eq:elliptic-norms} gives \(b\ll\sqrt D\). +Finally, \eqref{eq:j-y} bounds every archimedean conjugate of \(j_F\) +by \(O(1+\sqrt D)\); integrality removes all finite-place contributions. +\end{proof} + +\begin{remark}[An effective alternative for the CM bound] +\label{rem:effective-cm} +The Dirichlet assertion of \cite[Theorem~1.1]{OpenAIQuasiRH2026} +makes the implied constant in \(D\ll d^4\) effective. Indeed, its +zero-free half-plane \(\operatorname{Re}s>7/8\) supplies the short real +zero-free interval used in +\cite[proof of Theorem~2, pp.~363--364]{Chen2007}. The resulting estimate is +\(L(1,\chi_{-D_0})\gg1/\log(2D_0)\), with an effective absolute +constant, after handling finitely many small conductors. The Dirichlet +class-number formula therefore gives +\(h(\mathcal O_{D_0})\gg\sqrt{D_0}/\log(2D_0)\gg D_0^{1/4}\) +effectively. The conductor estimate in the preceding proof then yields +\(d\ge h(\mathcal O)\gg D^{1/4}\) with an effective constant for +every order. This concerns only the CM discriminant estimate: it does +not assert effective Galois-orbit constants or effective finiteness in +Theorems~\ref{thm:orbits} and~\ref{thm:main}. +\end{remark} + +\section{Canonical gluing and boundary lengths}\label{sec:gluing} + +\subsection{The canonical product isogeny} + +We first identify the product and gluing integer intrinsically, so +that the subsequent height bounds use the same complexity as the +Galois-orbit theorem. The description below follows +\cite[Lemmas~2.1 and~3.2]{DO}; we include the argument to make the +minimality and field of definition explicit. + +\begin{lemma}\label{lem:gluing} +Let \((A,\lambda)\) be a principally polarized abelian surface over +a field of characteristic zero, and suppose that \(A\) is isogenous +geometrically to the product of a non-CM and a CM elliptic curve. +There are unique elliptic subvarieties \(E,F\subset A\), with \(E\) +non-CM and \(F\) CM. They are defined over the field of \(A\). +For some positive integer \(n\), addition induces an isogeny +\[ + \phi:E\times F\longrightarrow A +\] +of degree \(n^2\), whose pullback polarization is \(n\) times the +product polarization and whose kernel is the graph of an isomorphism +\(\theta:F[n]\to E[n]\). This isomorphism is defined over the field of +\(A\). The minimum degree of an isogeny from an elliptic product to +\(A\) is \(n^2\), and its factors are then isomorphic to \(E,F\). +\end{lemma} + +\begin{proof} +The two simple isogeny factors are nonisogenous, so there are no +homomorphisms between them. Their images in \(A\) are therefore +unique elliptic subvarieties. Uniqueness gives descent. +The pullback of \(\lambda\) to \(E\times F\) is diagonal, say with +positive elliptic multipliers \(n_E,n_F\). Comparing degrees of +polarizations gives \(|\ker\phi|=n_En_F\). +Each projection of the kernel is injective, since \(E,F\) are +embedded subvarieties, and +\[ + \ker\phi\subset E[n_E]\times F[n_F]. + \] +Consequently \(n_En_F\le n_E^2\) and \(n_En_F\le n_F^2\), which force +\(n_E=n_F=n\). The projections then identify the kernel with each +full \(n\)-torsion group. This gives the graph, including its descent. + +Any isogeny \(P\times Q\to A\) from an elliptic product maps the two +factors, in some order, onto \(E,F\). It factors through their +addition isogeny, so its degree is at least \(n^2\). Equality forces +both factor maps to have degree one. +\end{proof} + +The Galois action on the endomorphism ring of a CM elliptic curve +has image of order at most two: it acts on its imaginary quadratic +endomorphism algebra. Thus, when \(A\) is defined over a number field +\(k_1\), all endomorphisms of \(F\) are defined after an extension of +\(k_1\) of degree at most two. Defining the graph itself requires +no such extension. + +\subsection{The degeneration input}\label{subsec:degeneration} + +The new local issue is how the integer \(n\) changes the degeneration +of the non-CM factor. We first compute the resulting length from +semiabelian uniformization, then compare it with the boundary of a +fixed integral compactification. + +Suppose a principally polarized abelian variety over a complete +discretely valued field has split semiabelian reduction. Its +uniformization consists of an extension of a good-reduction abelian +variety by a split torus, modulo a period lattice \(Y\). +If \(X\) is the character lattice of the torus, a polarization gives +a map \(Y\to X\), an isomorphism for a principal polarization. +The degeneration data carry a functorial valuation pairing +\[ + Y\times X\longrightarrow\R. +\] +We use the extended absolute value in defining this pairing. In toric +rank one, its value on generators identified by the principal +polarization is a positive number, denoted \(\ell(A)\). + +\begin{lemma}\label{lem:length} +In the situation of Lemma~\ref{lem:gluing} over a number field, let +\(v\) be a finite place. If \(E\) has potential good reduction, so +does \(A\). Otherwise, after an extension giving split semiabelian +reduction, +\begin{equation}\label{eq:length} + \ell(A)=\frac{\log|j_E|_v}{n}. +\end{equation} +\end{lemma} + +\begin{proof} +The CM factor has potential good reduction +\cite[Theorem~6(a)]{ST}; for an elliptic curve this also follows from +the integrality of its \(j\)-invariant. After a finite local extension, +it has good reduction and \(E\) has either good reduction or split +Tate reduction \cite[Theorem~1 and Applications]{Tate}. +Potential good reduction and rational toric rank are isogeny +invariants. It remains to determine the length in the Tate case. + +Functoriality of semiabelian uniformization gives maps of tori and +period lattices for \(\phi:E\times F\to A\) +\cite[Corollary~4.5.5.5]{Lan}. The map of one-dimensional tori is an +isomorphism. Indeed, a nontrivial kernel would contain nontrivial +roots of unity. Torus torsion injects into the Tate elliptic curve, +because its period lattice is torsion free. Such a kernel would +therefore give a nonzero point of \(\ker\phi\) in \(E\times\{0\}\), +contradicting the graph description. +This reasoning takes place on characteristic-zero generic fibers +and does not require \(v(n)=0\). + +Write \(Y',X'\) for the source lattices of \(E\times F\), with +its product principal polarization. Pulling back the polarization +means that the composite +\[ + Y'\xrightarrow{\phi_Y}Y_A\xrightarrow{\lambda_A}X_A + \xrightarrow{\phi_X^*}X' +\] +is \(n\lambda_{E\times F}\). +The last map is an isomorphism by the torus calculation, and the two +principal-polarization maps are isomorphisms. Hence \(\phi_Y\) has +absolute multiplier \(n\). Functoriality of the valuation pairing, +\[ + \langle\phi_Yy,x\rangle=\langle y,\phi_X^*x\rangle, +\] +now gives \(n\ell(A)=\ell(E\times F)\). +For the Tate parameter \(q_E\), the latter equals +\(-\log|q_E|_v=\log|j_E|_v\), by the expansion of \(j\) +\cite[Equation~(18)]{Tate}. +All absolute values are extended from the original field; this is +why the calculation is unchanged by the local extension. +\end{proof} + +To use this reciprocal length in a height estimate on the curve, +we need to measure it by a boundary divisor on the moduli space. +The following comparison provides this link uniformly at finite places. + +We specify the part of integral toroidal compactification theory used +in the proof. At full symplectic level \(m\ge3\), choose a smooth +projective toroidal compactification of the Siegel moduli scheme over +\(\Z[\zeta_m,1/m]\), with reduced boundary divisor \(D_m\). +The boundary is an effective relative Cartier divisor. + +\begin{lemma}[Integral boundary comparison]\label{lem:boundary-comparison} +Fix the level \(m\) and compactification above. There are constants +\(c_m,C_m>0\), depending only on these choices, with the following +property. Let \(k_v/\Q_p\) be a finite extension with \(p\nmid m\), +and let \(A/k_v\) be a principally polarized abelian surface with +full level \(m\). +Write \(\lambda_{D_m,v}(A)\) for the model local height of the boundary. +\begin{enumerate}[label=(\roman*)] +\item If \(A\) has potential good reduction, then +\(\lambda_{D_m,v}(A)=0\). +\item If \(A\) has potential split semiabelian reduction of toric +rank one, then +\[ + c_m\ell(A)\le\lambda_{D_m,v}(A)\le C_m\ell(A). +\] +\end{enumerate} +Here \(\ell(A)\) is evaluated after an extension giving split +semiabelian reduction. With the absolute value extended from the +original field, both sides are unchanged by every further local +extension. +\end{lemma} + +\begin{proof} +Here is the precise chart calculation behind these assertions. +The integral toroidal construction is developed in \cite{FC}; we use +\cite[Construction~6.3.1.1 and Theorem~6.4.1.1]{Lan}, with projectivity +supplied by \cite[Theorem~7.3.3.4]{Lan}. +By \cite[Theorem~6.4.1.1(2), (5)]{Lan}, the completed neighborhood of +a stratum is the prescribed torus embedding over an abelian-scheme +torsor over a finite \'etale cover of a lower-dimensional moduli +space. The boundary is the reduced relative Cartier normal-crossings +divisor of that embedding \cite[Theorem~6.4.1.1(3)]{Lan}. +Its strata pull back scheme-theoretically to the toroidal strata, +and their normal coordinates are the toric monomials +\cite[Proposition~6.3.1.6(1)]{Lan}. +After trivializing the torus torsor on a smooth cone chart, extend +the primitive ray generators to a lattice basis. In the resulting +coordinates \(q_1,\ldots,q_r\), its reduced-boundary ideal is +\((q_1\cdots q_r)\); this is the relative toric divisor of +\cite[Theorem~6.1.2.8(5)]{Lan}. + +For a rank-one degeneration, the abelian quotient has good reduction. +Its moduli point therefore lies in the interior of this lower-dimensional +base. The extension coordinates belong to the abelian-scheme torsor; +after a local extension they extend over the valuation ring by +properness. This applies equally at ordinary and supersingular points +of that base. Thus there is no additional boundary coordinate from +the base or extension data. + +The valuation form lies on a rank-one ray of the cone. If a rank-one +positive semidefinite form is a positive sum of such forms, each +summand vanishes on its kernel and lies on the same ray. Consequently +the ray is a face of every cone containing the form. In a smooth +cone chart only its primitive normal monomial, say \(q_c\), has +positive valuation; the other normal monomials are units. Locally +the reduced-boundary ideal is therefore \((q_c)\), up to a unit. +There is no vertical factor: this is the scheme-theoretic toric +boundary ideal in a relative normal-crossings chart over +\(\Z[\zeta_m,1/m]\), not a comparison only on the generic fiber. + +Finally, \cite[Propositions~6.2.5.8(3), 6.2.5.11, and~6.3.1.6(4)]{Lan} +identify the tautological +ideal-valued toric pairing with the pairing of the semiabelian +degeneration. This identifies \(-\log|q_c|\) with \(\ell(A)\) times +the fixed rational factor comparing the primitive normal character +and the rank-one polarized cusp lattice. Those lattices and their +indices are part of the fixed level datum. They do not depend on +an isogeny presenting \(A\), or on the ramification of its field. +There are finitely many cusp types at fixed level, giving the stated +two fixed comparison constants. For good reduction the point is +interior and every boundary monomial is a unit. Computing these +values with the extended absolute value preserves them under every +local field extension. + +The integral Siegel model here requires only that \(m\) be invertible, +including at residue characteristic two. This also follows from +the good-prime conditions in \cite[Definition~1.4.1.1]{Lan}: +for the self-dual symplectic datum over \(\Z\), the algebra is +\(\Q\), the order is \(\Z\), and there is no type-D or discriminant +exclusion. Later we will use levels \(3\) and \(4\), so every finite +prime is handled by an invertible level. +\end{proof} + +\section{A uniform height estimate on a curve}\label{sec:height} + +Fix a smooth curve \(S\) over a number field \(K_0\), carrying a +principally polarized abelian scheme of relative dimension two with +full level divisible by \(12\). Let \(\bar S\) be its smooth +projective completion. The two level-forgetting maps extend, by +properness, to morphisms +\[ + f_m:\bar S\longrightarrow\overline{\A}_{2,m}\qquad(m=3,4). +\] +Let \(\Delta\) be the reduced inverse image of the boundary. +Its support is the same at the two levels, because it is characterized +by nonzero potential toric rank. It may be empty. + +\begin{proposition}\label{prop:height} +There are constants \(c,n_0>0\), depending only on this fixed family, +such that at every mixed point \(t\in S(\Qbar)\), with the notation +of Lemma~\ref{lem:gluing} and \(D=|\disc\End(F)|\), +\[ + n\ge n_0\quad\Longrightarrow\quad h(j_E)\le c\,n(1+\sqrt D). +\] +\end{proposition} + +No Hodge-genericity hypothesis is needed for this height estimate. +When the boundary is nonempty, a height associated with it controls +\(h(j_E)\), with a logarithmic isogeny error. We will obtain a reverse +bound in which \(h(j_E)\) occurs with coefficient \(O(1/n)\), allowing +absorption for large \(n\). When the boundary is empty, the finite-place +estimates and compactness at infinity give the bound directly. + +\subsection{Uniform local height comparisons}\label{subsec:model} + +We first make explicit the uniformity in the height machine. +An \(M_{K_0}\)-bound means constants \(c_v\), zero at all but finitely +many finite places of \(K_0\), inherited at places above \(v\) by +extension of its absolute value. Their weighted sum is bounded +independently of any finite extension of \(K_0\). + +If \(\Delta\ne\varnothing\), write +\(f_m^*D_m=\Delta+R_m\), where \(R_m\) is effective. Local heights of +effective divisors on a proper variety are bounded below. +Functoriality therefore gives +\begin{equation}\label{eq:local-comparison} + \lambda_{\Delta,v}(t) + \le\lambda_{D_m,v}(f_m(t))+O_v(1). +\end{equation} +If \(\Delta=\varnothing\), the generic pullback divisor is zero, so +instead +\begin{equation}\label{eq:zero-comparison} + \lambda_{D_m,v}(f_m(t))=O_v(1). +\end{equation} +These errors are \(M_{K_0}\)-bounds. At finite places prime to \(m\), +we may use exactly the model local height described in +Lemma~\ref{lem:boundary-comparison}. + +Here is a model justification that also handles the finitely many +bad primes of the curve and of its maps. Over +\(\mathcal O_{K_0}[1/m]\), take a proper model of \(\bar S\), +then the closure of the graph of \(f_m\) in its product with the +fixed integral toroidal compactification. If needed, pass to a +fixed further model on which the relevant divisors are Cartier. Models of the same generic divisor differ by +vertical divisors at finitely many primes. An effective vertical +Cartier divisor \(V\) is bounded above by a fixed multiple of the +full special fiber: locally a power of the uniformizer lies in the +ideal of \(V\), and quasi-compactness supplies a common power. +Taking positive and negative bounds for vertical discrepancies +shows that changing models introduces an error bounded in the +extended absolute value. This proves the asserted uniformity even +when the point and the ramification of its field vary. + +Combining \eqref{eq:local-comparison} with +Lemmas~\ref{lem:boundary-comparison} and~\ref{lem:length}, +and choosing \(m\in\{3,4\}\) prime to the residue characteristic, +gives, when \(\Delta\ne\varnothing\), +\begin{equation}\label{eq:finite-bound} + \lambda_{\Delta,v}(t) + \le c\,\frac{\log^+|j_E|_v}{n}+O_v(1), +\end{equation} +with fixed \(c\). If \(\Delta=\varnothing\), the two-sided comparison in +Lemma~\ref{lem:boundary-comparison} and \eqref{eq:zero-comparison} instead give +\begin{equation}\label{eq:finite-empty} + \frac{\log^+|j_E|_v}{n}\le O_v(1). +\end{equation} +The bounds include the potential-good-reduction case, where the +boundary value and \(\log^+|j_E|_v\) both vanish. + +\subsection{An archimedean boundary estimate} + +At a complex point, let \(\rho(A)\) be the length of the shortest +nonzero period of \(A\) for its principal polarization. + +\begin{lemma}\label{lem:arch-boundary} +If \(\Delta\ne\varnothing\), then at each archimedean place of \(K_0\), +\[ + \lambda_\Delta(t)\ll 1+\rho(A_t)^{-2}. +\] +\end{lemma} + +\begin{proof} +Work near one point of \(\Delta\), with disk parameter \(z\). +After a fixed ramified base change, monodromy on \(H_1\) is unipotent +and has logarithm \(N\) with \(N^2=0\); this is the monodromy +description for a semistable degeneration of abelian varieties +\cite{SGA7}. +The image of \(N\) is isotropic. Choose a rational Lagrangian +containing it and contained in \(\ker N\), and adapt an integral +symplectic basis to its primitive lattice. On the universal cover +of the punctured disk, the resulting period matrix \(Z\) has monodromy +\[ + Z\longmapsto Z+M,\qquad M\in\operatorname{Sym}_2(\Z). +\] + +For every \(u\in\Z^2\), the holomorphic function +\(q_u=\exp(2\pi i\,u^tZu)\) is single-valued and bounded in absolute +value by one. It extends across \(z=0\); its order there is +\(u^tMu\), by the monodromy of its logarithm. +Thus \(M\) is positive semidefinite. Writing +\(q_u=z^{u^tMu}\) times a holomorphic unit, and using the finitely +many vectors \(e_i,e_i+e_j\), shows that +\begin{equation}\label{eq:period-boundary} + Y:=\Im Z=\frac{-\log|z|}{2\pi}M+R(z), +\end{equation} +where \(R\) extends harmonically across zero. + +On \(\ker M\), the form \(R(0)\) is positive definite. For any +nonzero real \(v\) in that kernel, \(v^tR(z)v=v^tYv>0\) on the +punctured disk. If its value at zero were zero, the minimum +principle would force this harmonic function to vanish identically. +If \(M=0\), the same argument and the holomorphic extension just +constructed would extend \(Z\) to the interior of Siegel space. +Uniqueness of extension to the separated toroidal compactification +would contradict the choice of a boundary point. Hence \(M\ne0\). + +Choose a nonzero integral \(w\in\im M\), possible because \(M\) is +integral. On the orthogonal decomposition +\(\im M\oplus\ker M\), the first block of \(Y\) grows as a positive +definite matrix times \(-\log|z|\), the off-diagonal blocks are +bounded, and the kernel block stays positive definite. +The Schur-complement formula gives +\[ + w^tY^{-1}w=O\bigl((-\log|z|)^{-1}\bigr). +\] +This is the squared norm of the period \(w\) in +\(\Z^2+Z\Z^2\). Therefore +\(\rho(A_t)^2\ll(-\log|z|)^{-1}\), and +\(-\log|z|\ll\rho(A_t)^{-2}\). +The boundary local height is a fixed multiple of \(-\log|z|\) up to +a bounded term, also after the fixed ramification. +Finitely many such disks, followed by compactness on their +complement, prove the lemma. The argument includes both ranks +one and two for \(M\). +\end{proof} + +\begin{lemma}\label{lem:rho} +At a mixed point, choose reduced elliptic periods at a complex +embedding and write \(y_E=\Im\tau_E\), \(y_F=\Im\tau_F\). Then +\[ + \rho(A)^{-2}\le ny_F+\frac{y_E}{n}, + \qquad + \rho(A)^2\le\frac n{y_E}. +\] +\end{lemma} + +\begin{proof} +Identify Lie spaces using \(E\times F\to A\). +The period lattice of \(A\) is an overlattice of +\(\Lambda_E\oplus\Lambda_F\) contained in its \(1/n\) multiple, and +its Hermitian form is \(n\) times the product form. +Its intersection with \(\Lie E\) is exactly \(\Lambda_E\), +because the graph kernel has trivial intersection with +\(E\times\{0\}\). + +A period with nonzero \(F\)-projection has squared norm at least +\(n\cdot n^{-2}/y_F=1/(ny_F)\). +A nonzero period with zero \(F\)-projection has squared norm at +least \(n/y_E\), and the first reduced period of \(E\) attains this +value. The two assertions follow. +\end{proof} + +\subsection{Proof of Proposition~\ref{prop:height}} + +Suppose first that \(\Delta=\varnothing\). +At every embedding of \(K_0\), the image of \(\bar S\) is compact +in the interior moduli space. The shortest period length has a +positive lower bound on this image. Lemma~\ref{lem:rho} therefore +gives \(y_E\ll n\) at every embedding of a field of definition of +the point. Equations~\eqref{eq:j-y} and \eqref{eq:finite-empty}, +summed with the normalized local weights, show +\[ + h(j_E)\ll n. +\] +This proves the proposition in this case. + +Now let \(\Delta\ne\varnothing\). Choose \(h_\Delta\) shifted to be +nonnegative. It is an ample height because \(\Delta\) is a nonzero +effective divisor on a projective curve. +Lemmas~\ref{lem:arch-boundary} and \ref{lem:rho}, together with +\eqref{eq:b-index}, give at the archimedean places +\[ + \lambda_\Delta(t)\ll + 1+n\sqrt D+\frac{y_E}{n}. +\] +Use \eqref{eq:j-y} for \(y_E\), add \eqref{eq:finite-bound}, and sum. +The \(M_{K_0}\)-bounds give a constant independent of the field. +We obtain +\begin{equation}\label{eq:height-upper} + h_\Delta(t)\le B\left(1+n\sqrt D+\frac{h(j_E)}n\right) +\end{equation} +for a fixed \(B>0\). + +On the other hand, \eqref{eq:faltings-isogeny} gives +\(\hF(E)+\hF(F)\le\hF(A)+\log n\). +The lower bound for elliptic Faltings height, +\eqref{eq:elliptic-height}, and Lemma~\ref{lem:family-height} imply +\begin{equation}\label{eq:height-lower} + h(j_E)\le A_0\bigl(1+h_\Delta(t)+\log n\bigr) +\end{equation} +for fixed \(A_0>0\). Combining \eqref{eq:height-upper} and +\eqref{eq:height-lower}, the coefficient of \(h(j_E)\) on the +right is \(A_0B/n\). +For \(n\ge2A_0B\) it can be absorbed. +Since \(\log n\ll n\), this gives +\(h(j_E)\ll n(1+\sqrt D)\), as claimed. +\qed + +\section{An essential period on a threefold}\label{sec:threefold} + +We now extract a stronger power of the gluing index by repeating the +non-CM factor. This lets one period vector contain both independent +periods of \(E\), so no nonzero integer row on \(E^2\) can annihilate +its \(E\)-coordinates. Repeated elliptic factors and polarizations +adapted to reduced period bases are used in +\cite[\S\S7.2--7.3]{GR}; here the gluing data produce a threefold +isogenous to \(E^2\times F\). +All constructions in this section are over a number +field \(k\) containing the definitions of \(E,F,\theta\), and all +endomorphisms of \(F\). Put \(d=[k:\Q]\), and fix one complex +embedding. Take a reduced basis \(e_1,e_2\) of \(\Lambda_E\), and +a first reduced period \(f\) of \(F\). Use \(b,y_F\) from +Lemma~\ref{lem:cm}, and put \(y=y_E\). + +The two squared period lengths of \(E\) have sizes \(1/y\) and +\(y\). Weighting the first copy of \(E\) by approximately \(y^2\) +makes their contributions comparable. Weighting \(F\) by approximately +\(y\) then makes the cube root of the product polarization degree +grow at the same rate as the squared norm. The remaining gain is +the power \(n^{4/3}\) contributed by the quotient. + +\begin{proposition}\label{prop:threefold} +There is a polarized abelian threefold \((B,L)/k\), isogenous to +\(E^2\times F\), such that +\begin{equation}\label{eq:threefold-estimate} + n^{4/3}\ll + (b^2+1)d\,\max\{1,\hF(B),\log\deg_L B\}, +\end{equation} +and +\begin{equation}\label{eq:B-height-degree} + \hF(B)\le2\hF(E)+\hF(F)+\log n,\qquad + \deg_LB=6n^4uv, + \quad u=\ceil{y^2},\quad v=\ceil y. +\end{equation} +\end{proposition} + +\begin{proof} +We use the index \(b\) to realize both points +\(\theta^{-1}(be_i/n)\) as endomorphism images of the common +torsion point \(f/n\). For \(i=1,2\), choose \(f_i\in\Lambda_F\) +so that the class of \(f_i/n\) is \(\theta^{-1}(e_i/n)\). +Since \(b\Lambda_F\subset\End(F)f\), there is \(\beta_i\in\End(F)\) +with +\begin{equation}\label{eq:beta} + \beta_if=bf_i. +\end{equation} +There is no need for \(\beta_i\) to be invertible on \(F[n]\). + +Define +\[ + \Gamma=\{(\theta\beta_1(p),\theta\beta_2(p),p):p\in F[n]\}, + \qquad B=(E^2\times F)/\Gamma. +\] +All maps in this expression are defined over \(k\). +Indeed, the analytically selected \(\beta_i\) belong to \(\End(F)\), +all of which is defined over \(k\). The last-coordinate projection +is inverse to the displayed graph embedding as a group-scheme map. +It therefore identifies \(\Gamma\) with \(F[n]\), without any +coprimality requirement on \(n,b,\beta_1,\beta_2\); hence +the quotient map \(\pi:E^2\times F\to B\) has degree \(n^2\). +The quotient and its map are defined over \(k\). Since \(\Gamma\) +is killed by \(n\), the quotient property factors multiplication +by \(n\), over the same field, as +\[ + E^2\times F\xrightarrow{\pi}B\xrightarrow{\psi}E^2\times F, + \qquad\psi\pi=[n]. +\] +Since \(\deg[n]=n^6\), one has \(\deg\psi=n^4\). + +Give the product the line bundle +\[ + M=\mathcal O_E(u[0])\boxtimes + \mathcal O_E([0])\boxtimes\mathcal O_F(v[0]), + \qquad L=\psi^*M. +\] +These ample line bundles are defined over \(k\). +Although the integers \(u,v\) were chosen using one embedding, +they require no extension of the field. Since +\(\deg_M(E^2\times F)=6uv\), the degree assertion follows. +The height assertion follows from \(\deg\pi=n^2\) and +\eqref{eq:faltings-isogeny}. + +In the product Lie space, the vector +\[ + \omega=\frac{(be_1,be_2,f)}n +\] +is a period of \(B\). Indeed \eqref{eq:beta} gives +\(\theta\beta_i(f/n)=be_i/n\), precisely the membership condition +for the quotient lattice. + +This period is essential with respect to every geometric abelian +subvariety. If it lay in the Lie algebra of a proper abelian +subvariety of \(B_\C\), the identity component of its inverse image +under \(\pi\) would be a proper abelian subvariety of +\(E^2\times F\) whose Lie algebra contains the displayed vector. By semisimplicity up to isogeny, such a +subvariety is annihilated by a nonzero homomorphism to \(E\) or \(F\). +A homomorphism to \(E\) is an integer row \((a_1,a_2,0)\), since +\(\End(E)=\Z\) and \(\Hom(F,E)=0\); its value on the vector cannot +vanish, because \(e_1,e_2\) are independent over \(\Z\). +A homomorphism to \(F\) acts by a nonzero endomorphism on \(f\), +and again cannot vanish. A rational homomorphism may be made +integral by clearing denominators, which does not affect this +argument. + +The identity \(\pi^*L=[n]^*M\) gives the norm +\[ + \|\omega\|_L^2 + =b^2u\|e_1\|^2+b^2\|e_2\|^2+v\|f\|^2 + \ll(b^2+1)y. +\] +Here \eqref{eq:elliptic-norms} bounds \(u/y\) and +\(|\tau_E|^2/y\) by \(O(y)\), and \(v/y_F\ll y\). +The factor \(n^2\) in the pulled-back metric has cancelled the +denominator \(n^2\) in the squared norm. +At the same time, \((uv)^{1/3}\ge y\), so +\[ + \frac{(\deg_LB)^{1/3}}{\|\omega\|_L^2} + \gg\frac{n^{4/3}}{b^2+1}. +\] +Apply \eqref{eq:period-theorem} to this essential period. +\end{proof} + +\begin{remark} +The large endomorphisms \(\beta_i\) do not appear in the bound. +Their role is to put a specified vector into the quotient lattice, +not to bound the degree of a homomorphism. The projection of +\(\Gamma\) to \(F[n]\) fixes its order even if either \(\beta_i\) +has a large kernel on \(F[n]\). +\end{remark} + +\section{The Galois orbit bound}\label{sec:orbits} + +We apply the preceding estimates to the curve of +Theorem~\ref{thm:orbits}. Pass to a component of a full-level cover +with level divisible by \(12\), and normalize it. After choosing a +number field \(K_0\), enlarged once and for all, this gives a smooth +curve \(S/K_0\) with a family as in Section~\ref{sec:height}, and a +map \(S\to C\) of fixed bounded degree over a dense open. +Further fixed covers or restrictions used to describe the family +have the same property. Only finitely many points of \(C\) are +omitted. This passage is permitted for a singular \(C\) as well. + +If \(s\) belongs to that dense open, choose a lift \(t\) with +\[ + [K_0(t):\Q]\ll [K(s):K]. +\] +Such a lift exists because the finite fiber has bounded degree, +and \(K_0/K\) is fixed. By Lemma~\ref{lem:gluing}, \(E,F,\theta\) +are defined over the field of the fiber. After an extension of +degree at most two, all CM endomorphisms are defined too. +For the resulting field \(k\), put \(d=[k:\Q]\); then +\begin{equation}\label{eq:field-cost} + d\ll [K(s):K]. +\end{equation} +It remains to bound \(\max\{n^2,D\}\) by a fixed power of \(d\). + +For \(n\ge n_0\), Lemma~\ref{lem:cm} and +Proposition~\ref{prop:height} give +\begin{equation}\label{eq:coarse-height} + D\ll d^4,\qquad b^2+1\ll d^4,\qquad + h(j_E)\ll n d^2,\qquad h(j_F)\ll d^2. +\end{equation} +Let \((B,L)\) be the threefold from Proposition~\ref{prop:threefold}. +Its height satisfies +\[ + \hF(B)\le2\hF(E)+\hF(F)+\log n\ll nd^2. +\] +To bound its polarization degree, use \eqref{eq:j-y} at the chosen +embedding. A single archimedean contribution to absolute height +is at most \(d\) times the total, so +\[ + y\ll1+d\,h(j_E)\ll nd^3. +\] +It follows that +\(\log(6n^4uv)\ll1+\log n+\log d\ll nd^2\). +Thus \eqref{eq:threefold-estimate} and \eqref{eq:coarse-height} give +\[ + n^{4/3}\ll d^4\cdot d\cdot nd^2=nd^7. +\] +Dividing by \(n\) and cubing proves +\begin{equation}\label{eq:coarse-orbit} + n\ll d^{21},\qquad + \max\{n^2,D\}\ll d^{42}. +\end{equation} +For the remaining \(n7/8\)}, +OpenAI Math Release preprint +\href{https://github.com/openai/math/blob/main/preprints/The-Quasi-Riemann-Hypothesis-September-30-2026/paper.pdf}{OAI:The-Quasi-Riemann-Hypothesis-September-30-2026}, +2026. +\par\endgroup + +\bibitem{OS} +M. Orr and C. Schnell, +\emph{Correction to the article Height bounds and the Siegel property}, +Algebra Number Theory \textbf{17} (2023), 1231--1237. +\href{https://doi.org/10.2140/ant.2023.17.1231}{doi:10.2140/ant.2023.17.1231}. + +\bibitem{Papas} +G. Papas, +\emph{On the \(v\)-adic values of \(G\)-functions I: Splittings in \(\A_2\)}, +\href{https://arxiv.org/abs/2510.11806v1}{arXiv:2510.11806v1}, +13 October 2025. + +\bibitem{Pazuki} +F. Pazuki, +\emph{Theta height and Faltings height}, +Bull. Soc. Math. France \textbf{140} (2012), 19--49. +\href{https://doi.org/10.24033/bsmf.2623}{doi:10.24033/bsmf.2623}. + +\bibitem{PT} +J. Pila and J. Tsimerman, +\emph{The Andr\'e--Oort conjecture for the moduli space of abelian surfaces}, +Compos. Math. \textbf{149} (2013), 204--216. +\href{https://doi.org/10.1112/S0010437X12000589} +{doi:10.1112/S0010437X12000589}. + +\bibitem{PW} +J. Pila and A. J. Wilkie, +\emph{The rational points of a definable set}, +Duke Math. J. \textbf{133} (2006), 591--616. +\href{https://doi.org/10.1215/S0012-7094-06-13336-7} +{doi:10.1215/S0012-7094-06-13336-7}. + +\bibitem{PZ} +J. Pila and U. Zannier, +\emph{Rational points in periodic analytic sets and the Manin--Mumford conjecture}, +Rend. Lincei Mat. Appl. (9) \textbf{19} (2008), 149--162. +\href{https://doi.org/10.4171/RLM/514}{doi:10.4171/RLM/514}. + +\bibitem{Pink} +R. Pink, +\emph{A common generalization of the conjectures of Andr\'e--Oort, +Manin--Mumford, and Mordell--Lang}, preprint, April 17, 2005. +\href{https://people.math.ethz.ch/~pink/ftp/AOMMML.pdf}{Author's manuscript}. + +\bibitem{ST} +J.-P. Serre and J. Tate, +\emph{Good reduction of abelian varieties}, +Ann. of Math. (2) \textbf{88} (1968), 492--517. +\href{https://doi.org/10.2307/1970722}{doi:10.2307/1970722}. + +\bibitem{Siegel} +C. L. Siegel, +\emph{\"Uber die Classenzahl quadratischer Zahlk\"orper}, +Acta Arith. \textbf{1} (1935), 83--86. +\href{https://doi.org/10.4064/aa-1-1-83-86}{doi:10.4064/aa-1-1-83-86}. + +\bibitem{Silverman} +J. H. Silverman, +\emph{Heights and elliptic curves}, in \emph{Arithmetic Geometry} +(G. Cornell and J. H. Silverman, eds.), Springer, New York, 1986, +253--265. +\href{https://doi.org/10.1007/978-1-4613-8655-1_10} +{doi:10.1007/978-1-4613-8655-1\_10}. + +\bibitem{Tate} +J. Tate, +\emph{A review of non-Archimedean elliptic functions}, in +\emph{Elliptic Curves, Modular Forms, \& Fermat's Last Theorem} +(Hong Kong, 1993), Series in Number Theory, vol.~I, +International Press, Cambridge, MA, 1995, 162--184. +\href{https://people.math.harvard.edu/~ctm/home/text/others/tate/tate-unif/tate-unif.pdf} +{Author's reprint}. + +\bibitem{Tsimerman} +J. Tsimerman, +\emph{The Andr\'e--Oort conjecture for \(\A_g\)}, +Ann. of Math. (2) \textbf{187} (2018), 379--390. +\href{https://doi.org/10.4007/annals.2018.187.2.2} +{doi:10.4007/annals.2018.187.2.2}. + +\bibitem{Zilber} +B. Zilber, +\emph{Exponential sums equations and the Schanuel conjecture}, +J. London Math. Soc. (2) \textbf{65} (2002), 27--44. +\href{https://doi.org/10.1112/S0024610701002861} +{doi:10.1112/S0024610701002861}. + +\end{thebibliography} +\end{document} diff --git a/preprints/The-E-times-CM-Component-of-Zilber-Pink-for-Curves-in-A2-September-24-2026/paper.pdf b/preprints/The-E-times-CM-Component-of-Zilber-Pink-for-Curves-in-A2-September-24-2026/paper.pdf new file mode 100644 index 0000000000000000000000000000000000000000..0fd4c7d5b1e713125e86b2485c9844cf714e2596 --- /dev/null +++ b/preprints/The-E-times-CM-Component-of-Zilber-Pink-for-Curves-in-A2-September-24-2026/paper.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:fd6298d9c106fea90c945dd51c2358fcebe61dae44c7a297f43e1385f71e32df +size 413183 diff --git a/preprints/The-Euclidean-Steinitz-Bergstrom-theorem-September-24-2026/README.md b/preprints/The-Euclidean-Steinitz-Bergstrom-theorem-September-24-2026/README.md new file mode 100644 index 0000000000000000000000000000000000000000..d0c00dd1f1c7f05bedcee0787eca7e80d60b4e67 --- /dev/null +++ b/preprints/The-Euclidean-Steinitz-Bergstrom-theorem-September-24-2026/README.md @@ -0,0 +1,18 @@ +# [The Euclidean Steinitz–Bergström theorem](The-Euclidean-Steinitz-Bergstrom-theorem-September-24-2026.pdf) + +**Author:** OpenAI + +**Date:** September 24, 2026 + +## Citation + +```bibtex +@misc{OAI:The-Euclidean-Steinitz-Bergstrom-theorem-September-24-2026, + author = {{OpenAI}}, + title = {{The Euclidean Steinitz--Bergstr{\"o}m theorem}}, + howpublished = {OpenAI Math Release preprint + \href{https://github.com/openai/math/blob/main/preprints/The-Euclidean-Steinitz-Bergstrom-theorem-September-24-2026/The-Euclidean-Steinitz-Bergstrom-theorem-September-24-2026.pdf}{OAI:The-Euclidean-Steinitz-Bergstrom-theorem-September-24-2026}}, + year = {2026} +} +``` + diff --git a/preprints/The-Euclidean-Steinitz-Bergstrom-theorem-September-24-2026/The-Euclidean-Steinitz-Bergstrom-theorem-September-24-2026.pdf b/preprints/The-Euclidean-Steinitz-Bergstrom-theorem-September-24-2026/The-Euclidean-Steinitz-Bergstrom-theorem-September-24-2026.pdf new file mode 100644 index 0000000000000000000000000000000000000000..d6d6dd47f2e95e2e2f26caab1c0a559edefcf403 --- /dev/null +++ b/preprints/The-Euclidean-Steinitz-Bergstrom-theorem-September-24-2026/The-Euclidean-Steinitz-Bergstrom-theorem-September-24-2026.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:d8924ac01b6d4dc0e70bd3fcb025132dc7fb66aa239dd7cb77d2310aa8f7b326 +size 494022 diff --git a/preprints/The-Euclidean-Steinitz-Bergstrom-theorem-September-24-2026/build/assembly.tex b/preprints/The-Euclidean-Steinitz-Bergstrom-theorem-September-24-2026/build/assembly.tex new file mode 100644 index 0000000000000000000000000000000000000000..9c6a3601ce93aa02b3fa3a69d85548eaa24e5092 --- /dev/null +++ b/preprints/The-Euclidean-Steinitz-Bergstrom-theorem-September-24-2026/build/assembly.tex @@ -0,0 +1,278 @@ +\section{Constructing the signed walk}\label{sec:construction} + +We will encode the signed walk by one point in a convex body in a larger +coefficient space. The body controls a contracted version of every prefix. +Its extra coordinates will be chosen so that their contributions exactly +restore the parts removed by the contractions. This section states the two +analytic inputs and proves Theorem~\ref{thm:signing} from them. + +\subsection{The two analytic inputs} + +Let $D\subset\R^m$ be a nonempty bounded open convex set, and let $Q$ be a +positive definite diagonal $m\times m$ matrix. Define its diagonal Dirichlet +energy by +\begin{equation}\label{eq:dirichlet-energy} + \lambda_Q(D)= + \inf_{0\ne g\in H^1_0(D)} + \frac{\displaystyle\int_D\langle Q\nabla g,\nabla g\rangle\,dx} + {\displaystyle\int_Dg^2\,dx}. +\end{equation} +Here $H^1_0(D)$ is the closure of $C_c^\infty(D)$ in the Sobolev norm, and +derivatives are weak derivatives. This normalization corresponds to +Brownian motion with covariance rate $2Q$. Central symmetry will always +mean symmetry about the origin. Fix +\begin{equation}\label{eq:signing-constants} + \kappa=\frac1{100},\qquad \delta=\frac14. +\end{equation} + +Our first input turns small coordinate energies into signs while allowing a +bounded shift at each step. The shifts may depend on previously assigned +coordinates. + +\begin{proposition}[Signs with adaptive shifts]\label{prop:adaptive} +Let $K\subset\R^m$ be nonempty, bounded, open, convex and centrally +symmetric. Suppose that +\begin{equation}\label{eq:all-diagonal-energy} + \lambda_Q(K)\le\kappa^2\tr Q + \quad\text{for every positive definite diagonal }Q. +\end{equation} +Let $j_1,\ldots,j_s$ be distinct elements of $\{1,\ldots,m\}$, and let +$q_i:\R^{i-1}\to[-\delta,\delta]$ be arbitrary functions, for +$1\le i\le s$. There exist signs $\varepsilon_1,\ldots,\varepsilon_s$ +such that the recursion +\[ + x^{(0)}=0,\qquad + x^{(i)}=x^{(i-1)}+ + \bigl(q_i(x^{(i-1)}_{j_1},\ldots,x^{(i-1)}_{j_{i-1}}) + +\varepsilon_i\bigr)e_{j_i} +\] +has $x^{(s)}\in K$. For $i=1$, the function $q_1$ has a one-point domain. +\end{proposition} + +This is an existence statement; the choice of signs can depend on the whole +body $K$ and all the prescribed functions. Its proof uses backward +transformations of $K$, followed by forward choices of the signs. + +The second input supplies a body satisfying a uniform energy bound. Its +linear maps retain a $d$-dimensional state and read one new coordinate at +each step. + +\begin{proposition}[Energy of a filtered path body]\label{prop:filtered} +There are positive absolute constants $A_0,B_0,H_0$ with the following +property. Let $d\ge1$ and $n\ge0$ be integers and put $m=d+n$. For +$1\le t\le n$, let $C_t$ be an invertible real $d\times d$ matrix and let +$b_t\in\R^d$ satisfy +\[ + C_tC_t^T+b_tb_t^T=I_d. +\] +Define linear maps $R_t:\R^m\to\R^d$ by +\begin{equation}\label{eq:filter-recursion} + R_0x=(x_1,\ldots,x_d),\qquad + R_tx=C_tR_{t-1}x+b_tx_{d+t}\quad(1\le t\le n). +\end{equation} +Then the bounded open convex centrally symmetric set +\begin{equation}\label{eq:filtered-body} + \mathcal D= + \left\{x\in\R^m: + \begin{array}{ll} + |x_j|0$, $\delta>0$, and let $m_1,\ldots,m_n$ be real +rows of length $m$. Set +\[ + K=(2D_*D)\cap\bigcap_{i=1}^n\{x:|m_i x|<\delta\}. +\] +For every positive definite diagonal $Q$, +\begin{equation}\label{eq:slab-energy} + \lambda_Q(K) + \le D_*^{-2}\lambda_Q(D) + +\pi^2\delta^{-2}\sum_{i=1}^n m_iQm_i^T. +\end{equation} +\end{lemma} + +\subsection{The filter, the shifts, and their cancellation} + +We now prove the signed-prefix theorem from +Propositions~\ref{prop:adaptive} and~\ref{prop:filtered}. Their proofs in the +remaining sections are independent of this application. + +\begin{proof}[Proof of Theorem~\ref{thm:signing}] +Write the prescribed sequence as $v_1,\ldots,v_n\in\R^d$, with +$\lVert v_i\rVert_2\le1$. The hypotheses give $d,n\ge1$. + +\paragraph{The symmetric contractions.} +For a positive absolute $\sigma<1$, to be chosen below, set +\begin{equation}\label{eq:symmetric-filter} + b_i=\sigma v_i,\qquad + \eta_i=\frac1{1+\sqrt{1-\lVert b_i\rVert_2^2}},\qquad + C_i=I-\eta_ib_ib_i^T. +\end{equation} +On the span of $b_i$, the matrix $C_i$ has eigenvalue +$\sqrt{1-\lVert b_i\rVert_2^2}$, and on its orthogonal complement it is the +identity. This includes $b_i=0$, for which $\eta_i=1/2$ and $C_i=I$. +Consequently +\[ + C_i=(I-b_ib_i^T)^{1/2}=C_i^T\succ0, + \qquad C_i^2+b_ib_i^T=I, + \qquad \frac12\le\eta_i\le1. +\] +Use these matrices in~\eqref{eq:filter-recursion}, and let $\mathcal D$ be +the body in Proposition~\ref{prop:filtered}. + +For $x\in\R^{d+n}$, let $x^0$ be obtained by setting its first $d$ +coordinates to zero. Define row vectors $m_i$ by +\begin{equation}\label{eq:predictor-rows} + m_ix=\eta_i b_i^TR_{i-1}x^0. +\end{equation} +Each row $m_i$ is supported only on the noise coordinates +$d+1,\ldots,d+i-1$. In particular it is unaffected by any coordinate +assigned at step $i$ or later. + +\paragraph{The cost of the predictor constraints.} +Let $M$ be the matrix with rows $m_i$. We claim that every column of $M$ +has squared Euclidean norm at most $\sigma^2$. Its initial $d$ columns are +zero. For a noise coordinate $d+j$, follow its contribution to the state: +\[ + r_j=b_j,\qquad r_i=C_ir_{i-1}\quad(jj$. Symmetry of $C_i$ gives the exact loss +identity +\[ + \lVert r_{i-1}\rVert_2^2-\lVert r_i\rVert_2^2 + =(b_i^Tr_{i-1})^2. +\] +Thus the losses telescope: +\begin{equation}\label{eq:predictor-columns} + \begin{aligned} + \lVert Me_{d+j}\rVert_2^2 + &=\sum_{i=j+1}^n\eta_i^2(b_i^Tr_{i-1})^2\\ + &\le\sum_{i=j+1}^n + \bigl(\lVert r_{i-1}\rVert_2^2-\lVert r_i\rVert_2^2\bigr) + \le\lVert b_j\rVert_2^2\le\sigma^2. + \end{aligned} +\end{equation} +In particular, for every positive definite diagonal $Q$, +\begin{equation}\label{eq:predictor-weighted-cost} + \sum_{i=1}^n m_iQm_i^T + =\sum_{j=1}^{d+n}Q_{jj}\lVert Me_j\rVert_2^2 + \le\sigma^2\tr Q. +\end{equation} + +Choose an absolute dilation $D_*>0$ and define +\begin{equation}\label{eq:application-body} + K=(2D_*\mathcal D)\cap + \{x:|m_ix|<\delta\text{ for }1\le i\le n\}. +\end{equation} +This is a bounded open convex centrally symmetric set containing a +neighborhood of zero. Lemma~\ref{lem:slab-energy}, +Proposition~\ref{prop:filtered}, and~\eqref{eq:predictor-weighted-cost} give +\begin{equation}\label{eq:application-energy} + \lambda_Q(K) + \le\left(H_0D_*^{-2}+\pi^2\delta^{-2}\sigma^2\right)\tr Q. +\end{equation} +We can therefore fix, once and for all, +\begin{equation}\label{eq:absolute-parameters} + D_*\ge\max\{1,\sqrt{2H_0}/\kappa\},\qquad + \sigma=\min\{1/2,\kappa\delta/(\sqrt2\pi)\}. +\end{equation} +The same body $K$ then satisfies~\eqref{eq:all-diagonal-energy} for every +positive definite diagonal $Q$. These choices are independent of $d$, $n$ +and the vectors. + +\paragraph{Choosing the coefficients.} +For $u\in\R$ write +$\operatorname{clip}_\delta(u)=\max\{-\delta,\min\{u,\delta\}\}$. +Apply Proposition~\ref{prop:adaptive} to the distinct axes +$e_{d+1},\ldots,e_{d+n}$. At step $i$ prescribe the shift +\[ + q_i=\operatorname{clip}_\delta(m_ix^{(i-1)}). +\] +By the support of $m_i$, this is a function of the previously assigned +coordinates, as required. The proposition supplies signs and a final point +$x=x^{(n)}\in K$. Its first $d$ coordinates remain zero. Also, coordinates +assigned before step $i$ never change afterward, so +\[ + m_ix^{(i-1)}=m_ix. +\] +The final membership $x\in K$ implies $|m_ix|<\delta$. Hence each shift +was already within the permitted interval: clipping was inactive, and +\begin{equation}\label{eq:unclipped-coordinates} + x_{d+i}=m_ix+\varepsilon_i + =\eta_i b_i^TR_{i-1}x+\varepsilon_i. +\end{equation} +Here $x^0=x$ because the initial block is zero. Clipping is used only to +make the application of Proposition~\ref{prop:adaptive} unconditional; +causality and the final constraints subsequently remove it. + +Figure~\ref{fig:cancellation} illustrates the identity that completes the +construction. + +\paragraph{Cancellation.} +For this single final coefficient vector, $R_0x=0$ and +\begin{align*} + R_tx + &=C_tR_{t-1}x+b_tx_{d+t}\\ + &=(I-\eta_tb_tb_t^T)R_{t-1}x + +b_t(\eta_tb_t^TR_{t-1}x+\varepsilon_t)\\ + &=R_{t-1}x+\varepsilon_tb_t. +\end{align*} +Thus, simultaneously for every $0\le t\le n$, +\begin{equation}\label{eq:feedback-cancellation} + R_tx=\sum_{i=1}^t\varepsilon_ib_i. +\end{equation} +Since $x\in2D_*\mathcal D$, all these states have norm less than +$2D_*A_0\sqrt d$. Dividing~\eqref{eq:feedback-cancellation} by $\sigma$ +proves Theorem~\ref{thm:signing} with the absolute constant +$C=2D_*A_0/\sigma$. +\end{proof} + +\begin{figure}[htbp] + \centering + \resizebox{0.96\textwidth}{!}{% + \begin{tikzpicture}[ + >=stealth, + every node/.style={font=\small}, + state/.style={draw,rounded corners,align=center,minimum height=9mm}] + \node[state] (r) at (0,0) {$r$}; + \node[state] (contract) at (3.7,0) + {$C_tr=r-\eta_tb_tb_t^Tr$}; + \node[draw,circle,inner sep=2pt] (sum) at (7.5,0) {$+$}; + \node[state] (out) at (10.5,0) {$r+\varepsilon_tb_t$}; + \node[state] (incoming) at (3.7,-1.5) + {$b_tx_{d+t}=b_t(\eta_tb_t^Tr+\varepsilon_t)$}; + \draw[->] (r) -- (contract); + \draw[->] (contract) -- (sum); + \draw[->] (sum) -- (out); + \draw[->] (r.south) -- (0,-1.5) -- (incoming.west); + \draw[->] (incoming.east) -- (7.5,-1.5) -- (sum.south); + \end{tikzpicture}} + \caption{For the final coefficient vector, put $r=R_{t-1}x$. + The predictor term in the incoming coordinate restores exactly the + component removed by $C_t$, leaving the signed increment + $\varepsilon_tb_t$.} + \label{fig:cancellation} +\end{figure} + +\input{brownian} + +It remains to prove the two analytic inputs. We first construct the domain +transformations behind Proposition~\ref{prop:adaptive}, and then establish +the uniform energy bound in Proposition~\ref{prop:filtered}. diff --git a/preprints/The-Euclidean-Steinitz-Bergstrom-theorem-September-24-2026/build/brownian.tex b/preprints/The-Euclidean-Steinitz-Bergstrom-theorem-September-24-2026/build/brownian.tex new file mode 100644 index 0000000000000000000000000000000000000000..7fe6c395778a075d264587b84a8f3ae5eecb78eb --- /dev/null +++ b/preprints/The-Euclidean-Steinitz-Bergstrom-theorem-September-24-2026/build/brownian.tex @@ -0,0 +1,139 @@ +\subsection{Brownian proofs of the slab bound} + +We now prove Lemma~\ref{lem:slab-energy}, which bounded the cost of the +predictor constraints. Brownian survival converts intersection with slabs +into an event to which Gaussian correlation applies. The two Brownian +facts below will also be used in the remaining analytic arguments. + +\begin{lemma}[Brownian survival rate]\label{lem:brownian-rate} +Let $D\subset\R^m$ be bounded, nonempty, open and convex, and let $Q$ be +positive definite and diagonal. If $W$ is Brownian motion with covariance +rate $2Q$, started at $x\in D$, then +\begin{equation}\label{eq:brownian-rate} + \lim_{T\to\infty}-\frac1T + \log\mathbb P_x\{W(s)\in D\text{ for all }0\le s\le T\} + =\lambda_Q(D). +\end{equation} +In particular, the starting point may be $0$ when $D$ is centrally +symmetric. +\end{lemma} + +\begin{proof} +The generator of $W$ is $\tr(Q\nabla^2)$. Let +$\tau_D=\inf\{s\ge0:W(s)\notin D\}$, and let +\[ + P_s^D h(x)=\mathbb E_x[h(W(s))\mathbf 1_{\{s<\tau_D\}}] +\] +be its killed semigroup. The corresponding nonnegative self-adjoint +operator is the Dirichlet realization of $-\tr(Q\nabla^2)$, with form +$\int_D\nabla g^TQ\nabla g$ on $H_0^1(D)$. +These standard Dirichlet facts apply here because bounded convex domains +are connected and Lipschitz; see, for example, +\cite[Section~2 and Appendix~C]{DaviesSimon1984}, after the linear change +of coordinates for $Q$ and the time normalization above. +The killed representation can also be obtained first on smooth interior +domains and then by exhaustion: their compactly supported smooth +functions exhaust the form domain, while any continuous path remaining +in $D$ on a compact time interval has compact image in $D$. + +Write $\lambda=\lambda_Q(D)$. Compactness of +$H_0^1(D)\hookrightarrow L^2(D)$ and the spectral theorem give +$\lVert P_s^D\rVert_{2\to2}=e^{-s\lambda}$ and a first eigenfunction +$\varphi$ that can be chosen nonnegative by taking the absolute value +of a Rayleigh minimizer. Interior elliptic regularity and the strong +maximum principle make $\varphi$ strictly positive at every point of $D$. +It is bounded: free heat +kernel domination and +$\varphi=e^\lambda P_1^D\varphi$ give this directly by +Cauchy--Schwarz. Consequently, +\[ + e^{-T\lambda}\varphi(x) + =P_T^D\varphi(x) + \le \lVert\varphi\rVert_\infty\, + \mathbb P_x\{\tau_D>T\}. +\] +For the reverse bound, let $p_Q$ denote the free heat kernel. For $T\ge1$, +the semigroup property and free kernel domination yield +\begin{align*} + \mathbb P_x\{\tau_D>T\} + &\le \lVert p_Q(1,x,\cdot)\rVert_2 + \lVert P_{T-1}^D\mathbf 1_D\rVert_2\\ + &\le \lVert p_Q(1,x,\cdot)\rVert_2\,|D|^{1/2} + e^{-(T-1)\lambda}. +\end{align*} +Both prefactors are finite for the fixed $D,Q,x$. +Taking logarithms and dividing by $T$ proves +\eqref{eq:brownian-rate}. Finally, a nonempty open convex set symmetric +about $0$ contains $0$ as an interior point. +\end{proof} + +We will use the following path form of Gaussian correlation. Its +finite-grid proof also records how closed constraints are handled. + +\begin{lemma}[Gaussian correlation for path constraints] +\label{lem:gaussian-path-correlation} +Let $X:[0,T]\to\R^m$ be a centered Gaussian process with continuous paths. +For $i=1,\ldots,r$, let $E_i$ be an event of the form +\[ + E_i=\bigcap_{j=1}^{r_i} + \{A_{ij}X(s)\in K_{ij}\text{ for all }s\in I_{ij}\}, +\] +where $A_{ij}$ is linear, $K_{ij}$ is a closed centrally symmetric convex +set in its target space, and $I_{ij}\subset[0,T]$ is a closed interval. +Then +\[ + \mathbb P\Bigl(\bigcap_{i=1}^r E_i\Bigr) + \ge \prod_{i=1}^r\mathbb P(E_i). +\] +\end{lemma} + +\begin{proof} +Replace each interval by a finite grid. All the constraints are then +closed centrally symmetric convex conditions on one finite-dimensional +centered Gaussian vector. Represent that vector as a linear image of a +standard Gaussian on its linear support; thus a possibly singular +covariance creates no difficulty. Royen's Gaussian correlation +inequality~\cite{Royen2014}, iterated over the finitely many events, +gives the desired product bound on the grids. +Now refine nested grids whose unions are dense in the respective +intervals. Closedness of each $K_{ij}$ and path continuity identify the +decreasing limit with the stated event. Continuity of probability from +above proves the assertion. +\end{proof} + +\begin{proof}[Proof of Lemma~\ref{lem:slab-energy}] +The set $K$ is again a bounded open centrally symmetric convex +neighborhood of $0$. Take Brownian motion $W$ of covariance rate $2Q$ +started at $0$. On $[0,T]$, impose the closed constraints +\[ + W(s)\in\overline{D_*D},\qquad + |m_iW(s)|\le\delta/2\quad(1\le i\le n). +\] +Their simultaneous occurrence implies survival in $K$: +$\overline{D_*D}\subset 2D_*D$, since $0$ is interior to the convex set +$D$, and the half-threshold slabs lie inside the open full-threshold +slabs. By Lemma~\ref{lem:gaussian-path-correlation}, and then by replacing +each closed-event probability by its smaller open-event probability, +\begin{align*} + &\mathbb P_0\{W([0,T])\subset K\}\\ + &\quad\ge + \mathbb P_0\{W([0,T])\subset D_*D\} + \prod_{i=1}^n + \mathbb P_0\{|m_iW(s)|<\delta/2\text{ for all }0\le s\le T\}. +\end{align*} +For a nonzero row $m_i$, the process $m_iW$ is one-dimensional Brownian +motion with covariance rate $2c_i$, where $c_i=m_iQm_i^T>0$. +The first Dirichlet eigenfunction of $-c_i\,d^2/dx^2$ on +$(-\delta/2,\delta/2)$ is $\cos(\pi x/\delta)$, with eigenvalue +$\pi^2c_i/\delta^2$. Lemma~\ref{lem:brownian-rate} gives the corresponding +survival exponent. A zero row gives the identically zero process, so +its probability is one and its cost is zero. + +Take negative logarithms, divide by $T$, and let $T\to\infty$. +Lemma~\ref{lem:brownian-rate} applies to $K$ and $D_*D$, and the change +of variables in the Rayleigh quotient gives +$\lambda_Q(D_*D)=D_*^{-2}\lambda_Q(D)$. +This proves~\eqref{eq:slab-energy}. +The argument used only inclusions between closed and open events; it +requires no assertion about equality of their survival rates. +\end{proof} diff --git a/preprints/The-Euclidean-Steinitz-Bergstrom-theorem-September-24-2026/build/consequences.tex b/preprints/The-Euclidean-Steinitz-Bergstrom-theorem-September-24-2026/build/consequences.tex new file mode 100644 index 0000000000000000000000000000000000000000..38a72b5ca4b9d07aa578653f72e4a83a77dbf9e9 --- /dev/null +++ b/preprints/The-Euclidean-Steinitz-Bergstrom-theorem-September-24-2026/build/consequences.tex @@ -0,0 +1,137 @@ +\section{Further consequences}\label{sec:consequences} + +The signed-prefix theorem gives bounds for other finite-dimensional norms +and, through matrix transference, for arrays whose rows can be permuted +separately. + +\subsection{Finite-dimensional \texorpdfstring{$\ell_p$}{lp} bounds} + +\begin{corollary}[Finite-dimensional $\ell_p$ bounds]\label{cor:lp-steinitz} +Let $1\le p\le\infty$, with $1/\infty=0$, and let $d,N\ge1$ be integers. +Every sequence $v_1,\ldots,v_N\in\R^d$ with $\lVert v_i\rVert_p\le1$ +admits signs $\varepsilon_i\in\{-1,1\}$ such that +\[ + \max_{0\le k\le N} + \left\lVert\sum_{i=1}^k\varepsilon_i v_i\right\rVert_p + \le C d^{\max\{1/p,\,1-1/p\}}. +\] +If additionally $\sum_{i=1}^N v_i=0$, there is a permutation +$\pi\in\mathfrak S_N$ such that +\[ + \max_{0\le k\le N} + \left\lVert\sum_{i=1}^k v_{\pi(i)}\right\rVert_p + \le C d^{\max\{1/p,\,1-1/p\}}. +\] +Here $C$ is the same absolute constant as in +Theorems~\ref{thm:signing} and~\ref{thm:steinitz}, independent of $p,d,N$. +For $1\le p\le2$, both bounds have sharp dimension order $d^{1/p}$: +for every $d$ there is a unit-ball sequence for which every signing has +a prefix of norm at least $d^{1/p}$, and a finite indexed zero-sum +unit-ball family for which every permutation has a prefix of norm at +least $\tfrac13d^{1/p}$. +\end{corollary} + +\begin{proof} +For $1\le p\le2$, the norm comparisons +\[ + \lVert x\rVert_2\le\lVert x\rVert_p + \le d^{1/p-1/2}\lVert x\rVert_2 +\] +allow direct application of Theorems~\ref{thm:signing} +and~\ref{thm:steinitz}, followed by the bound $C d^{1/p}$ on all prefixes. +For $2\le p\le\infty$, use +\[ + \lVert x\rVert_p\le\lVert x\rVert_2 + \le d^{1/2-1/p}\lVert x\rVert_p. +\] +Divide the input vectors by $d^{1/2-1/p}$, apply the corresponding +Euclidean theorem, and scale back to obtain $C d^{1-1/p}$. +Scaling preserves the zero sum, and in each case the one signing or +permutation controls all prefixes. At $p=\infty$ this gives $Cd$. + +For the lower bounds, let $1\le p\le2$. The ordered standard basis +$e_1,\ldots,e_d$ has final signed sum of norm $d^{1/p}$ for every signing. +For the ordering problem, take $e_1,\ldots,e_d$ and $d$ copies of +$w=-\tfrac1d(1,\ldots,1)$. This indexed family sums to zero and lies in +the unit ball, since $\lVert w\rVert_p=d^{1/p-1}\le1$. +If $d\ge2$, consider any permutation and its prefix just after the +$k=\lfloor d/2\rfloor$-th basis vector. If $t$ copies of $w$ have +appeared, put $a=t/d\in[0,1]$. The prefix sum $s$ has $k$ coordinates +equal to $1-a$ and $d-k$ equal to $-a$. Since both $k$ and $d-k$ are +at least $d/3$, convexity gives +\[ + \lVert s\rVert_p^p + =k(1-a)^p+(d-k)a^p + \ge\frac d3\bigl((1-a)^p+a^p\bigr) + \ge\frac d3\,2^{1-p}. +\] +Thus $\lVert s\rVert_p\ge\tfrac12(2d/3)^{1/p} +\ge\tfrac13d^{1/p}$, for every permutation. When $d=1$, the family +is $(1,-1)$ and either permutation has a prefix of norm one. +\end{proof} + +\subsection{A colorful row-permutation consequence} + +B\'ar\'any's matrix transference principle turns the signed-prefix bound +into a row-permutation bound for arrays. Each row may have its own +permutation, while the controlled sums aggregate the same number of +entries from every row. + +\begin{corollary}[Euclidean colorful Steinitz bound]\label{cor:colorful-steinitz} +Let $d,k,n\ge1$ be integers and let $a_i^j\in\R^d$, for +$1\le j\le k$ and $1\le i\le n$, satisfy +\[ + \lVert a_i^j\rVert_2\le1, + \qquad + \sum_{j=1}^k\sum_{i=1}^n a_i^j=0. +\] +There is one tuple of permutations +$(\pi_1,\ldots,\pi_k)\in\mathfrak S_n^k$ such that +\[ + \max_{0\le m\le n} + \left\lVert\sum_{j=1}^k\sum_{i=1}^m a_{\pi_j(i)}^j\right\rVert_2 + \le 2C\sqrt d, +\] +where $C$ is the absolute constant in Theorem~\ref{thm:signing}. +The same tuple works for every $m$ and may be chosen jointly from the +whole array. Only the global total sum is assumed to vanish; the bound +controls the displayed aggregate synchronous prefixes, not each row +separately. +\end{corollary} + +\begin{proof} +For an arbitrary $k\times n$ array $B=(b_i^j)$ in the Euclidean unit +ball $B_2^d$, with no zero-sum condition, list its entries column by +column: +\[ + v_{(i-1)k+j}=b_i^j + \qquad(1\le i\le n,\ 1\le j\le k). +\] +Apply Theorem~\ref{thm:signing} to this sequence of length $kn$. With +$\varepsilon_i^j=\varepsilon_{(i-1)k+j}$, every complete-column prefix +satisfies +\[ + \left\lVert\sum_{j=1}^k\sum_{i=1}^m\varepsilon_i^j b_i^j\right\rVert_2 + = + \left\lVert\sum_{\ell=1}^{km}\varepsilon_\ell v_\ell\right\rVert_2 + \le C\sqrt d. +\] +This is the column-major reduction in the proof of +\cite[Lemma~2.1]{Barany2024Matrix}, with the present signed-prefix bound +in place of its dimension-linear bound. Thus B\'ar\'any's matrix signed +constant satisfies $V(B_2^d)\le C\sqrt d$, uniformly in $k,n$. +Here $V$ allows an entrywise signing of an arbitrary array, whereas the +row-permutation constant $U$ ranges over total-zero arrays and controls +the same aggregate prefixes. His transference theorem +\cite[Theorem~2.2]{Barany2024Matrix} gives $U(B_2^d)\le2V(B_2^d)$ for +every number of rows. Shorter auxiliary arrays may be padded with +trailing zero columns when using the bound uniformly in the number of +columns. Since $B_2^d$ is a centrally symmetric convex body, this gives +the asserted bound. +\end{proof} + +Taking $k=1$ in the regular-simplex example in the introduction shows that no +$o(\sqrt d)$ bound can hold uniformly over these arrays. Thus the order +in the dimension is sharp; the numerical constant $2C$ is not asserted +to be optimal. The row permutations are existential, with no online or +efficient algorithm claimed. diff --git a/preprints/The-Euclidean-Steinitz-Bergstrom-theorem-September-24-2026/build/introduction.tex b/preprints/The-Euclidean-Steinitz-Bergstrom-theorem-September-24-2026/build/introduction.tex new file mode 100644 index 0000000000000000000000000000000000000000..a2744e9bdb2d3928a96c6b3b76716755ee58c81b --- /dev/null +++ b/preprints/The-Euclidean-Steinitz-Bergstrom-theorem-September-24-2026/build/introduction.tex @@ -0,0 +1,206 @@ +\section{Introduction} + +Let $v_1,\ldots,v_N$ be vectors in the Euclidean unit ball of $\R^d$. +There are two natural ways to keep their partial sums small: choose signs +in a prescribed order, or, when the family sums to zero, choose an order +without changing any signs. The difficulty in the first problem is to +control every prefix by one choice of signs, with a bound independent of +$N$. We prove such a bound on the square-root scale in the dimension. + +\begin{theorem}[Prescribed-order signed prefixes]\label{thm:signing} +There is an absolute constant $C$ such that, for all integers $d,N\ge1$ and +every sequence $v_1,\ldots,v_N\in\R^d$ with $\lVert v_i\rVert_2\le1$, +there are signs $\varepsilon_1,\ldots,\varepsilon_N\in\{-1,1\}$ for which +\[ + \max_{0\le k\le N} + \left\lVert\sum_{i=1}^k\varepsilon_i v_i\right\rVert_2\le C\sqrt d. +\] +The same signs control all prefixes, and $C$ is independent of $d$, $N$ +and the sequence. +\end{theorem} + +The zero-sum ordering problem is quantified by the Euclidean Steinitz +constant. For an indexed family with $\sum_{i=1}^N v_i=0$, define +\[ + \beta(v_1,\ldots,v_N) + =\min_{\pi\in\mathfrak S_N}\; + \max_{0\le k\le N}\left\lVert\sum_{i=1}^k v_{\pi(i)}\right\rVert_2, +\] +where $\mathfrak S_N$ is the set of permutations of $\{1,\ldots,N\}$ +and an empty sum is zero. Let $S_2(d)$ be the supremum of $\beta$ over +all finite indexed zero-sum families of Euclidean unit-ball vectors in +$\R^d$. The Euclidean Steinitz--Bergstr\"om conjecture asks whether +$S_2(d)=O(\sqrt d)$. Theorem~\ref{thm:signing} gives its affirmative resolution. + +\begin{theorem}[Euclidean Steinitz--Bergstr\"om bound]\label{thm:steinitz} +There is an absolute constant $C$ such that, for all integers $d,N\ge1$ and +every indexed family $v_1,\ldots,v_N\in\R^d$ satisfying +$\lVert v_i\rVert_2\le1$ and $\sum_{i=1}^N v_i=0$, some permutation +$\pi\in\mathfrak S_N$ satisfies +\[ + \max_{0\le k\le N}\left\lVert\sum_{i=1}^k v_{\pi(i)}\right\rVert_2 + \le C\sqrt d. +\] +The constant in Theorem~\ref{thm:signing} also suffices here. +\end{theorem} + +Both statements allow repetitions and zero vectors. The empty-sequence +and dimension-zero extensions have every partial sum equal to zero. The construction is +existential: its choices may depend on the complete input sequence. Its +successive choice of signs does not supply an online rule or an efficient +algorithm. + +The order of growth in Theorem~\ref{thm:steinitz} is necessary. In the +$d$-dimensional hyperplane orthogonal to the all-ones vector +$\mathbf1\in\R^{d+1}$, let $e_i$ denote the standard coordinate vectors and take +\[ + u_i=\sqrt{\frac{d+1}{d}}\left(e_i-\frac{\mathbf1}{d+1}\right), + \qquad 1\le i\le d+1. +\] +These unit vectors sum to zero and satisfy $\langle u_i,u_j\rangle=-1/d$ +for $i\ne j$. Hence every sum of $k$ distinct members has squared norm +$k(d+1-k)/d$. With $k=\lfloor(d+1)/2\rfloor$, this is at least $d/4$, +regardless of the ordering. Thus +\[ + \tfrac12\sqrt d\le S_2(d)\le C\sqrt d\qquad(d\ge1). +\] + +\subsection{From signs to an ordering} + +We give the short transference argument at once. Chobanyan's +transference principle~\cite{Chobanyan1994} relates rearrangements and +signed sums. We use its finite positive-forward, negative-reverse form +\cite[Theorem~2.1 and Remark~1]{ChobanyanEtAl2023}. + +\begin{proof}[Proof of Theorem~\ref{thm:steinitz} from Theorem~\ref{thm:signing}] +Fix an indexed zero-sum family and choose an order attaining its minimum +$\beta$; this exists because there are only finitely many permutations. +Relabel the vectors in that order. Put +\[ + A_j=\sum_{i=1}^jv_i, + \qquad B_j=\sum_{i=1}^j\varepsilon_i v_i, + \qquad 0\le j\le N, +\] +where the signs come from Theorem~\ref{thm:signing}. Hence +$\lVert A_j\rVert_2\le\beta$ and $\lVert B_j\rVert_2\le C\sqrt d$. +The positive-sign and negative-sign selections from an original prefix +sum respectively to +\[ + P_j=\frac{A_j+B_j}{2},\qquad M_j=\frac{A_j-B_j}{2}. +\] +Place all positive-sign indices in their original order, followed by all +negative-sign indices in reverse original order. A partial sum in the +first block is some $P_j$. In the second block, the unused indices are +the negative-sign indices of an original prefix. Since the total sum +is zero, the current partial sum is therefore some $-M_j$. This also +covers either block being empty. Every new partial sum has norm at most +$(\beta+C\sqrt d)/2$. By minimality, +\[ + \beta\le\frac{\beta+C\sqrt d}{2}, + \qquad\text{and hence}\qquad \beta\le C\sqrt d. +\] +This permutes indices and so preserves all multiplicities. +\end{proof} + +The signed-prefix bound also yields finite-dimensional $\ell_p$ estimates +and a colorful row-permutation theorem. We develop these distinct +consequences in Section~\ref{sec:consequences}. + +\subsection{Historical context} + +The finite rearrangement theorem goes back to Steinitz's work on +conditionally convergent vector series; Ambrus and Heck give a historical +account in \cite[Section~2]{AmbrusHeck2026}. +The dimension bound for zero-sum rearrangement holds for arbitrary norms: +Grinberg and Sevast'yanov~\cite[Theorem~1]{GrinbergSevastyanov1980} prove +that the partial sums can be kept within $d$ times the unit ball. +The smaller square-root scale in Euclidean space has a long history; +Behrend discusses this expected growth in +\cite[p.~108]{Behrend1954}. The modern formulation and its attribution +to Bergstr\"om are recorded by Ambrus and Heck +\cite[Conjecture~5]{AmbrusHeck2026}. Their Theorem~7 reduces the +asymptotic question to a relaxed rearrangement problem for vectors with +norms in $[1-\varepsilon,1]$, with $0<\varepsilon\le1$ fixed independently +of the dimension. In that problem each prefix is compared with its +proportional share of the total sum, rather than with zero. +For prescribed-order Euclidean signing, the square-root-dimension +question is stated explicitly by Bansal, Jiang, Meka, Singla and +Sinha~\cite[Section~6, Conjecture~6.3]{BansalEtAl2021Full}, following +Banaszczyk~\cite{Banaszczyk2012}. Their discussion records Banaszczyk's +bound $O(\sqrt d+\sqrt{\log N})$; the length term becomes significant when +$N$ is large compared with the dimension. In a 2026 preprint, Dutta, Jha +and Jiang state an efficient prescribed-order signing bound +\[ + O\bigl(\sqrt d+d^{1/4}\log^{7/4}N\bigr), +\] +and the corresponding zero-sum ordering bound +\cite[Theorem~1.1 and Corollary~1.2]{DuttaJhaJiang2026}. +Theorem~\ref{thm:signing} gives an absolute multiple of $\sqrt d$ +for every finite length. + +Terminal vector balancing has a different quantifier. For columns in the +Euclidean unit ball, Li~\cite[Theorem~1.1]{Li2026} bounds the final +signed sum in infinity norm by an absolute constant and gives a +deterministic polynomial-bit algorithm on rational input. That terminal +statement implies a Euclidean bound of order $\sqrt d$ for one sum. +Applying it separately to each prefix permits the signs to change with +the prefix; Theorem~\ref{thm:signing} instead supplies one common signing. +The present all-prefix estimate is Euclidean and does not assert a +constant infinity-norm all-prefix bound. + +\subsection{Proof strategy} + +Gaussian convex bodies are a central setting for vector balancing, notably +in the theorem of Banaszczyk~\cite{Banaszczyk1998}. Our proof takes place +in a space of coefficients: one coordinate is +reserved for each incoming vector, together with $d$ initial state +coordinates. A sequence of linear maps sends these coefficients to +states in $\R^d$. At each step an invertible contraction acts on the old +state and the next coefficient injects a multiple of the incoming vector. +The body of coefficients for which every state is bounded will provide +the simultaneous control of all prefixes. + +Two analytic ingredients make this construction possible. First, if a +bounded symmetric convex body has small Dirichlet energy for every +positive diagonal weight, it supports a single square root of a probability density +with small derivatives in every coordinate. A lifted Steiner +symmetrization preserves these bounds and permits a coordinate step +of the form $q+\varepsilon$, where $|q|\le1/4$ and +$\varepsilon\in\{-1,1\}$. This adapts the directional-density and +symmetrization method of Guo, Fang and Lu +\cite[Sections~3--4]{GuoFangLu2026}.\footnote{Guo, Fang and Lu +credit the Odin Automatic AI Research Agent for their proof.} +Bandeira~\cite{Bandeira2026} gives a related formulation for terminal +balancing in terms of Dirichlet eigenvalues. Akbas and Sra +\cite[Lemmas~2.2--2.3]{AkbasSra2026} develop the common quadratic-energy +witness and lifted $L^2$-symmetrization argument. Here the hypotheses use +diagonal weights, and the coordinate shifts may depend on earlier choices. +We prove the required energy and shifted-step statements in full. + +Second, the filtered body has a diagonal Dirichlet-energy bound +independent of the number of filters. To prove this, we follow a stationary Ornstein--Uhlenbeck process +in coefficient space and localize its filtered states to dyadic spectral +scales. A matrix estimate controls changes at one scale by an additive +trace budget, including gains inserted between noncommuting contractions. +At a fixed scale, this estimate gives a uniform positive probability +for all constraints in a group of indices with bounded accumulated energy, +over a matched time interval. Gaussian correlation combines these groups, +scales and time intervals; the exponential survival rate yields the energy +bound. + +The two ingredients meet through a scalar predictor. The allowed shift +in the incoming coefficient exactly compensates for the contraction of +the old state. Consequently the final states are the signed partial +sums themselves. We give this complete construction in +Section~\ref{sec:construction}, after stating the two analytic +propositions. Section~\ref{sec:shifted-signs} proves the shifted-sign proposition. +Section~\ref{sec:matrix} establishes the matrix variation estimate, and +Section~\ref{sec:survival} uses it to prove the filtered-body energy bound. +Section~\ref{sec:consequences} develops the norm-comparison and colorful +row-permutation consequences. + +Throughout, all unspecified constants are absolute. Vectors and matrices +are real; $\lVert\cdot\rVert_\op$ and $\lVert\cdot\rVert_\HS$ denote +operator and Hilbert--Schmidt norms. For symmetric matrices, $A\le B$ +means that $B-A$ is positive semidefinite. Functions of positive definite +matrices are defined by the spectral calculus. diff --git a/preprints/The-Euclidean-Steinitz-Bergstrom-theorem-September-24-2026/build/main.tex b/preprints/The-Euclidean-Steinitz-Bergstrom-theorem-September-24-2026/build/main.tex new file mode 100644 index 0000000000000000000000000000000000000000..a043d8287c6ba4d617b82160e63df3d80432df51 --- /dev/null +++ b/preprints/The-Euclidean-Steinitz-Bergstrom-theorem-September-24-2026/build/main.tex @@ -0,0 +1,60 @@ +\documentclass[11pt]{article} +\usepackage[T1]{fontenc} +\usepackage[utf8]{inputenc} +\usepackage{lmodern} +\usepackage[margin=1in]{geometry} +\usepackage{amsmath,amssymb,amsthm,mathtools} +\def\DoNotLoadEpstopdf{} +\usepackage{graphicx} +\usepackage{tikz} +\usetikzlibrary{arrows.meta,positioning,calc} +\usepackage{microtype} +\usepackage{xurl} +\usepackage[colorlinks=true,linkcolor=blue!45!black,citecolor=blue!45!black,urlcolor=blue!45!black]{hyperref} +\usepackage{bookmark} +\hypersetup{ + pdftitle={The Euclidean Steinitz--Bergstr\"om theorem}, + pdfauthor={OpenAI} +} +\newtheorem{theorem}{Theorem}[section] +\newtheorem{proposition}[theorem]{Proposition} +\newtheorem{lemma}[theorem]{Lemma} +\newtheorem{corollary}[theorem]{Corollary} +\theoremstyle{definition} +\newtheorem{definition}[theorem]{Definition} +\theoremstyle{remark} +\newtheorem{remark}[theorem]{Remark} +\newcommand{\R}{\mathbb R} +\newcommand{\Z}{\mathbb Z} +\newcommand{\HS}{\mathrm{HS}} +\newcommand{\op}{\mathrm{op}} +\DeclareMathOperator{\tr}{tr} +\DeclareMathOperator{\supp}{supp} +\DeclareMathOperator{\diag}{diag} +\title{The Euclidean Steinitz--Bergstr\"om theorem} +\author{OpenAI} +\date{September 24, 2026} +\begin{document} +\maketitle +\begin{abstract} +Every prescribed-order finite sequence of vectors in the Euclidean unit +ball of $\R^d$ has one signing for which every signed prefix has norm at +most $C\sqrt d$, with $C$ absolute and independent of the sequence length. +Consequently, every indexed zero-sum family of unit-ball vectors admits an +ordering with the same bound for its unsigned partial sums. This determines +the Euclidean Steinitz constant up to absolute factors, +$S_2(d)=\Theta(\sqrt d)$, and resolves the Euclidean +Steinitz--Bergstr\"om conjecture. +\end{abstract} +{\small\tableofcontents} +\clearpage +\input{introduction} +\input{assembly} +\input{signing} +\input{matrix} +\input{survival} +\input{consequences} +\clearpage +\bibliographystyle{plain} +\bibliography{references} +\end{document} diff --git a/preprints/The-Euclidean-Steinitz-Bergstrom-theorem-September-24-2026/build/matrix.tex b/preprints/The-Euclidean-Steinitz-Bergstrom-theorem-September-24-2026/build/matrix.tex new file mode 100644 index 0000000000000000000000000000000000000000..01ca97a86e99e24b239b139055b6f72378cd3aa3 --- /dev/null +++ b/preprints/The-Euclidean-Steinitz-Bergstrom-theorem-September-24-2026/build/matrix.tex @@ -0,0 +1,602 @@ +\section{Matrix variation along the filter} +\label{sec:matrix} + +The proof of Proposition~\ref{prop:filtered} must control all the states +of a filtered Gaussian vector, even when the sequence is arbitrarily +long. We first prove a deterministic estimate that measures variation +of the filter by a total energy, rather than by the number of steps. +Spectral localization will give separate bounds for changes in the +sequence index and in Ornstein--Uhlenbeck time. + +Throughout this section, fix a recursion as in +Proposition~\ref{prop:filtered} and a positive definite diagonal matrix +$Q\in\R^{m\times m}$. The sequence index is $t\in\{0,\ldots,n\}$. +For a real matrix $A$, write +$\lVert A\rVert_{\HS}^2=\tr(AA^\top)$. + +\subsection{Trace budgets and spectral scales} + +The support of $R_{t-1}$ is contained in the initial block and the first +$t-1$ new coordinates. Hence the new coordinate has no cross term in +$R_tR_t^\top$, and induction gives +\begin{equation} + R_tR_t^\top=I_d, + \qquad + L_t:=R_tQR_t^\top + =C_tL_{t-1}C_t^\top+Q_{d+t,d+t}b_tb_t^\top. + \label{eq:speed-recursion} +\end{equation} +In particular $L_t$ is positive definite. Every $C_t$ is a contraction, +since $C_tC_t^\top\preceq I_d$. + +Let $\ell_t$ be the trace lost in the contraction in +\eqref{eq:speed-recursion}, and let $g_t$ be the trace added afterward. +Both are nonnegative. Since $\lVert b_t\rVert\le1$, +\begin{equation} + \sum_{t=1}^n g_t\le\sum_{t=1}^n Q_{d+t,d+t}, + \qquad + \sum_{t=1}^n\ell_t + =\tr L_0+\sum_{t=1}^n g_t-\tr L_n + \le\tr Q. + \label{eq:trace-budgets} +\end{equation} +These two budgets are independent of the length of the sequence. + +Fix the following piecewise linear cutoff on $(0,\infty)$: +\[ + f_0(x)= + \begin{cases} + 0,&00$, +\[ + \sum_{\theta\in2^{\mathbb Z}} + \bigl(p'(x/\theta)+p_1'(x/\theta)\bigr)\le C. +\] +For $p'$, use the bounds $u^{1/2}$ for $u\le1$ and $u^{-2}$ for +$u\ge1$; for $p_1'$, use $u^{1/2}$ and $u^{-1}$ respectively. +All four dyadic sums are geometric. Pairing ordered eigenvalues +before and after each contraction or gain, and integrating the last +bound between each pair, proves +\begin{equation} + \sum_{\theta\in2^{\mathbb Z}}\theta E_\theta(0,n) + \le C\left(\sum_i\ell_i+\sum_i g_i\right) + \le C\tr Q. + \label{eq:scale-energy} +\end{equation} +The factor $\theta$ cancels the factor $1/\theta$ from differentiating +$p(x/\theta)$ or $p_1(x/\theta)$. + +Here is the variation estimate to be used in the next section. + +\begin{lemma}[Weighted increment estimate] +\label{lem:increment} +There are absolute constants $e_0>0$ and $C<\infty$ with the following +property. For every recursion in Proposition~\ref{prop:filtered}, +every positive definite diagonal $Q$, every dyadic $\theta$, and +$0\le s\le t\le n$, if $E=E_\theta(s,t)\le e_0$, then +\begin{equation} + \bigl\lVert(F_t^\theta-F_s^\theta) + (I_m+Q/\theta)^{1/2}\bigr\rVert_{\HS}^2 + \le C\bigl(1+k_t(\theta)\bigr)E^{1/a}, + \qquad a=\tfrac32. + \label{eq:increment} +\end{equation} +\end{lemma} + +We prove this in three steps. A scalar inequality first controls +changes of spectral cutoffs. Exact matrix expansions then compare +the two endpoints using the old part of the state and the accumulated +new gains. Finally, a separate estimate bounds those accumulated +gains by the additive energy, even through noncommuting contractions. + +\subsection{A scalar comparison} + +For $x,y>0$, define +\[ + \mathcal J(x,y)=p(x)-p(y) + +\left(\frac yx-1\right)\frac{w(x)+w(y)}2. +\] + +\begin{lemma} +\label{lem:scalar-comparison} +There is an absolute $c>0$ such that, for all $x,y>0$, +\begin{equation} + \begin{split} + \mathcal J(x,y)&\ge c(1+y)|f_0(x)-f_0(y)|^2,\\ + \frac yx\mathcal J(y,x)&\ge c(1+y)|f_0(x)-f_0(y)|^2. + \end{split} + \label{eq:scalar-comparison} +\end{equation} +\end{lemma} + +\begin{proof} +The function $u\mapsto\log w(e^u)=\min\{au,-u\}$ is +$a$-Lipschitz. Write $r=y/x$. For $r<1$, the two endpoint lower +bounds give +\[ + w(u)\ge + \max\{w(x)(u/x)^a,\,w(y)(y/u)^a\},\qquad y\le u\le x. +\] +The two expressions cross inside $[y,x]$, by the same Lipschitz +bound. Integrating them against $du/u$ yields +\[ + p(x)-p(y)\ge + \frac{w(x)+w(y)-2\sqrt{w(x)w(y)}\,r^{a/2}}a. +\] +For $r>1$, integration of the minimum of the corresponding endpoint +upper bounds on $[x,y]$ gives +\[ + p(y)-p(x)\le + \frac{2\sqrt{w(x)w(y)}\,r^{a/2}-w(x)-w(y)}a. +\] +Using $2\sqrt{w(x)w(y)}\le w(x)+w(y)$ in either case, we obtain +\begin{equation} + \mathcal J(x,y)\ge\frac{w(x)+w(y)}2 B(r), + \qquad + B(r)=r-1-\frac{r^{a/2}-1}{a/2}. + \label{eq:scalar-remainder} +\end{equation} +If $b=a/2=3/4$, then $B(1)=B'(1)=0$ and +$B''(r)=(1-b)r^{b-2}>0$. Moreover +$B(0+)=1/b-1>0$ and $B(r)/r\longrightarrow1$ at infinity. +Thus, with an absolute positive constant, +\[ + B(r)\ge c + \begin{cases} + 1,&01$. +The coefficient in this trace has the exact conjugation identity +\begin{align} + J^{1/2}S^{-1/2}A^{a-1}S^{-1/2}J^{1/2} + &=T^\top A^{a-1}T\notag\\ + &=H^{a-1}(I_d-H)^{-a}. + \label{eq:gain-conjugation} +\end{align} +The second equality follows from \eqref{eq:relative-spectrum} and +the polar decomposition. The factors on its right commute because +they are functions of $H$; no commutation of $Z_*$ and $V_*$ is +assumed. + +As long as $G\le1$, every eigenvalue of $A$ is at most one, so +\eqref{eq:relative-spectrum} gives $H\preceq I_d/2$. Since +$a-1=1/2$, spectral calculus in $H$ then gives +\[ + H^{a-1}(I_d-H)^{-a}\preceq2^aH^{1/2}. +\] +Also $V_*\preceq D$, and hence +\[ + H\preceq J^{-1/2}DJ^{-1/2}=D(I_d+D)^{-1}. +\] +The square root preserves positive semidefinite order. One direct +proof is the convergent integral formula +\[ + M^{1/2}=\frac1\pi\int_0^\infty + u^{-1/2}M(uI_d+M)^{-1}\,du, + \qquad M\succeq0: +\] +the integrand preserves order because +$M(uI_d+M)^{-1}=I_d-u(uI_d+M)^{-1}$ and inversion reverses order. +It follows that +\[ + H^{1/2}\preceq D^{1/2}(I_d+D)^{-1/2}. +\] +Conjugating \eqref{eq:gain-conjugation} back by $J^{-1/2}$ yields +\[ + S^{-1/2}A^{a-1}S^{-1/2} + \preceq2^aD^{1/2}(I_d+D)^{-3/2}=2^ap_1'(D). +\] +Pair this inequality with $\dot V_*\succeq0$ in +\eqref{eq:gain-derivative}. Since $\dot D=\dot V_*$ during a gain, +we obtain the conditional differential bound +\begin{equation} + \dot G\le c_a\tr\bigl(p_1'(D)\dot V_*\bigr) + =c_a\frac{d}{d\xi}\tr p_1(D), + \qquad G\le1. + \label{eq:gain-differential-bound} +\end{equation} + +\paragraph{Closing the bound.} +Initially $G=0$, and contractions cannot increase it. The sum of +all increases of $\tr p_1(D)$ during gains is at most $E$, by +\eqref{eq:energy-definition}. If $G$ first reached one during a +gain, integration of \eqref{eq:gain-differential-bound} up to that +point would give $G\le c_aE\le1/16$, a contradiction. +Thus the differential bound applies throughout, and +$G\le c_aE$ at the endpoint. In particular, +\[ + A\preceq qI_d,\qquad + V\preceq q(I_d+Z)\preceq q(I_d+X),\qquad + q\le16^{-2/3}<\tfrac14. +\] +If $E=0$, this already gives $V=0$ and the second conclusion in +\eqref{eq:direct-gain}. Suppose now that $q>0$. + +The ordered eigenvalues of $X$ and $Z$ obey +\[ + x_j\ge z_j\ge(1-q)x_j-q, +\] +because $X-Z=V\preceq q(I_d+X)$. For a corresponding pair $x,z$, +if $x\le4q\le1$, then +\[ + p(x)-p(z)\le p(x)=\frac{x^a}{a} + \le\frac4a qx^{a-1}=\frac4a q\phi(x). +\] +If $x>4q$, then $z\ge x/2$ and $x-z\le q(1+x)$. +On $[x/2,x]$ we have $w(u)\le2^aw(x)$ and $u^{-1}\le2/x$, +so +\[ + p(x)-p(z)\le Cq(1+x^{-1})w(x)\le Cq\phi(x). +\] +The last inequality holds for both $x\le1$ and $x\ge1$ directly +from \eqref{eq:scalar-potentials}. Summing over the eigenvalues +proves the trace bound in \eqref{eq:direct-gain}. +\end{proof} + +\begin{proof}[Proof of Lemma~\ref{lem:increment}] +The total variation of $\tr p$ along the separate contraction and +gain steps from $s$ to $t$ is at most $E$. Therefore +\[ + \bigl|\tr p(Y)-\tr p(X)\bigr|\le E. +\] +Together with Lemma~\ref{lem:direct-gain}, this bounds the sum on +the right of \eqref{eq:endpoint-comparison} by +\[ + E+2\tr\bigl(p(X)-p(Z)\bigr) + \le E+CE^{1/a}k_t. +\] +Since $0\le E\le e_0\le1$, we have $E\le E^{1/a}$. +Lemma~\ref{lem:endpoint-comparison} now proves +\eqref{eq:increment}, including $E=0$. +\end{proof} + +The two estimates needed for the probability argument are now in +place. Equation~\eqref{eq:scale-energy} bounds the sum of the +energies over all spectral scales, while +Lemma~\ref{lem:increment} turns a short interval in this energy into +a small Gaussian increment. In particular, zero energy implies +$F_t^\theta=F_s^\theta$ exactly, so repeated indices at zero +distance will not increase the size of a covering in the next section. diff --git a/preprints/The-Euclidean-Steinitz-Bergstrom-theorem-September-24-2026/build/references.bib b/preprints/The-Euclidean-Steinitz-Bergstrom-theorem-September-24-2026/build/references.bib new file mode 100644 index 0000000000000000000000000000000000000000..7194810de6037741f7fc09bf5c9d3f8ed0d045f5 --- /dev/null +++ b/preprints/The-Euclidean-Steinitz-Bergstrom-theorem-September-24-2026/build/references.bib @@ -0,0 +1,200 @@ +@misc{GuoFangLu2026, + author = {Guo, Shengtao and Fang, Ethan X. and Lu, Junwei}, + title = {Vector Balancing via Directional Total Variation}, + year = {2026}, + howpublished = {arXiv:2609.11189}, + note = {Version 1, 10 September 2026}, + url = {https://arxiv.org/abs/2609.11189v1} +} +@article{AmbrusHeck2026, + author = {Ambrus, Gergely and Heck, Rainie}, + title = {A note on the {Steinitz} Lemma}, + journal = {Mathematika}, + volume = {72}, + number = {2}, + pages = {e70085}, + year = {2026}, + doi = {10.1112/mtk.70085}, + url = {https://arxiv.org/abs/2505.09465v3} +} +@misc{DuttaJhaJiang2026, + author = {Dutta, Kunal and Jha, Agastya Vibhuti and Jiang, Haotian}, + title = {Near-Optimal Constructive Bounds for {$\ell_2$} Prefix Discrepancy and {Steinitz} Problems via Affine Spectral Independence}, + year = {2026}, + howpublished = {arXiv:2604.13355}, + note = {Version 1, 14 April 2026}, + url = {https://arxiv.org/abs/2604.13355v1} +} +@article{Royen2014, + author = {Royen, Thomas}, + title = {A simple proof of the {Gaussian} correlation conjecture extended to some multivariate gamma distributions}, + journal = {Far East Journal of Theoretical Statistics}, + volume = {48}, + number = {2}, + pages = {139--145}, + year = {2014}, + url = {https://arxiv.org/abs/1408.1028v2} +} +@article{Borell1975, + author = {Borell, Christer}, + title = {The {Brunn--Minkowski} inequality in {Gauss} space}, + journal = {Inventiones Mathematicae}, + volume = {30}, + pages = {207--216}, + year = {1975}, + doi = {10.1007/BF01425510} +} +@article{Prekopa1973, + author = {Pr{\'e}kopa, Andr{\'a}s}, + title = {On logarithmic concave measures and functions}, + journal = {Acta Scientiarum Mathematicarum (Szeged)}, + volume = {34}, + pages = {335--343}, + year = {1973}, + url = {https://acta.bibl.u-szeged.hu/14411/} +} +@article{DaviesSimon1984, + author = {Davies, E. B. and Simon, Barry}, + title = {Ultracontractivity and the heat kernel for {Schr\"odinger} operators and {Dirichlet} {Laplacians}}, + journal = {Journal of Functional Analysis}, + volume = {59}, + number = {2}, + pages = {335--395}, + year = {1984}, + doi = {10.1016/0022-1236(84)90076-4}, + url = {https://math.caltech.edu/SimonPapers/173.pdf} +} +@article{Bergstrom1931, + author = {Bergstr{\"o}m, Viktor}, + title = {Zwei S{\"a}tze {\"u}ber ebene Vektorpolygone}, + journal = {Abhandlungen aus dem Mathematischen Seminar der Universit{\"a}t Hamburg}, + volume = {8}, + pages = {206--214}, + year = {1931}, + doi = {10.1007/BF02941002} +} +@article{Behrend1954, + author = {Behrend, F. A.}, + title = {The {Steinitz--Gross} theorem on sums of vectors}, + journal = {Canadian Journal of Mathematics}, + volume = {6}, + pages = {108--124}, + year = {1954}, + doi = {10.4153/CJM-1954-013-0} +} +@article{ChobanyanEtAl2023, + author = {Chobanyan, Sergei and Chobanyan, Levon and Gorgadze, Zaza and Ghlonti, Giorgi}, + title = {An Algorithm for Finding a Near-Optimal Rearrangement in the {Steinitz} Functional}, + journal = {Bulletin of TICMI}, + volume = {27}, + number = {1}, + pages = {21--27}, + year = {2023}, + url = {https://www.viam.science.tsu.ge/others/ticmi/blt/vol27_1/3.pdf} +} +@incollection{Chobanyan1994, + author = {Chobanyan, Sergej}, + title = {Convergence A.S. of Rearranged Random Series in {Banach} Space and Associated Inequalities}, + booktitle = {Probability in Banach Spaces, 9}, + series = {Progress in Probability}, + volume = {35}, + publisher = {Birkh{\"a}user}, + pages = {3--29}, + year = {1994}, + doi = {10.1007/978-1-4612-0253-0_1} +} +@article{Banaszczyk1998, + author = {Banaszczyk, Wojciech}, + title = {Balancing vectors and {Gaussian} measures of {$n$}-dimensional convex bodies}, + journal = {Random Structures \& Algorithms}, + volume = {12}, + number = {4}, + pages = {351--360}, + year = {1998}, + doi = {10.1002/(SICI)1098-2418(199807)12:4<351::AID-RSA3>3.0.CO;2-S} +} + +@misc{Li2026, + author = {Li, Xiaoyu}, + title = {Fast Spectral Signing for Vector Balancing}, + year = {2026}, + howpublished = {arXiv:2609.30044}, + note = {Version 1, 24 September 2026}, + url = {https://arxiv.org/abs/2609.30044v1} +} + +@misc{BansalEtAl2021Full, + author = {Bansal, Nikhil and Jiang, Haotian and Meka, Raghu and Singla, Sahil and Sinha, Makrand}, + title = {Prefix Discrepancy, Smoothed Analysis, and Combinatorial Vector Balancing}, + year = {2021}, + howpublished = {arXiv:2111.07049}, + note = {Version 1, 13 November 2021}, + url = {https://arxiv.org/abs/2111.07049v1} +} +@article{GrinbergSevastyanov1980, + author = {Grinberg, V. S. and Sevast'yanov, S. V.}, + title = {Value of the {Steinitz} constant}, + journal = {Functional Analysis and Its Applications}, + volume = {14}, + number = {2}, + pages = {125--126}, + year = {1980}, + doi = {10.1007/BF01086559} +} +@misc{Bandeira2026, + author = {Bandeira, Afonso S.}, + title = {A simplification of the solution to the {Koml\'os} Conjecture}, + year = {2026}, + howpublished = {Randomstrasse 101}, + note = {September 21, 2026}, + url = {https://randomstrasse101.math.ethz.ch/posts/komlos/} +} + +@article{Banaszczyk2012, + author = {Banaszczyk, Wojciech}, + title = {On series of signed vectors and their rearrangements}, + journal = {Random Structures \& Algorithms}, + volume = {40}, + number = {3}, + pages = {301--316}, + year = {2012}, + doi = {10.1002/rsa.20373} +} +@article{BrascampLieb1976, + author = {Brascamp, Herm Jan and Lieb, Elliott H.}, + title = {On extensions of the {Brunn--Minkowski} and {Pr\'ekopa--Leindler} theorems, including inequalities for log concave functions, and with an application to the diffusion equation}, + journal = {Journal of Functional Analysis}, + volume = {22}, + number = {4}, + pages = {366--389}, + year = {1976}, + doi = {10.1016/0022-1236(76)90004-5} +} +@misc{BryanClutterbuckRankin2026, + author = {Bryan, Paul and Clutterbuck, Julie and Rankin, Cale}, + title = {Convexity inequalities for eigenvalues and log-concavity of eigenfunctions}, + year = {2026}, + howpublished = {arXiv:2605.01334}, + note = {Version 1, May 2, 2026}, + url = {https://arxiv.org/abs/2605.01334v1} +} + +@article{Barany2024Matrix, + author = {B{\'a}r{\'a}ny, Imre}, + title = {A matrix version of the {Steinitz} lemma}, + journal = {Journal f{\"u}r die reine und angewandte Mathematik}, + volume = {809}, + pages = {261--267}, + year = {2024}, + doi = {10.1515/crelle-2024-0008}, + url = {https://arxiv.org/abs/2308.10102v2} +} + +@misc{AkbasSra2026, + author = {Akbas, Emrullah and Sra, Suvrit}, + title = {Tighter bounds on {Koml\'os} discrepancy: existence and algorithmic results}, + year = {2026}, + howpublished = {arXiv:2609.27172}, + note = {Version 2, 29 September 2026}, + url = {https://arxiv.org/abs/2609.27172v2} +} diff --git a/preprints/The-Euclidean-Steinitz-Bergstrom-theorem-September-24-2026/build/signing.tex b/preprints/The-Euclidean-Steinitz-Bergstrom-theorem-September-24-2026/build/signing.tex new file mode 100644 index 0000000000000000000000000000000000000000..a76880aa79bc61de05a8e4da03fd13983027754b --- /dev/null +++ b/preprints/The-Euclidean-Steinitz-Bergstrom-theorem-September-24-2026/build/signing.tex @@ -0,0 +1,332 @@ +\section{From low energy to shifted signs}\label{sec:shifted-signs} + +We prove Proposition~\ref{prop:adaptive}. Its spectral hypothesis will first +give a single normalized function with small derivatives in every coordinate. A lifted +Steiner symmetrization then produces a new domain from which one can take +one of two signed steps, even after a prescribed small shift. Repeating +this construction backwards will allow us to choose the signs forwards. +The construction adapts the common-density and prescribed-section arguments +of Guo, Fang and Lu~\cite[Proposition~2.2, Lemmas~3.3--3.4 and~4.1--4.3]{GuoFangLu2026}. +Quadratic-energy versions of this approach appear in +Bandeira~\cite{Bandeira2026} and Akbas and Sra +\cite[Lemmas~2.2--2.3]{AkbasSra2026}. Here we retain only coordinate Sobolev +energies and enough geometric slack for bounded adaptive shifts. +The coordinate-energy and shifted-step arguments needed here are proved +in full below. + +Throughout this section, +\[ + \kappa=\frac1{100},\qquad \delta=\frac14,\qquad \ell=\frac52. +\] + +\subsection{One function for all coordinate energies} + +We first record a useful boundary convention. If a bounded open convex +domain $D\subset\R^m$ contains zero, then +\begin{equation}\label{eq:zero-extension} + H_0^1(D)=\{f|_D:f\in H^1(\R^m),\ f=0\text{ a.e. outside }D\}, +\end{equation} +where functions on the left are extended by zero. Indeed, for a function on +the right, $f_r(x)=f(x/r)$ converges to $f$ in $H^1(\R^m)$ as $r\uparrow1$. +For $r<1$ its support lies in the compact subset $r\overline D$ of $D$. +Mollification with a sufficiently small radius therefore approximates $f_r$ +by functions in $C_c^\infty(D)$. Conversely, zero extension preserves the +$H^1$ norm on $C_c^\infty(D)$ and hence on its completion. + +\begin{lemma}\label{lem:simultaneous-density} +Let $K\subset\R^m$ be a nonempty bounded open convex set symmetric about +zero. If +\[ + \lambda_Q(K)\le\kappa^2\tr Q + \qquad\text{for every positive definite diagonal }Q, +\] +then there is a nonnegative $f\in H^1(\R^m)$, vanishing almost everywhere +outside $K$, such that +\[ + \int f^2=1,\qquad \int|\partial_i f|^2\le\kappa^2 + \quad(1\le i\le m). +\] +\end{lemma} + +\begin{proof} +Let $\mathcal F$ be the nonnegative functions in $H^1(\R^m)$ that vanish +almost everywhere outside $K$ and have $L^2$ norm one, and put +$E_i(f)=\int|\partial_i f|^2$. Consider the set of simultaneous upper bounds +\[ + \mathcal A=\{u\in\R^m:E_i(f)\le u_i\text{ for every }i + \text{ for some }f\in\mathcal F\}. +\] +It is nonempty and is closed under addition of nonnegative vectors. It is +also convex. For $f,g\in\mathcal F$ and $0\le t\le1$, the function +$h=(tf^2+(1-t)g^2)^{1/2}$ belongs to $\mathcal F$ and satisfies +\[ + |\partial_i h|^2\le t|\partial_i f|^2+(1-t)|\partial_i g|^2 + \quad\text{a.e.} +\] +This follows from the Sobolev chain rule for the Euclidean norm and +Cauchy--Schwarz. + +To see that $\mathcal A$ is closed, let $u^{(j)}\in\mathcal A$ converge to +$u$, and choose corresponding $f_j\in\mathcal F$. Their $H^1$ norms are +bounded. Since all supports lie in a fixed bounded set, weak $H^1$ +compactness and compactness in $L^2$ give a subsequence converging weakly in +$H^1$ and strongly in $L^2$ to a member $f$ of $\mathcal F$. Lower +semicontinuity gives $E_i(f)\le u_i$ for every $i$. + +If $c=\kappa^2(1,\ldots,1)$ did not belong to $\mathcal A$, strict +separation would give a nonzero $w\in\R^m$ with +\[ + w\cdot c<\inf_{u\in\mathcal A}w\cdot u. +\] +The upper-set property forces every $w_i$ to be nonnegative. For $\epsilon>0$ +let $Q_\epsilon=\diag(w_i+\epsilon)$. By \eqref{eq:zero-extension}, taking +absolute values and normalizing does not change the Rayleigh infimum, so +\[ + \inf_{u\in\mathcal A}w\cdot u + =\inf_{f\in\mathcal F}\sum_iw_iE_i(f) + \le\lambda_{Q_\epsilon}(K) + \le\kappa^2\sum_i(w_i+\epsilon). +\] +Letting $\epsilon\downarrow0$ contradicts separation. This argument also +covers zero components of $w$. Thus $c\in\mathcal A$, as required. +\end{proof} + +\subsection{Convexity under Minkowski averages} + +We will apply Jensen's inequality to the energies of horizontal sections. +The needed property is a diagonal version of the classical +Brascamp--Lieb convexity inequality for the first Dirichlet eigenvalue +\cite[Theorem~6.2]{BrascampLieb1976}; see also the statement in +Bryan, Clutterbuck and Rankin~\cite[Theorem~1.2]{BryanClutterbuckRankin2026}. +We give the proof from Gaussian log-concavity and the Brownian survival +rate, including the open-domain approximation used here. + +\begin{lemma}\label{lem:minkowski} +Let $D_0,D_1\subset\R^m$ be nonempty bounded open convex sets symmetric +about zero, and let $Q$ be positive definite diagonal. For $0\le\alpha\le1$, +\[ + \lambda_Q((1-\alpha)D_0+\alpha D_1) + \le(1-\alpha)\lambda_Q(D_0)+\alpha\lambda_Q(D_1). +\] +\end{lemma} + +\begin{proof} +The endpoints are immediate, so suppose $0<\alpha<1$ and write +$D_\alpha=(1-\alpha)D_0+\alpha D_1$. Fix $0<\eta<1$, set $r=1-\eta$, +and put +\[ + A_i=r\overline{D_i},\qquad + A_\alpha=(1-\alpha)A_0+\alpha A_1. +\] +These are compact convex sets, and $A_\alpha\subset D_\alpha$: indeed, +$A_\alpha\subset r\overline{D_\alpha}$, which is compactly contained in +$D_\alpha$. + +Let $W$ start at zero with Brownian covariance rate $2Q$. On a finite grid +of positive times in $[0,T]$, its values form a Gaussian vector. Gaussian +log-concavity~\cite[Theorem~2]{Prekopa1973} shows that +\[ + \mathbb P\{W\text{ lies in }A_\alpha\text{ on the grid}\} + \ge\prod_{i=0}^1 + \mathbb P\{W\text{ lies in }A_i\text{ on the grid}\}^{\alpha_i}, + \qquad (\alpha_0,\alpha_1)=(1-\alpha,\alpha). +\] +Here the Minkowski average of the two grid events is contained in the grid +event for $A_\alpha$. Increase the grids to a dense subset of $[0,T]$. +Closedness of the $A_i$ and continuity of Brownian paths allow passage to +the decreasing limits of these events. Since $rD_i\subset A_i$ and +$A_\alpha\subset D_\alpha$, it follows that +\[ + \mathbb P_0\{W([0,T])\subset D_\alpha\} + \ge\prod_{i=0}^1 + \mathbb P_0\{W([0,T])\subset rD_i\}^{\alpha_i}. +\] +Lemma~\ref{lem:brownian-rate}, followed by the scaling of the Dirichlet +quotient, now gives +\[ + \lambda_Q(D_\alpha) + \le r^{-2}\bigl((1-\alpha)\lambda_Q(D_0)+\alpha\lambda_Q(D_1)\bigr). +\] +Finally let $\eta\downarrow0$. +\end{proof} + +\subsection{A section that permits shifted signed steps} + +The next lemma is the geometric step in the signing construction. Its +conclusion holds for every allowed shift after the point in the new domain +has been chosen. + +\begin{lemma}\label{lem:shift-section} +Let $K\subset\R^m$ be a nonempty bounded open convex set symmetric about +zero and satisfying $\lambda_Q(K)\le\kappa^2\tr Q$ for every positive +definite diagonal $Q$. For every coordinate vector $e$, there is another +domain $T_e(K)$ with the same properties such that, for all $y\in T_e(K)$ +and $|q|\le\delta$, some $\varepsilon\in\{-1,1\}$ satisfies +\[ + y+(q+\varepsilon)e\in K. +\] +\end{lemma} + +\begin{proof} +Lift $K$ to the bounded open convex set +\[ + B=\{(y,s)\in\R^m\times\R:y+se\in K,\ |s|<\ell\}. +\] +Let $P$ be its projection onto $\R^m$. For $y\in P$, its vertical fiber +$I_y=\{s:(y,s)\in B\}$ is a nonempty open interval. Write $L(y)=|I_y|$. +Convexity gives +\[ + (1-t)I_y+tI_z\subset I_{(1-t)y+tz}, + \qquad + L((1-t)y+tz)\ge(1-t)L(y)+tL(z). +\] +Thus $L$ is positive, concave and continuous on the open convex set $P$. +The vertical Steiner symmetrization of $B$ is +\[ + B^*=\{(y,s):y\in P,\ |s|2s\},\qquad + b=\sup\{s\ge0:D_s\ne\varnothing\}. +\] +Then $0u\}|$ by replacing each superlevel set with its +centered interval of length $\mu(y,u)$. This preserves the $L^2$ norm. +Since $\Psi$ vanishes almost everywhere outside $B$, its rearrangement vanishes +almost everywhere outside $B^*$. + +For nonnegative $a,b\in L^2(\R)$, the layer-cake formula gives +\[ + \int ab + =\int_0^\infty\!\int_0^\infty + |\{a>u\}\cap\{b>v\}|\,du\,dv + \le\int a^*b^*. +\] +The intersection is bounded by the smaller of the two superlevel measures, +and centered intervals attain that bound. Since rearrangement preserves +the individual norms, $\|a^*-b^*\|_2\le\|a-b\|_2$. Applying this to the +fibers at $y$ and $y+he_i$, then integrating over $y$, bounds the horizontal +difference quotients of $\Psi^*$ by those of $\Psi$. Their weak limits yield +\begin{equation}\label{eq:rearranged-energy} + \int|\partial_{y_i}\Psi^*|^2\le\int|\partial_i f|^2\le\kappa^2. +\end{equation} +Sobolev slicing and Fubini therefore imply that, for almost every $s$, +$\Psi^*(\cdot,s)\in H^1(\R^m)$ and vanishes almost everywhere outside +$D_{|s|}$. Each nonzero such slice belongs to $H_0^1(D_{|s|})$ by +\eqref{eq:zero-extension}. + +We next show that the rearranged density puts enough mass at heights above +one. Its mean absolute height is +\begin{equation}\label{eq:mean-height} + \begin{split} + M&=\iint |s|\Psi^*(y,s)^2\,dy\,ds\\ + &=\frac1{8\ell}\int_{-\ell}^{\ell}\int_{-\ell}^{\ell} + \int_{\R^m}\min\{f(y+re)^2,f(y+r'e)^2\}\,dy\,dr\,dr'. + \end{split} +\end{equation} +Indeed, a centered interval of length $u$ has integral of $|s|$ equal to +$u^2/4$. Apply this to the superlevel intervals of $\Psi^{*2}$ and use Tonelli; +the factor $(2\ell)^{-1}$ in $\Psi^2$ gives the displayed normalization. +For nonnegative translates $u,v$ of $f$, both of norm one, +\[ + \int\min(u^2,v^2) + =1-\tfrac12\|u^2-v^2\|_1 + \ge1-\|u-v\|_2. +\] +Moreover, +\[ + \|f(\cdot+re)-f(\cdot+r'e)\|_2 + \le |r-r'|\,\|\partial_e f\|_2\le2\ell\kappa. +\] +It follows from \eqref{eq:mean-height} that +\begin{equation}\label{eq:height-lower} + M\ge\frac\ell2(1-2\ell\kappa)=\frac{19}{16}>1. +\end{equation} +Under the probability density $\Psi^{*2}(y,s)$, we have $|s|0$, +\begin{equation}\label{eq:ou-comparison} + \mathbb P\{U([0,T])\subset D\} + \le e^{-T\lambda_Q^{\mathrm{OU}}(D)}, + \qquad + \lambda_Q(D)\le\lambda_Q^{\mathrm{OU}}(D)+\tfrac12\tr Q. +\end{equation} +\end{lemma} + +\begin{proof} +The Gaussian density and its reciprocal are bounded on $D$, so the +Gaussian and Lebesgue $H_0^1$ form domains agree. Integration by parts +identifies the displayed form with the killed Ornstein--Uhlenbeck +semigroup $P_T^D$. Its $L^2(\gamma_m)$ operator norm is +$e^{-T\lambda_Q^{\mathrm{OU}}(D)}$. The killed-semigroup representation +used in Lemma~\ref{lem:brownian-rate} therefore gives +\[ + \mathbb P\{U([0,T])\subset D\} + =\langle\mathbf1_D,P_T^D\mathbf1_D\rangle_{L^2(\gamma_m)} + \le\gamma_m(D)e^{-T\lambda_Q^{\mathrm{OU}}(D)}. +\] +This proves the first inequality. Equivalently, the representation follows +from the Dirichlet Feynman--Kac formula after the conjugation below; its +potential is bounded on $D$ \cite{DaviesSimon1984}. + +Write $\rho(x)=(2\pi)^{-m/2}e^{-|x|^2/2}$ and put $g=h\sqrt\rho$. +For $h\in C_c^\infty(D)$, expansion and integration by parts give +\[ + \int_D\langle Q\nabla h,\nabla h\rangle\,d\gamma_m + =\int_D\left[ + \langle Q\nabla g,\nabla g\rangle + +\left(\tfrac14 x^TQx-\tfrac12\tr Q\right)g^2 + \right]dx. +\] +The identity extends by form closure to $H_0^1(D)$. Multiplication by +$\sqrt\rho$ maps this space bijectively to itself and preserves the +corresponding $L^2$ norms. Since the potential is at least +$-\tr Q/2$, taking Rayleigh infima proves the second inequality. +\end{proof} + +\subsection{One energy interval at one spectral scale} + +Fix the data of Proposition~\ref{prop:filtered} and a positive diagonal +$Q$. We use the matrices and scalar quantities from the preceding +section: +\[ + L_t=R_tQR_t^T,\qquad X_t=L_t/\theta,\qquad + F_t=F_t^\theta=R_t^Tf_0(X_t)R_t,\qquad + k_t=k_t(\theta)=\tr\phi(X_t). +\] +In this subsection the dyadic scale $\theta$ is fixed. Recall that +$a=3/2$ and that $E_\theta(s,t)$ is additive and nonnegative for $s\le t$. +Put $h_t=E_\theta(0,t)$. + +\begin{lemma}\label{lem:epoch-prob} +There is an absolute $c_1$ with the following property. Let +$\mathcal I\subset\{0,\ldots,n\}$ be an index set whose values $h_t$ +lie in an interval of length at most $e_0$, where $e_0$ is the constant +in Lemma~\ref{lem:increment}. For every $u\ge0$, +\begin{equation}\label{eq:epoch-prob} + \mathbb P\left\{ + \lVert F_tU(\tau)\rVert\le c_1\sqrt{k_t} + \text{ for every }t\in\mathcal I, + \ u\le\tau\le u+\theta^{-1} + \right\}\ge\tfrac12. +\end{equation} +The same conclusion holds on any shorter time interval. +\end{lemma} + +\begin{proof} +We may omit indices with $F_t=0$. By +\eqref{eq:cutoff-moments}, every remaining index has $k_t\ge1/4$. +Divide these indices into the classes +\[ + \mathcal I_j=\{t\in\mathcal I:K_j\le k_t<2K_j\}, + \qquad K_j=2^j/4,\qquad j=0,1,\ldots. +\] +We first bound the expected supremum within one nonempty class, writing +$k=K_j$. + +Stationarity, coisometry and \eqref{eq:cutoff-moments} give +\begin{equation}\label{eq:vector-second-moment} + \mathbb E\lVert F_tU(\tau)\rVert^2 + =\tr f_0(X_t)^2\le4k_t\le8k. +\end{equation} +For two indices in the class, Lemma~\ref{lem:increment}, with the indices +put in chronological order, implies +\begin{equation}\label{eq:index-increment} + \mathbb E\lVert(F_t-F_s)U(\tau)\rVert^2 + \le Ck|h_t-h_s|^{1/a}. +\end{equation} +Here we dropped the positive right weight in that lemma and used +$1+k_t\le6k$. For time increments at a fixed index, +\[ + \operatorname{Cov}(U(\tau)-U(\sigma)) + =2(I-e^{-Q|\tau-\sigma|})\preceq2|\tau-\sigma|Q, +\] +so that +\begin{align} + \mathbb E\lVert F_t(U(\tau)-U(\sigma))\rVert^2 + &\le2|\tau-\sigma|\tr(F_t^2Q)\notag\\ + &=2\theta|\tau-\sigma|\tr(X_tf_0(X_t)^2) + \le Ck\theta|\tau-\sigma|.\label{eq:time-increment} +\end{align} +This trace identity uses $R_tR_t^T=I$; it does not require $Q$ and +$F_t$ to commute. + +We give the elementary chaining argument for the two parameters. Choose +$H$ so that $h_t\in[H,H+e_0]$ for every index under consideration, and put +$\alpha=(h_t-H)/e_0$ and $\beta=\theta(\tau-u)$ in $[0,1]$. +Split a mixed increment into an +index increment and a time increment. Equations~\eqref{eq:index-increment} +and \eqref{eq:time-increment} bound its second moment by +\[ + Ck\bigl(|\alpha-\alpha'|^{1/a}+|\beta-\beta'|\bigr), +\] +where $\alpha$ is the normalized energy coordinate; the constant absorbs +$e_0\le1$. At depth $l$, divide both parameter intervals into dyadic +cells of width $2^{-l}$ and choose a representative from each occupied +product cell. There are at most $C4^l$ representatives. The increment +from each representative to that of its parent cell is a centered +Gaussian vector with second moment at most +\[ + v_l=Ck2^{-l/a}. +\] + +For use here, if $G_1,\ldots,G_M$ are arbitrarily dependent centered +Gaussian vectors and $\mathbb E\lVert G_i\rVert^2\le v$, then +\begin{equation}\label{eq:gaussian-vector-max} + \mathbb E\max_{i\le M}\lVert G_i\rVert + \le2\sqrt{v(\log M+1/2)}. +\end{equation} +Indeed, for $v>0$, diagonalizing the covariance of $G_i$ gives +$\mathbb E e^{\lVert G_i\rVert^2/(4v)}\le e^{1/2}$, because its +eigenvalues are nonnegative and sum to at most $v$. Bound the exponential +of the maximum by the sum of the exponentials, then use Jensen's +inequality and Cauchy--Schwarz. The case $v=0$ is immediate. + +Apply \eqref{eq:gaussian-vector-max} to the increments at each depth. +Starting with \eqref{eq:vector-second-moment} and summing the parent +increments gives an absolute $C_0$ such that +\begin{equation}\label{eq:class-mean} + \mathbb E\sup_{\substack{t\in\mathcal I_j\\ + u\le\tau\le u+\theta^{-1}}} + \lVert F_tU(\tau)\rVert + \le C\sqrt k\left(1+\sum_{l\ge1} + 2^{-l/(2a)}(1+\sqrt l)\right) + \le C_0\sqrt k. +\end{equation} +To justify convergence of the representatives, note that there are only +finitely many distinct energy values. If $h_t=h_s$, +Lemma~\ref{lem:increment} gives $F_t=F_s$ exactly. At sufficiently fine +depth a target index therefore has the same filter as its representative, +and the representative times converge to its target time. Continuity +of $U$ completes the passage to the supremum. Equivalently, one may +first use finite time grids and then increase them to a dense set. +Neither argument uses a lower bound on the positive gaps between energy +values. In particular, the number of indices does not enter +\eqref{eq:class-mean}. + +It remains to control all the $k$ classes at once. On a finite time +grid the supremum $S_j$ in \eqref{eq:class-mean} is a maximum of Gaussian +vector norms. Represent the joint Gaussian vectors as linear images of +one standard Gaussian vector. Each corresponding linear map has norm at +most one because its output covariance is $F_t^2\preceq I$, by +\eqref{eq:cutoff-moments}; consequently +$S_j$ is a $1$-Lipschitz function of that standard Gaussian vector. +Equivalently, it is a supremum of scalar Gaussian functionals, each +with variance at most one. + +Markov's inequality and \eqref{eq:class-mean} give +$\mathbb P\{S_j\le2C_0\sqrt{K_j}\}\ge1/2$. +Borell's Gaussian enlargement inequality +\cite[Theorem~3.1]{Borell1975}, applied to this sublevel set, gives +\[ + \mathbb P\{S_j>2C_0\sqrt{K_j}+r\} + \le1-\Phi(r)\le e^{-r^2/2},\qquad r\ge0, +\] +where $\Phi$ is the standard normal distribution function. The same +bound holds for the full time supremum by increasing dense grids and +path continuity. Choose $c_1>2C_0$. The probability that any constraint in class +$j$ exceeds $c_1\sqrt{k_t}$ is at most +\[ + \exp\bigl(-(c_1-2C_0)^2K_j/2\bigr). +\] +With $A=(c_1-2C_0)^2/8$, the sum of these bounds is at most +\[ + \sum_{j\ge0}e^{-A2^j} + \le\frac{e^{-A}}{1-e^{-A}}\le\tfrac12 +\] +once $c_1$ is an absolute constant large enough that $A\ge\log3$. +This proves \eqref{eq:epoch-prob}. Restricting the time interval can +only increase its probability. +\end{proof} + +The two bounds in this proof have different roles: the trace of the +covariance controls the expected vector norm, while its operator norm +gives the absolute concentration parameter. This distinction is what +allows the class failure probabilities to be summed without a factor +depending on the dimension. + +\subsection{Combining the constraints} + +\begin{proof}[Proof of Proposition~\ref{prop:filtered}] +Fix $d,n,Q$ and the entire recursion. Let $\Theta$ be the finite set of +dyadic scales for which some $F_t^\theta$ is nonzero. If $q_{\min}$ and +$q_{\max}$ are the smallest and largest diagonal entries of $Q$, then +coisometry gives $q_{\min}I\preceq L_t\preceq q_{\max}I$, and the +support of $f_0$ gives +\begin{equation}\label{eq:used-scales} + \Theta\subset[q_{\min}/4,2q_{\max}], + \qquad \sum_{\theta\in\Theta}\theta\le4q_{\max}\le4\tr Q. +\end{equation} +At each scale, partition the sequence indices by the half-open intervals +\[ + [r e_0,(r+1)e_0),\qquad r=0,1,\ldots, +\] +containing $E_\theta(0,t)$. These are the energy intervals to which +Lemma~\ref{lem:epoch-prob} applies. There are at most +$C(1+E_\theta(0,n))$ nonempty intervals, including any repetitions of +an energy value in the same interval. + +For each fixed $t$, the scale constraints control the original state +norm. To see this, diagonalize $L_t$ and write $y=R_tU(\tau)$ in this +eigenbasis, with eigenvalues $\lambda_i>0$. Each $\lambda_i$ admits a +dyadic scale with $\lambda_i/\theta\in[1,2]$, where $f_0=1$; that scale +belongs to $\Theta$. Since $R_t^T$ is an isometry, +\begin{align} + \sum_{\theta\in\Theta}\lVert F_t^\theta U(\tau)\rVert^2 + &=\sum_i |y_i|^2\sum_{\theta\in\Theta} + f_0(\lambda_i/\theta)^2 + \ge\lVert R_tU(\tau)\rVert^2.\label{eq:spectral-reconstruction} +\end{align} +Thus all the scale constraints at a given time imply, by +\eqref{eq:scale-size}, +\begin{equation}\label{eq:state-survival-bound} + \lVert R_tU(\tau)\rVert^2 + \le c_1^2\sum_{\theta\in\Theta}k_t(\theta) + \le c_1^2 C_\phi d + \qquad(0\le t\le n), +\end{equation} +with an absolute $C_\phi$. + +We also impose coordinate bounds. A stationary scalar +Ornstein--Uhlenbeck process of rate $q>0$, restricted to an interval of +length at most $1/q$, stays in $[-2,2]$ with probability at least an +absolute $p_0>0$. By stationarity and time rescaling, it suffices to +consider rate one on $[0,1]$. In distribution that process is +\[ + e^{-s}\bigl(Z+B(e^{2s}-1)\bigr),\qquad 0\le s\le1, +\] +where $Z$ is standard normal and $B$ is an independent standard Brownian +motion. The event $|Z|\le1$ and +$\sup_{r\le e^2-1}|B(r)|\le1$ has a fixed positive probability and +implies the asserted bound. + +Choose absolute constants +\[ + A_0>c_1\sqrt{C_\phi},\qquad B_0>2, +\] +and let $\mathcal D$ be the domain in Proposition~\ref{prop:filtered} +with these constants. It is bounded, open, centrally symmetric and +convex, and contains a neighborhood of zero. The closed constraints +\eqref{eq:state-survival-bound} and $|U_j(\tau)|\le2$ lie strictly inside +$\mathcal D$. + +For a fixed $T>0$, cover $[0,T]$ by at most $1+T\theta$ consecutive +closed intervals of length at most $1/\theta$ at scale $\theta$. +For every such interval and every nonempty energy interval impose the +event in Lemma~\ref{lem:epoch-prob}. Also cover $[0,T]$ by at most +$1+TQ_{jj}$ intervals of length at most $1/Q_{jj}$ for coordinate $j$ +and impose its box event. All these events are closed symmetric convex +constraints on linear evaluations of the centered Gaussian path. +There are finitely many events for this fixed input and $T$. +Lemma~\ref{lem:gaussian-path-correlation} therefore multiplies their +probability lower bounds. In particular, its finite-grid argument and +the continuity of the paths apply even when some constraints are +duplicates. The strict slack just chosen ensures that their intersection +is an event of survival in the open domain $\mathcal D$. + +It follows that +\begin{align} + -\log\mathbb P\{U([0,T])\subset\mathcal D\} + &\le C\sum_{\theta\in\Theta} + (1+T\theta)(1+E_\theta(0,n)) + +C\sum_{j=1}^m(1+TQ_{jj})\notag\\ + &\le A(Q,R,d,n)+C'T\tr Q.\label{eq:survival-rate-bound} +\end{align} +Here \eqref{eq:scale-energy} and \eqref{eq:used-scales} bound the +coefficient of $T$ by an absolute constant times $\tr Q$, and +\[ + A(Q,R,d,n) + =C\left[\sum_{\theta\in\Theta}(1+E_\theta(0,n))+m\right]<\infty. +\] +This additive constant may depend on the input and on the smallest +entry of $Q$; no uniform bound on it is required. + +By Lemma~\ref{lem:ou-energy}, +\[ + \lambda_Q^{\mathrm{OU}}(\mathcal D) + \le C'\tr Q+\frac{A(Q,R,d,n)}{T}. +\] +Let $T$ tend to infinity while keeping the input fixed. This removes +the additive constant, including the number of spectral scales. +The second inequality in \eqref{eq:ou-comparison} now gives +\[ + \lambda_Q(\mathcal D)\le(C'+1/2)\tr Q. +\] +Thus $H_0=C'+1/2$ is absolute. The constants $A_0,B_0,H_0$ are +independent of $d,n,Q$ and of the recursion, as required. +\end{proof} diff --git a/preprints/The-Euclidean-plane-is-not-five-colorable-September-23-2026/README.md b/preprints/The-Euclidean-plane-is-not-five-colorable-September-23-2026/README.md new file mode 100644 index 0000000000000000000000000000000000000000..9e7b5045ab8f8c7e949b794a5dc2450aa2f2c549 --- /dev/null +++ b/preprints/The-Euclidean-plane-is-not-five-colorable-September-23-2026/README.md @@ -0,0 +1,17 @@ +# [The Euclidean plane is not five-colorable](paper.pdf) + +**Author:** OpenAI + +**Date:** September 23, 2026 + +## Citation + +```bibtex +@misc{OAI:The-Euclidean-plane-is-not-five-colorable-September-23-2026, + author = {{OpenAI}}, + title = {{The Euclidean plane is not five-colorable}}, + howpublished = {OpenAI Math Release preprint + \href{https://github.com/openai/math/blob/main/preprints/The-Euclidean-plane-is-not-five-colorable-September-23-2026/paper.pdf}{OAI:The-Euclidean-plane-is-not-five-colorable-September-23-2026}}, + year = {2026} +} +``` diff --git a/preprints/The-Euclidean-plane-is-not-five-colorable-September-23-2026/build/figures/exclusion-continuum.tex b/preprints/The-Euclidean-plane-is-not-five-colorable-September-23-2026/build/figures/exclusion-continuum.tex new file mode 100644 index 0000000000000000000000000000000000000000..1b1d95c093ee02fcb09ae06e81123d642b2353bb --- /dev/null +++ b/preprints/The-Euclidean-plane-is-not-five-colorable-September-23-2026/build/figures/exclusion-continuum.tex @@ -0,0 +1,40 @@ +\begin{tikzpicture}[x=2.65cm,y=2.65cm,font=\small, + point/.style={circle,fill=black,inner sep=1.45pt}] + \coordinate (X) at (0,0); + \coordinate (Z) at (1.25,0); + \coordinate (Q) at (.65,.45); + \coordinate (P) at (.3,{sqrt(1-.95*.95)}); + \coordinate (F) at ({cos(40)},{sin(40)}); + + \fill[black!2] (Z) circle[radius=1]; + \draw[black!35,densely dashed] (X) circle[radius=1]; + \node[black!60,anchor=south east] at (-.58,.83) + {$\partial B(x,\rho)$}; + \draw[black!65,densely dotted,semithick] (Z) circle[radius=1]; + \node[black!65,anchor=north west] at (1.78,-.85) + {$\{y:|y-z|=1\}$}; + + \draw[black!65] (X) -- node[below=5pt] {$|z-x|>1$} (Z); + \draw[black!65] (Q) -- node[above right=2pt] {$|z-q|<1$} (Z); + + \draw[blue!55!black,line width=1.45pt,line cap=round] + (X) .. controls (.045,.01) and (.025,.13) .. (.085,.14) + .. controls (.135,.15) and (.12,.07) .. (.17,.085) + .. controls (.245,.105) and (.18,.28) .. (.235,.295) + .. controls (.26,.32) and (.28,.30) .. (P) + .. controls (.34,.34) and (.35,.48) .. (.405,.475) + .. controls (.48,.47) and (.48,.385) .. (.545,.40) + .. controls (.59,.41) and (.62,.46) .. (Q) + .. controls (.70,.43) and (.66,.575) .. (.715,.59) + .. controls (.74,.605) and (.75,.625) .. (F); + \node[blue!55!black,anchor=south] at (.43,.55) {$K_j$}; + + \node[point,label={[label distance=3pt]left:$x$}] at (X) {}; + \node[point,label={[label distance=3pt]right:$z$}] at (Z) {}; + \node[point,label={[label distance=3pt]below left:$q$}] at (Q) {}; + \node[point,label={[label distance=3pt]above left:$p$}] at (P) {}; + \node[point,fill=blue!55!black] at (F) {}; + \node[align=center] at (.65,-1.31) + {$z\in\Delta(K_j)$\quad and\quad + $z\notin H_{i_j}\cup H_{i_{j+1}}$}; +\end{tikzpicture} diff --git a/preprints/The-Euclidean-plane-is-not-five-colorable-September-23-2026/build/figures/moser-placement.tex b/preprints/The-Euclidean-plane-is-not-five-colorable-September-23-2026/build/figures/moser-placement.tex new file mode 100644 index 0000000000000000000000000000000000000000..4242ab773d16494ec3a5d98a752dfbfcdef39dbc --- /dev/null +++ b/preprints/The-Euclidean-plane-is-not-five-colorable-September-23-2026/build/figures/moser-placement.tex @@ -0,0 +1,62 @@ +\begin{tikzpicture}[x=3.8cm,y=3.8cm,font=\small, + vertex/.style={circle,fill=black,inner sep=1.7pt}, + first/.style={blue!55!black,line width=.9pt}, + second/.style={black!65,line width=.9pt,densely dashed}] + \begin{scope} + \clip (-1.68,-.24) rectangle (.68,1.93); + \foreach \sector in {0,60,120,180,240,300} { + \path[fill=teal!12,draw=teal!45!black,line width=.25pt] + plot[domain=0:1,samples=101,variable=\r] + ({\r*cos(\sector+acos(\r/2)-60)}, + {\r*sin(\sector+acos(\r/2)-60)}) + -- plot[domain=1:0,samples=101,variable=\r] + ({\r*cos(\sector-acos(\r/2)+60)}, + {\r*sin(\sector-acos(\r/2)+60)}) -- cycle; + \path[fill=teal!12,draw=teal!45!black,line width=.25pt] + plot[domain=1:sqrt(3),samples=101,variable=\r] + ({\r*cos(\sector+60-acos(\r/2))}, + {\r*sin(\sector+60-acos(\r/2))}) + -- plot[domain=sqrt(3):2,samples=101,variable=\r] + ({\r*cos(\sector+acos(\r/2))}, + {\r*sin(\sector+acos(\r/2))}) + -- plot[domain=2:sqrt(3),samples=101,variable=\r] + ({\r*cos(\sector-acos(\r/2))}, + {\r*sin(\sector-acos(\r/2))}) + -- plot[domain=sqrt(3):1,samples=101,variable=\r] + ({\r*cos(\sector-60+acos(\r/2))}, + {\r*sin(\sector-60+acos(\r/2))}) -- cycle; + } + \foreach \sector in {0,60,120,180,240,300} + \draw[black!25,densely dotted,line width=.35pt] + (0,0) -- (\sector:2); + \draw[fill=white,draw=black!45,line width=.3pt] + (0,0) circle[radius=.012]; + \end{scope} + \foreach \name/\gx/\gy in { + O/0/0, + A/{sqrt(3)/2}/{1/2}, + B/{sqrt(3)/2}/{-1/2}, + T/{sqrt(3)}/0, + UA/{(5*sqrt(3)-sqrt(11))/12}/{(sqrt(33)+5)/12}, + UB/{(5*sqrt(3)+sqrt(11))/12}/{(sqrt(33)-5)/12}, + UT/{5*sqrt(3)/6}/{sqrt(33)/6}} + { + \coordinate (\name) at + ({(35*(\gx-290/250)-12*(\gy+149/250))/37}, + {(12*(\gx-290/250)+35*(\gy+149/250))/37}); + } + \draw[first] (O) -- (A) -- (T) -- (B) -- (O); + \draw[first] (A) -- (B); + \draw[second] (O) -- (UA) -- (UT) -- (UB) -- (O); + \draw[second] (UA) -- (UB); + \draw[orange!70!black,line width=1.5pt] (T) -- + node[midway,right=5pt,text=black] {$1$} (UT); + + \node[vertex,label=left:$z_0$] at (O) {}; + \node[vertex,label=above:$z_A$] at (A) {}; + \node[vertex,label=below:$z_B$] at (B) {}; + \node[vertex,label=right:$z_T$] at (T) {}; + \node[vertex,label=left:$z_{uA}$] at (UA) {}; + \node[vertex,label=below right:$z_{uB}$] at (UB) {}; + \node[vertex,label=above:$z_{uT}$] at (UT) {}; +\end{tikzpicture} diff --git a/preprints/The-Euclidean-plane-is-not-five-colorable-September-23-2026/build/figures/proof-flow.tex b/preprints/The-Euclidean-plane-is-not-five-colorable-September-23-2026/build/figures/proof-flow.tex new file mode 100644 index 0000000000000000000000000000000000000000..5dd8cec7ef3eb39a61d0669accdab7f8a41759d8 --- /dev/null +++ b/preprints/The-Euclidean-plane-is-not-five-colorable-September-23-2026/build/figures/proof-flow.tex @@ -0,0 +1,32 @@ +\begin{tikzpicture}[ + font=\small, + stage/.style={draw=black!55, rounded corners=2pt, fill=black!3, + align=center, text width=2.48cm, minimum height=1.05cm, + inner sep=5pt}, + link/.style={-{Stealth[length=2mm]}, semithick, draw=black!70}, + node distance=0.48cm] + \node[stage] (proper) {Arbitrary proper\\five-coloring}; + \node[stage, right=of proper] (law) + {Invariant probability\\on proper labelings}; + \node[stage, right=of law] (factor) + {Continuous\\translation factor}; + \node[stage, right=of factor] (weak) + {Weak measurable\\five-coloring}; + \node[stage] (palettes) at ([yshift=-2.28cm]weak.center) + {Angular palettes\\of size at most two}; + \node[stage] (cycle) at ([yshift=-2.28cm]factor.center) + {Common exclusion\\continua on a cycle\\of length $3$, $4$, or $5$}; + \node[stage, text width=2.48cm, + font=\footnotesize, minimum height=1.27cm] (cases) + at ([yshift=-2.28cm]law.center) + {$3$: Moser graph\\$4$: parity obstruction\\$5$: six-step words}; + \node[stage, fill=black!9] (end) at ([yshift=-2.28cm]proper.center) + {Contradiction}; + \draw[link] (proper) -- (law); + \draw[link] (law) -- (factor); + \draw[link] (factor) -- (weak); + \draw[link] (weak) -- (palettes); + \draw[link] (palettes) -- (cycle); + \draw[link] (cycle) -- (cases); + \draw[link] (cases) -- (end); +\end{tikzpicture} diff --git a/preprints/The-Euclidean-plane-is-not-five-colorable-September-23-2026/build/main.tex b/preprints/The-Euclidean-plane-is-not-five-colorable-September-23-2026/build/main.tex new file mode 100644 index 0000000000000000000000000000000000000000..6e155c2f604cfba424562b0b3742a79a15cc294e --- /dev/null +++ b/preprints/The-Euclidean-plane-is-not-five-colorable-September-23-2026/build/main.tex @@ -0,0 +1,27 @@ +\documentclass[11pt]{article} +\input{preamble} +\title{The Euclidean plane is not five-colorable} +\author{OpenAI} +\date{September 23, 2026} +\begin{document} +\maketitle +\begin{abstract} +We prove that every coloring of the Euclidean plane with five colors has a +monochromatic unit-distance pair, with no regularity assumption on the color +classes. Consequently, the chromatic number of the plane is either six or seven. +\end{abstract} +\input{sections/introduction} +\input{sections/spectral} +\input{sections/rigidity} +\input{sections/transfer} +\input{sections/palettes} +\input{sections/transitions} +\input{sections/interfaces} +\input{sections/angular} +\begingroup +\raggedright +\setlength{\bibsep}{5pt} +\bibliographystyle{abbrvnat} +\bibliography{references} +\endgroup +\end{document} diff --git a/preprints/The-Euclidean-plane-is-not-five-colorable-September-23-2026/build/preamble.tex b/preprints/The-Euclidean-plane-is-not-five-colorable-September-23-2026/build/preamble.tex new file mode 100644 index 0000000000000000000000000000000000000000..d6fce50f33d14e1edefa6b8f90f35f70307924fa --- /dev/null +++ b/preprints/The-Euclidean-plane-is-not-five-colorable-September-23-2026/build/preamble.tex @@ -0,0 +1,39 @@ +\pdftrailerid{} +\usepackage[T1]{fontenc} +\usepackage[utf8]{inputenc} +\usepackage{mathpazo} +\usepackage[margin=1in]{geometry} +\usepackage{amsmath,amssymb,amsthm,mathtools} +\usepackage{booktabs,array,graphicx,mathrsfs} +\usepackage[numbers,sort&compress]{natbib} +\usepackage{tikz} +\usetikzlibrary{arrows.meta,positioning,calc} +\usepackage{url} +\urlstyle{same} +\usepackage[unicode,colorlinks=true,linkcolor=blue,citecolor=blue,urlcolor=blue]{hyperref} +\hypersetup{pdftitle={The Euclidean plane is not five-colorable},pdfauthor={OpenAI},pdfsubject={The Hadwiger--Nelson problem, arbitrary colorings, spectral rigidity, and measurable geometric obstructions},pdflang={en}} +\providecommand{\doi}[1]{doi: \href{https://doi.org/#1}{\nolinkurl{#1}}} +\renewcommand{\bibsection}{\section*{\refname}\addcontentsline{toc}{section}{\refname}} +\newtheorem{theorem}{Theorem}[section] +\newtheorem{lemma}[theorem]{Lemma} +\newtheorem{proposition}[theorem]{Proposition} +\newtheorem{corollary}[theorem]{Corollary} +\theoremstyle{definition} +\newtheorem{definition}[theorem]{Definition} +\theoremstyle{remark} +\newtheorem{remark}[theorem]{Remark} +\numberwithin{equation}{section} +\newcommand{\C}{\mathbb C} +\newcommand{\R}{\mathbb R} +\newcommand{\Q}{\mathbb Q} +\newcommand{\Z}{\mathbb Z} +\newcommand{\N}{\mathbb N} +\newcommand{\T}{\mathbb T} +\newcommand{\ind}{\mathbf 1} +\newcommand{\norm}[1]{\lVert#1\rVert} +\newcommand{\abs}[1]{\lvert#1\rvert} +\DeclareMathOperator{\id}{id} +\DeclareMathOperator{\dist}{dist} +\DeclareMathOperator{\supp}{supp} +\allowdisplaybreaks[1] +\setlength{\emergencystretch}{1em} diff --git a/preprints/The-Euclidean-plane-is-not-five-colorable-September-23-2026/build/references.bib b/preprints/The-Euclidean-plane-is-not-five-colorable-September-23-2026/build/references.bib new file mode 100644 index 0000000000000000000000000000000000000000..c2ba9afd79cd50036db29c9ac22b1bfe8fde353a --- /dev/null +++ b/preprints/The-Euclidean-plane-is-not-five-colorable-September-23-2026/build/references.bib @@ -0,0 +1,368 @@ +@book{Bekka2008, + author = {Bekka, Bachir and de la Harpe, Pierre and Valette, Alain}, + title = {{Kazhdan}'s Property ({T})}, + series = {New Mathematical Monographs}, + volume = {11}, + publisher = {Cambridge University Press}, + year = {2008}, + doi = {10.1017/CBO9780511542749}, + note = {Theorem locators refer to the author manuscript dated February 23, 2007}, + url = {https://ncatlab.org/nlab/files/BekkaHarpeValetteOnKashdanPropertyT.pdf} +} + +@misc{BourgeatHeinrichMelottiRobert2015, + author = {Bourgeat, Thomas and Heinrich, Marc and Melotti, Paul and Robert, Jean-Marc}, + title = {A probabilistic {Hadwiger--Nelson} problem}, + year = {2015}, + eprint = {1501.02441}, + archivePrefix = {arXiv}, + primaryClass = {math.CO}, + url = {https://arxiv.org/abs/1501.02441v1} +} + +@article{ExooIsmailescu2020, + author = {Exoo, Geoffrey and Ismailescu, Dan}, + title = {The Chromatic Number of the Plane Is at Least 5: A New Proof}, + journal = {Discrete \& Computational Geometry}, + volume = {64}, + year = {2020}, + pages = {216--226}, + doi = {10.1007/s00454-019-00058-1}, + eprint = {1805.00157}, + archivePrefix = {arXiv}, + url = {https://arxiv.org/abs/1805.00157v1} +} + +@article{Crum1956, + author = {Crum, M. M.}, + title = {On Positive-Definite Functions}, + journal = {Proceedings of the London Mathematical Society}, + series = {3}, + volume = {6}, + number = {4}, + year = {1956}, + pages = {548--560}, + doi = {10.1112/plms/s3-6.4.548} +} + +@article{Falconer1981, + author = {Falconer, K. 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Part 1}, + journal = {Mathematics Competitions}, + volume = {16}, + number = {1}, + year = {2003}, + pages = {9--41}, + url = {https://www.wfnmc.org/Journal%202003%201.pdf} +} + +@misc{SokolovVoronov2025, + author = {Sokolov, Georgy and Voronov, Vsevolod}, + title = {On the Chromatic Number of the Plane for Map-Type Colorings}, + year = {2025}, + eprint = {2502.01958v1}, + archivePrefix = {arXiv}, + primaryClass = {math.CO}, + note = {Version 1, submitted February 4, 2025}, + url = {https://arxiv.org/abs/2502.01958v1} +} + +@misc{TaoDisintegration2008, + author = {Tao, Terence}, + title = {{254A}, Lecture 9: Ergodicity}, + year = {2008}, + note = {February 4; Theorem 4 and Remark 5}, + url = {https://terrytao.wordpress.com/2008/02/04/254a-lecture-9-ergodicity/} +} + +@article{Townsend1981, + author = {Townsend, S. P.}, + title = {Every 5-Coloured Map in the Plane Contains a Monochrome Unit}, + journal = {Journal of Combinatorial Theory, Series A}, + volume = {30}, + number = {1}, + year = {1981}, + pages = {114--115}, + doi = {10.1016/0097-3165(81)90046-7}, + url = {https://www.jennysteve.net/MapColouring/1981note.pdf} +} + +@article{Townsend2005, + author = {Townsend, S. P.}, + title = {Colouring the Plane with No Monochrome Units}, + journal = {Geombinatorics}, + volume = {14}, + number = {4}, + year = {2005}, + pages = {184--193}, + url = {https://www.jennysteve.net/MapColouring/2005paper.pdf} +} + +@misc{Voronov2025Interval, + author = {Voronov, Vsevolod}, + title = {The Chromatic Number of the Plane with an Interval of Forbidden Distances Is at Least 7}, + year = {2025}, + eprint = {2304.10163v3}, + archivePrefix = {arXiv}, + primaryClass = {math.CO}, + note = {Version 3, submitted April 13, 2025; first version April 20, 2023}, + url = {https://arxiv.org/abs/2304.10163v3} +} + +@article{Woodall1973, + author = {Woodall, D. R.}, + title = {Distances Realized by Sets Covering the Plane}, + journal = {Journal of Combinatorial Theory, Series A}, + volume = {14}, + number = {2}, + year = {1973}, + pages = {187--200}, + doi = {10.1016/0097-3165(73)90020-4} +} + +@article{ZimmerDiscrete1976, + author = {Zimmer, Robert J.}, + title = {Ergodic actions with generalized discrete spectrum}, + journal = {Illinois Journal of Mathematics}, + volume = {20}, + number = {4}, + pages = {555--588}, + year = {1976}, + doi = {10.1215/ijm/1256049648} +} + +@article{ZimmerExtensions1976, + author = {Zimmer, Robert J.}, + title = {Extensions of ergodic group actions}, + journal = {Illinois Journal of Mathematics}, + volume = {20}, + number = {3}, + pages = {373--409}, + year = {1976}, + doi = {10.1215/ijm/1256049780} +} + +@article{deBruijnErdos1951, + author = {de Bruijn, N. G. and Erd{\H{o}}s, P.}, + title = {A colour problem for infinite graphs and a problem in the theory of relations}, + journal = {Indagationes Mathematicae}, + volume = {13}, + pages = {371--373}, + year = {1951}, + url = {https://users.renyi.hu/~p_erdos/1951-01.pdf} +} + +@article{deGrey2018, + author = {de Grey, Aubrey D. N. J.}, + title = {The Chromatic Number of the Plane Is at Least 5}, + journal = {Geombinatorics}, + volume = {28}, + number = {1}, + year = {2018}, + pages = {18--31}, + eprint = {1804.02385}, + archivePrefix = {arXiv}, + url = {https://arxiv.org/abs/1804.02385v3} +} + +@article{vonNeumann1929, + author = {von Neumann, John}, + title = {Zur allgemeinen {Theorie} des {Masses}}, + journal = {Fundamenta Mathematicae}, + volume = {13}, + pages = {73--116}, + year = {1929}, + doi = {10.4064/fm-13-1-73-116} +} + + +@misc{Reed2026, + author = {Reed, Jonathan {\ensuremath{f(n)}}}, + title = {The {Hadwiger--Nelson} Problem: Formal Verification of the 7-Color Chromatic Number of the Plane via Toroidal Projection and the Irrationality of {$2\pi$}}, + year = {2026}, + note = {Public manuscript version present May 14, 2026}, + howpublished = {\href{https://github.com/AEjonanonymous/Hadwiger-Nelson/blob/97210989cd34e0f884f2635f8cfb9822b33388ad/The\%20Hadwiger-Nelson\%20Problem\%20-\%20Formal\%20Verification\%20of\%20the\%207-Color\%20Chromatic\%20Number\%20of\%20the\%20Plane\%20via\%20Toroidal\%20Projection\%20and\%20the\%20Irrationality\%20of\%202\%CF\%80.pdf}{Public manuscript, May 14 version}} +} + +@misc{DeroinKleptsynNavas2008, + author = {Deroin, Bertrand and Kleptsyn, Victor and Navas, Andr{\'e}s}, + title = {On the question of ergodicity for minimal group actions on the circle}, + year = {2008}, + eprint = {0806.1974v1}, + archivePrefix = {arXiv}, + url = {https://arxiv.org/abs/0806.1974v1}, + note = {\url{https://arxiv.org/abs/0806.1974v1}} +} diff --git a/preprints/The-Euclidean-plane-is-not-five-colorable-September-23-2026/build/sections/angular.tex b/preprints/The-Euclidean-plane-is-not-five-colorable-September-23-2026/build/sections/angular.tex new file mode 100644 index 0000000000000000000000000000000000000000..2c4fe7c9a3af06de5fb707d8280395c971dca5d8 --- /dev/null +++ b/preprints/The-Euclidean-plane-is-not-five-colorable-September-23-2026/build/sections/angular.tex @@ -0,0 +1,773 @@ +\section{Angular obstructions and the final contradiction} +\label{sec:angular} + +We now show that none of the three cycle lengths supplied by +Proposition~\ref{prop:interface-cycle} is possible. Fix its common +center \(x\), cyclically index the cycle labels by \(j\), and let +edge \(j\) join labels \(j\) and \(j+1\). Write \(K_j\) for its +continuum and \(v_j\) for its limiting unit direction. Thus there +are points +\[ + q_{j,n}=x+r_{j,n}w_{j,n}\in K_j,\qquad + r_{j,n}>0,\quad r_{j,n}\longrightarrow0,\quad + w_{j,n}\in S,\quad w_{j,n}\longrightarrow v_j. +\] +The rays used below record only these selected limiting directions +\(v_j\); no straight ray segment is assumed to lie in \(K_j\). +All palettes in this section are at this fixed center. As before, +\(R\) rotates by \(\kappa=\pi/3\), and \(J\) by \(\pi/2\). + +\subsection{Restrictions on the two sides of the unit circle} + +The one-sided strips below are a counterpart of the complementary-color +strips in \citet[Proposition~9]{SokolovVoronov2025}. Related circle and +complementary-arc arguments also occur in +\citet[Sections~3--5]{Voronov2025Interval}, for a positive interval of +forbidden distances. We use only the single forbidden distance one; +the input is the exclusion continua just extracted. + +Let \(\mathcal B\subset S\) be the finite, antipodally symmetric +set of directions perpendicular to at least one \(v_j\). +At \(e\notin\mathcal B\), give edge \(j\) sign \(+\) or \(-\) +according as \(e\cdot v_j\) is positive or negative. Let \(O(e)\) +be the set of labels remaining after the endpoints of every positive +edge are removed, and let \(I(e)\) be the set remaining after the +endpoints of every negative edge are removed. Labels outside the +cycle belong to both sets. + +\begin{lemma}[One-sided strip restrictions]\label{lem:angular-strips} +Both \(O(e)\) and \(I(e)\) are nonempty for every +\(e\notin\mathcal B\). On an angular neighborhood of \(e\), an +open strip just outside radius one uses only \(O(e)\), and an open +strip just inside radius one uses only \(I(e)\), in both cases up +to plane-null sets. Moreover, +\begin{equation}\label{eq:source-25} + P_x(e)\cap O(e)\ne\varnothing, + \qquad P_x(e)\cap I(e)\ne\varnothing + \quad\text{for almost every }e\in S\setminus\mathcal B. +\end{equation} +\end{lemma} + +\begin{proof} +For a positive edge sign, the identity +\[ + \abs{x+e-q_{j,n}}^2 + =1+r_{j,n}^2-2r_{j,n}(e\cdot w_{j,n}) +\] +is strictly less than one for some sufficiently large fixed \(n\). +By continuity, the strict inequality holds when \(e\) varies in +a small angular neighborhood and its radius increases slightly +above one. Such points have distance greater than one from +\(x\in K_j\) and less than one from \(q_{j,n}\in K_j\); they lie +in \(\Delta(K_j)\). Proposition~\ref{prop:interface-cycle} excludes both +endpoint essential supports there. If the sign is negative, +the displayed distance is greater than one for large \(n\), and +the same argument at radii slightly below one gives the inner +strip. There are finitely many edges, so the angular neighborhoods +and positive radial widths can be intersected. An empty allowed +set would exclude every color from a nonempty open strip, contrary +to the measurable partition. + +Cover \(S\setminus\mathcal B\) by countably many smaller arcs +on each of which all signs are constant and both strip restrictions +hold with positive widths. For each such arc, polar-coordinate +Fubini gives the relevant restriction almost everywhere angularly +for almost every radius in its permitted interval. Impose no +condition from that arc outside its permitted interval; its failures +then exclude only a null set from the whole radial half-line. +Intersect these countably many full-measure radial conditions with the +full-measure condition of typical sampling at center \(x\). +Choose radii from this intersection tending to one from above. +For each fixed arc the outer restriction eventually holds along +the entire sequence. A weak-* subsequential trace in +\(\mathcal W_x\) has zero coordinates for the excluded labels +on that arc. Its coordinates sum to one, and its positive +coordinates belong to \(P_x\) almost everywhere, so it supplies +an outer allowed palette member. Countably many arcs suffice. +A separate sequence tending to one from below proves the inner +assertion. The two sides need not use the same trace. +\end{proof} + +\subsection{Exclusion of a five-cycle} + +\begin{proposition}\label{prop:angular-five} +The extracted interface cycle cannot have length five. +\end{proposition} + +\begin{proof} +Index its five labels modulo five. A label belongs to \(I(e)\) +exactly when its two incident signs are positive, and to \(O(e)\) +exactly when both are negative. Their nonemptiness requires a +pair of consecutive signs of each kind. If there were two or +more runs of each kind, the two required runs of length at least +two and the other two nonempty runs would occupy at least six +positions. Thus the five signs consist of one positive run and +one negative run, of lengths two and three in some order. + +The allowed sets are the internal vertices of these runs, of +sizes one and two. They are disjoint. The palette size bound +\eqref{eq:source-13} and \eqref{eq:source-25} imply that almost +every palette is a pair, containing one vertex from each allowed +set. Put \(D_i=\{i,i+2\}\). After a cyclic relabeling, the +length-two run occupies edges \(0,1\), with middle vertex \(1\), +and the length-three run occupies edges \(2,3,4\), with internal +vertices \(3,4\). The two possible palettes are therefore +\(D_1=\{1,3\}\) and \(D_4=\{4,1\}\), whose indices differ by +two modulo five. Reversing every sign interchanges the two +allowed sets and preserves these options. Consequently the +palette options at antipodal directions are identical. + +Choose a direction on a full six-position \(R\)-orbit for which +all the preceding almost-everywhere assertions and +\eqref{eq:source-12} hold. Write \(p_j\in\Z/5\Z\) for the +index of its palette \(D_{p_j}\) at \(R^je\). Two diagonals +\(D_a,D_b\) are disjoint exactly when \(b-a=\pm1\) modulo five. +Thus successive indices make steps \(\varepsilon_j\in\{1,-1\}\), +including the closing step from position five to position zero. +For antipodal positions \(j,j+3\), the common two-element option +set allows index difference only \(0\) or \(\pm2\). +But the possible sums of three steps are \(-3,-1,1,3\); none +is zero modulo five, and a sum is \(\pm2\) modulo five only if +all three steps have the same sign. Every consecutive triple +of steps must therefore be constant. Overlapping triples force +all six signs to agree, whereas six equal steps do not close +modulo five. This is the desired contradiction. +\end{proof} + +\subsection{Four-cycle words and projective ray geometry} + +Suppose next that the cycle labels are \(0,1,2,3\), with outsider +\(X\), and put \(U=\{0,2\}\), \(V=\{1,3\}\). +If \(X\notin P_x(e)\), the two requirements in +\eqref{eq:source-25} force a consecutive pair of positive signs +and a consecutive pair of negative signs. Hence the four signs +are a cyclic run of two of each sign, which we call a +\emph{split}. The two allowed sets, after removing \(X\), are +the opposite middle vertices of the runs. They determine a +diagonal \(d(e)\in\{U,V\}\), and \(d(-e)=d(e)\). + +Define the effective state \(p(e)\) to be \(X\) whenever +\(X\in P_x(e)\), and otherwise to be the forced diagonal +\(d(e)\). The group represented by this state is contained +in the palette: it is the singleton \(\{X\}\) in the first +case and the entire diagonal in the second. Consequently, +\begin{equation}\label{eq:source-26} + p(e)\ne p(Re),\qquad + p(e)=p(-e)\ \text{if both states are different from }X, + \quad\text{almost everywhere}. +\end{equation} +If a diagonal occurs only on a null set, the remaining +singleton-versus-pair state system contradicts +Lemma~\ref{lem:transitions-two-state}; a single remaining state +already contradicts the first assertion in \eqref{eq:source-26}. +We may therefore suppose that both diagonal states have positive +angular measure. + +For a nonparallel pair of density-one directions of the \(U\) +and \(V\) state sets, the density conclusion of +Lemma~\ref{lem:palette-transfer} excludes all four cycle labels +at their sum with \(x\). That sum has density one of \(X\). +Lemma~\ref{lem:transitions-strict-separation} now applies to these +three represented groups. If \(F_U,F_V\) denote the compact +angular essential supports of the diagonal state sets, then +\[ + F_D\cap RF_D=\varnothing\qquad(D=U,V). +\] +The same holds for \(R^{-1}\). Compactness provides a common +positive angular separation margin. + +Choose a dense conull set \(G\) of angular parameters, invariant +under addition of \(\kappa\), on which all six orbit positions +avoid \(\mathcal B\), satisfy the state rules, and belong to +\(F_U\) or \(F_V\) whenever the respective state occurs. Define +\[ + w(t)_j=p(e^{i(t+j\kappa)}),\qquad j\in\Z/6\Z, + \quad t\in G. +\] +There exists \(\eta>0\) such that whenever \(t,t'\in G\) +have circular angular distance less than \(\eta\), a diagonal +at any position of \(w(t)\) cannot equal a diagonal at an +adjacent position of \(w(t')\). Indeed, such equality would +give points of the same compact essential support at angular +distance less than the margin from an \(R\)- or \(R^{-1}\)-pair. +We refer to this as the cross-word restriction. + +\begin{lemma}[Six-position word alternatives]\label{lem:angular-words} +Every word \(w(t)\) contains \(X\), and all its \(X\) positions +have the same parity. Every direction outside \(\mathcal B\) +is split. +\end{lemma} + +\begin{proof} +A word without \(X\) alternates \(U,V\) by adjacent inequality, +contradicting the antipodal equality in \eqref{eq:source-26}. +Suppose that \(X\) occurs on both parities. Any pair of its +opposite-parity positions must be antipodal, since otherwise +they are adjacent on the six-cycle. There are exactly two +such occurrences: fixing one on each parity prevents any +additional occurrence on either parity. Cyclically shift +them to positions zero and three. Positions one and two +are distinct non-\(X\) states, and their antipodes four and +five have the same respective values. Thus the word is a +three-periodic permutation of \(X,U,V\). + +Such a word is isolated under the cross-word restriction. +In any compatible nearby word, each of its positions zero +and three has neighbors \(U,V\) in the original word; it +must therefore be \(X\). The remaining four positions are +non-\(X\) by their own adjacent inequality. If the original +positions one and two are \(U,V\), the cross-word restrictions +against position two and position one, respectively, force +the new positions one and two to be \(U,V\); the other block +is forced in the same way. Hence the new word is identical. + +Any two points of a dense subset of the circle can be joined +by a finite chain within that subset whose steps have angular +distance less than \(\eta\): subdivide a connecting arc into +steps smaller than \(\eta/3\) and approximate its intermediate +vertices by points of the dense set. Isolation would therefore +propagate this word throughout \(G\). But +\(w(t+\kappa)_j=w(t)_{j+1}\), and a three-periodic permutation +word is not equal to its one-position shift. Since \(G\) is +shift invariant, this is impossible. + +It follows that the nonempty set of \(X\) positions has a +unique parity, denoted \(h(t)\in\Z/2\Z\). The three positions +of the other parity are non-\(X\), hence split. They contain +one representative from each antipodal pair of positions. +Since splitting is antipodally invariant, all six positions +are split. In particular almost every direction is split. +Signs are locally constant outside the finite set \(\mathcal B\). +A nonsplit direction there would have a nonsplit open +neighborhood of positive measure, which is impossible. +\end{proof} + +\begin{lemma}[Geometry of four split rays]\label{lem:angular-four-rays} +Under the conclusions of Lemma~\ref{lem:angular-words}, the four +rays consist of two distinct antipodal pairs. Opposite directions +occupy opposite edge positions. On the projective angular circle +of circumference \(\pi\), the forced diagonal \(d\) equals \(U\) +on one open arc and \(V\) on its complementary open arc. One +of these arcs has length at most \(\pi/2\). +\end{lemma} + +\begin{proof} +Every generic open semicircle \(\{v:e\cdot v>0\}\) selects +exactly two rays, counted with multiplicity, and the selected +edge positions are adjacent. At a boundary perpendicular to +one unoriented ray line, cross the boundary without crossing +any other such boundary. The selected multiplicity on that +line changes from the number of its rays in one direction +to the number in the opposite direction; all other selections +stay fixed. Equality of the total count before and after +shows that these two multiplicities coincide. Thus all rays +are antipodally paired. + +If only one unoriented line occurs, it has two rays in each +direction. The selected edge positions are a fixed adjacent +pair or its complement. These two split patterns give the +same diagonal. The other diagonal could not occur on a set +of positive measure, contrary to our assumption. There are +therefore two distinct lines, with one ray in each of their +two directions. + +A specified ray on one line can be positive simultaneously +with either ray on the other line: the corresponding two +nonopposite unit directions fit in an open semicircle. +Its edge position is consequently adjacent to both positions +from the other line. In a four-cycle the remaining position +is its opposite, so its antipodal ray occupies the opposite +edge position. This applies to both lines. + +Modulo antipodes, the two perpendicular boundaries divide +the angular circle into two open arcs. The signs, up to +global reversal, are constant on each arc. Crossing a +boundary replaces one selected edge by its opposite while +keeping the selected edge from the other line. The two +middle vertices of the resulting split are the other +diagonal. Thus the forced diagonal takes the two different +values on the two arcs. Their lengths sum to \(\pi\), which +proves the last assertion. +\end{proof} + +\begin{proposition}\label{prop:angular-four} +The extracted interface cycle cannot have length four. +\end{proposition} + +\begin{proof} +Suppose \(t,t'\in G\) are less than \(\eta\) apart and +\(h(t)\ne h(t')\). Select the three positions opposite to +the \(X\) parity in \(w(t)\), and the other three positions +opposite to the \(X\) parity in \(w(t')\). All selected +states are diagonals; around the six positions they alternate +between the two words. Cross-word adjacent states differ. +Since only \(U,V\) are available, all three selected entries +of the first word have one diagonal value, and all three +of the second have the other value. + +For either word the three selected directions, projected +modulo antipodes, are spaced equally by \(\kappa\). Each +is a non-\(X\) direction, so the forced diagonal there is +its selected value. Both of the open diagonal arcs in +Lemma~\ref{lem:angular-four-rays} would therefore contain +three equally spaced projective positions. An open arc +containing such three positions has length greater than +\(2\pi/3\): its closed complementary arc avoids the three +points, so lies strictly inside one of the three gaps and +has length less than \(\pi/3\). This contradicts +the diagonal arc of length at most \(\pi/2\). + +Thus \(h\) agrees at all sufficiently close points of \(G\). +The dense-set chain argument from +Lemma~\ref{lem:angular-words} makes it constant on \(G\). +Yet shifting by \(\kappa\) shifts every word by one position +and reverses this parity. The shift invariance of \(G\) +gives a contradiction. +\end{proof} + +\subsection{Triangle interfaces force alternating sectors} + +It remains to consider a triangle cycle. Call its labels the +\emph{triangle labels}, and write \(X,Y\) for the outsiders. +At every \(e\notin\mathcal B\), two of the three edge signs +agree. Those two edges cover all three triangle labels. +Lemma~\ref{lem:angular-strips} therefore provides an open +strip on an angular neighborhood of \(e\), approaching radius +one from one side, which uses only \(X,Y\) almost everywhere. +Equation~\eqref{eq:source-25} gives an outsider in \(P_x(e)\) +almost everywhere. There cannot be two outsiders in such +a palette, since \eqref{eq:source-12} would leave no outsider +at \(Re\). We obtain a measurable binary state +\[ + s(e)\in\{X,Y\},\qquad s(Re)\ne s(e) + \quad\text{almost everywhere}. +\] + +The use of two complementary colors near a unit circle has a +predecessor in Townsend's annulus argument +\citep[Theorems~1 and~4]{Townsend2005}; here the restrictions hold +almost everywhere and need not form an annulus. The next lemma +extracts local constancy directly from the open strips. + +\begin{lemma}[Local binary rigidity]\label{lem:angular-binary-local} +If \(e_0,Re_0\notin\mathcal B\), then \(s\) is almost +everywhere constant on an angular neighborhood of \(e_0\). +\end{lemma} + +\begin{proof} +Let \(\mathcal U,\mathcal V\) be the open strips restricted +to \(X,Y\) near \(e_0,Re_0\), respectively. Each approaches +radius one from its prescribed side, and these sides may +differ. Put \(z_0=x+e_0\). There are two nonparallel unit +directions \(v'_1,v'_2\) such that +\(z_0+v'_j\in\mathcal V\). To verify this, rotate so +\(e_0=1\) and consider +\[ + F(r,\theta)=\abs{r e^{i\theta}-1}^2-1 + =r^2-2r\cos\theta. +\] +At \((1,\pi/3)\), \(F=0\) and +\(\partial_\theta F=\sqrt3\ne0\). The implicit function +theorem permits \(r\) to move onto either desired radial +side while \(\theta\) stays in the angular arc of +\(\mathcal V\). This gives a point strictly inside +\(\mathcal V\) at unit distance from \(z_0\). Openness +then gives an open set of such unit directions, containing +two nonparallel choices. + +Choose a smaller connected polar product strip +\(\mathcal U'\subset\mathcal U\), still approaching radius +one on an angular neighborhood of \(e_0\), close enough to +\(z_0\) that +\(q+v'_1,q+v'_2\in\mathcal V\) for every +\(q\in\mathcal U'\). We claim that \(c\) is almost +everywhere constant on \(\mathcal U'\). + +Fix a point of \(\mathcal U'\). On a sufficiently small +ball around it, all starting points \(q\) and sufficiently +small shifts \(h\) admit the following construction inside +\(\mathcal U\) and \(\mathcal V\). Since +\(Jv'_1,Jv'_2\) form a basis, write +\(h=t_1Jv'_1+t_2Jv'_2\), with coefficients linear in \(h\). +Move first by \(t_1Jv'_1\) and then by \(t_2Jv'_2\). +For either move, starting at \(q'\) and using \(t=t_j\), +\(v'=v'_j\), its two endpoints in \(\mathcal U\) have +the common unit neighbor +\begin{equation}\label{eq:angular-even-path} + q'+\frac{t}{2}Jv'+\sqrt{1-t^2/4}\,v'\in\mathcal V. +\end{equation} +The two distances are one because \(v'\) and \(Jv'\) are +orthogonal unit vectors. Smallness keeps \(\abs{t}<2\) +and every constructed point in the stated open set. The +combined path has four unit edges, with repetitions allowed. + +Fix such an \(h\). Every vertex of the path is a translation +of \(q\), so for almost every starting point all five vertices +are typical and have colors in \(\{X,Y\}\). Typical-point +properness makes the colors alternate along the path. +Consequently \(c(q)=c(q+h)\) for almost every \(q\) in +the small ball, for each fixed permissible \(h\). +All path maps are jointly measurable in \((q,h)\): the +coefficients are linear and the square-root terms continuous. +Fubini gives this equality for almost every \((q,h)\). +In a still smaller ball, every ordered pair \((q,q')\) +has \(q'-q\) within the permissible shift range; the +invertible change \((q,h)\mapsto(q,q+h)\) shows that +almost every pair in that ball has equal colors. Choosing +one typical first point by Fubini proves local essential +constancy of \(c\). + +The sets of points of \(\mathcal U'\) with local essential +color \(X\), or local essential color \(Y\), are disjoint +open sets covering \(\mathcal U'\); they are disjoint because +overlapping open neighborhoods have positive area. +Connectedness forces just one of these sets to be nonempty. +A countable subcover of monochromatic neighborhoods then +shows that \(\mathcal U'\) uses one color almost everywhere. + +Finally choose typical radial samplings at \(x\) tending +to one from the side of \(\mathcal U'\). Polar Fubini +ensures that the samples equal this color almost everywhere +on its angular arc. A weak-* trace puts that outsider in +\(P_x\) almost everywhere on the arc. It is the unique +outsider there, so \(s\) is essentially constant near +\(e_0\), as asserted. +\end{proof} + +\begin{lemma}[The boundary is one orbit]\label{lem:angular-boundary} +The essential boundary of \(s\) consists of exactly one +six-position \(R\)-orbit. Its complementary sectors have +angle \(\pi/3\), with alternating essential values \(X,Y\). +\end{lemma} + +\begin{proof} +Let \(E_s\) be the set of directions at which \(s\) is not +almost everywhere constant on any open neighborhood. +Its complement is open, so \(E_s\) is closed. If it were +empty, local essential constancy and connectedness of the +circle, followed by a countable subcover, would make \(s\) +globally constant almost everywhere. The flip under \(R\) +excludes this. That same almost-everywhere flip transports +local essential constancy in both directions, so \(E_s\) +is \(R\)-invariant. + +Consider any six-position orbit contained in \(E_s\). +By Lemma~\ref{lem:angular-binary-local}, every consecutive +pair in it has at least one vertex in \(\mathcal B\). +Because \(\mathcal B\) is antipodally symmetric, project +the orbit modulo antipodes. Its three positions form a +triangle, and every edge has an endpoint among the +projective classes of \(\mathcal B\). At least two such +classes are needed. There are at most three projective +classes in \(\mathcal B\), one for each interface ray. +Distinct \(R\)-orbits have disjoint projective positions: +a shared class means a shared direction or its antipode, +and the antipode is already the third rotation in the orbit. +Thus two distinct boundary orbits would require at least +four projective classes in \(\mathcal B\). There can be +at most one orbit, and nonemptiness supplies exactly one. + +On each of the six complementary open sectors, local +essential constancy and connectedness give one essential +value. Rotation maps each sector to the next and flips +that value, proving the assertion. +\end{proof} + +\subsection{An open three-label region and the triangle contradiction} + +Rotate coordinates about \(x\) so that the sector boundaries +of Lemma~\ref{lem:angular-boundary} have angles that are +multiples of \(\pi/3\). The outsider state is then determined +by the sign of \(\sin(3\theta)\), in one of the two possible +orders of \(X,Y\). + +\begin{lemma}[Polynomial three-label region]\label{lem:angular-region} +Put \(z=\xi+i\upsilon\), \(l=\xi^2+\upsilon^2\). If +\(00\) so that each point remains in +its corresponding neighborhood after any common translation +\(t\in B(0,\rho)\). For each vertex, the translations +that put it outside the typical set, or in the null subset +where a nontriangle label occurs, form a null set. The +union of these seven exceptional translation sets is null. +Choose \(t\) outside that union. All seven translated +vertices are then typical and use the three triangle labels. +Their unit edges are preserved by the placement and common +translation, and typical-point properness gives a proper +three-coloring of the graph in +\eqref{eq:angular-seven-vectors}. The two shared-edge +triangle pairs proved that this graph has no such coloring. +\end{proof} + +\begin{proof}[Proof of Theorem~\ref{thm:weak-five}] +If a weak measurable five-coloring existed, +Proposition~\ref{prop:interface-cycle} would produce an +interface cycle of length three, four, or five with the +common-center and exclusion properties used above. +Propositions~\ref{prop:angular-five}, \ref{prop:angular-four}, +and \ref{prop:angular-three} exclude all three possibilities. +Thus no weak measurable five-coloring exists. +\end{proof} + +\subsection{Rational verification of the seven-point certificate} +\label{sec:angular-certificate-verification} + +All terminating decimals in this verification denote exact rational +numbers. + +\begin{proof}[Proof of Lemma~\ref{lem:angular-certificate}] +Table~\ref{tab:angular-coordinates} gives coordinate centers +\((\xi_0,\upsilon_0)\), each within \(h=0.00011\) of +the corresponding exact coordinate of \(z_g\), and strict +upper and lower polynomial bounds. +\begin{table}[htbp] +\centering +\begin{tabular}{@{}crrrr@{}} +\toprule +\(g\)&\(\xi_0\)&\(\upsilon_0\)&\(P_+\)&\(P'_-\)\\ +\midrule +\(0\)&\(-1.2906\)&\(0.1876\)&\(0.90\)&\(1.35\)\\ +\(A\)&\(-0.6335\)&\(0.9414\)&\(0.095\)&\(0.110\)\\ +\(B\)&\(-0.3092\)&\(-0.0045\)&\(0.000002\)&\(0.00065\)\\ +\(T\)&\(0.3478\)&\(0.7493\)&\(0.023\)&\(0.026\)\\ +\(uA\)&\(-1.1598\)&\(1.1790\)&\(9.8\)&\(19.0\)\\ +\(uB\)&\(-0.3666\)&\(0.5700\)&\(0.0022\)&\(0.024\)\\ +\(uT\)&\(-0.2358\)&\(1.5614\)&\(12.65\)&\(12.9\)\\ +\bottomrule +\end{tabular} +\caption{Coordinate boxes and bounds \(Pa^3(1-b/4)\min(\abs{a-1},\abs{b-1})^2>P'_-. +\end{align*} +The last inequalities are direct rational products of +the second table, with the target bounds in the first. +For example, its final row gives +\[ + 2.44(2.273)^2<12.65, + \qquad + (2.493)^3(1-2.495/4)(1.493)^2>12.9. +\] +The other six rows give the corresponding displayed strict +bounds by the same two products. The squared-radius +intervals also lie respectively in +\[ + (1.70,1.71),\ (1.28,1.29),\ (0.095,0.096),\ (0.68,0.69), + \ (2.73,2.74),\ (0.459,0.46),\ (2.49,2.50). +\] +Finally \(P_+0$ with the following properties. For each edge +$i_ji_{j+1}$ of the simple label cycle, with indices modulo $\ell$, +there are a compact connected set +$K_j\subset\overline{B(x,\rho)}$, a direction $v_j\in S$, and a +sequence $q_{j,k}\in K_j\setminus\{x\}$ such that +\begin{enumerate} +\item $x\in K_j$ and $K_j\cap\partial B(x,\rho)\ne\varnothing$; +\item $q_{j,k}\to x$ and +$(q_{j,k}-x)/|q_{j,k}-x|\to v_j$; +\item both endpoint supports are excluded from the same open set: +\[ + H_{i_j}\cap\Delta(K_j)= + H_{i_{j+1}}\cap\Delta(K_j)=\varnothing. +\] +\end{enumerate} +The extracted cycle need not consist of edges of $\Gamma_x$. +\end{proposition} + +The cyclic transition centers determine which small disks remain +unfilled in our approximation; the cycle in the proposition comes from a +loop of the resulting continuous graph-valued map. We first construct +that map by averaging the color indicators and removing disks whose +total radii tend to zero. A planar cover obstruction forces a nontrivial +loop around one remaining disk. Connected components of the inverse +images of its edge midpoints then cross an annulus, and their limits give +the sets $K_j$. + +Figure~\ref{fig:exclusion-continuum} illustrates the straddling mechanism +for one of these continua. + +\begin{figure}[htbp] +\centering +\input{figures/exclusion-continuum} +\caption{The strict inequalities $|z-q|<1<|z-x|$, with $x,q\in K_j$, +put $z$ in $\Delta(K_j)$. Connectedness supplies a point $p\in K_j$ +with $|z-p|=1$. Both endpoint essential supports are excluded from $\Delta(K_j)$. The shape is schematic: the proof assumes neither an arc nor +a smooth interface nor a tangent at $x$.} +\label{fig:exclusion-continuum} +\end{figure} + +\subsection{Disk averages and missing edges} + +We begin with continuous probability coordinates. Their unit-distance +inequality will yield color exclusions from the straddling regions of +the limiting sets. Small overlaps for missing transition edges will +control the disks removed from the approximation. + +For $\epsilon>0$, set +\[ + p_i^\epsilon(q)=\frac1{\pi\epsilon^2} + \int_{B(0,\epsilon)}a_i(q+v)\,dv. +\] +These are continuous probability coordinates: +$p_i^\epsilon\ge0$ and $\sum_i p_i^\epsilon=1$ everywhere. +The symmetric-difference estimate for two disks gives +$\operatorname{Lip}(p_i^\epsilon)\le C/\epsilon$, with an absolute +constant $C$. Approximation by local averages and Lebesgue +differentiation give $p_i^\epsilon\to a_i$ in $L^1_{\rm loc}$ and +almost everywhere as $\epsilon\downarrow0$. + +\begin{lemma}[Disk inequalities and support detection] +\label{lem:transitions-disk-averages} +For every pair $q,z\in\R^2$ with $|q-z|=1$ and every label $i$, +\begin{equation} + p_i^\epsilon(q)+p_i^\epsilon(z)\le1. + \label{eq:source-17} +\end{equation} +If $q_n\to q$, $\epsilon_n\to0$, and +$\liminf_n p_i^{\epsilon_n}(q_n)>0$, then $q\in H_i$. +\end{lemma} + +\begin{proof} +Translate $q$ and $z$ by the same vector uniformly distributed in +$B(0,\epsilon)$. Almost every such translation puts both points in +the typical set. They cannot then both have color $i$; averaging +proves \eqref{eq:source-17}. Thus this inequality has no exceptional +unit pairs. For the last assertion, every neighborhood of $q$ +eventually contains $B(q_n,\epsilon_n)$, and the stated lower bound +makes its intersection with $A_i$ have positive area. +\end{proof} + +\begin{lemma}[A missing edge has small overlap] +\label{lem:transitions-missing-edge} +If $ij\notin\Gamma_x$, there is a ball $B$ about $x$ such that, for +every smaller ball $B'$ with $\overline{B'}\subset B$, +\begin{equation} + \int_{B'}p_i^\epsilon(q)p_j^\epsilon(q)\,dq=o(\epsilon). + \label{eq:source-18} +\end{equation} +For any ball $B''$ with $\overline{B''}\subset B'$ and any fixed +$\alpha>0$, the set +\[ + \{q\in B'':p_i^\epsilon(q)>\alpha,\quad + p_j^\epsilon(q)>\alpha\} +\] +can be covered by finitely many open disks whose total radii tend to +zero as $\epsilon\downarrow0$. Every point of the set lies in a disk +interior. +\end{lemma} + +\begin{proof} +The negation of \eqref{eq:source-14}, and nonnegativity of $D_{ij}$, +give a ball $B$ with $D_{ij}(B,h)=o(|h|)$. More explicitly, for all +sufficiently small $r>0$ put +\[ + \omega(r)=\sup_{0<|h|\le r}\frac{D_{ij}(B,h)}{|h|}; + \qquad \omega(r)\longrightarrow0. +\] +For independent uniform vectors $h_1,h_2\in B(0,\epsilon)$, expand the +product of the two disk averages and integrate over $q\in B'$. When +$\epsilon<\dist(\overline{B'},\R^2\setminus B)$, the change of root +$r=q+h_1$ bounds the area where $c(q+h_1)=i$ and $c(q+h_2)=j$ by +\[ + D_{ij}(B,h_2-h_1)\le 2\epsilon\omega(2\epsilon). +\] +For $h_1=h_2$ the simultaneous event is empty. Fubini now gives +\eqref{eq:source-18}, with a bound uniform over the two shifts. + +Take a square grid with side $\ell=c_\alpha\epsilon$, where +$c_\alpha>0$ is fixed small enough that the Lipschitz oscillation of +either probability on a square is at most $\alpha/2$. Select all +squares touching the displayed violation set in $B''$. Their number +$N_\epsilon$ is finite, and for small $\epsilon$ each lies in $B'$ and +has both probabilities at least $\alpha/2$ throughout. Disjointness +of their interiors and \eqref{eq:source-18} imply +\[ + N_\epsilon\ell^2\frac{\alpha^2}{4}=o(\epsilon), + \qquad N_\epsilon\epsilon=o(1). +\] +Cover every selected closed square by the open disk with the same +center and radius $\ell$. These disks strictly contain the squares +and have total radii $N_\epsilon\ell=o(1)$. +\end{proof} + +\subsection{Graph-valued maps outside small core disks} + +The preceding overlap bound allows us to remove all simultaneous +threshold violations associated with missing edges. Fix +$0<\alpha<1/5$, and define +\begin{equation} + q_i^\epsilon(z)= + \frac{(p_i^\epsilon(z)-\alpha)_+} + {\sum_{k\in\mathcal I}(p_k^\epsilon(z)-\alpha)_+}, + \qquad i\in\mathcal I. + \label{eq:source-22} +\end{equation} +The denominator is positive because the five probabilities sum to one. +Thus $q^\epsilon$ is a continuous map into the simplex of probability +vectors on $\mathcal I$. Let $\mathscr G$ be the one-skeleton of that +simplex, realized as the complete graph on its five vertices. In +particular, $q_i^\epsilon(z)>0$ if and only if +$p_i^\epsilon(z)>\alpha$. + +Choose a closed square $Q$ of side ten with +$\partial Q\cap\mathcal L=\varnothing$. Such a square is obtained by +a small translation: local finiteness leaves only finitely many points +to avoid near the boundary of all squares in that translation family. +Choose a larger closed square $Q'$ containing $Q$ in its interior, and +write $L_0=\mathcal L\cap Q'$, a finite set by +Theorem~\ref{thm:cyclic-local-finiteness}. + +\begin{lemma}[Small holes and tree fillings] +\label{lem:interfaces-hole-maps} +There are $\epsilon_n\downarrow0$ and finite families $\mathcal H_n$ +of open disks with pairwise disjoint closures such that their total +radii tend to zero and the following hold for all sufficiently large +$n$. +\begin{enumerate} +\item For each $x\in L_0$, precisely one disk of $\mathcal H_n$ is +designated the core disk $B_{n,x}$ and contains $x$ in its interior. +Different points give different core disks. If $x\in Q$, then +$\overline{B_{n,x}}\subset\operatorname{int}(Q)$; if $x\notin Q$, then +$\overline{B_{n,x}}\cap Q=\varnothing$. +\item The map $q^{\epsilon_n}$ takes values in $\mathscr G$ on +$Q\setminus\bigcup_{B\in\mathcal H_n}B$. +\item There is a continuous map +\begin{equation} + g_n:Q\setminus\bigcup_{x\in L_0\cap Q}B_{n,x} + \longrightarrow\mathscr G + \label{eq:source-23} +\end{equation} +equal to $q^{\epsilon_n}$ outside all the open disks of +$\mathcal H_n$. It is obtained by filling the non-core disks through +trees, preserving their boundary values. +\end{enumerate} +\end{lemma} + +\begin{proof} +Fix a small $\delta>0$. Start with the original core disks +$B(x,\delta)$, $x\in L_0$, and put +\[ + M_\delta=Q'\setminus\bigcup_{x\in L_0}B(x,\delta). +\] +This is compact, and every $\Gamma_z$, $z\in M_\delta$, is a forest, +including any isolated vertices. For each nonedge of $\Gamma_z$, +Lemma~\ref{lem:transitions-missing-edge} supplies a ball on which the +corresponding simultaneous threshold violations admit finite open +disk covers with total radii tending to zero as $\epsilon\to0$. +There are finitely many nonedges, so choose one radius $s_z>0$ +working for all of them. More explicitly, choose it small enough +to have the transition estimate on $B(z,3s_z)$, the integral +estimate on $B(z,2s_z)$, and the disk covers for violations in +$B(z,s_z)$. The covers include the violation points in the +interiors of their disks. + +Choose finitely many balls $B(z,s_z)$ covering $M_\delta$, and let +$\lambda_\delta>0$ be a Lebesgue number of this cover on $M_\delta$. +For each selected ball, mark all the disks covering violations for +its forest's nonedges. These additional disks form a finite family +whose total radii $m_{\delta,\epsilon}$ tend to zero for this fixed +$\delta$. At a point of $M_\delta$ outside all marked disks, the +positive coordinates of \eqref{eq:source-22} form a clique in the +forest of every selected ball containing the point. They therefore +number at most two, and a two-element set is a forest edge. This +proves the required local graph-valued assertion. + +We next merge disks until their closures are pairwise disjoint. +Whenever two current disks have intersecting closures, replace them +by an open disk strictly containing both closures, with radius at +most the sum of their radii plus a prescribed positive error. +An enclosing disk with radius at most the sum exists by choosing its +center on the line of centers; in the containment case the larger +disk suffices. A slight enlargement makes containment strict. +Each merge reduces the finite number of disks, so the sum of all +enlargement errors can be made less than any specified +$\eta>0$, using the initial number as a bound on the number of merges. + +A final disk is called core if its merger class contains an original +core disk. Every final boundary avoids every original disk interior. +The total final radii are at most +\[ + |L_0|\delta+m_{\delta,\epsilon}+\eta, +\] +and each final non-core radius is at most +$m_{\delta,\epsilon}+\eta$. Take $\delta_n\downarrow0$. +After fixing $\delta_n$, its finite cover, and its Lebesgue number, +choose $\epsilon_n$ and the merger errors so that +\[ + \epsilon_n<\min\{1/n,\epsilon_{n-1}/2\},\qquad + m_{\delta_n,\epsilon_n}+\eta_n + <\min\{1/n,\lambda_{\delta_n}/3\}, +\] +omitting the preceding-$\epsilon$ condition at $n=1$. +The total final radii now tend to zero. + +For large $n$, no final disk can contain two distinct points of $L_0$: +its diameter is smaller than their minimum positive separation. +Every original core disk belongs to some final disk, so this gives +exactly one core disk for each point. Since a disk of radius $r$ +containing $x$ lies in $B(x,2r)$, the positive distances between the +finitely many points of $L_0$ and $\partial Q$ give the asserted +positions of the core disks. + +Consider a final non-core disk whose closure meets $Q$. Its vanishing +diameter puts its entire closed disk inside $Q'$ for large $n$. +Moreover, its closure is disjoint from every final core closure, +and these final core disks contain all the original core disks. +It therefore lies outside all original core interiors, hence in +$M_{\delta_n}$. Its diameter is less than $\lambda_{\delta_n}$, +so its entire closed disk lies in one selected forest ball. +Its boundary avoids all original marked interiors. The original +map $q^{\epsilon_n}$ thus maps that boundary into the selected +forest, and its connected boundary image lies in one component +tree, possibly a single vertex. A finite tree is contractible, +so the boundary map extends continuously over the closed disk +with values in that tree. If the disk crosses $\partial Q$, +make the extension on the whole disk and restrict it to $Q$. + +Outside all final open disks, every point of $Q$ belongs to +$M_{\delta_n}$ and avoids all original marked disks, so the original +map is graph-valued there. Paste it to the finitely many tree +extensions. Their closed disks have disjoint closures and agree +with the original map on their boundaries, so finite pasting proves +\eqref{eq:source-23}. Empty disk lists require no operation. +\end{proof} + +\subsection{A nontrivial core boundary} + +We will repeatedly use the following area estimate. For a closed +disk $\overline{B(u,r)}$ with $r<1$, every point at distance one from +some point of that disk belongs to +\[ + E(u,r)=\{z:1-r\le |z-u|\le1+r\}. +\] +The triangle inequality proves this containment, and the annulus +has area $4\pi r$. For the families in +Lemma~\ref{lem:interfaces-hole-maps}, write +\begin{equation} + E_n=\bigcup_{B(u,r)\in\mathcal H_n}E(u,r), + \qquad + |E_n|\le4\pi\sum_{B(u,r)\in\mathcal H_n}r\longrightarrow0. + \label{eq:interfaces-annulus-area} +\end{equation} +Here all radii are less than one for large $n$. If $z\notin E_n$, +no exact unit-distance point from $z$ lies in any closed hole disk, +regardless of how many disks there are. + +\begin{lemma}[A cover obstruction on a square] +\label{lem:interfaces-cover} +A closed square of side ten has no relatively open cover of +multiplicity at most two and mesh at most two, where mesh is the +supremum of the Euclidean diameters of its members. +\end{lemma} + +\begin{proof} +Compactness gives a finite subcover $U_1,\ldots,U_N$ with the same +bounds. Choose $z_a\in U_a$. There is a continuous partition of +unity $\phi_a$ with $\phi_a(z)>0$ only when $z\in U_a$; explicitly, +use $\dist(z,Q\setminus U_a)$ and normalize their sum. None of the +$U_a$ equals $Q$, since its diameter would exceed two, so these +distance functions are well defined. Set +\[ + F(z)=\sum_{a=1}^N\phi_a(z)z_a. +\] +Convexity gives $F(Q)\subset Q$. At most two weights are positive +at any point, so the image is contained in the finite union of +segments $[z_a,z_b]$ and singleton vertices. Also +\[ + |F(z)-z| + \le\sum_a\phi_a(z)|z_a-z|\le2. +\] +For any $y\in Q$ with $\dist(y,\partial Q)>2$, the continuous map +$z\mapsto y+z-F(z)$ takes $Q$ into $Q$, since its image lies in +the closed radius-two disk about $y$. The Brouwer fixed point +theorem, in its planar form +\citep[Theorem~1.9]{Hatcher2002}, therefore supplies a fixed point, +and at that point $F(z)=y$. Thus $F(Q)$ contains the nonempty open +central region $\{y\in Q:\dist(y,\partial Q)>2\}$, impossible for a +finite union of segments. +\end{proof} + +\begin{lemma}[Non-null core loops] +\label{lem:interfaces-core-loop} +For all sufficiently large $n$, some core disk $B_{n,x}$ contained +in $Q$ has a boundary loop whose image under $g_n$ is not null +homotopic in $\mathscr G$. +\end{lemma} + +\begin{proof} +Otherwise, along an infinite subsequence all the core boundary +loops are null homotopic. Extend each such loop over its disk +in $\mathscr G$, and paste to obtain a continuous map +$\widetilde g_n:Q\to\mathscr G$, still equal to $q^{\epsilon_n}$ +outside all holes. For each $i\in\mathcal I$, take the components +of the relatively open set +\[ + U_{n,i}=\{z\in Q:(\widetilde g_n(z))_i>0\}. +\] +The square and its relatively open subsets are locally path +connected; hence these components are open and path connected. +They form a cover of $Q$ of multiplicity at most two, since every +point of $\mathscr G$ has at most two positive coordinates. + +We claim that the mesh of this component cover is at most $1+o(1)$. +If not, after restricting to a subsequence there are a fixed +constant $c>0$, a fixed label $i$, and paths in $U_{n,i}$ joining +points at distance greater than $1+c$. Replace either endpoint +lying in a hole by a boundary point on the path. The path must +exit that disk because the other endpoint is distant, and the +replacement changes the endpoint by at most the disk's diameter. +If both endpoints require replacement, their disks are distinct +for large $n$. On every hole boundary the original map is +preserved. Thus the replacement endpoints $a_n,b_n$ satisfy +\[ + p_i^{\epsilon_n}(a_n)>\alpha,\qquad + p_i^{\epsilon_n}(b_n)>\alpha +\] +and remain at distance at least $1+c-o(1)$. Pass to limits +$a_n\to a$, $b_n\to b$ in $Q$, with $|a-b|>1$. +Lemma~\ref{lem:transitions-disk-averages} gives $a\in H_i$. + +Choose a sufficiently small fixed open plane ball $U$ about $a$. +For all large $n$ and every $z\in U$, the distances to $a_n$ and +$b_n$ are strictly below and strictly above one, respectively. +Continuity along the path supplies a point at exact distance one +from $z$. If $z\notin E_n$, every such crossing point is outside +all closed hole disks by \eqref{eq:interfaces-annulus-area}. +At this point $\widetilde g_n=q^{\epsilon_n}$ and its $i$th +coordinate is positive, so $p_i^{\epsilon_n}>\alpha$. +The pointwise inequality \eqref{eq:source-17} therefore gives +$p_i^{\epsilon_n}(z)\le1-\alpha$. + +It follows that +\begin{equation} + \alpha |A_i\cap U| + \le + \norm{p_i^{\epsilon_n}-a_i}_{L^1(U)} + +\alpha|E_n\cap U| + \longrightarrow0. + \label{eq:interfaces-mesh-measure} +\end{equation} +The convergence uses local $L^1$ convergence of disk averages and +\eqref{eq:interfaces-annulus-area}. This contradicts $a\in H_i$, +which requires positive $A_i$-measure in $U$. The mesh claim +follows. For large $n$ the mesh is therefore at most two, +contradicting Lemma~\ref{lem:interfaces-cover}. This argument also +rules out the possibility of having no core disks in $Q$. +\end{proof} + +\subsection{Graph words and annular crossings} + +We now have a graph-valued map with a non-null loop around a core disk. +To turn that loop into connected crossing sets, we identify edge +midpoints that every homotopic loop must meet. We first recall the +reduced-path argument for graphs; compare +\citet[Section~1.A]{Hatcher2002}. + +\begin{lemma}[Midpoint detection by a reduced circuit] +\label{lem:interfaces-graph-word} +Let $G$ be a finite connected simple graph. Every nontrivial free +homotopy class of loops in $G$ has a nonempty cyclically reduced closed +edge word $W$. Such a word contains a simple cycle. Moreover, if $m$ +is the midpoint of any edge occurring in $W$, every loop freely +homotopic to a nonzero power of $W$ meets $m$. +\end{lemma} + +\begin{proof} +An edge path is reduced when consecutive edges never reverse each +other, and a reduced closed edge path is cyclically reduced when its +last and first edges do not reverse each other either. Paths between +vertices, up to homotopy fixing their endpoints, have unique reduced +edge representatives. One way to see this is to construct the covering +tree based at a vertex $v$: its vertices are reduced paths starting at +$v$, and its edges append one edge and then reduce. Deleting the last +edge gives the unique path toward the empty word, so this covering +graph is a tree. The map recording the endpoint is a bijection on +each incident-edge star and hence a covering map. Lifting a path to +this tree and contracting it to the unique simple path between its +lifted endpoints proves existence and uniqueness of the reduced +representative. These contractions apply to arbitrary continuous +paths: their compact images in the locally finite covering tree lie +in finite subtrees. + +Move a loop's basepoint to a vertex along a path, reduce its edge +word, and successively trim inverse first and last edges. The result +is cyclically reduced and is empty precisely when the loop is null +homotopic. Moving the basepoint, or following the basepoint during a +free homotopy, changes the based class by conjugation. + +We record how conjugation affects a nonempty cyclically reduced word. +First reduce the conjugating path $P$ in $PWP^{-1}$. At the last edge +of $P$, cancellation against one end of $W$ shortens $P$ and moves +one edge of $W$ to its other end. This only rotates $W$ cyclically. +Cancellation at both ends of $W$ at once is impossible because $W$ +is cyclically reduced. If there is no cancellation at either end, +the conjugating path is simply removed when the whole closed word is +cyclically trimmed. Induction on the length of $P$ shows that +conjugation leaves the cyclically reduced circuit unchanged up to +cyclic rotation. + +A nonzero power of $W$ is already cyclically reduced: concatenate +copies of $W$ for a positive power, or copies of its reversed inverse +for a negative power. Its cyclic reduction therefore uses every edge +of $W$. On the other hand, $G\setminus\{m\}$ deformation retracts onto +the graph with the edge containing $m$ deleted: contract its two +remaining half-open stubs to their endpoint vertices. A loop avoiding +$m$ consequently has a reduced circuit using no such edge, and cannot +be freely homotopic to a nonzero power of $W$. + +Finally, take a shortest nonempty subloop between repeated vertices +of $W$. It has no repeated interior vertex. Its length is not one, +since $G$ has no loop edges, and is not two, since that would be an +immediate reversal in a simple graph. Thus it is a simple cycle. +\end{proof} + +\begin{lemma}[A connected midpoint preimage crosses the annulus] +\label{lem:interfaces-annular-crossing} +Let $B$ be an open disk containing $x$, with +$\overline B\subset B(x,\rho)$, and set +$\mathcal A=\overline{B(x,\rho)}\setminus B$. +Suppose $g:\mathcal A\to G$ is continuous, where $G$ is a finite +connected simple graph, and the image of the counterclockwise inner +boundary has nonempty cyclically reduced word $W$. If $m$ is the +midpoint of an edge occurring in $W$, then $g^{-1}(\{m\})$ has a +compact connected component meeting both boundary circles of +$\mathcal A$. +\end{lemma} + +\begin{proof} +Every ray from $x$ meets $\partial B$ once, at a positive radius +$r(\theta)$ depending continuously on the angle. Radial interpolation +between $r(\theta)$ and $\rho$ identifies $\mathcal A$ with a closed +annulus. In particular, winding number about $x$ identifies its +fundamental group with $\Z$, with the counterclockwise inner boundary +as generator. + +Put $Z=g^{-1}(\{m\})$. This is a compact metric space. Suppose no +component of $Z$ meets both boundaries, and write +$I=Z\cap\partial B$ and $O=Z\cap\partial B(x,\rho)$ for its compact +inner and outer contact sets. We first obtain a disjoint compact partition +$Z=Z_1\sqcup Z_2$ with $I\subset Z_1$ and $O\subset Z_2$. + +In any compact metric space, a point's component is the intersection +of all of its clopen neighborhoods. Indeed, this intersection +contains the component. If it were disconnected, its two nonempty +compact parts could be enclosed in disjoint open sets. A finite +intersection of clopen neighborhoods already lies in their union, +by compactness of the complement of that union. Intersecting this +clopen set with the open set containing the original point gives +a smaller clopen neighborhood omitting the other part, a +contradiction. The intersection is therefore connected and equals +the component. + +Thus, for each point of $I$ and each point of $O$, some clopen +neighborhood of the first omits the second. Compactness of $O$ +permits a finite intersection producing a clopen neighborhood of +the first point disjoint from all of $O$. Compactness of $I$ +then permits a finite union containing all of $I$ and still +disjoint from $O$. Take this union as $Z_1$ and its complement in +$Z$ as $Z_2$. If either $I$ or $O$ is empty, use $Z_1=\varnothing$ +or $Z_1=Z$, respectively. This proves the partition assertion. + +The set +\[ + C_1=Z_1\cup\overline B +\] +is compact, lies in the open outer disk, and is disjoint from $Z_2$. +Choose a sufficiently fine translated square grid, with $x$ on no +grid line, and take every closed grid cell meeting $C_1$. Their +finite union $V$ lies in the open outer disk and avoids $Z_2$, by the +positive compact separations. Also $C_1\subset\operatorname{int}(V)$: +at a point of $C_1$ on a grid edge or vertex, all adjacent cells +are included. Consequently $\partial V$ lies in $\mathcal A$ +and avoids $Z_1\cup Z_2=Z$. + +Orient each selected square boundary counterclockwise and cancel +shared edges. The remaining edges lie on $\partial V$ and form +a balanced finite directed chain, so they decompose into closed +polygonal loops. Their total winding about $x$ is one. Indeed, +winding is additive before and after cancellation, and exactly one +selected square contains $x$ in its interior; its winding is one +and that of every other selected square is zero. At least one +of the resulting loops has nonzero winding, say $d$. Its image +under $g$ avoids $m$, but the annulus calculation makes that image +freely homotopic to $W^d$. This contradicts +Lemma~\ref{lem:interfaces-graph-word}. +\end{proof} + +\subsection{Limiting continua and simultaneous color exclusions} + +\begin{proof}[Proof of Proposition~\ref{prop:interface-cycle}] +Use the maps of Lemma~\ref{lem:interfaces-hole-maps} and the +non-null loops from Lemma~\ref{lem:interfaces-core-loop}. +Since $L_0\cap Q$ is finite, pass to a subsequence with a fixed +$x\in L_0\cap\operatorname{int}(Q)$ whose core boundary loop +is non-null at every index. Choose $\rho>0$ so small that +$\overline{B(x,\rho)}\subset\operatorname{int}(Q)$ and contains no +other point of $L_0$. Decreasing $\rho$ if necessary, the other +core closed disks are disjoint from $\overline{B(x,\rho)}$ for +large $n$, while +$\overline{B_{n,x}}\subset B(x,\rho)$. The map $g_n$ is therefore +defined on the closed annulus +\[ + \mathcal A_n=\overline{B(x,\rho)}\setminus B_{n,x}. +\] +The inner disk contains $x$ strictly; its possibly eccentric +position causes no difficulty for the radial annulus coordinates +used in Lemma~\ref{lem:interfaces-annular-crossing}. + +Let $W_n$ be a nonempty cyclically reduced circuit representing the +image of the counterclockwise inner boundary. By +Lemma~\ref{lem:interfaces-graph-word}, it contains a simple cycle. +There are only finitely many simple cycles in the complete graph +on five vertices. Pass to a subsequence with one chosen cycle, +of length $\ell\in\{3,4,5\}$, and label its vertices +$i_0,\ldots,i_{\ell-1}$ in cyclic order. For each of its edge +midpoints $m_j$, Lemma~\ref{lem:interfaces-annular-crossing} gives +a compact connected component +\[ + K_{n,j}\subset g_n^{-1}(\{m_j\})\cap\mathcal A_n +\] +meeting both annulus boundaries. + +Take Hausdorff limits along a common further subsequence for the +finitely many indices $j$: +\[ + K_{n,j}\longrightarrow K_j\subset\overline{B(x,\rho)}. +\] +The nonempty compact subsets of a compact metric space are compact +in the Hausdorff metric. For completeness, finite nets give total +boundedness of this hyperspace, and for a Hausdorff-Cauchy sequence +the set of limits of convergent point selections is its nonempty +compact Hausdorff limit, proving completeness. Connectedness also +passes to the limit: a disconnection of $K_j$ into two nonempty +compact parts gives disjoint separated neighborhoods containing +every sufficiently close $K_{n,j}$, with each neighborhood met, +contradicting connectedness of $K_{n,j}$. + +Each $K_{n,j}$ meets $\partial B_{n,x}$. These contact points tend +to $x$, because the disk contains $x$ and its diameter tends to +zero. Hence $x\in K_j$. Outer-boundary contact and compactness +likewise give $K_j\cap\partial B(x,\rho)\ne\varnothing$. +In particular each $K_j$ is nontrivial. + +We prove the support exclusion for either endpoint +$i\in\{i_j,i_{j+1}\}$. At a point of $K_{n,j}$ outside all holes, +$g_n=q^{\epsilon_n}$ has value $m_j$, whose two endpoint +coordinates are $1/2$. Formula~\eqref{eq:source-22} therefore gives +$p_i^{\epsilon_n}>\alpha$ there. + +Fix any closed ball $U$ of positive radius contained in +$\Delta(K_j)$. The strict inequalities defining the straddling +region hold with a positive uniform margin on $U$. For every +$z$, either distance extremum changes by at most +$d_{\rm H}(K_{n,j},K_j)$ when $K_j$ is replaced by $K_{n,j}$. +Thus, for all sufficiently large $n$, the distances from every +$z\in U$ to points of $K_{n,j}$ straddle one. The continuous +image of the connected set $K_{n,j}$ under $q\mapsto|z-q|$ +is an interval, so some point of $K_{n,j}$ is at exact distance +one from $z$. + +For $z\in U\setminus E_n$, that point is outside every closed hole +disk. The preceding endpoint probability bound and +\eqref{eq:source-17} give $p_i^{\epsilon_n}(z)\le1-\alpha$. +Exactly as in \eqref{eq:interfaces-mesh-measure}, +\[ + \alpha |A_i\cap U| + \le \norm{p_i^{\epsilon_n}-a_i}_{L^1(U)} + +\alpha |E_n\cap U|\longrightarrow0. +\] +Thus $A_i$ has zero measure in $U$. Every point of the open set +$\Delta(K_j)$ lies in the interior of such a ball, and consequently +has a neighborhood of zero $A_i$-measure. By the definition of +$H_i$, this proves $H_i\cap\Delta(K_j)=\varnothing$ for both +endpoint labels. + +Finally, connectedness and the two contact conditions imply +\[ + \{|q-x|:q\in K_j\}=[0,\rho]. +\] +Choose points in $K_j$ at radii $\rho/k$, $k\ge2$, and pass to a +subsequence of their unit directions converging to some $v_j\in S$. +Relabel these points as $q_{j,k}$. They supply the required +sequences and complete the proof. +\end{proof} diff --git a/preprints/The-Euclidean-plane-is-not-five-colorable-September-23-2026/build/sections/introduction.tex b/preprints/The-Euclidean-plane-is-not-five-colorable-September-23-2026/build/sections/introduction.tex new file mode 100644 index 0000000000000000000000000000000000000000..72b2b6b557713a79155b9d7306ab995dcadc0863 --- /dev/null +++ b/preprints/The-Euclidean-plane-is-not-five-colorable-September-23-2026/build/sections/introduction.tex @@ -0,0 +1,324 @@ +\section{Introduction}\label{sec:introduction} + +A proper $k$-coloring of the Euclidean plane is a function +$c:\R^2\to\{1,\ldots,k\}$ such that $c(x)\ne c(y)$ whenever +$\norm{x-y}=1$. The Hadwiger--Nelson problem asks for the least such +$k$, denoted by $\chi(\R^2)$, with no regularity imposed on the color +classes. Equivalently, it asks for the chromatic number of the graph +whose vertices are all points of the plane and whose edges are the +unit-distance pairs. Edges in a finite drawing of this graph may cross. + +The problem dates to Edward Nelson in 1950. Nelson obtained the lower +bound four, and John Isbell found the upper bound seven by a hexagonal +coloring; the historical record of these unpublished observations is +recounted by \citet[pp.~15--19]{Soifer2003History}. Hadwiger's published +formulation \citep{Hadwiger1961} asks for a covering by sets that avoid +unit distance and gives the bounds four and seven. The seven-point +construction of \citet{MoserMoser1961}, now called the Moser spindle, +gives a particularly small finite obstruction to three colors. We use +its two overlapping-triangle constraints again in the final step of our +proof and verify the required placement explicitly. + +The lower bound remained four until \citet{deGrey2018} constructed a +finite unit-distance graph that is not four-colorable in 2018. +His corrected construction has 1,581 vertices. Subsequent work made +these finite obstructions smaller and their verification more accessible: +\citet{Heule2018} used satisfiability proofs and extraction of +unsatisfiable cores to obtain 553-vertex examples, and +\citet{Parts2020Minimization} obtained a 509-vertex graph. +\citet{ExooIsmailescu2020} gave an alternative proof of the lower bound five, +and Parts gave a human-verifiable proof \citep{Parts2020Human}. +These advances concern finite obstructions to four colors. The +compactness theorem of \citet[Theorem~1]{deBruijnErdos1951} says that a +graph is $k$-colorable, for fixed finite $k$, if and only if every finite +subgraph is $k$-colorable. Consequently an unrestricted lower bound can +also be established without first displaying a finite graph witnessing +it. + +The problem with measurable color classes has a different history. +Write $\chi_m(\R^2)$ for the least number of colors when each class must +be Lebesgue measurable. \citet{Falconer1981} proved +$\chi_m(\R^2)\ge5$ using density points and rotations. +\citet{Payne2009} developed this method for translation-invariant +unit-distance subgraphs. In particular, the graph on all of $\R^2$ +whose edges have rational unit displacement vectors is two-colorable, +while every measurable coloring requires at least five colors. Thus +measurable and unrestricted colorings cannot be identified merely +because their distance constraints look the same. + +For colorings by regular regions, \citet{Woodall1973} studied +distance-avoiding closed covers and region decompositions. Townsend +corrected Woodall's argument for a six-color lower bound in the +planar-map setting, announcing the repair in 1981 and giving its details +in 2005 \citep{Townsend1981,Townsend2005}. More recently, +\citet{SokolovVoronov2025} proved a lower bound seven for polygonal +colorings in a locally finite map framework. Their interface argument +excludes both adjacent colors wherever distances to the interface take +values both below and above one. We derive corresponding exclusions +from weak measurable data, without assuming regular color boundaries. + +We prove the following result, working throughout in ZFC. + +\begin{theorem}\label{thm:main} +The Euclidean plane has no proper five-coloring, even when arbitrary +color classes are allowed. Consequently, +\[ + 6\le \chi(\R^2)\le 7. +\] +\end{theorem} + +The remaining alternatives six and seven are unresolved.\footnote{ +The May 14, 2026 manuscript of \citet{Reed2026} announces +$\chi(\R^2)=7$ through a circle-density argument. Its proposed strict +bound of $\pi/3$ on the angular measure of a unit-independent subset +of the unit circle fails for the half-open arc +$\{e^{it}:0\le t<\pi/3\}$: this arc has angular measure $\pi/3$ +and contains no unit pair. The displayed formal theorem assumes that +density bound as a hypothesis, so the argument does not establish the +announced equality.} +The proof has two parts: a transfer from unrestricted proper colorings to a +measurable condition, and a geometric obstruction to five labels under that condition. +The transfer holds for every finite number of colors. + +\begin{definition}[Weak measurable coloring]\label{def:weak-coloring} +Let $k$ be a positive integer, let $S^1\subset\R^2$ be the unit circle, +and let $\sigma$ be its normalized arc-length measure. A +\emph{weak measurable $k$-coloring} is a Lebesgue measurable function +$c:\R^2\to\{1,\ldots,k\}$, with color classes $A_i=c^{-1}(\{i\})$, +such that, for every $R>0$, +\[ + \sum_{i=1}^k\int_{B(0,R)}\int_{S^1} + \ind_{A_i}(x)\ind_{A_i}(x+u)\,d\sigma(u)\,dx=0. +\] +The integrals use the completed product measure; equivalently, one +may first choose Borel representatives of the color classes forming +a partition of the plane. +\end{definition} + +This definition permits exceptional unit pairs. It is unchanged by +modifying the coloring on a plane Lebesgue null set: by Fubini, the +pairs for which either $x$ or $x+u$ belongs to that set have product +measure zero. The required bridge to the unrestricted problem is the +following equivalence. + +\begin{theorem}[Transfer of colorability]\label{thm:transfer} +For every positive integer $k$, in ZFC, +\begin{equation}\label{eq:source-1} + \begin{gathered} + \text{a proper $k$-coloring of $\R^2$ exists} + \\ \Longleftrightarrow \\ + \text{a weak measurable $k$-coloring of $\R^2$ exists}. + \end{gathered} +\end{equation} +\end{theorem} + +\begin{theorem}\label{thm:weak-five} +There is no weak measurable five-coloring of the Euclidean plane. +\end{theorem} + +\begin{proof}[Deduction of Theorem~\ref{thm:main}] +A proper five-coloring would give a weak measurable five-coloring by +Theorem~\ref{thm:transfer}, contrary to Theorem~\ref{thm:weak-five}. + +For the upper bound, use the hexagonal construction described by +\citet{Hadwiger1961}. +Take Voronoi hexagons of a triangular lattice, each with circumradius +$r=2/5$. Their centers have nearest-neighbor distance $\sqrt3r$. +Write the lattice as a scaled copy of $\Z+\Z\omega$, where +$\omega=e^{2\pi i/3}$. Multiplication by $2-\omega$ gives a similar +sublattice of index seven, since $|2-\omega|^2=7$. Color centers by +its seven cosets, and give each point the color of its hexagon, +assigning boundary points to any incident hexagon. Distances inside +one hexagon are at most $2r<1$. Centers of distinct same-color +hexagons are at least $\sqrt{21}r$ apart, so points in those hexagons +are at least $(\sqrt{21}-2)r>1$ apart. This is a proper seven-coloring, +including all boundaries. +\end{proof} + +\subsection{Proof overview and the role of earlier methods} + +The two parts of the proof address different obstacles. A coloring with +arbitrary classes provides no density-point structure to use, while a measurable +coloring may have boundaries far too irregular to analyze as a planar +map. Theorem~\ref{thm:transfer} supplies the measurable input. The proof +of Theorem~\ref{thm:weak-five} then constructs the connected interfaces +it needs instead of assuming them. Figure~\ref{fig:proof-route} shows how +these two parts combine. + +\paragraph{From arbitrary labels to measurable fields.} +Let $F$ be the field of real algebraic numbers, $E=F(i)$ the algebraic +plane, and $K=\{u\in E:|u|=1\}$ its group of algebraic rotations. +Restricting a proper coloring to the countable set $E$ and averaging +its translates and rotations gives an invariant probability law on +proper labelings of $E$. This is a standard amenable averaging +construction. Probabilistic approaches to the Hadwiger--Nelson problem +already use invariant randomization and finite-graph compactness +\citep{BourgeatHeinrichMelottiRobert2015,GwynStavrianos2020}; +F\o lner averaging over countable groups of plane isometries also occurs +in the fractional-coloring work of +\citet{MatolcsiRuzsaVargaZsamboki2025}. These methods supply symmetry. +Here a further argument must retain the unit-distance exclusions while +producing ordinary measurable functions on the plane. + +For this purpose let $D$ be the compact group of all characters of the +additive discrete group $E$. A character is a homomorphism from $E$ to +the unit complex circle. The ordinary continuous characters form the +Borel subgroup +\[ + C=\{z\mapsto\exp(i\xi\cdot z):\xi\in\R^2\}\subset D. +\] +Rotations act on characters by $(ud)(z)=d(uz)$. +The spectral theorem, in $L^2$ of the invariant labeling law, +separates each label indicator into the part +supported on $C$ and its orthogonal remainder. The central rigidity +statement, Theorem~\ref{thm:rigidity}, says that every $K$-invariant +probability on $D$ assigning zero mass to $C$ is Haar probability. The +remainder therefore has zero correlation under every nonzero algebraic +translation, so removing it preserves the zero unit correlations. +The $C$-supported subspace is the space of functions on a probability +factor, so projection onto it is conditional expectation and preserves +nonnegativity and keeps the sum of the label coordinates equal to one. +Translation operators on this factor extend continuously to $\R^2$. +The translated coordinates admit jointly measurable versions, giving +fields on the plane and the probability space. For a suitable sample, +choose the least label with positive coordinate at each point +where the coordinates sum to one, and assign arbitrary labels on the +remaining plane-null set. The zero unit correlations give the weak +measurable coloring condition. Section~\ref{sec:transfer} carries out this passage, including +the measurable-version and exceptional-pair details. + +The rigidity proof occupies Sections~\ref{sec:spectral} +and~\ref{sec:rigidity}. Its structural ingredient is the +Furstenberg--Zimmer compact-extension method +\citep{Furstenberg1977,ZimmerExtensions1976,ZimmerDiscrete1976}. +We give the particular compact-factor tower and conditional averaging +arguments used here; the corrected modern account of +\citet{JamneshanStructure,JamneshanCorrigendum} supplies further +context. For an ergodic law, a relative singularity lemma along that +tower shows that the +law of the difference of two characters, sampled conditionally independently +over its terminal factor, still gives $C$ measure zero. Its Fourier +coefficient is nonnegative and radial, and its line averages defined +by an invariant mean vanish. +Two identities between sums of algebraic unit directions give a difference +inequality for this coefficient. A finite number-field argument shows +that a positive value at any nonzero radius would force a uniform positive +lower bound on an interval of algebraic radii, contradicting those averages. +This proves Haar rigidity without a regularity assumption on the original +coloring. + +For the converse transfer, density points of a weak color class contain no unit pair +(Lemma~\ref{lem:palettes-density-proper}). This adapts the familiar +density-point mechanism of Falconer and Payne to product-almost-everywhere +unit-pair exclusion. A common +translation places any fixed finite configuration among points having +density one in their assigned color, called typical points, +and graph compactness then gives a proper coloring of the whole plane. +The weak condition uses Lebesgue measure in position and arc length in +direction; it is distinct from the invariant-mean notion of +monochromatic-pair frequency in \citet{GwynStavrianos2020}. + +\paragraph{From weak labels to connected exclusions.} +For the five-color obstruction, fix a center $x$ and take weak-* limits +of the typical angular samples +$e\mapsto(\ind_{A_i}(y+re))_{i=1}^5$, where $e\in S^1$, +$y\to x$, and $r\to1$. The \emph{palette} at $(x,e)$ collects, +up to angular null sets, all labels having positive weight at $e$ in +these limits. Smooth transport of two unit directions gives exclusions +between palettes. Section~\ref{sec:palettes} +shows that almost every palette contains at most two labels, at every +fixed center. Section~\ref{sec:transitions} records abundant transitions +between labels in a finite graph at each center and proves that centers +where this graph contains a cycle form a closed locally finite set. +The estimates use the decay of the Fourier transform of circle measure; +no finite perimeter or smoothness of a color boundary is assumed. + +Section~\ref{sec:interfaces} smooths the color indicators by disk +averages, then thresholds and renormalizes the resulting probability +vectors. Outside disks with vanishing total radii, these modified +vectors have at most two positive coordinates, so they lie in the +complete graph on the five labels. The forest +structure away from cyclic centers lets us fill the holes in those regions +continuously in that graph. A planar cover obstruction forces a +nontrivial loop around one cyclic-center disk. A simple cycle in the +reduced edge path of that loop supplies the label cycle. For each edge +midpoint, a connected component of its inverse image crosses the annulus. +Limits of these components give compact connected sets $K_j$ with a +common point $x$. +Each $K_j$ is assigned the two endpoint labels of its edge. +Both endpoint labels are absent almost everywhere from the open region +\[ + \Delta(K_j)=\{z:\min_{q\in K_j}|z-q|<1<\max_{q\in K_j}|z-q|\}. +\] +This connected-set exclusion has a geometric predecessor in the +interface arguments of \citet[Propositions~1--2]{SokolovVoronov2025}. +Here the continua and their common point are consequences of the +measure estimates and the topological extraction. + +A simple cycle on five labels has length three, four, or five. +Section~\ref{sec:angular} excludes each possibility. Chosen limiting +directions of sequences in the continua approaching $x$ determine +allowed labels just inside and just outside the unit circle about $x$. A five-cycle gives +incompatible six-position angular words. A four-cycle gives two antipodal +pairs of directions and an impossible parity rule. A three-cycle forces six +alternating sectors for the two outside labels. The last configuration +leaves an open region using only the three cycle labels almost everywhere; a rational +placement certificate puts all seven vertices of the Moser obstruction +in that region. These contradictions complete the proof of +Theorem~\ref{thm:weak-five}. + +\begin{figure}[htbp] +\centering +\input{figures/proof-flow} +\caption{The contradiction argument for five colors. The transfer from +arbitrary proper colorings to weak measurable colorings is proved for +every finite number of colors; the geometric obstruction uses five labels.} +\label{fig:proof-route} +\end{figure} + +\subsection{A probabilistic consequence} + +\begin{samepage} +\begin{corollary}[Positive five-color badness]\label{cor:positive-badness} +There is a constant $\delta>0$ with the following properties. +\begin{enumerate} +\item[(i)] For every coloring $c:\R^2\to\{1,\ldots,5\}$ and every +normalized finitely additive left-invariant measure $\mu$ on all subsets +of the plane-isometry group $E(2)$, the invariant-mean badness satisfies +\[ + p_5^\mu(c):=\mu\bigl(\{T\in E(2):c(T^{-1}0)=c(T^{-1}1)\}\bigr) + \ge\delta, +\] +where $\R^2$ is identified with $\C$. +\item[(ii)] If $c$ is a Lebesgue-measurable five-coloring with two +linearly independent periods, then +$\mathbb P(c(A)=c(A+U))\ge\delta$, where $A$ is uniform in a +fundamental parallelogram and $U$ is an independent uniform unit +direction. For any Lebesgue-measurable five-coloring, if the conditional +square-table frequencies +\[ + \mathbb P\bigl(c(A_R)=c(A_R+U)\mid A_R+U\in[-R,R]^2\bigr) + \qquad(R>1) +\] +converge as $R\to\infty$, their limit is at least $\delta$; here +$A_R$ is uniform in $[-R,R]^2$ and independent of $U$. +\end{enumerate} +\end{corollary} +\end{samepage} + +\begin{proof} +Theorem~\ref{thm:main} and graph compactness +\citep[Theorem~1]{deBruijnErdos1951} give a finite unit-distance graph +$H$ that is not five-colorable. Set $m=|E(H)|$ and $\delta=1/m$. +Every five-coloring of $H$ has at least one monochromatic edge. +Averaging this count with any of the measures in (i) gives +$mp_5^\mu(c)\ge1$, by isometry invariance and finite additivity; +this is the finite-graph bound of +\citet[Lemma~2.10 and Theorem~2.11]{GwynStavrianos2020}. +The periodic and asymptotic needle frequencies in (ii) are bounded +below by the minimum monochromatic-edge fraction of $H$, hence by +$1/m$, by \citet[Theorems~1--2]{BourgeatHeinrichMelottiRobert2015}. +Borel representatives give the same needle frequencies by Fubini; +for a periodic coloring choose them in one cell and extend periodically. +The graph and the positive constant here are existential. +\end{proof} diff --git a/preprints/The-Euclidean-plane-is-not-five-colorable-September-23-2026/build/sections/palettes.tex b/preprints/The-Euclidean-plane-is-not-five-colorable-September-23-2026/build/sections/palettes.tex new file mode 100644 index 0000000000000000000000000000000000000000..d1d09cbce68ece64cbf7f10b5e7e1adf8f387eee --- /dev/null +++ b/preprints/The-Euclidean-plane-is-not-five-colorable-September-23-2026/build/sections/palettes.tex @@ -0,0 +1,663 @@ +\section{Circle palettes and their cardinality}\label{sec:palettes} + +We now assume a weak measurable five-coloring and develop angular +palettes at every center. Their labels record all possible limiting +circle samples. The main result of this section is that almost every +palette contains at most two labels, at every fixed center. We obtain +this bound from exclusions between circle samples and a separation +property for centers whose palettes meet one fixed pair of labels. + +\subsection{Typical samples, traces, and maximal palettes} + +For this section and the subsequent contradiction, the label set is +\(\{1,\ldots,5\}\); write \(A_i\) for the classes, +\(a_i=\ind_{A_i}\), and \(c\) for the coloring. +We choose Borel representatives forming a partition. +Changing a coloring on a plane-null set leaves the weak condition +unchanged, because that set is hit at either endpoint only on a +\(dx\,d\sigma(u)\)-null set. + +Call a point \emph{typical} if it has density one of its actual color. +Typical points form a conull set. By +Lemma~\ref{lem:palettes-density-proper}, any unit pair of typical +points has different colors. The same lemma applies to two +density-one points of a class regardless of the values assigned +to the coloring at those individual points. +For each label let +\[ + H_i=\{q\in\R^2: + \abs{A_i\cap B(q,r)}>0\text{ for every }r>0\} +\] +be its closed spatial essential support. + +Let \(S=S^1\) carry uniform angular probability \(\sigma\). +Let \(R\) denote rotation by \(\kappa=\pi/3\), and \(J\) rotation +by \(\pi/2\). We identify unit directions with angles modulo +\(2\pi\). Two directions are called parallel if they are equal +or opposite; all angular density and essential-support statements +refer to angular Lebesgue measure. + +Call \((y,r)\), with \(y\in\R^2\) and \(r>0\), a \emph{typical circle +sampling} if \(y+re\) is typical for almost every \(e\in S\). +For each fixed \(r\), this holds for almost every \(y\) by Fubini +and translation invariance of plane measure. +For each fixed \(y\), it holds for almost every \(r>0\) by polar +coordinates and Fubini. + +\begin{definition}[Circle traces and maximal palettes] +\label{def:palettes-traces} +For each fixed center \(x\), let \(\mathcal W_x\) be the family of +vector functions in \(L^\infty(S;\R^5)\) that can be obtained as +\[ + b=\operatorname{w}^*\!-\!\lim_{n\to\infty} + (a_i(y_n+r_ne))_{i=1}^5,\qquad + y_n\to x,\quad r_n\to1, +\] +where every \((y_n,r_n)\) is a typical circle sampling. +Constant sequences are permitted. +For a label \(i\), form the essential measurable union +\[ + D_{x,i}=\mathop{\operatorname{ess\,union}}_{b\in\mathcal W_x} + \{e:b_i(e)>0\}. +\] +The maximal palette is the measurable set-valued function +\[ + P_x(e)=\{i:e\in D_{x,i}\}, +\] +defined up to angular null sets. +\end{definition} + +Here are the compactness and countability facts in this definition. +Typical samplings with radius one can be chosen at centers approaching +any prescribed \(x\). Their bounded vectors have weak-* convergent +subsequences, since \(L^1(S)\) is separable and the bounded ball +of its dual is weak-* compact and metrizable. Thus \(\mathcal W_x\) +is nonempty. Every \(b\in\mathcal W_x\) has nonnegative coordinates +and \(\sum_i b_i=1\) almost everywhere, since these assertions pass +to a weak-* limit by integration against nonnegative test functions. + +The family also has the following closure under varying centers: +\begin{equation}\label{eq:palettes-trace-closure} + x_n\to x,\quad b^n\in\mathcal W_{x_n},\quad + b^n\rightharpoonup^* b + \quad\Longrightarrow\quad b\in\mathcal W_x . +\end{equation} +To see this, fix a metric for the bounded-ball weak-* topology. +From a sequence realizing \(b^n\), choose one sample within \(1/n\) +of \(b^n\) in that metric, with center within \(1/n\) of \(x_n\) +and radius within \(1/n\) of one. These chosen samples realize \(b\) +at \(x\). In particular \(\mathcal W_x\) is compact. + +For completeness, the essential union for a fixed label can always +be taken over a countable subfamily. Let \(M_i\) be the supremum +of the measures of finite unions of the sets \(\{b_i>0\}\). +Choose finite unions approaching \(M_i\), and let \(D_{x,i}\) be +their countable union. Its measure is \(M_i\). If another member +\(\{b_i>0\}\) had positive measure outside this union, adjoining it +to sufficiently large finite partial unions would contradict the +definition of \(M_i\). This proves that every individual trace +has its positive coordinates in the palette almost everywhere. +It also gives nonemptiness of \(P_x(e)\) almost everywhere, by +using any one trace and its sum-one property. + +The null set in the last assertion may depend on the individual +trace, and the angular null sets may depend on \(x\). We shall use +countable generating trace families whenever a conclusion is to be +asserted for the whole palette. No joint measurability of \(P_x\) +in its center is assumed. + +\subsection{Exclusion, density, and smooth transport} + +\begin{lemma}[Palette transfer]\label{lem:palette-transfer} +The following assertions hold at every fixed center \(x\). + +\begin{enumerate} +\item For almost every independent-angle pair \((e,v)\in S\times S\), +\begin{equation}\label{eq:source-10} + c(x+e+v)\notin P_x(e)\cup P_x(v). +\end{equation} + +\item If \(e\) is an angular density-one point of +\(D_{x,i}=\{e':i\in P_x(e')\}\), then for each \(v\) nonparallel +to \(e\), the spatial density of \(A_i\) at \(x+e+v\) is zero. + +\item Let \(z\) be any fixed center, possibly \(z=x\). Suppose on an open angular +arc, or locally on a union of such arcs, there is a smooth representation +\[ + z+f=x+s(f)+t(f),\qquad s(f),t(f)\in S, +\] +with \(s(f),t(f)\) nonparallel and \(s\) a local diffeomorphism in +the angular coordinate. Then almost everywhere on that domain, +\begin{equation}\label{eq:source-11} + P_z(f)\cap P_x(s(f))=\varnothing. +\end{equation} +\end{enumerate} +Each assertion permits an exceptional angular set depending on the +fixed centers and on the specified maps. +\end{lemma} + +\begin{proof} +Fix \(b\in\mathcal W_x\), with realizing samplings \((y_n,r_n)\). +For every \(n\), the point \(y_n+r_ne\) is typical for almost every +\(e\). The second point \(y_n+r_ne+v\) is typical for almost every +pair \((e,v)\): in angular coordinates the map +\[ + G_n(e,v)=y_n+r_ne+v +\] +has Jacobian of absolute value +\(r_n\abs{\sin(\arg v-\arg e)}\), which is nonsingular off a +product-null set. On countably many smaller coordinate patches +away from that set, change of variables pulls back any plane-null +set to a product-null set. The two typical points have distance +one. Lemma~\ref{lem:palettes-density-proper} therefore gives +\begin{equation}\label{eq:palettes-sample-disjoint} + a_i(y_n+r_ne)a_i(y_n+r_ne+v)=0 + \quad\text{for almost every }(e,v). +\end{equation} + +We need the following strong convergence for the second factor: +\begin{equation}\label{eq:palettes-strong-circle-sum} + a_i(y_n+r_ne+v)\longrightarrow a_i(x+e+v) + \quad\text{in }L^1(S\times S). +\end{equation} +Here is a change-of-variables proof which also covers arbitrary +measurable color boundaries. For \(\delta>0\), discard the angular +pairs with +\(\abs{\sin(\arg v-\arg e)}\le\delta\). Their measure tends to +zero with \(\delta\). On the remaining set, for large \(n\), the +Jacobian is bounded below by \(\delta/2\). The multiplicity of +\(G_n\) there is at most two: an inverse image of a specified +sum amounts to an intersection of the circle of radius \(r_n\) +with a translated circle of radius one. A coincident-circle +case can occur only at the antiparallel, hence discarded, directions. +The same assertions hold for \(G(e,v)=x+e+v\). +Their images lie in one fixed compact plane region for all large \(n\). + +It follows that on the retained angular set the pushforwards of +angular product measure have densities bounded by a constant +depending only on \(\delta\), uniformly in \(n\). +Approximate \(a_i\) on a compact neighborhood of those images by +a continuous function taking values in \([0,1]\), in plane \(L^1\). +The density bounds control its composition errors uniformly. +For the continuous approximant the compositions converge uniformly, +since \(y_n\to x\) and \(r_n\to1\). +First making the approximation error small, then letting \(n\to\infty\), +and finally letting \(\delta\downarrow0\), proves +\eqref{eq:palettes-strong-circle-sum}. + +Set \(B_n(e)=a_i(y_n+r_ne)\) and +\(Q(e,v)=a_i(x+e+v)\). The error in replacing the second factor +of \eqref{eq:palettes-sample-disjoint} by \(Q\), after integration, +is at most the \(L^1\) error in +\eqref{eq:palettes-strong-circle-sum}. Weak-* convergence of \(B_n\) +then applies to the fixed \(L^1(S)\) test +\(\int_S Q(e,v)\,d\sigma(v)\). We obtain +\[ + \int_{S\times S}b_i(e)a_i(x+e+v)\,d\sigma(e)\,d\sigma(v)=0. +\] +Both factors are nonnegative, so their product is zero almost +everywhere. Taking the countable generating trace families in +Definition~\ref{def:palettes-traces}, for all five labels, proves +exclusion of \(P_x(e)\). Interchanging \(e,v\) proves exclusion +of \(P_x(v)\), and gives \eqref{eq:source-10}. +In particular this passage used a strongly convergent factor +against a weak-* limit, not a product of two weak-* limits. + +Next fix an angular density-one point \(e\) of \(D_{x,i}\) and +any nonparallel direction \(v\). The map +\[ + (e',v')\longmapsto x+e'+v' +\] +is a smooth local diffeomorphism near \((e,v)\). +Choose real angular coordinates near \(e,v\) and an inverse branch +on a plane neighborhood of \(x+e+v\). +For small \(h\), the inverse image under this branch of +\(B(x+e+v,h)\) lies in a product of angular intervals of length +\(O(h)\) about \(e,v\), and its change-of-variables factors are +bounded. On this branch, \eqref{eq:source-10} implies that a +point of \(A_i\) has first coordinate outside \(D_{x,i}\), apart +from a spatial null set. Therefore +\[ + \abs{A_i\cap B(x+e+v,h)} + \le C h\, + \abs{[e-Ch,e+Ch]\setminus D_{x,i}} + =o(h^2). +\] +The constants can depend on the fixed nonparallel pair. This +establishes assertion 2 for each such \(v\). It is a +two-dimensional neighborhood argument and does not restrict +\eqref{eq:source-10} to a possibly exceptional single-angle fiber. + +Finally fix the centers and representation in assertion 3, and work +first on a smaller compact arc on which one inverse branch of the +two-angle sum map is available. Let a trace \(b\in\mathcal W_z\) +be realized by \((z_n,r_n)\). Applying this inverse branch to +\(z_n+r_nf-x\) gives perturbed representations +\[ + z_n+r_nf=x+s_n(f)+t_n(f), +\] +where \(s_n,t_n\) are nonparallel and converge to \(s,t\) in \(C^1\) +on the smaller arc. Shrinking the arc if necessary, \(s\) is +a diffeomorphism with derivative bounded away from zero, and the +\(s_n\) have uniformly bounded inverse derivatives for large \(n\). + +For \(D_i=D_{x,i}\), these facts imply +\begin{equation}\label{eq:palettes-pullback-continuity} + \ind_{D_i}(s_n(f))\longrightarrow\ind_{D_i}(s(f)) + \quad\text{in }L^1\text{ on the smaller arc}. +\end{equation} +To verify it, approximate \(\ind_{D_i}\) on a common image interval +by a continuous function in angular \(L^1\). Change of variables +and the inverse-derivative bounds control the errors of composition +uniformly for \(s_n\) and for \(s\). Uniform convergence of \(s_n\) +handles the continuous approximant. + +Almost every \(f\) with \(s_n(f)\in D_i\) has \(s_n(f)\) a +density-one point of \(D_i\). Indeed the set of exceptions in +\(D_i\) is angular null by differentiation, and \(s_n\) pulls +it back to a null set. Assertion 2 makes the density of \(A_i\) +at \(z_n+r_nf\) zero for all these \(f\). The sampling +\((z_n,r_n)\) is typical on its circle, so for almost every such +\(f\) its actual color cannot be \(i\). Consequently +\[ + a_i(z_n+r_nf)\ind_{D_i}(s_n(f))=0 + \quad\text{almost everywhere on the smaller arc}. +\] +Use \eqref{eq:palettes-pullback-continuity} to replace the second +factor strongly in \(L^1\), and then use weak-* convergence of the +first factor. Nonnegativity gives +\[ + b_i(f)\ind_{D_i}(s(f))=0 + \quad\text{almost everywhere}. +\] +Taking a countable generating family of traces at \(z\) and the +five labels proves \eqref{eq:source-11} on this arc. +A countable cover by smaller arcs proves it on the stated domain. +\end{proof} + +\begin{remark}\label{rem:palettes-transport-scope} +In assertion 3 of Lemma~\ref{lem:palette-transfer}, the second endpoint +map \(t\) may be constant. Nonparallelness makes the two-dimensional +sum map invertible, while the local diffeomorphism requirement on +\(s\) alone controls preimages of angular null sets. +In particular, for any fixed direction \(a\), the representation +\[ + (x+a)+f=x+f+a +\] +is admissible off \(f=\pm a\). The proof applies at every fixed +pair of centers and never needs a common exceptional angular set +over an uncountable family of centers. +\end{remark} + +\begin{corollary}[Sixty-degree exclusion] +\label{cor:palettes-sixty-degree} +At every fixed center \(x\), +\begin{equation}\label{eq:source-12} + P_x(f)\cap P_x(Rf)=\varnothing + \qquad\text{for almost every }f\in S. +\end{equation} +\end{corollary} + +\begin{proof} +Use \(x+f=x+Rf+R^{-1}f\) in +Lemma~\ref{lem:palette-transfer}. The endpoint directions are +nonparallel and the first endpoint map is a rotation. +\end{proof} + +\subsection{Separation of binary centers} +\label{subsec:palettes-binary} + +Fix two distinct labels \(i_+,i_-\). Call a center \(x\) +\emph{binary for \(\{i_+,i_-\}\)} if +\[ + P_x(e)\cap\{i_+,i_-\}\ne\varnothing + \qquad\text{for almost every }e\in S. +\] +The exclusion \eqref{eq:source-12} makes this intersection a singleton +almost everywhere. Indeed, at almost every \(e\), the intersections +at \(e\) and \(Re\) are nonempty disjoint subsets of a two-element set; +they must therefore be the two different singletons. Thus there is a +unique measurable selection from this pair, up to a null set, and it +alternates under \(R\). Conversely, the existence of an alternating +selection from the pair plainly implies the displayed intersection +condition. + +Write this selection as a measurable \(2\pi\)-periodic sign function +\(s_x:\R\to\{-1,1\}\), assigning the signs \(+1,-1\) to \(i_+,i_-\) +respectively and choosing arbitrary values on the exceptional null +set. With \(\kappa=\pi/3\), it satisfies +\begin{equation}\label{eq:palettes-binary-signs} + s_x(t+\kappa)=-s_x(t) + \qquad\text{for almost every }t. +\end{equation} +In particular, either sign occupies angular measure \(\pi\) in a +period. + +\begin{lemma}[Binary-center separation]\label{lem:binary-separation} +There is an absolute constant \(\delta_*>0\) such that, for every +fixed pair of labels, any two distinct centers binary for that pair +are at distance at least \(\delta_*\). +\end{lemma} + +\begin{proof} +Suppose that \(x,y\) are distinct binary centers for the same pair, +and put \(d=|y-x|\). We will derive a contradiction whenever +\(00\), with \(d_1<1/2\), such that whenever +\(01\), we therefore have +\(\ell_{r+1}\ge\lambda\ell_r\). The process stops after finitely +many steps, at the first \(N\) for which \(\ell_N\ge L\), and +\[ + L\le\ell_N\le(1+C_0d)L. +\] + +Let \(G=F_{j_{N-1}}\circ\cdots\circ F_{j_0}\). +For each elementary map, +\[ + |(\log F_j')'|=\left|\frac{F_j''}{F_j'}\right|\le2C_0d. +\] +Applying the chain rule at two points of \(I_0\), whose images +before step \(r\) lie in \(I_r\), gives +\[ + \operatorname{osc}_{I_0}\log G' + \le2C_0d\sum_{r=0}^{N-1}\ell_r. +\] +The geometric growth bounds the sum independently of \(N\): +\[ + \sum_{r=0}^{N-1}\ell_r + \le\ell_N\sum_{q=1}^{N}\lambda^{-q} + \le\frac{\ell_N}{\lambda-1}. +\] +Consequently, +\begin{equation}\label{eq:palettes-binary-distortion} + \operatorname{osc}_{I_0}\log G' + \le\frac{4C_0(1+C_0d)L}{c_0} + \le\frac{6C_0L}{c_0}. +\end{equation} +In particular, +\[ + \frac{\sup_{I_0}G'}{\inf_{I_0}G'} + \le D_{\rm dist},\qquad D_{\rm dist}=\exp(6C_0L/c_0), +\] +uniformly in the initial interval, the number of steps, and +\(00$, +the spectral projection +\[ + Q_y=\ind_{(\delta,\infty)}(A_y^*A_y) +\] +is nonzero on a positive-measure set of fibers. Its rank is finite and +bounded above by $\norm{L}_\infty^2/\delta$. The rank is invariant on +$Y$, and $Y$ is ergodic as a factor of $X$. It is therefore a constant +integer $d\geq1$ almost everywhere. The range of $Q_y$ is orthogonal +to the fiber constants. + +Here are the measurability details for this construction. Work with +standard Borel models and fiber-supported conditional probabilities. +Choose a countable algebra generating the Borel structure of $X$ and +its simple functions with complex-rational coefficients. Their +restrictions are dense in each $L^2(\mu_y)$. Conditional integration +against the bounded kernel represents the images of these sections under +$A_y$ and $A_y^*$ by measurable functions on $X$; the same holds for +polynomials in $A_y^*A_y$. On their common bounded spectral interval, +there are uniformly bounded polynomials converging pointwise to +$\ind_{(\delta,\infty)}$: first choose bounded continuous approximations +to this indicator, with value zero at $\delta$, and then uniformly +approximate them by polynomials. Spectral calculus gives convergence +in each fiber Hilbert norm. Domination gives convergence in global +$L^2(X)$ as well. A subsequence converges in fiber norm for almost every +$y$, identifying the global limit with the projected section. Thus the +projected dense sections are measurable. Their fiber scalar products +make rank and linear-independence tests measurable. Selecting the first +available independent section at each step and applying fiberwise +Gram--Schmidt gives measurable fiber-orthonormal sections +$e_1,\ldots,e_d$ spanning the range of $Q_y$. + +For each $k\in K$ the sections $U_ke_i$ differ from the $e_i$ by a +unitary coefficient matrix over $Y$. Consequently +$\sum_{i=1}^d|e_i(x)|^2$ is invariant on $X$. Its integral is $d$, so +ergodicity gives +\begin{equation}\label{eq:rigidity-bounded-basis} + \sum_{i=1}^d|e_i(x)|^2=d\quad\text{almost everywhere}. +\end{equation} +In particular, the sections are bounded. Adjoin them to $Y$ to generate +an invariant factor $Y'$. This is a strict extension: every $e_i$ has +conditional mean zero over $Y$ and has unit fiber norm, and hence is not +$Y$-measurable. + +Bounded polynomials in the $e_i,\bar e_i$, with bounded coefficients +from $Y$, are dense in $L^2(Y')$. Indeed these functions form a bounded +unital conjugation-closed algebra generating $Y'$, and approximation +first by functions of finitely many generators and then by polynomials +gives the density. A rotation of a fixed polynomial expands into +finitely many fixed bounded monomials in the $e_i,\bar e_i$. Its +coefficients are functions on $Y$, uniformly bounded independently of +the rotation: the original coefficient sup norms are preserved, and +the entries of the unitary matrices have modulus at most one. For a +bounded $H$ on $Y'$, choose one polynomial approximation in $L^2$. +Rotating preserves its approximation error. The finite expansion of +the rotated polynomial proves \eqref{eq:source-4} for $Y'\to Y$. + +Starting at the trivial factor, repeat this step whenever the relative +product is not ergodic, and take generated joins at limit ordinals. +Every countable stage is a standard factor modulo null sets. The +process stops at a countable ordinal. Otherwise, at every successor +through all countable ordinals, choose a unit vector in the new factor +orthogonal to the preceding factor. Nestedness makes these vectors +pairwise orthogonal. There would be $\aleph_1$ such vectors, +contradicting separability of $L^2(X)$. At the +stopping stage the relative product is ergodic by the stopping rule. +\end{proof} + +\subsection{Conditional multiple averages} + +A F\o lner sequence in $K$ means nonempty finite sets $\Phi_N\subset K$ +such that +\[ + \frac{|h\Phi_N\mathbin{\triangle}\Phi_N|}{|\Phi_N|}\longrightarrow0 + \qquad(h\in K). +\] +Such sequences exist because $K$ is countable abelian. Write +$\operatorname{Avg}_{u\in\Phi}b_u=|\Phi|^{-1}\sum_{u\in\Phi}b_u$. +Every average below is a finite-set average in this discrete group. + +We use the Hilbert-space mean ergodic theorem in its every-F\o lner +form. For a unitary representation $V$ of $K$, these averages converge +strongly to the invariant projection along every F\o lner sequence. +For completeness, the orthogonal complement of the invariant vectors +is the closed span of vectors $V_hv-v$. The average of such a vector +has norm at most +$|h\Phi_N\triangle\Phi_N|\norm{v}/|\Phi_N|$, and density proves the +claim. In particular, the representation $u\mapsto U_{u^d}$, for +$d\in\Z\setminus\{0\}$, has the same invariant vectors as $U$, because +the power map $u\mapsto u^d$ is onto. No assertion about unweighted +images of F\o lner sets under a power map is needed. + +The proof below uses the relative weak-mixing induction for multiple +averages, with a Hilbert-space van der Corput estimate; compare +\citet{Furstenberg1977}. We give the argument for the power maps of +$K$ along every F\o lner sequence, since both features are used below. + +\begin{lemma}[Conditional multiple averages]\label{lem:multiple-averages} +Suppose that $X$ is an ergodic standard probability-preserving +$K$-system, that $Y$ is an invariant factor, and that $X\times_YX$ is +ergodic. Let $P=\mathbb E(\,\cdot\mid Y)$. For pairwise distinct +integers $n_1,\ldots,n_l$ and bounded functions $v_1,\ldots,v_l$ on $X$, +\begin{equation}\label{eq:source-6} + \operatorname{Avg}_{u\in\Phi_N} + \left\|P\left(\prod_{j=1}^lU_{u^{n_j}}v_j\right) + -\prod_{j=1}^lU_{u^{n_j}}Pv_j\right\|_2^2 + \longrightarrow0 +\end{equation} +along every F\o lner sequence $(\Phi_N)$ in $K$. The pair criterion +used in the proof is, for bounded $v,w$ and $d\ne0$, +\begin{equation}\label{eq:source-7} + \operatorname{Avg}_{u\in\Phi_N} + \norm{P(vU_{u^d}w)-Pv\,U_{u^d}Pw}_2^2\longrightarrow0. +\end{equation} +\end{lemma} + +\begin{proof} +Put $\mathcal R=X\times_YX$ and denote its conditional expectation onto +$Y$ by $P_{\mathcal R}$. First prove \eqref{eq:source-7}. With +$v_0=v-Pv$ and $w_0=w-Pw$, its difference equals +$P(v_0U_{u^d}w_0)$. Conditional independence gives +\begin{equation}\label{eq:rigidity-pair-tensor} + \norm{P(v_0U_{u^d}w_0)}_2^2 + =\int_{\mathcal R}(v_0\otimes\bar v_0) + U_{u^d}(w_0\otimes\bar w_0)\,d\mu_{\mathcal R}. +\end{equation} +The integral of $w_0\otimes\bar w_0$ is +$\int_Y|Pw_0|^2\,d\nu=0$. The mean ergodic theorem on the ergodic +system $\mathcal R$, applied through the surjective power map, makes +the average of the right side tend to zero. This proves the pair +criterion along every F\o lner sequence. + +The same criterion holds on $\mathcal R$ over $Y$. To see this first +for simple tensors $v=a_1\otimes a_2$ and $w=b_1\otimes b_2$, use +\[ + P_{\mathcal R}(vU_{u^d}w) + =P(a_1U_{u^d}b_1)P(a_2U_{u^d}b_2). +\] +Subtract the analogous product of the factor parts. The difference is +a sum of two terms, each containing one error from +\eqref{eq:source-7} on $X$ and one uniformly bounded factor. Its +averaged squared $L^2(Y)$ norm therefore tends to zero. Linearity gives +the result for finite sums of tensors. Such sums can approximate any +bounded function on $\mathcal R$ in $L^2$, with a common sup bound: +use conditional expectations onto finite partitions generated by +rectangles in the two coordinates. Approximate one of $v,w$ at a time; +conditional-expectation contraction and measure preservation bound the +error uniformly in $u$. The pair criterion follows for all bounded +functions on $\mathcal R$. + +We next prove the following vector assertion on either $X$ or +$\mathcal R$. On either system use $P$ for its expectation onto $Y$. +If $n_1,\ldots,n_l$ are distinct \emph{nonzero} integers, the $v_j$ are +bounded, and $Pv_{j_*}=0$ for some $j_*$, then +\begin{equation}\label{eq:rigidity-centered-vector} + \left\|\operatorname{Avg}_{u\in\Phi_N} + \prod_{j=1}^lU_{u^{n_j}}v_j\right\|_2\longrightarrow0 +\end{equation} +for every F\o lner sequence. Both systems are ergodic and satisfy the +pair criterion, so the same induction proves the assertion on each. +By rescaling we may assume $\norm{v_j}_\infty\leq1$. + +For $l=1$, the assertion is mean ergodicity and +$\int v_1=\int Pv_1=0$. Suppose it is proved for $l-1$, and set +\[ + V_u=\prod_{j=1}^lU_{u^{n_j}}v_j, + \qquad F_{j,h}=U_{h^{n_j}}v_j\,\bar v_j\quad(h\in K). +\] +For each fixed $h$, invariance of the integral gives +\begin{align} + \int V_{hu}\overline{V_u}\,d\mu + &=\int\prod_{j=1}^lU_{u^{n_j}}F_{j,h}\,d\mu\notag\\ + &=\int F_{l,h}\prod_{j0$ and every F\o lner sequence in $K$, +\begin{equation}\label{eq:source-9} + \operatorname{Avg}_{u\in\Phi_N}m(2r|\operatorname{Re}u|) + \longrightarrow m(r)^2. +\end{equation} +\end{lemma} + +\begin{proof} +First take $A,B>0$ and put $L=\sqrt{AB}\in F$. For every $u\in K$, +\begin{equation}\label{eq:rigidity-equal-moduli} + |A-Bu^2|^2 + =(A-B)^2+4AB(\operatorname{Im}u)^2 + =|-L+(A-B)u+Lu^2|^2. +\end{equation} +For the second equality, factor the last expression as +$u((A-B)+L(u-u^{-1}))$ and use $u-u^{-1}=2i\operatorname{Im}u$. + +On the first side, +$\psi_{A-Bu^2}=\psi_AU_{u^2}\psi_{-B}$. By +\eqref{eq:source-6} the projection of this product may be replaced, +in averaged squared $L^2$ error, by $g_AU_{u^2}g_{-B}$. Both have +$L^2$ norm at most one, so their squared norms differ in averaged +absolute value by a quantity tending to zero. Indeed the absolute +difference is at most twice their $L^2$ distance, and then +Cauchy--Schwarz applies to the finite average. Ergodicity of $Y$ and +surjectivity of the square map now give +\begin{align} + \operatorname{Avg}_{u\in\Phi_N}m(A-Bu^2) + &\longrightarrow + \left(\int_Y|g_A|^2\,d\nu\right) + \left(\int_Y|g_{-B}|^2\,d\nu\right)\notag\\ + &=m(A)m(B). + \label{eq:rigidity-two-term-limit} +\end{align} +Here the intervening squared norm is +$\int_Y|g_A|^2U_{u^2}|g_{-B}|^2\,d\nu$. + +On the second side of \eqref{eq:rigidity-equal-moduli}, apply +\eqref{eq:source-6} with exponents $0,1,2$. The projected character is +approximated by +\[ + g_{-L}\,U_ug_{A-B}\,U_{u^2}g_L. +\] +Its squared norm is at most $\norm{g_{A-B}}_2^2=m(|A-B|)$, since the +two outside factors have modulus at most one. Equal moduli give equal +$m$-values, so \eqref{eq:rigidity-two-term-limit} implies +$m(A)m(B)\leq m(|A-B|)$. Taking negative logarithms proves +\eqref{eq:source-8}, including zero values of $m$ with the stated +extended-real convention. If $A=0$ or $B=0$, the inequality is an +identity because $w(0)=0$. + +For \eqref{eq:source-9}, use +$ru+r\bar u=2r\operatorname{Re}u$ and the distinct exponents $1,-1$. +The same approximation reduces the average to squared norms of +$U_ug_r\,U_{u^{-1}}g_r$. Those squared norms equal +\[ + \int_Y|g_r|^2U_{u^{-2}}|g_r|^2\,d\nu. +\] +Mean ergodicity on $Y$ through the surjective power map $u\mapsto u^{-2}$ +gives the limit $m(r)^2$. Radiality replaces the real argument by its +absolute value, proving \eqref{eq:source-9}. +\end{proof} + +\subsection{General-position rotations and bounded residues} + +The remaining task is to show that $m(r)=0$ for every positive $r\in F$. +Assuming instead that $m(r)>0$, we will obtain a uniform lower bound +$m(s)\ge c>0$ on some interval $(0,\varepsilon)\cap F$, contradicting +the vanishing line average. By \eqref{eq:source-9}, a suitable set +of rotations with bounded $w(2r|\operatorname{Re}u|)$ has finitely many +translates covering $K$. The first lemma below provides rotations in +rational general position from which the proof will extract bases. +The second controls the resulting integer combinations of nonnegative +lengths under a fixed cap, using only the difference inequality +\eqref{eq:source-8}. + +\begin{lemma}[General position in a number field] +\label{lem:rigidity-general-position} +Let $N_0\subset F$ be a number field, let $N=N_0(i)$, and let +$d_0=[N:\Q]$. There are arbitrarily large finite sets $S\subset K\cap N$ +such that every $d_0$ distinct elements of $S$ form a $\Q$-basis of $N$. +\end{lemma} + +\begin{proof} +For $t\in N_0$ put +\[ + u(t)=\frac{1+it}{1-it}\in K\cap N. +\] +Choose a $\Q$-basis of $N_0$. In the rational coordinates of $t$, the +coordinates of $u(t)$ in a $\Q$-basis of $N$ are rational functions over +$\Q$. This follows, for example, by inverting the matrix for +multiplication by $1-it$. Its denominator is nonzero for all parameters +in $N_0$, since $t$ is real. + +No nonzero $\Q$-linear functional $\ell:N\to\Q$ vanishes on all $u(t)$. +Suppose one did. For every fixed $a\in N_0$, the rational function +$\ell(u(ha))$ vanishes at every rational $h$, and hence is identically +zero. Its formal expansion at $h=0$ comes from +\[ + u(ha)=1+2iha-2h^2a^2+\cdots. +\] +The first three coefficients imply +$\ell(1)=\ell(ia)=\ell(a^2)=0$ for every $a\in N_0$. Squares span +$N_0$ over $\Q$, as +$4a=(a+1)^2-(a-1)^2$. Thus $\ell$ vanishes on $N_0+iN_0=N$, a +contradiction. This is a calculation in rational function coordinates +and formal series; it uses no usual-topology continuity of $\ell$. + +For any proper rational subspace of $N$, choose a nonzero rational +linear functional annihilating it. Its value on $u(t)$ has a nonzero +numerator polynomial in the rational coordinates of $t$. A finite +union of such proper subspaces therefore cannot contain all $u(t)$: +the product of their nonzero numerator polynomials is nonzero, and a +nonzero polynomial over the infinite field $\Q$ cannot vanish on all +of rational affine space. Inductively choose each new element outside +the spans of all subsets of at most $d_0-1$ earlier elements. Every +subset of at most $d_0$ chosen elements remains independent. This +constructs finite sets of every required size. +\end{proof} + +\begin{lemma}[Addition under a cap]\label{lem:rigidity-capped-residues} +Let $w:F_{\geq0}\to[0,\infty]$ satisfy $w(0)=0$ and +\eqref{eq:source-8}. Let $p_1,\ldots,p_q\in F_{\geq0}$ with +$p=\max_jp_j>0$ and $w(p_j)\leq H<\infty$. If +\[ + z=\sum_{j=1}^q a_jp_j\in[0,p),\qquad + a_j\in\Z,\qquad\sum_j|a_j|\leq M, +\] +then $w(z)\leq4MH$. +\end{lemma} + +\begin{proof} +Keep every partial sum as its residue in $[0,p)$. A positive input +$p_jp$, the new residue is +$a-(p-b)$, whose cost is at most $w(a)+w(p)+w(b)$. If $a+b=p$, its +cost is zero. Adding a signed input therefore increases the current +cost by at most $4H$. Expand the integer combination into at most $M$ +signed inputs and start from residue zero. Its final residue is $z$ +itself, proving the bound. +\end{proof} + +\subsection{Completion of the rigidity theorem} + +\begin{proof}[Proof of Theorem~\ref{thm:rigidity}] +First let $\mu$ be ergodic and wild, and use the factor and radial +function constructed above. Suppose that $m(r)>0$ for some $r\in F$, +$r>0$. Choose $H>0$ finite so that $e^{-H}0$ such that +\begin{equation}\label{eq:rigidity-selection-transversality} + \max_{k\in\mathcal T}|\operatorname{Re}(kv)|\geq c_0 + \qquad(v\in\T) +\end{equation} +for every selection. To prove this, for a fixed selection the maximum +is a continuous function of the ordinary unit direction $v$. It never +vanishes: the rational span of the selection contains $1$ and $i$, so +vanishing would imply both $\operatorname{Re}v=0$ and +$\operatorname{Re}(iv)=0$. Take a positive minimum on the compact unit +circle and then the minimum over the finitely many selections. We may +and do take $c_0\leq1$. + +Now let $v\in K$ satisfy $|\operatorname{Re}v|0 + \qquad\bigl(s\in F\cap(0,2rc_0)\bigr). +\] +Choose a nonnegative $\phi\in C_c(\R)$ supported in this open interval +with positive integral. Positivity of the functional $I$ and its +agreement with Lebesgue integration on continuous test functions give +\[ + I\bigl(\phi(t)m(t)\bigr) + \geq e^{-4M_0H}I(\phi) + =e^{-4M_0H}\int_\R\phi(t)\,dt>0. +\] +This contradicts \eqref{eq:source-2} for the wild law $\nu_Y$, with +$a=1$, in view of \eqref{eq:rigidity-difference-fourier}. This step +uses no ordinary-measure measurability of the radial function: $I$ is +defined on all the bounded, boundedly supported functions in question. + +Thus $m(r)=0$ for every positive $r\in F$, and radiality gives $m(z)=0$ +for all nonzero $z\in E$. Consequently +\[ + \left|\int_D\psi_z\,d\mu\right| + =\left|\int_Yg_z\,d\nu\right| + \leq\norm{g_z}_2=0\qquad(z\ne0). +\] +Fourier uniqueness on the compact abelian group $D$ identifies $\mu$ +with Haar probability. + +Finally, an arbitrary wild invariant probability has an ergodic +decomposition for the countable group $K$ on the standard compact space +$D$. Since $C$ is invariant Borel and has measure zero, almost every +ergodic component also gives $C$ measure zero. The ergodic case just +proved makes each such component Haar probability. Their integral is +therefore Haar probability as well. This proves the theorem. +\end{proof} diff --git a/preprints/The-Euclidean-plane-is-not-five-colorable-September-23-2026/build/sections/spectral.tex b/preprints/The-Euclidean-plane-is-not-five-colorable-September-23-2026/build/sections/spectral.tex new file mode 100644 index 0000000000000000000000000000000000000000..f7c3117d5daf50e71e1f6b0a808bf71a25de43e4 --- /dev/null +++ b/preprints/The-Euclidean-plane-is-not-five-colorable-September-23-2026/build/sections/spectral.tex @@ -0,0 +1,624 @@ +\section{Spectral preliminaries and relative singularity} +\label{sec:spectral} + +This section and Section~\ref{sec:rigidity} establish the probability-law +rigidity used in the coloring transfer of Section~\ref{sec:transfer}. +The laws live on the character group of the discrete algebraic plane. +We state the rigidity theorem after defining that group, then prove +that exclusion of ordinary continuous characters persists in the +conditional difference laws arising along compact towers. This +relative singularity is the input to the Fourier argument in +Section~\ref{sec:rigidity}. + +\subsection{Algebraic rotations and continuous characters} + +\begin{definition}\label{def:spectral-groups} +Identify the plane with $\C$, and set +\[ + F=\overline{\Q}\cap\R,\qquad E=F(i),\qquad + K=\{u\in E:u\bar u=1\}. +\] +We regard $E$ as an additive discrete group and $K$ as a multiplicative +discrete group. Let +\[ + D=\widehat{E_{\mathrm{disc}}} +\] +be the compact character group of $E$. A point $d\in D$ is a homomorphism +from the additive group $E$ to the unit complex group $\T$. We use additive +notation on $D$, so $(d+d')(z)=d(z)d'(z)$. For $a,z\in E$, put +\[ + (ad)(z)=d(az),\qquad \psi_z(d)=d(z). +\] +The subgroup of ordinary continuous characters is +\begin{equation}\label{eq:spectral-continuous-characters} + C=j(\R^2),\qquad + j(\xi)(z)=\exp(i\xi\cdot z), +\end{equation} +where the dot product uses the real coordinates of $z$. +\end{definition} + +Both $E$ and $K$ are countable abelian groups. Every nonzero power map +$u\mapsto u^n$ on $K$ is onto: a complex $n$th root of an algebraic number +of modulus one is again algebraic and has modulus one; negative powers +also cause no difficulty. Thus $K$ is divisible. The compact group $D$ is +metrizable because $E$ is countable. + +The map $j$ is continuous and injective. Indeed, a continuous plane +character that is one on the dense subgroup $E$ is one everywhere, and +its frequency is zero. Its image is the union of the compact images of +closed balls in $\R^2$, so $C$ is Borel. The inverse of $j$ on each such +compact image is continuous; consequently $j^{-1}:C\to\R^2$ is Borel. +For $a\ne0$, multiplication by $a$ is an automorphism of $D$ and carries +$C$ onto itself. On the frequency parameters it acts by the transpose +of the real linear map of multiplication by $a$. In particular, an +element of $K$ acts on these parameters by an orthogonal map. + +\begin{definition}\label{def:spectral-wild} +A $K$-invariant Borel probability measure $\nu$ on $D$ is called +\emph{wild} if $\nu(C)=0$. Its Fourier coefficients are denoted by +\[ + f_\nu(z)=\int_D\psi_z(d)\,d\nu(d),\qquad z\in E. +\] +\end{definition} + +\begin{theorem}[Rigidity of wild character laws]\label{thm:rigidity} +Every $K$-invariant probability measure $\nu$ on $D$ with $\nu(C)=0$ +is Haar measure. Equivalently, writing $m_D$ for normalized Haar measure, +\begin{equation}\label{eq:source-3} + \nu\text{ is wild}\quad\Longrightarrow\quad\nu=m_D. +\end{equation} +\end{theorem} + +The Fourier coefficients of Haar probability vanish at every nonzero +$z\in E$. This is the conclusion needed in Section~\ref{sec:transfer}: +a rotation-fixed label indicator has a $K$-invariant finite spectral +measure, and its part off $C$ is again invariant. If that part is +nonzero, normalization allows the theorem to be applied; its correlations +at every nonzero algebraic translation then vanish. The factor argument in +Lemma~\ref{lem:palettes-continuous-factor} will make the $C$-supported +projections nonnegative, with $L^2$-continuous translation orbits. + +There is a classical analogue for measurable radial positive-definite +functions on the full real plane: each is the sum of a continuous +positive-definite function and a nonnegative multiple of +$\ind_{\{0\}}$ \citep{Crum1956,GneitingSasvari1999}. +Here the Fourier coefficients are initially defined only on $E$, with +invariance under its algebraic rotations $K$; no measurable radial +positive-definite extension to $\R^2$ is assumed. +We prove the required rigidity directly on $D$. + +The proof of the theorem is completed in Section~\ref{sec:rigidity}. +Here we first obtain null-event and line-average consequences of +$\nu(C)=0$. We then show that this condition survives when the law is +replaced by the difference of two conditionally independent samples +over a factor in a compact tower. + +\begin{lemma}\label{lem:spectral-null-events} +Let $\nu$ be wild. Then every coset of $C$ has $\nu$-measure zero, and +the law of the difference of two independent $\nu$-samples gives $C$ +measure zero. Moreover, for each $a\in E\setminus\{0\}$, the event +\[ + L_a=\{d\in D:t\mapsto d(at)\text{ is continuous on }F + \text{ in its usual topology}\} +\] +is Borel and has $\nu$-measure zero. +\end{lemma} + +\begin{proof} +If a coset $d+C$ has positive mass, its $K$-orbit is finite, since the +cosets in that orbit are disjoint and have the same mass. The action on +this finite orbit gives a finite image of the divisible abelian group +$K$. Such an image is trivial: a finite abelian group cannot be divisible +unless it is trivial. Hence every $u\in K$ fixes the coset, and +$(u-1)d\in C$. Taking $u\ne1$ and applying the inverse scalar shows +$d\in C$, contrary to $\nu(C)=0$. The difference assertion now follows +by integrating $\nu(d+C)=0$ against $\nu(d)$; the relation +$d'-d\in C$ is Borel. + +Restriction to the line $aF$ is a continuous map from $D$ to +$\widehat{F_{\mathrm{disc}}}$. The ordinary continuous characters on $F$ +are exactly $t\mapsto e^{i\lambda t}$, $\lambda\in\R$. Their +parametrization is continuous and injective, with Borel image by compact +exhaustion, as for $j$. This proves the Borel assertion for $L_a$. + +If $aF,bF$ are nonparallel, then $a,b$ form an $F$-basis of $E$. A +character continuous on both lines has the form +\[ + d(at+bs)=e^{i(\lambda t+\eta s)}\qquad(t,s\in F) +\] +for real $\lambda,\eta$. The corresponding real coordinate change is +invertible and continuous, so this character belongs to $C$. Thus +$L_a\cap L_b\subset C$ for nonparallel lines. Rotations give infinitely +many distinct unoriented lines $uaF$, and their events have the same +$\nu$-measure. Any finite collection of them is pairwise disjoint modulo +$\nu$-null sets. Their common measure must therefore be zero. +\end{proof} + +\subsection{A line average without a measurability assumption} + +Fix a translation-invariant mean $\mathfrak m$ on the bounded functions +on the discrete abelian group $F/\Z$. Such a normalized positive mean +exists by amenability of abelian groups; see +\citet[Theorem~G.2.1]{Bekka2008}. Because the group is discrete, +the mean is defined on every bounded function. Invariant means and +the finite-set averaging criterion originate in the work of +\citet{vonNeumann1929} and \citet{Folner1955}. The restriction of +$\mathfrak m$ to ordinary continuous periodic functions is integration +over the usual circle. To check this, every nonconstant ordinary circle +character is multiplied by a nontrivial scalar under some translation +from $F/\Z$, and hence has mean zero. Trigonometric-polynomial +approximation gives the assertion for all continuous periodic functions. + +For a bounded function $h:F\to\C$ supported in a bounded interval, define +\begin{equation}\label{eq:spectral-invariant-integral} + I(h)=\mathfrak m\left(t+\Z\longmapsto + \sum_{k\in\Z}h(t+k)\right). +\end{equation} +The periodization is well defined and bounded: only a uniformly bounded +number of summands can be nonzero. The functional $I$ is positive and +translation invariant under $F$. For a continuous compactly supported +function $\phi$ on $\R$, it satisfies +\[ + I(\phi|_F)=\int_\R\phi(t)\,dt. +\] +We also use $I$ to integrate bounded, boundedly supported functions with +values in a Hilbert space, in the weak sense. Scalar pairings define a +bounded functional on the Hilbert space, and Hilbert-space duality gives +the resulting vector. In particular, if $V(t)$ is such a function, then +\[ + \norm{I(V)}\le I\bigl(\norm{V(t)}\bigr). +\] +This construction does not impose ordinary real-variable measurability +on $V$ or on a Fourier coefficient restricted to $F$. + +\begin{lemma}[Vanishing line average]\label{lem:line-average} +If $\nu$ is wild, then +\begin{equation}\label{eq:source-2} + I\bigl(\phi(t)f_\nu(at)\bigr)=0 + \qquad\bigl(a\in E\setminus\{0\},\ \phi\in C_c(\R)\bigr). +\end{equation} +\end{lemma} + +\begin{proof} +On $L^2(\nu)$ consider the unitary representation of the additive group +$F$ given by +\[ + T_tv=\psi_{at}v. +\] +First, this representation has no nonzero vector that is strongly +continuous for the usual topology on $F$. Suppose that $v$ is such a +vector. Put +\[ + c_v(t)=\int_D|v|^2\psi_{at}\,d\nu. +\] +It is positive definite and uniformly continuous on $F$, since +\[ + |c_v(t+s)-c_v(t)|\le\norm{v}_2\norm{T_sv-v}_2. +\] +It therefore extends continuously to $\R$. The extension remains +positive definite, by approximation of each finite set of real +arguments by elements of $F$. + +Apply the ordinary Bochner theorem to this continuous positive-definite +function on $\R$; see \citet[Theorem~D.2.2]{Bekka2008}. +It is important here that the coefficient has first been extended to +$\R$; no local compactness of $F$ in its usual topology is assumed. +Push its finite positive +representing measure to $\widehat{F_{\mathrm{disc}}}$ using the +continuous-character parametrization. Its Fourier coefficients agree +with those of the restriction-map pushforward of $|v|^2\nu$. Uniqueness +of Fourier coefficients for finite measures on the compact character +group identifies these two measures. Indeed, the characters form a +unital conjugation-closed algebra separating its points, so their +uniform density in the continuous functions gives this uniqueness. The pushforward of $|v|^2\nu$ is +thus concentrated on the ordinary continuous characters, so $|v|^2\nu$ +is concentrated on $L_a$. Lemma~\ref{lem:spectral-null-events} gives +$\nu(L_a)=0$, and hence $v=0$. + +Now form the weak integral +\[ + V_\phi=I\bigl(\phi(t)T_t1\bigr)\in L^2(\nu). +\] +Translation invariance of $I$ gives +\[ + T_sV_\phi-V_\phi + =I\bigl((\phi(t-s)-\phi(t))T_t1\bigr), +\] +and consequently +\[ + \norm{T_sV_\phi-V_\phi}_2 + \le I(|\phi(\cdot-s)-\phi|) + =\int_\R|\phi(t-s)-\phi(t)|\,dt\longrightarrow0 +\] +as $s\in F$ tends to zero. Thus $V_\phi$ is strongly continuous and +must be zero by the preceding argument. Pairing with the constant +function proves \eqref{eq:source-2}. +\end{proof} + +\subsection{A measurable equivariant center} + +The subgroup $C$ need not be closed. We will nevertheless choose a +canonical representative of a positive-mass $C$-coset when a probability +kernel on that coset is available. The following elementary location +rule avoids any moment assumption on that kernel. + +\begin{lemma}\label{lem:spectral-location} +For every Borel probability measure $\lambda$ on $\R^2$, the function +\begin{equation}\label{eq:spectral-location-objective} + J_\lambda(s)=\int_{\R^2} + \left(\sqrt{1+|s-t|^2}-\sqrt{1+|t|^2}\right)\,d\lambda(t) +\end{equation} +has a unique minimizer $L(\lambda)$. The map $\lambda\mapsto L(\lambda)$ +is Borel for the usual Borel structure on the space of probability +measures. It is equivariant under translations and orthogonal maps. +\end{lemma} + +\begin{proof} +The difference in \eqref{eq:spectral-location-objective} is bounded in +absolute value by $|s|$. The objective is therefore finite without a +first-moment assumption. It is Lipschitz in $s$ and strictly convex, +because $s\mapsto\sqrt{1+|s-t|^2}$ is strictly convex for every $t$. + +Choose $R>0$ with $p=\lambda(\overline{B(0,R)})>1/2$. On this ball the +integrand is at least $|s|-R-\sqrt{1+R^2}$, and elsewhere it is at least +$-|s|$. Hence +\[ + J_\lambda(s)\ge + (2p-1)|s|-p\bigl(R+\sqrt{1+R^2}\bigr)\longrightarrow\infty. +\] +Coercivity and continuity give a minimizer, and strict convexity gives +uniqueness. + +For each fixed $s$, the integrand is bounded and continuous in $t$, so +$\lambda\mapsto J_\lambda(s)$ is Borel. The infimum over rational +points equals the minimum. Enumerate $\Q^2$ and, for each positive +integer $n$, choose its first point whose objective is less than that +infimum plus $1/n$. These choices are measurable. For a fixed $\lambda$ +they converge to the unique minimizer: coercivity confines all sufficiently +accurate choices to a compact set, and every cluster point minimizes +$J_\lambda$. Their limit is therefore the Borel map $L$. + +If $\lambda'$ is the law of $t-r$ for $t$ of law $\lambda$, then an +identity of bounded differences gives +\[ + J_{\lambda'}(s)=J_\lambda(s+r)-J_\lambda(r). +\] +Thus $L(\lambda')=L(\lambda)-r$. Orthogonal covariance follows from +$J_{O_*\lambda}(Os)=J_\lambda(s)$. These identities prove both asserted +equivariances. +\end{proof} + +\subsection{Compact towers and relative singularity} + +We use standard probability spaces and probability-preserving actions +of the countable group $K$. Factors are countably generated modulo null +sets. For an action write $U_kH(x)=H(kx)$, and identify functions on a +factor with their pullbacks. Conditional kernels, factor actions, and +equivariant maps may be chosen on common conull sets whenever needed; +the acting group is countable. + +We use disintegration in the following precise form. A factor map +$\pi:X\to Y$ between standard Borel probability models admits a +measurable probability kernel $y\mapsto\mu_y$, concentrated on +$\pi^{-1}(y)$ for almost every $y$, such that +$\mu=\int\mu_y\,d\nu(y)$. The kernel gives conditional expectations +by integration and is unique almost everywhere; see +\citet[Theorem~4 and Remark~5]{TaoDisintegration2008}. +For an equivariant factor map, uniqueness makes the kernel equivariant +for each group element. Intersecting the corresponding conull sets +makes equivariance simultaneous for the countable group $K$. +This is the fiber-measure input to all relative products below. + +Consider a tower $(Y_\alpha)_{\alpha\le\theta}$ of factors of an +ergodic system $X$, where $\theta$ is a countable ordinal. Start with the +trivial factor $Y_0$, and at a limit ordinal take the factor generated +by the preceding ones. Require the following approximation property at +each successor $Y$ over its predecessor $W$: for every bounded $H$ on +$Y$ and every $\varepsilon>0$, there are bounded functions +$H_1,\ldots,H_q$ on $Y$ and a finite constant $B$ such that, for every +$k\in K$, +\begin{equation}\label{eq:source-4} + \left\|U_kH-\sum_{r=1}^q b_r^k H_r\right\|_2<\varepsilon + \quad\text{for some }b_r^k\in L^\infty(W) + \text{ with }\|b_r^k\|_\infty\le B. +\end{equation} +The basis functions and bounds are fixed before $k$ is chosen. + +\begin{lemma}[Relative singularity]\label{lem:relative-singularity} +Let $X$ be an ergodic standard probability-preserving $K$-system, and let +$d:X\to D$ be an equivariant measurable map with wild law. For every +countable tower of factors as above, and every factor $Y$ in that tower, +the law $\nu_Y$ of +\[ + d(x)-d(x') +\] +for two conditionally independent samples of $X$ over $Y$ satisfies +$\nu_Y(C)=0$. +\end{lemma} + +\begin{proof} +All relative difference laws in the statement are $K$-invariant, by +equivariance of conditional kernels. We use transfinite induction in +its universal form: at each ordinal the assertion ranges over +\emph{all} ambient ergodic standard systems, equivariant maps with wild +law, and towers satisfying \eqref{eq:source-4}. This allows an earlier +induction assertion to be applied below to a new factor system and a +new equivariant map. + +At the trivial factor the assertion is +Lemma~\ref{lem:spectral-null-events}. Suppose that it holds at every +earlier ordinal, and that it fails at a noninitial stage $Y$ of a tower +in a system $X$. Write $\kappa_y$ for the conditional law of $d$ over +$y\in Y$. + +\paragraph{Positive-mass cosets and their centers.} +The function +\[ + r(y,z)=\kappa_y(z+C) + =\int_D\ind_C(d'-z)\,d\kappa_y(d') +\] +is Borel in $(y,z)$, by Borel parameter integration. This uses only that +$C$ is Borel, not that it is closed. The function +$x\mapsto r(y(x),d(x))$ is invariant and has integral $\nu_Y(C)>0$. +Ergodicity makes it a positive constant $a$ almost everywhere. + +For almost every $y$, therefore, $\kappa_y$-almost every sampled coset +has mass $a$. Such cosets are disjoint and there are only finitely many; +they exhaust the probability. Their number $m$ satisfies $ma=1$, and +is consequently independent of $y$. Thus $\kappa_y$ is concentrated +on $m$ distinct cosets of mass $1/m$ each. None is $C$ almost surely, +because the original marginal law gives $C$ measure zero. + +On the Borel set $r(y,z)>0$, define the probability kernel +$\lambda_{y,z}$ on $\R^2$ by +\[ + \lambda_{y,z}(A)=\frac{1}{r(y,z)} + \int_D\ind_C(d'-z)\ind_A\bigl(j^{-1}(d'-z)\bigr) + \,d\kappa_y(d') +\] +for Borel $A\subset\R^2$, interpreting the integrand as zero off $C$. +The Borel inverse of $j$ makes this a measurable kernel. Set +\[ + b(y,z)=z+j\bigl(L(\lambda_{y,z})\bigr). +\] +If $z$ is replaced by $z+j(r_0)$, the kernel is translated by $-r_0$; +Lemma~\ref{lem:spectral-location} shows that $b(y,z)$ is unchanged. +It is therefore constant on each positive-mass coset, and remains in +that coset. The same lemma and the orthogonal action of $K$ on frequency +parameters show +\[ + b(ky,kz)=k b(y,z) +\] +on the relevant conull sets. + +Let $Z$ be the factor of $X$ generated by $Y$ and +$b(x)=b(y(x),d(x))$. It is a standard ergodic factor. The map +$b:Z\to D$ is equivariant and has wild law. Its conditional law +$\beta_y$ over $Y$ is uniform on the $m$ distinct centers of the +positive-mass cosets. We have converted the hypothetical failure of +singularity into a finite set of equivariant centers over each fiber. +The earlier induction hypothesis and the line average will give +selected real algebraic frequencies with small conditional collision +probabilities on a common fiber. Compact approximation, however, will +force many pairs of unit-direction phase lists to match one pair of +lists. Writing each selected frequency as a sum of its two directions +then places its phases near at most $m^2$ products, forcing a larger +collision probability. + +\paragraph{A finite list of phases.} +For each $y$, consider the unordered list of $m$ phases $\psi_1$ at +the points of $\beta_y$, retaining repetitions when phases coincide. +Encode this list by the vector $H(y)\in\C^m$ of its elementary +symmetric functions. The vector is measurable: its power sums are +$m\int\psi_1^n\,d\beta_y$, and Newton's identities recover the +elementary symmetric functions. It is bounded. + +The encoding is a continuous injection of the compact space +$\T^m/S_m$ into $\C^m$. On unordered lists use maximum matching +distance, +\[ + d_{\mathrm{match}}([z_1,\ldots,z_m],[z'_1,\ldots,z'_m]) + =\min_{\pi\in S_m}\max_{1\le r\le m}|z_r-z'_{\pi(r)}|. +\] +The inverse encoding is uniformly continuous for this metric. +Equivariance of $\beta_y$ and $\psi_1(kd)=\psi_k(d)$ show that +$U_kH$ encodes the list of phases at $\psi_k$. + +Choose $\rho>0$ so small that uniform-circle probability of +$\{|z-1|\le5\rho\}$ is less than $1/(16m^2)$. There are an integer +$N\ge1$ and a constant $c>0$ with the following property: if a +probability $\omega$ on $\T$ satisfies +\[ + \left|\int z^n\,d\omega(z)\right|0$ such that two encoding vectors at distance less than $s$ +have phase lists matching within $\rho$. Put $\tau=s/100$, and fix +$\gamma>0$ such that +\begin{equation}\label{eq:spectral-error-choice} + \frac{2\gamma^2}{\tau^2}<10^{-4}. +\end{equation} + +\paragraph{Return to an earlier factor.} +The rotations of $H$ admit vector-valued approximations of the form +\begin{equation}\label{eq:spectral-vector-approximation} + \left\|U_kH-\sum_{r=1}^q b_r^k H_r\right\|_2<\gamma, + \qquad \|b_r^k\|_\infty\le B, +\end{equation} +with fixed bounded $\C^m$-valued functions $H_r$ on $Y$ and +coefficients over some earlier factor $W$. At a successor this follows +from \eqref{eq:source-4}, applied to the finitely many coordinates and +combining their data. At a limit, first approximate $H$ in $L^2$, +within $\gamma/2$, by bounded functions on an earlier successor +factor. Such approximation follows from generation of the limit +factor, for example by conditional expectations. Its error remains +unchanged under rotation. Then use \eqref{eq:source-4} at that +successor, with its predecessor as $W$, to obtain a further error less +than $\gamma/2$. Combining coordinates gives +\eqref{eq:spectral-vector-approximation}. The coordinate count, +number of basis functions, their uniform bounds, and $B$ are all fixed +independently of $k$. + +The earlier tower through $W$ is also a tower of factors of $Z$: +$Z$ contains $Y$ and hence every predecessor of $Y$. All measures, +actions, limit joins, and approximation properties of those factors +are unchanged. The universal earlier induction assertion therefore +applies in $Z$ to the map $b$. It says that the difference law of $b$ +over $W$ is wild. Its Fourier coefficient at $a\in E$ is +\begin{equation}\label{eq:spectral-conditional-coefficient} + p(a)=\left\|\mathbb E_Z(\psi_a(b)\mid W)\right\|_2^2\ge0, +\end{equation} +by conditional independence. Lemma~\ref{lem:line-average} applies to +this coefficient function. + +\paragraph{Fixing the number of tests before choosing a small parameter.} +At any fixed fiber of $W$, the two coefficient arrays for two +approximations in \eqref{eq:spectral-vector-approximation} belong to +a fixed bounded subset of $\C^{2q}$. Partition this subset into a +finite number $L$ of bins so fine that, within one bin, the corresponding +approximating functions differ pointwise by at most $\tau$ for each +of the two vectors. For example, coordinate diameters less than +\[ + \frac{\tau}{1+\sum_{r=1}^q\|H_r\|_\infty} +\] +suffice. The number $L$ is independent of the fiber and of the chosen +rotations. Fix an integer $M\ge2$ so large that +\[ + \frac{0.9M}{L}>10m^2. +\] +All approximation data have been fixed before this choice. + +There is a nonzero $t\in F$ with $|t|<1/M$ for which every value +\[ + p\bigl(n(j-l)t\bigr),\qquad + 1\le n\le N,\quad 1\le j,l\le M,\quad j\ne l, +\] +is as small as we prescribe. To see this, choose a nonnegative +$\phi\in C_c((-1/M,1/M))$ with positive integral. Each indicated +nonnegative test has zero $\phi$-weighted $I$-integral by +\eqref{eq:source-2}. Their finite sum does too. If that sum were +bounded below by a prescribed positive number at every point where +$\phi>0$, positivity of $I$ would contradict $I(\phi)>0$. Thus some +such point makes their sum, and hence every test, as small as required. +Choosing this bound below one also ensures $t\ne0$, since $p(0)=1$. + +Set $x_j=jt$. Choose the prescribed smallness so that Markov's +inequality and a finite union bound give, on a set of $W$-probability +greater than $0.99$, all of the bounds +\[ + \left|\mathbb E_Z\bigl(\psi_{n(x_j-x_l)}(b)\mid W=w\bigr)\right|10m^2$. Fix $j_0\in J$. For either type of +encoding the conditional $L^2$ distance from index $j$ to index $j_0$ +is at most $3\tau$: the two approximation errors each cost less than +$\tau$, and the approximants differ pointwise by at most $\tau$. + +For a uniform $j\in J$ and a conditional point $y$ over $w$, the +probability that either encoding distance is at least $s$ is at most +\[ + \frac{18\tau^2}{s^2}=0.0018. +\] +Thus, with probability greater than $0.95$, both phase lists for +index $j$ match their respective representative lists for $j_0$ +within $\rho$. Let $q(y)$ be the fraction of indices in $J$ for +which both matches hold. We have +\begin{equation}\label{eq:spectral-matching-fraction} + \mathbb E(q\mid W=w)>0.95. +\end{equation} + +\paragraph{The collision contradiction.} +Fix such a fiber point $y$ and any point $b_0$ in the support of +$\beta_y$. For every matching index $j$, the two phases +$\psi_{k_j}(b_0)$ and $\psi_{k'_j}(b_0)$ are within $\rho$ of entries +of their respective representative lists. Consequently +\[ + \psi_{x_j}(b_0)=\psi_{k_j}(b_0)\psi_{k'_j}(b_0) +\] +is within $2\rho$ of one of the at most $m^2$ products of entries +from those two lists. The matchings need not identify the same center +in the two lists; allowing all $m^2$ products accounts for this. + +Assign each of the $q(y)|J|$ matching indices to one such product. +By the sum-of-squares inequality, the number of ordered pairs assigned +to the same product is at least $q(y)^2|J|^2/m^2$. Removing diagonal +pairs discards at most $|J|$ pairs. Hence, for independent uniform +$j,l\in J$, +\[ + \mathbb P_{j,l}\bigl(j\ne l,\ + |\psi_{x_j}(b_0)-\psi_{x_l}(b_0)|\le4\rho\bigr) + \ge\frac{q(y)^2}{m^2}-\frac1{|J|}. +\] +This bound holds for every possible center $b_0$, so it can be +integrated first over $\beta_y$ and then over the conditional law of +$y$ given $w$. Jensen's inequality and +\eqref{eq:spectral-matching-fraction} give a lower bound strictly +greater than +\[ + \frac{0.95^2}{m^2}-\frac1{|J|} + >\frac{0.95^2-0.1}{m^2} + >\frac1{4m^2}. +\] +But \eqref{eq:source-5} bounds the conditional collision probability +for every distinct pair in the original set of $M$ indices by +$1/(4m^2)$. Averaging those bounds over the selected $J$ gives an +upper bound smaller than $1/(4m^2)$. Choosing $J$ after fixing $w$ +is harmless because all those pair bounds hold simultaneously at that +fiber. This contradiction completes the induction step, and hence the +universal transfinite induction. +\end{proof} diff --git a/preprints/The-Euclidean-plane-is-not-five-colorable-September-23-2026/build/sections/transfer.tex b/preprints/The-Euclidean-plane-is-not-five-colorable-September-23-2026/build/sections/transfer.tex new file mode 100644 index 0000000000000000000000000000000000000000..318a6d1fb1b5be85972af6be9bf6961fba7983e2 --- /dev/null +++ b/preprints/The-Euclidean-plane-is-not-five-colorable-September-23-2026/build/sections/transfer.tex @@ -0,0 +1,316 @@ +\section{Transfer between proper and weak measurable colorings} +\label{sec:transfer} + +We prove Theorem~\ref{thm:transfer} for every positive integer $k$. +Starting from an arbitrary proper coloring, we first average its +translations and rotations on the algebraic plane. The resulting +probability space carries label indicators, but its translations need +not be strongly continuous in the usual plane topology. We project those +indicators onto the functions whose translation orbits are continuous in $L^2$. +The projection must preserve nonnegativity and the sum of the label +coordinates; the next lemma establishes this by identifying it as +conditional expectation. Haar rigidity then preserves the zero unit +correlations, and one measurable sample yields a weak coloring. +The converse uses density points and finite graph compactness. + +\subsection{The continuous part of a stationary labeling system} + +Retain the groups \(E,K,D\), the Borel subgroup \(C\subset D\), and +the parametrization \(j:\R^2\to C\) from +Definition~\ref{def:spectral-groups}. In the following lemma, the topology +on \(E\) used to define continuity is its usual subspace topology in +\(\R^2\), although its measure-preserving action is initially only an +action of the countable discrete group. + +\begin{lemma}[Continuous vectors and positivity] +\label{lem:palettes-continuous-factor} +Let a standard probability space carry a measure-preserving action of +the additive group \(E\), with Koopman operators \(T_z\). The subspace +\[ + \mathcal H_c + =\left\{v\in L^2: + \lim_{\substack{z\to0\\z\in E}}\norm{T_zv-v}_2=0\right\} +\] +is \(L^2\) of a factor. It is also the spectral subspace on \(C\) +for the representation of \(E_{\rm disc}\). +The operators on this subspace extend to a strongly continuous +unitary representation \(T_x\), \(x\in\R^2\), which preserves the positive +cone and the constant one. + +If the system also has rotations \(V_u\), \(u\in K\), satisfying +\(V_uT_zV_u^{-1}=T_{uz}\), then \(\mathcal H_c\) and its orthogonal +projection are rotation invariant. +\end{lemma} + +\begin{proof} +The subspace is closed. Indeed, if \(v\) is close in \(L^2\) to a +continuous vector \(w\), then +\[ + \norm{T_zv-v}_2 + \le 2\norm{v-w}_2+\norm{T_zw-w}_2. +\] +It contains the constants and is invariant under the translations. + +Apply the spectral theorem for the discrete abelian group \(E\), with +dual group \(D\). For \(v\in L^2\), write \(\lambda_v\) for its finite +scalar spectral measure, so +\[ + \langle v,T_zv\rangle + =\int_D\psi_z(d)\,d\lambda_v(d),\qquad + \norm{T_zv-v}_2^2 + =\int_D\abs{\psi_z(d)-1}^2\,d\lambda_v(d). +\] +If \(\lambda_v(C^c)=0\), the second integral tends to +zero as \(z\to0\) in the usual topology, by dominated convergence and +the continuous-character parametrization. + +Conversely, suppose \(v\in\mathcal H_c\). Its coefficient +\(F_v(z)=\langle v,T_zv\rangle\) is uniformly continuous on \(E\): +\[ + \abs{F_v(z+h)-F_v(z)} + \le \norm{v}_2\,\norm{T_hv-v}_2 . +\] +It extends to a continuous positive-definite function on \(\R^2\). +Positive definiteness follows by approximating each finite list of +real-plane points by points of \(E\) and passing to the limit in its +defining quadratic form. Bochner's theorem supplies a finite measure +on \(\R^2\) representing this extension. Its pushforward under \(j\) +has the same Fourier coefficients on \(D\) as \(\lambda_v\), so Fourier +uniqueness identifies the two measures. Thus \(\lambda_v(C^c)=0\). +The representation result is the abelian spectral theorem +(also called the Stone--Naimark--Ambrose--Godement theorem); it and +Bochner's theorem are given in +\citet[Theorems~D.3.1 and~D.2.2]{Bekka2008}. The spectral theorem +is applied to the discrete locally compact group $E$, while +Bochner's theorem is applied to $\R^2$. This identifies +\(\mathcal H_c\) with the range of the spectral projection on the Borel +set \(C\). + +To obtain the factor assertion, it is essential that the original +\(T_z\) are Koopman operators. They commute with complex conjugation +and with pointwise composition. For bounded \(v,w\in\mathcal H_c\), +\[ + \norm{T_z(vw)-vw}_2 + \le \norm{v}_\infty\norm{T_zw-w}_2 + +\norm{w}_\infty\norm{T_zv-v}_2, +\] +so \(vw\in\mathcal H_c\). For a Lipschitz function +\(\Phi:\C\to\C\), +\[ + \norm{T_z\Phi(v)-\Phi(v)}_2 + \le {\rm Lip}(\Phi)\norm{T_zv-v}_2 . +\] +In particular, radial projection of \(v\) onto a bounded complex disk +is still in \(\mathcal H_c\). Such projections converge to \(v\) +in \(L^2\) as the disk radius tends to infinity. + +Let \(\mathcal F_c\) be the completed factor generated by the bounded +members of \(\mathcal H_c\). Those members form a bounded unital +star-algebra. Its \(L^2\) closure is \(L^2(\mathcal F_c)\): +for a finite list of its bounded generators, polynomials in their +coordinates and conjugates approximate continuous functions on the +compact range by Stone--Weierstrass; continuous functions are dense +in \(L^2\) of the corresponding finite-dimensional distribution. +Finite-coordinate measurable functions are dense in the generated +factor, by the monotone-class theorem. The closedness already proved +therefore gives \(L^2(\mathcal F_c)\subset\mathcal H_c\). +Conversely, the bounded truncations of every \(v\in\mathcal H_c\) +are \(\mathcal F_c\)-measurable and converge to \(v\) in \(L^2\). +Hence \(\mathcal H_c=L^2(\mathcal F_c)\). Its orthogonal projection +is conditional expectation. + +For \(x\in\R^2\), choose \(z_n\in E\) with \(z_n\to x\). +For \(v\in\mathcal H_c\), +\[ + \norm{T_{z_n}v-T_{z_m}v}_2 + =\norm{T_{z_n-z_m}v-v}_2\longrightarrow0 . +\] +Define \(T_xv\) by this limit. The definition is independent of the +approximating sequence, and strong limits together with the original +group law give the group law on \(\R^2\), inverses, and strong +continuity. The operators are unitary. The positive cone in \(L^2\) +is closed, so \(T_x\) preserves positivity, and \(T_x1=1\). +No pointwise action of \(\R^2\) is needed for these conclusions. + +Finally, for \(u\in K\), covariance gives +\[ + \norm{T_zV_uv-V_uv}_2 + =\norm{T_{u^{-1}z}v-v}_2\longrightarrow0 . +\] +Thus rotations preserve \(\mathcal H_c\); their inverses do as well, +so its orthogonal projection commutes with them. +\end{proof} + +\begin{proposition}[From proper to weak measurable colorings] +\label{prop:palettes-forward} +Let \(k\) be a positive integer. If the plane admits an unrestricted +proper \(k\)-coloring, then it admits a weak measurable \(k\)-coloring. +\end{proposition} + +\begin{proof} +\emph{An invariant law on proper labelings.} +Restrict the given coloring to \(E\). In the compact metrizable space +\(\{1,\ldots,k\}^E\), let \(\Omega\) be the closed set of proper +labelings of the unit-distance graph on \(E\). It is nonempty. +The countable group \(E\rtimes K\) acts continuously on \(\Omega\); +translations and rotations preserve the unit-edge constraints. +This group is amenable, being an extension of two abelian groups +\citep[Theorem~G.2.1 and Proposition~G.2.2(ii)]{Bekka2008}. +The F{\o}lner criterion \citep[Theorem~G.5.1]{Bekka2008}, +applied to successive finite subsets of this countable group, gives +a finite-set F{\o}lner sequence. +Averages of a point mass over a finite-set F{\o}lner sequence have a +weakly convergent subsequence, and every limit is invariant. This +gives an invariant Borel probability \(\mu\) on \(\Omega\). + +Let \(f_i\) be the indicator that the origin has label \(i\). +Choose the translation convention so \(T_zf_i\) is the indicator +of label \(i\) at \(z\). The rotations about the origin fix each +\(f_i\), and properness gives +\begin{equation}\label{eq:palettes-proper-correlation} + \int_\Omega f_i\,T_uf_i\,d\mu=0 + \qquad (u\in K,\ 1\le i\le k). +\end{equation} +\smallskip +\noindent\emph{The continuous factor and zero correlations.} +Use Lemma~\ref{lem:palettes-continuous-factor} to write +\[ + p_i=\mathbb E(f_i\mid\mathcal F_c),\qquad b_i=f_i-p_i. +\] +Then \(0\le p_i\le1\) and \(\sum_i p_i=1\). +Both \(p_i\) and \(b_i\) are rotation-fixed. The scalar translation +spectral measure of \(b_i\) is \(K\)-invariant: covariance gives +equality of its Fourier coefficients at \(z\) and \(uz\), and +Fourier uniqueness gives invariance of the measure. It gives \(C\) measure zero, by the spectral-subspace assertion of the lemma. +If it is nonzero, normalize it and apply Theorem~\ref{thm:rigidity}. +The scalar spectral measure is therefore a multiple of Haar measure, +including the zero multiple. It follows that +\[ + \langle b_i,T_zb_i\rangle=0\qquad(z\in E\setminus\{0\}). +\] +The subspace \(\mathcal H_c\) and its orthogonal complement are +translation invariant, so both cross terms in +\eqref{eq:palettes-proper-correlation} vanish. Consequently +\[ + \int_\Omega p_i\,T_up_i\,d\mu=0\qquad(u\in K). +\] +The algebraic unit directions are dense in the usual unit circle, +as is seen already from the rational tangent-half-angle parametrization. +Strong continuity of the extended operators therefore gives this +identity for every usual unit direction \(u\in S^1\). +For every \(x\in\R^2\), unitarity and the group law then give +\begin{equation}\label{eq:palettes-field-correlation} + \int_\Omega (T_xp_i)(T_{x+u}p_i)\,d\mu + =\langle T_xp_i,T_xT_up_i\rangle + =\langle p_i,T_up_i\rangle=0 . +\end{equation} +The functions here are nonnegative equivalence classes. +This calculation uses no pointwise realization or multiplicativity +of the extended \(\R^2\) action. + +\smallskip +\noindent\emph{A measurable sample.} +The continuous map +\(x\mapsto T_xp_i\) takes values in the separable space \(L^2(\mu)\). +On a bounded spatial domain, approximate it by finite-valued simple +maps in the \(L^2(\mu)\) norm, choosing the integrals of their squared +approximation errors to be summable. Choose measurable representatives +for their finitely many values. The resulting jointly measurable +functions converge in the product \(L^2\) space. A subsequence converges +in \(L^2(\mu)\) for almost every \(x\), so its product-space limit represents +\(T_xp_i\) at almost every \(x\). Applying this on a countable +exhaustion, or on its disjoint measurable annuli, gives a jointly +measurable field \(H_i(x,\omega)\) on \(\R^2\times\Omega\), correct +as an \(L^2(\mu)\) representative for almost every \(x\). +Since the probability model is standard, Borel versions may be used. +Taking real parts and clamping each \(H_i\) to \([0,1]\) preserves this representation +property and makes all the fields everywhere nonnegative. + +There is a plane-null set \(N\) outside which all these representations +are correct. For almost every \(x\), their sum is one for +\(\mu\)-almost every \(\omega\). Moreover, on every bounded +spatial domain, +\[ + \int\!\int_{S^1} + \bigl(\ind_N(x)+\ind_N(x+u)\bigr)\,d\sigma(u)\,dx=0. +\] +The assertion about \(x+u\) follows by translation invariance of +Lebesgue measure for each fixed \(u\), followed by Fubini. +Thus \eqref{eq:palettes-field-correlation} applies to the chosen +representatives for almost every pair \((x,u)\). +Tonelli gives, for every positive integer \(m\), +\[ + \int_\Omega\sum_{i=1}^k + \int_{B(0,m)}\int_{S^1} + H_i(x,\omega)H_i(x+u,\omega)\,d\sigma(u)\,dx\,d\mu(\omega)=0. +\] +Choose one sample \(\omega\) for which all the inner integrals vanish, +and for which \(\sum_iH_i(x,\omega)=1\) for almost every \(x\). +There are only countably many conditions here. At every point +where the sum is one, assign the least index with positive field; +on the remaining plane-null set assign any index. This is measurable. +A same-color pair outside that null set would have positive product +of the corresponding fields, which occurs only on a +\(dx\,d\sigma(u)\)-null set. Hitting the spatial exceptional set at +either endpoint is also a product-null event. This proves the +weak measurable condition. +\end{proof} + +\subsection{Density points and the converse transfer} + +The next argument is a version of the density-point method used by +\citet{Falconer1981} and \citet[Lemma~1 and Proposition~1]{Payne2009}. +Here the hypothesis excludes unit pairs only in product measure, so we +first show directly that density-one points still form independent sets. + +\begin{lemma}[Density-point properness]\label{lem:palettes-density-proper} +In a weak measurable \(k\)-coloring with classes \(A_1,\ldots,A_k\), +the full-density-point set of each \(A_i\) contains no unit pair. +These density-point sets are disjoint and their union is conull. +\end{lemma} + +\begin{proof} +Suppose \(x\) and \(y=x+u\), with \(\abs{u}=1\), both have density one +of \(A_i\). For \(h>0\) small, choose \(x'\) uniformly in \(B(x,h)\) +and independently choose \(u'\) uniformly on an arc of positive +angular measure satisfying \(\abs{u'-u}0. + \label{eq:source-14} +\end{equation} +\end{definition} + +For each fixed pair $ij$, its edge-presence locus is closed. Indeed, a +ball centered at a limit of points of that locus contains a smaller +ball centered at one of those points, and $D_{ij}$ is monotone in its +ball argument. Moreover, +\[ + ij\in\Gamma_x\quad\Longrightarrow\quad x\in H_i\cap H_j. +\] +To check this in any neighborhood of $x$, use a smaller ball and then +choose $h$ small enough that both endpoints of every transition under +consideration remain in the neighborhood. A positive-area transition +set gives positive area of each endpoint color there; translation +preserves area. + +\begin{lemma}[An edge has an avoiding trace] +\label{lem:transitions-avoiding-trace} +If $ij\in\Gamma_x$, there is a trace $b^{ij}\in\mathcal W_x$ with +$b^{ij}_i=b^{ij}_j=0$ almost everywhere. Consequently, +\begin{equation} + P_x(e)\not\subset\{i,j\} + \quad\text{for almost every }e\in S, + \qquad ij\in\Gamma_x. + \label{eq:source-15} +\end{equation} +\end{lemma} + +\begin{proof} +Write +\[ + \mathcal Sg(q)=\int_S g(q+e)\,d\sigma(e). +\] +The estimate we need, for every fixed bounded ball $B$ and every label +$s$, is +\begin{equation} + \bigl\|\mathcal S\bigl(a_s-a_s(\,\cdot+h)\bigr) + \bigr\|_{L^2(B)}=o\bigl(\sqrt{|h|}\bigr) + \qquad(h\to0). + \label{eq:source-16} +\end{equation} +Here and in \eqref{eq:source-14}, the vector $h$ may approach zero in any +direction. + +Choose a boundedly supported $g\in L^2(\R^2)$ agreeing with $a_s$ on a +ball containing $B+S+h$ for all sufficiently small $h$, as well as +$B+S$. The Fourier transform of normalized circle measure satisfies +\[ + |\widehat\sigma(\xi)|\le C(1+|\xi|)^{-1/2}. +\] +For completeness, after rotating coordinates the relevant integral is +a constant multiple of $\int_0^{2\pi}\exp(ir\cos t)\,dt$, where +$r$ is proportional to $|\xi|$. For $r\ge1$, removing the arcs on +which $|\sin t|\le r^{-1/2}$ costs $O(r^{-1/2})$. On each remaining +interval, integration by parts using the derivative $-ir\sin t$ gives +endpoint terms $O(r^{-1/2})$, while the integral of +$|\cos t|/(r\sin^2t)$ is also $O(r^{-1/2})$. The bound for $r\le1$ +is immediate. + +By Plancherel, the square of the global $L^2$ norm of +$\mathcal S(g-g(\,\cdot+h))$, divided by $|h|$, is bounded by the +integral of $|\widehat g(\xi)|^2$ times +\[ + \frac{C}{1+|\xi|} + \frac{\min\{C|h|^2|\xi|^2,4\}}{|h|}. +\] +This multiplier is uniformly bounded: split at $|\xi|=|h|^{-1}$. +For each fixed $\xi$ it tends to zero as $h\to0$. Dominated convergence +therefore proves \eqref{eq:source-16}; the cutoff is fixed before $h$ +tends to zero. + +For a fixed displacement $h$, almost every root $q$ has both $q,q+h$ +typical, and both exact unit-circle samplings centered at these points +typical on their circles. At a transition from color $s$ to color $t$, +the first circle avoids $s$ and the second avoids $t$, by properness of +the coloring on typical points. Thus the mass of color $t$ on the +first circle equals +$\mathcal S(a_t-a_t(\,\cdot+h))(q)$. For a transition between $i$ and +$j$ in either order, the first-circle mass of these two colors together +is consequently bounded by +\[ + \bigl|\mathcal S(a_i-a_i(\,\cdot+h))(q)\bigr| + +\bigl|\mathcal S(a_j-a_j(\,\cdot+h))(q)\bigr|. +\] + +Fix a ball $B$ about $x$. Since $ij\in\Gamma_x$, there is a constant +$c_B>0$ and arbitrarily small nonzero $h$ for which the transition set +in $B$ has area at least $c_B|h|$. Cauchy--Schwarz and +\eqref{eq:source-16} show that the average of the preceding mass over +this transition set is at most +\[ + \frac{o(\sqrt{|h|})}{\sqrt{c_B|h|}}=o(1). +\] +Now take balls $B_m$ of radii tending to zero about $x$. After fixing +$B_m$ and its constant $c_{B_m}$, choose $h_m$ sufficiently small and +then a transition root $q_m\in B_m$ from the preceding full-measure set, +whose first-circle mass of $i,j$ is less than $1/m$. This order of choice requires no lower bound +on $c_{B_m}$ uniform in $m$. The circle samples at $q_m$ have radius +one; a weak-* subsequential limit belongs to $\mathcal W_x$ and has +zero mass in the two specified coordinates. Since it is nonnegative, +sums to one, and is supported within $P_x$ almost everywhere, it also +proves \eqref{eq:source-15}. +\end{proof} + +\subsection{Two angular state lemmas} + +A \emph{state} at a center $x$ will mean a measurable angular set +together with a specified nonempty group of labels contained in +$P_x(e)$ almost everywhere on that set. For a measurable set +$A\subset S$, its closed essential support is +\[ + \operatorname{ess\,supp}_S A + =\{a\in S:\sigma(A\cap I)>0\text{ for every open arc }I\ni a\}. +\] +It is empty exactly when $A$ is null. Almost every point of $A$ is +both a density-one direction for $A$ and a member of this support. +Changing the representative of a state on a null set changes neither +its support nor any assertion below. + +\begin{lemma}[Strict state separation] +\label{lem:transitions-strict-separation} +Suppose $A,B\subset S$ are states at $x$ representing disjoint label +groups whose union is $\mathcal I\setminus\{X\}$, for a fixed label +$X$. At any nonparallel density-one directions $a$ of $A$ and $b$ of +$B$, the point $x+a+b$ has spatial density one of $X$. + +If $\sigma(A)>0$, the closed essential support of $B$ contains no +pair at chordal distance one. In particular, it has a strict angular +separation from its image under $R$ and under $R^{-1}$, with the +empty-support case interpreted vacuously. + +The separation is uniform along a sequence of centers and states +$(x_n,A_n,B_n)$ with the same represented groups, provided +$\ind_{A_n}\to\ind_A$ in $L^1(S)$ for a set $A$ of positive measure. +More precisely, for all sufficiently large $n$, angular differences +between two points of $\operatorname{ess\,supp}_S B_n$ stay a fixed +positive distance from both $\kappa$ and $-\kappa$ modulo $2\pi$. +\end{lemma} + +\begin{proof} +The density conclusion of Lemma~\ref{lem:palette-transfer}, following +\eqref{eq:source-10}, gives density zero of each of the four represented +labels at the nonparallel sum. Their complement, the singleton $X$, +therefore has density one there. + +We prove the uniform assertion by contradiction; the fixed-center +case follows by using a constant sequence. If it fails, choose pairs +in the supports of $B_n$ with chordal distances tending to one. Each +support point is approximable by density-one directions of $B_n$. +We may thus choose density directions $b_n,d_n$ in those states such +that, after a subsequence, +\[ + b_n\to b,\qquad d_n\to d,\qquad + w_n=b_n-d_n\to w=b-d,\qquad |w|=1. +\] +Choose a small angular interval $I$ with $\sigma(A\cap I)>0$, whose +closure is separated from the directions parallel to any of $w,b,d$. +Such an interval exists because only finitely many directions are +excluded and $A$ has positive measure. + +For $a\in I$, the implicit function theorem gives a unit direction +$c_n(a)$, smoothly depending on $a$, with +\[ + |w_n+a-c_n(a)|=1, + \qquad c_n\longrightarrow\id\quad\text{in }C^1(I). +\] +Indeed, at $w_n=w$ and $c_n(a)=a$ the squared-length equation is +satisfied and its derivative in the angle of $c_n(a)$ is +$-2w\cdot Ja\ne0$. Shrinking $I$ around a density point of $A$ if +necessary makes this construction uniform on its closure. + +The maps $c_n$ have uniformly bounded inverse derivatives. It follows +that $\ind_{A_n}\circ c_n\to\ind_A$ in $L^1(I)$: the difference +between $\ind_{A_n}\circ c_n$ and $\ind_A\circ c_n$ is controlled by +the $L^1$ convergence under change of variables; continuous +approximation of $\ind_A$ and $c_n\to\id$ treat the remaining +difference. Together with $\ind_{A_n}\to\ind_A$ on $I$, this shows +that the set of $a\in I$ for which both $a$ and $c_n(a)$ lie in $A_n$ +has positive measure for large $n$. Excluding null sets, choose such +an $a$ for which both are density-one directions. The choice of $I$ +ensures that $(a,b_n)$ and $(c_n(a),d_n)$ are nonparallel. The points +\[ + x_n+a+b_n,\qquad x_n+c_n(a)+d_n +\] +both have density one of color $X$, yet their distance is +$|w_n+a-c_n(a)|=1$. This contradicts properness of density-one points. +At a single center, compactness of the support turns exclusion of +distance-one pairs into the asserted strict margin. +\end{proof} + +\begin{lemma}[Singleton--pair alternation is impossible] +\label{lem:transitions-two-state} +At a fixed center, there is no measurable state function with only the +two values $X,D$, representing respectively a singleton $\{X\}$ and +a fixed disjoint pair $D$, and changing state under $R$ almost +everywhere. +\end{lemma} + +\begin{proof} +Put +\[ + \gamma=2\arcsin\frac1{2\sqrt3}, + \qquad \cos\gamma=\frac56. +\] +The ratio $\gamma/\pi$ is irrational. Otherwise $e^{i\gamma}$ would +be a root of unity, making $2\cos\gamma=5/3$ a rational algebraic +integer, hence an integer, a contradiction. + +There is a positive-measure set of angles $t$ whose states at $t$ and +$t+\gamma$ are equal. If not, the state would flip almost everywhere +under $\gamma$, and hence be invariant under $2\gamma$. A measurable +function invariant under an irrational circle rotation is constant +almost everywhere: its nonconstant Fourier coefficients vanish, since +the corresponding rotation eigenvalues differ from one. This would +contradict the flip under $\kappa$. + +Choose $t$ from the positive-measure equality set, also imposing all +finitely many needed $R$-alternation identities and density-direction +conditions. For $j=0,1,2$, form the points +\[ + z_j=x+e^{i(t+2j\kappa)}+e^{i(t+\pi+\gamma+2j\kappa)} + =x+e^{i(t+2j\kappa)}(1-e^{i\gamma}). +\] +They have distance $|1-e^{i\gamma}|=1/\sqrt3$ from $x$ and form an +equilateral triangle of side one. In each sum the two nonparallel +endpoints have opposite states: $2\kappa$ preserves the state and +$\pi=3\kappa$ flips it. The density consequence of +Lemma~\ref{lem:palette-transfer} therefore gives density one at each +$z_j$ to the union of the two labels outside $\{X\}\cup D$. +A uniform common translation in a sufficiently small disk puts all +three vertices in that union and in the typical set with positive +probability; the three failure probabilities tend to zero by density +one. The resulting unit equilateral triangle would be properly +colored with two labels, a contradiction. +\end{proof} + +\subsection{The cyclic-center locus} + +\begin{theorem}[Local finiteness of cyclic centers] +\label{thm:cyclic-local-finiteness} +The set +\begin{equation} + \mathcal L=\{x\in\R^2:\Gamma_x\text{ contains a cycle}\} + \label{eq:source-19} +\end{equation} +is closed and locally finite: it has finite intersection with every +compact subset of the plane. +\end{theorem} + +For any specified simple label cycle, its locus is the intersection +of its finitely many closed edge-presence loci, hence is closed. +There are only finitely many simple cycles on five labels. It +therefore suffices to prove local finiteness separately for each +triangle, four-cycle, and five-cycle. + +For a triangle, \eqref{eq:source-15} for its three edges and +Proposition~\ref{prop:palette-size} force $P_x(e)$ to intersect the +pair of outside labels almost everywhere. Indeed, a nonempty subset +of the three triangle labels of size at most two lies in some +triangle edge, which is forbidden. By \eqref{eq:source-12} the +intersection with the outside pair is unique almost everywhere and +alternates under $R$. Thus every such $x$ is binary for that fixed +pair. Lemma~\ref{lem:binary-separation} proves local finiteness of +the triangle locus. + +\subsubsection{Four-cycles: continuity of states} + +Fix a four-cycle and relabel its vertices $0,1,2,3$ in cyclic order, +with fifth label $X$. Let $L$ be its closed locus and write +\[ + U=\{0,2\},\qquad V=\{1,3\}. +\] +For $x\in L$ define the canonical state $w_x(e)$ as follows: it is +$X$ if $X\in P_x(e)$, and otherwise it is $U$ or $V$ according to +which of these two pairs equals $P_x(e)$. This is exhaustive almost +everywhere. A palette not containing $X$ has size at most two, and +\eqref{eq:source-15} excludes every singleton and every adjacent pair +of the four-cycle, leaving precisely the two diagonals. The states +represent the groups $\{X\},U,V$ contained in their palettes, and +\eqref{eq:source-12} gives +\[ + w_x(e)\ne w_x(Re)\quad\text{almost everywhere}. +\] +Both diagonal states have positive angular measure. If either were +null, the other and $X$ would give the impossible alternation of +Lemma~\ref{lem:transitions-two-state}. + +\begin{lemma}[Continuity of four-cycle states] +\label{lem:transitions-four-continuity} +For each $s\in\{X,U,V\}$, the map +$x\mapsto\ind_{\{w_x=s\}}$ is continuous from $L$ into $L^1(S)$. +\end{lemma} + +\begin{proof} +At each $y\in L$ choose an avoiding trace $b_y^{ij}$ for each of its +four specified edges. If $k$ belongs to the group represented by a +state $s$, then +\begin{equation} + \ind_{\{w_y=s\}} + \le\sum_{ij\text{ in the four-cycle}} b^{ij}_{y,k} + \quad\text{almost everywhere}. + \label{eq:source-20} +\end{equation} +To see this, fix a generic direction in state $s$. If the palette +has another member besides $k$, some cycle edge contains that member +but does not contain $k$. For a diagonal state, choose an edge +incident to its other member; for $k=X$, choose an edge incident to +the possible additional label. If no other member exists, any +cycle edge suffices. The corresponding trace avoids its edge +endpoints and is supported within the palette, so its $k$-coordinate +equals one. There are only finitely many chosen traces, allowing one +common conull set of directions for this argument. + +Let $y_n\to x$ in $L$. Along any subsequence, pass further to weak-* +limits of the three state indicators and all four chosen traces. +Denote the former limits by $f_X,f_U,f_V$. The trace limits belong +to $\mathcal W_x$ by its closed-graph property. Inequality +\eqref{eq:source-20} passes to the weak-* limits, and its nonnegative +right-hand side shows that wherever $f_s>0$, every label represented +by $s$ belongs to $P_x$, almost everywhere. Two different state +groups would require at least three palette labels, contradicting +Proposition~\ref{prop:palette-size}. Since the $f_s$ are nonnegative +and sum to one, exactly one equals one at almost every direction. +Its group determines precisely the canonical state $w_x$: a diagonal +already fills both palette slots, while a positive singleton state +requires $X\in P_x$. Thus $f_s=\ind_{\{w_x=s\}}$ almost everywhere. + +Weak-* convergence of indicators to an indicator implies $L^1$ +convergence: for indicators $f_n,f$, test against $1$ and $f$ in +$\int|f_n-f|=\int f_n+\int f-2\int f_nf$. Every subsequence has a +further subsequence with this same $L^1$ limit, which proves the +claimed continuity. +\end{proof} + +Let $S_{U,x},S_{V,x}$ be the closed essential supports of the two +diagonal states, and put $N_x=S_{U,x}\cup S_{V,x}$. Both supports +are nonempty. Across their nonparallel density directions, the +four represented labels force singleton density one of $X$. +Lemma~\ref{lem:transitions-strict-separation} therefore gives a strict +margin between each support and its own $R$-shift. These margins +are uniform along any convergent sequence in $L$, by +Lemma~\ref{lem:transitions-four-continuity} and positivity of both +limiting diagonal states. We also have the following consequence +of that continuity: +\[ + b\in S_{s,x},\quad y_n\to x\text{ in }L + \quad\Longrightarrow\quad + \text{there are }b_n\in S_{s,y_n}\text{ with }b_n\to b + \quad(s=U,V). +\] +Indeed, every neighborhood of $b$ has positive limiting state +measure, hence positive state measure at all sufficiently large +$n$. Successively smaller neighborhoods give the sequence. + +The set $N_x$ is a proper subset of $S$. Otherwise a point in the +intersection of the two supports would rotate under $R$ outside +both, by their strict separations. The two nonempty supports would +therefore have to be disjoint and cover the connected circle, which +is impossible. On every component gap of $S\setminus N_x$ the +state is $X$ almost everywhere. Such a gap has angular length at +most $\kappa$, since an interval of greater length overlaps its +$R$-translate in an open interval, contrary to the state-change +rule. + +\subsubsection{Four-cycles: transport and the accumulation contradiction} + +For nonparallel $a,b\in S$ and $h$ sufficiently small, the local +inverse of the sum map defines +\[ + T_h(a,b)=(c,d),\qquad c+d=a+b-h, + \qquad (c,d)\text{ near }(a,b). +\] +It is smooth in the two angular variables and in $h$, and for fixed +$h$ it is a local diffeomorphism of the angular pair. Its branch at +$h=0$ is the identity. + +We will use the following support transport rule. If $y=x+h\in L$ +and $a\in S_{U,x}$, $b\in S_{V,x}$ are nonparallel, then +\[ + T_h(a,b)\in N_y\times N_y +\] +whenever the indicated nearby branch is defined. In arbitrarily +small product neighborhoods of $(a,b)$ there is a positive-measure +set of pairs of density directions for $U,V$ at $x$. The spatial +sums of every such nonparallel pair have density one of $X$. +The local diffeomorphism $T_h$ permits removal of all null sets +needed to make both output directions density directions of their +states at $y$ and members of the corresponding essential supports. +Neither output state can be $X$: the density consequence of +Lemma~\ref{lem:palette-transfer} at $y$ would then give density zero +of $X$ at the same spatial sum. Both outputs consequently belong +to $N_y$. Approximating the initial support pair by these generic +pairs and using closedness proves the rule. It also holds with +$U,V$ interchanged, and in the reverse direction under $T_{-h}$. +All null sets here concern the two fixed centers in a single +application; no joint measurability in the center is used. + +Suppose now that distinct points $y_n=x+h_n$ of $L$ tend to $x$. +After a subsequence, write +\[ + \delta_n=|h_n|>0,\qquad H_n=h_n/\delta_n\longrightarrow H\in S. +\] +Take any endpoint $a$ of a gap of $N_x$, first with the gap just to +its right in real angular coordinates. It belongs to at least one +diagonal support, say $S_{U,x}$; the other case is symmetric. +Assume $H\cdot a\ne0$. + +The first output angle of $T_h(a,b)$ has directional velocity +\[ + V_H(a;b)=-\frac{H\cdot b}{a\times b} + =-H\cdot Ja-(H\cdot a)\cot(\arg b-\arg a), + \qquad a\times b=\sin(\arg b-\arg a). +\] +Indeed, differentiating $c+d=a+b-h$ at $h=0$ in direction $H$ gives +$Ja\,\dot\theta+Jb\,\dot\phi=-H$; take the dot product with $b$. +Since $H\cdot a\ne0$, this velocity takes different values at +projectively different nonparallel directions $b$. The positive +measure of the $V$ state allows choices $b,b'\in S_{V,x}$, neither +parallel to $a$, with $V_H(a;b)>V_H(a;b')$. + +Set $c_n=T_{h_n}^{1}(a,b)$. By support transport, $c_n\in N_{y_n}$, +and smoothness gives +\[ + \arg c_n=\arg a+\delta_n V_H(a;b)+o(\delta_n). +\] +If $c_n\in S_{U,y_n}$ along an infinite subsequence, choose +$b'_n\in S_{V,y_n}$ with $b'_n\to b'$. Transporting +$(c_n,b'_n)$ back by $T_{-h_n}$ gives a first endpoint in $N_x$ +whose angle is +\[ + \arg a+\delta_n\bigl(V_H(a;b)-V_H(a;b')\bigr)+o(\delta_n). +\] +This lies strictly inside the gap for large $n$, a contradiction. +To justify the error term without any convergence rate for $b'_n$, +let $\tau(h,\theta,\beta)$ be a real lift of the first output angle +on a fixed branch near $(0,\arg a,\arg b')$. The identity +$\tau(0,\theta,\beta)=\theta$ gives, uniformly on a smaller +neighborhood, +\[ + \tau(h,\theta,\beta) + =\theta+D_h\tau(0,\theta,\beta)h+O(|h|^2). +\] +Thus the change in the auxiliary angle $\beta$ affects only the +coefficient multiplied by $h$. Continuity of that coefficient, +$c_n\to a$, $b'_n\to b'$, and $H_n\to H$ give the claimed +$o(\delta_n)$ remainder. + +It follows that $c_n\in S_{V,y_n}$ for all large $n$. The strict +separation for $U$ leaves a neighborhood of $Ra$ free of $S_{U,x}$. +Immediately to the right of $a$ the state is $X$ almost everywhere, +so immediately to the right of $Ra$ it is non-$X$ almost everywhere. +It must therefore be $V$ on a one-sided neighborhood of $Ra$, giving +$Ra\in S_{V,x}$. There are points of $S_{V,y_n}$ tending to $Ra$. +Together with $c_n\to a$, these violate the uniform strict separation +of $S_{V,y_n}$ from its own $R$-shift. + +For a gap lying to the left of $a$, choose +$V_H(a;b)=Stealth, every node/.style={font=\small}] +\node[draw,circle,minimum size=9mm,fill=blue!8] (k) at (0,0) {$k$}; +\node[draw,rounded corners,align=center] (j) at (-4,1.2) {position $j$\\question $Q_{\{k,j\},j}$ fixed}; +\node[draw,rounded corners,align=center] (l) at (4,1.2) {position $j' $\\question $Q_{\{k,j'\},j'}$ fixed}; +\node[draw,rounded corners,align=center] (a) at (0,-1.8) {every other position $j''$\\question $Q_{\{k,j''\},j''}$ fixed}; +\draw[<->,thick] (k)--node[below,sloped,font=\scriptsize] {$c_{\{k,j\}}$} (j); +\draw[<->,thick] (k)--node[below,sloped,font=\scriptsize] {$c_{\{k,j'\}}$} (l); +\draw[<->,thick] (k)--(a); +\end{tikzpicture} +\caption{The star operation centered at $k$. Each incident seed is +resampled within the fiber of its question at the opposite endpoint. +All nonincident seeds and all weights stay fixed. Thus every query +except the query at $k$ stays fixed. The schematic depicts the +conditional averaging operator $D_k$, not additional graph edges.} +\label{fig:star} +\end{figure} diff --git a/preprints/The-Factor-Two-Hardness-Threshold-for-Vertex-Cover-September-23-2026/build/main.tex b/preprints/The-Factor-Two-Hardness-Threshold-for-Vertex-Cover-September-23-2026/build/main.tex new file mode 100644 index 0000000000000000000000000000000000000000..fd9b92e24c154b28865909dfacab5f028fba0941 --- /dev/null +++ b/preprints/The-Factor-Two-Hardness-Threshold-for-Vertex-Cover-September-23-2026/build/main.tex @@ -0,0 +1,82 @@ +\pdfoutput=1 +\pdfinfoomitdate=1 +\pdftrailerid{} +\pdfsuppressptexinfo=15 +\documentclass[11pt]{article} +\makeatletter +\def\input@path{{vendor/}} +\makeatother +\usepackage[utf8]{inputenc} +\usepackage[margin=1in]{geometry} +\usepackage{amsmath,amssymb,amsthm,mathtools} +\usepackage{booktabs,enumitem,microtype} +\usepackage{xcolor} +\usepackage{tikz} +\usetikzlibrary{arrows.meta,positioning} +\usepackage[colorlinks=true,linkcolor=blue!45!black,citecolor=blue!45!black,urlcolor=blue!45!black]{hyperref} +\usepackage[nameinlink,noabbrev]{cleveref} +\usepackage{bookmark} +\crefname{theorem}{Theorem}{Theorems} +\crefname{lemma}{Lemma}{Lemmas} +\crefname{proposition}{Proposition}{Propositions} +\crefname{corollary}{Corollary}{Corollaries} +\crefname{section}{Section}{Sections} +\crefname{equation}{Equation}{Equations} +\crefname{figure}{Figure}{Figures} +\hypersetup{pdftitle={The Factor-Two Hardness Threshold for Vertex Cover},pdfauthor={OpenAI},pdfsubject={A deterministic gap reduction from Label Cover}} +\numberwithin{equation}{section} +\newtheorem{theorem}{Theorem}[section] +\newtheorem{lemma}[theorem]{Lemma} +\newtheorem{proposition}[theorem]{Proposition} +\newtheorem{corollary}[theorem]{Corollary} +\theoremstyle{definition} +\newtheorem{definition}[theorem]{Definition} +\theoremstyle{remark} +\newtheorem{remark}[theorem]{Remark} +\newcommand{\E}{\mathbb E} +\DeclareMathOperator{\Var}{Var} +\DeclareMathOperator{\Ent}{H} +\DeclareMathOperator{\KL}{D} +\DeclareMathOperator{\TV}{TV} +\DeclareMathOperator{\val}{val} +\DeclareMathOperator{\supp}{supp} +\DeclareMathOperator{\conv}{conv} +\DeclareMathOperator{\dist}{dist} +\newcommand{\norm}[1]{\left\|#1\right\|_*} +\newcommand{\abs}[1]{\left|#1\right|} +\newcommand{\ceil}[1]{\left\lceil#1\right\rceil} +\newcommand{\floor}[1]{\left\lfloor#1\right\rfloor} +\newcommand{\ind}{\mathbf 1} +\newcommand{\R}{\mathbb R} +\newcommand{\Ftwo}{\mathbb F_2} +\setlist[enumerate]{itemsep=3pt,topsep=5pt} +\setlist[itemize]{itemsep=3pt,topsep=5pt} +\emergencystretch=1.2em +\clubpenalty=10000 +\widowpenalty=10000 +\displaywidowpenalty=10000 + +\title{The Factor-Two Hardness Threshold for Vertex Cover} +\author{OpenAI} +\date{September 23, 2026} + +\begin{document} +\maketitle +\begin{abstract} +We prove that minimum Vertex Cover is NP-hard to approximate within +every fixed factor below two, even on simple unweighted graphs. +\end{abstract} + +\input{sections/introduction} + +\input{sections/source} +\input{sections/graph} +\input{sections/analysis} +\input{sections/restriction} +\input{sections/lists} +\input{sections/decoding} +\input{sections/conclusion} + +\bibliographystyle{plain} +\bibliography{references} +\end{document} diff --git a/preprints/The-Factor-Two-Hardness-Threshold-for-Vertex-Cover-September-23-2026/build/references.bib b/preprints/The-Factor-Two-Hardness-Threshold-for-Vertex-Cover-September-23-2026/build/references.bib new file mode 100644 index 0000000000000000000000000000000000000000..a76c50e3627fb4ec4a129aa1d672ff48fb7b14e2 --- /dev/null +++ b/preprints/The-Factor-Two-Hardness-Threshold-for-Vertex-Cover-September-23-2026/build/references.bib @@ -0,0 +1,199 @@ +@article{Hastad2001, + author = {Johan H{\aa}stad}, + title = {Some Optimal Inapproximability Results}, + journal = {Journal of the ACM}, + volume = {48}, + number = {4}, + pages = {798--859}, + year = {2001}, + month = jul, + note = {\url{https://doi.org/10.1145/502090.502098}}, + doi = {10.1145/502090.502098}, + url = {https://johanhastad.se/optimalinap.pdf} +} + +@article{DinurSafra2005, + author = {Irit Dinur and Samuel Safra}, + title = {On the hardness of approximating minimum vertex cover}, + journal = {Annals of Mathematics}, + volume = {162}, + number = {1}, + pages = {439--485}, + year = {2005}, + note = {\url{https://doi.org/10.4007/annals.2005.162.439}}, + doi = {10.4007/annals.2005.162.439}, + url = {https://annals.math.princeton.edu/2005/162-1/p08} +} + +@article{KhotMinzerSafra2023, + author = {Subhash Khot and Dor Minzer and Muli Safra}, + title = {Pseudorandom sets in {Grassmann} graph have near-perfect expansion}, + journal = {Annals of Mathematics}, + volume = {198}, + number = {1}, + pages = {1--92}, + year = {2023}, + doi = {10.4007/annals.2023.198.1.1}, + note = {\url{https://doi.org/10.4007/annals.2023.198.1.1}. Preliminary version in FOCS 2018} +} + +@article{KhotRegev2008, + author = {Subhash Khot and Oded Regev}, + title = {Vertex cover might be hard to approximate to within {$2-\epsilon$}}, + journal = {Journal of Computer and System Sciences}, + volume = {74}, + number = {3}, + pages = {335--349}, + year = {2008}, + month = may, + note = {\url{https://doi.org/10.1016/j.jcss.2007.06.019}}, + doi = {10.1016/j.jcss.2007.06.019}, + url = {https://cims.nyu.edu/~regev/papers/vc_hard.pdf} +} + +@article{Raz1998, + author = {Ran Raz}, + title = {A Parallel Repetition Theorem}, + journal = {SIAM Journal on Computing}, + volume = {27}, + number = {3}, + pages = {763--803}, + year = {1998}, + note = {\url{https://doi.org/10.1137/S0097539795280895}}, + doi = {10.1137/S0097539795280895}, + url = {https://epubs.siam.org/doi/10.1137/S0097539795280895} +} + +@inproceedings{DinurSteurer2014, + author = {Irit Dinur and David Steurer}, + title = {Analytical Approach to Parallel Repetition}, + booktitle = {Proceedings of the Forty-Sixth Annual ACM Symposium on Theory of Computing}, + series = {STOC '14}, + pages = {624--633}, + publisher = {Association for Computing Machinery}, + year = {2014}, + note = {\url{https://doi.org/10.1145/2591796.2591884}. Full version: \url{https://arxiv.org/abs/1305.1979v3}, Theorem~6.2}, + doi = {10.1145/2591796.2591884}, + url = {https://arxiv.org/abs/1305.1979v3} +} + +@article{DKKMS2025, + author = {Irit Dinur and Subhash Khot and Guy Kindler and Dor Minzer and Muli Safra}, + title = {Towards a Proof of the {2-to-1} Games Conjecture?}, + journal = {Theory of Computing}, + volume = {21}, + number = {11}, + pages = {1--50}, + year = {2025}, + note = {\url{https://doi.org/10.4086/toc.2025.v021a011}. Preliminary version in STOC 2018}, + doi = {10.4086/toc.2025.v021a011}, + url = {https://theoryofcomputing.org/articles/v021a011/v021a011.pdf} +} + +@article{FGLSS96, + author = {Feige, Uriel and Goldwasser, Shafi and Lov{\'a}sz, L{\'a}szl{\'o} and Safra, Shmuel and Szegedy, Mario}, + title = {Interactive Proofs and the Hardness of Approximating Cliques}, + journal = {Journal of the ACM}, + volume = {43}, number = {2}, pages = {268--292}, year = {1996}, + doi = {10.1145/226643.226652}, + url = {https://www.cs.tau.ac.il/~safra/PapersAndTalks/FGLSS.pdf} +} + +@article{AroraSafra98, + author = {Arora, Sanjeev and Safra, Shmuel}, + title = {Probabilistic Checking of Proofs: A New Characterization of {NP}}, + journal = {Journal of the ACM}, + volume = {45}, number = {1}, pages = {70--122}, year = {1998}, + doi = {10.1145/273865.273901}, + url = {https://www.cs.umd.edu/~gasarch/TOPICS/pcp/AS.pdf}, + note = {Preliminary version in FOCS 1992} +} + +@article{ALMSS98, + author = {Arora, Sanjeev and Lund, Carsten and Motwani, Rajeev and Sudan, Madhu and Szegedy, Mario}, + title = {Proof Verification and the Hardness of Approximation Problems}, + journal = {Journal of the ACM}, + volume = {45}, number = {3}, pages = {501--555}, year = {1998}, + doi = {10.1145/278298.278306}, + url = {https://people.csail.mit.edu/madhu/papers/1992/almss-journ.pdf}, + note = {Preliminary version in FOCS 1992} +} + +@inproceedings{Khot02, + author = {Khot, Subhash}, + title = {On the Power of Unique 2-Prover 1-Round Games}, + booktitle = {Proceedings of the Thirty-Fourth Annual ACM Symposium on Theory of Computing}, + pages = {767--775}, publisher = {ACM}, year = {2002}, + doi = {10.1145/509907.510017} +} + +@article{EfronStein1981, + author = {Efron, Bradley and Stein, Charles}, + title = {The Jackknife Estimate of Variance}, + journal = {The Annals of Statistics}, + volume = {9}, number = {3}, pages = {586--596}, year = {1981}, + doi = {10.1214/aos/1176345462}, + note = {\url{https://doi.org/10.1214/aos/1176345462}} +} + +@article{Friedgut1998, + author = {Ehud Friedgut}, + title = {{Boolean} Functions With Low Average Sensitivity Depend On Few Coordinates}, + journal = {Combinatorica}, + volume = {18}, + number = {1}, + pages = {27--35}, + year = {1998}, + doi = {10.1007/PL00009809}, + note = {\url{https://doi.org/10.1007/PL00009809}} +} + +@incollection{Karp1972, + author = {Richard M. Karp}, + title = {Reducibility among Combinatorial Problems}, + booktitle = {Complexity of Computer Computations}, + editor = {Raymond E. Miller and James W. Thatcher}, + publisher = {Plenum Press}, + pages = {85--103}, + year = {1972}, + doi = {10.1007/978-1-4684-2001-2_9}, + note = {\url{https://doi.org/10.1007/978-1-4684-2001-2_9}} +} + +@article{Karakostas2009, + author = {George Karakostas}, + title = {A better approximation ratio for the {Vertex Cover} problem}, + journal = {ACM Transactions on Algorithms}, + volume = {5}, + number = {4}, + pages = {41:1--41:8}, + year = {2009}, + doi = {10.1145/1597036.1597045}, + note = {\url{https://doi.org/10.1145/1597036.1597045}. Preliminary version in ICALP 2005} +} + +@article{KhotMinzerSafra2025, + author = {Subhash Khot and Dor Minzer and Muli Safra}, + title = {On Independent Sets, {2-to-2} Games and {Grassmann} Graphs}, + journal = {Theory of Computing}, + volume = {21}, + number = {10}, + pages = {1--55}, + year = {2025}, + doi = {10.4086/toc.2025.v021a010}, + note = {\url{https://doi.org/10.4086/toc.2025.v021a010}. Preliminary version in STOC 2017} +} + +@inproceedings{BarakKothariSteurer2019, + author = {Boaz Barak and Pravesh K. Kothari and David Steurer}, + title = {Small-Set Expansion in {Shortcode} Graph and the {2-to-2} Conjecture}, + booktitle = {10th Innovations in Theoretical Computer Science Conference}, + series = {Leibniz International Proceedings in Informatics}, + volume = {124}, + editor = {Avrim Blum}, + pages = {9:1--9:12}, + publisher = {Schloss Dagstuhl--Leibniz-Zentrum f{\"u}r Informatik}, + year = {2019}, + doi = {10.4230/LIPIcs.ITCS.2019.9}, + note = {\url{https://doi.org/10.4230/LIPIcs.ITCS.2019.9}} +} diff --git a/preprints/The-Factor-Two-Hardness-Threshold-for-Vertex-Cover-September-23-2026/build/sections/analysis.tex b/preprints/The-Factor-Two-Hardness-Threshold-for-Vertex-Cover-September-23-2026/build/sections/analysis.tex new file mode 100644 index 0000000000000000000000000000000000000000..87a906d336c6ee7c49edd99adff2baf2ae25ed6c --- /dev/null +++ b/preprints/The-Factor-Two-Hardness-Threshold-for-Vertex-Cover-September-23-2026/build/sections/analysis.tex @@ -0,0 +1,128 @@ +\section{A fixed function from an independent set} +\label{ana:section} + +The soundness argument begins by turning a large independent set into +one fixed Lipschitz function. We need a variance lower bound on fresh +random inputs and control of how much the function changes when one +position changes. Set +\begin{equation}\label{src:variance-parameters} + b_0=\frac1{2m},\qquad + \nu=\frac{b_0a^2}{4}=\frac1{2^7m^3}. +\end{equation} +Thus the independent-set density we must rule out is \(2b_0\). +First discard the low-norm vertices. All vertices +whose vectors have norm at most \(t\) form a clique: for any two of +them the triangle inequality gives +\(\norm{S_v+S_w}\leq2t\). An independent set therefore contains at +most one such vertex. Since +\(n\geq p>1/b_0\), an independent set of size at least +\(n/m=2b_0n\) contains more than \(b_0n\) vertices of norm greater +than \(t\). The following lemma turns these remaining vectors into +the required function. + +\begin{lemma}[Separation, variance, and oscillation] +\label{ana:function} +Suppose an independent set contains at least \(b_0n\) vertices with +\(\norm{S_v}>t\). There is a fixed nonempty finite set +\(A_*\subset\R^{\mathcal P}\) such that +\[ + \Pr_{v\in V(G)}[S_v\in A_*]\geq b_0, + \qquad + \norm{x+y}>2t\quad(x,y\in A_*), +\] +where the vertex is uniform. For a nonempty finite set \(B\), write +\(\dist_*(z,B)=\min_{b\in B}\norm{z-b}\). The function +\begin{equation} + F(z)=\frac12\bigl(\dist_*(z,-A_*)-\dist_*(z,A_*)\bigr) + \label{ana:distance-function} +\end{equation} +is odd and \(1\)-Lipschitz for \(\norm{\cdot}\), and \(F(z)>t\) +on \(A_*\). + +Let \(\mathcal E=\binom{[d]}2\). Draw independent uniform occurrence +seeds \(\boldsymbol c=(c_e)_{e\in\mathcal E}\in[M]^{\mathcal E}\) +and independent weight blocks \(s_j\) uniform on \([-1,1]^r\), +independently of the seeds. Put +\begin{equation} + f(\boldsymbol c,\boldsymbol s) + =F\left(\sum_{j=1}^d z(i_j(\boldsymbol c),s_j)\right). + \label{ana:f} +\end{equation} +This bounded function satisfies +\begin{equation} + \E_{\boldsymbol s}f(\boldsymbol c,\boldsymbol s)=0 + \quad\hbox{for every }\boldsymbol c, + \qquad + \E f^2\geq\nu d. + \label{ana:variance} +\end{equation} +On the weight cubes it has the following three oscillation bounds: +\begin{enumerate}[label=\textup{(\roman*)}] + \item Replacing one weight block \(s_j\), with all other data fixed, + changes \(f\) by at most \(2\). + \item Replacing one seed \(c_e\), with all other data fixed, + changes \(f\) by at most \(4\). + \item Fix \(k\in[d]\). Simultaneously replacing any seeds incident + to \(k\), while keeping all nonincident seeds, all weights, + and the question received at every opposite position + \(j\ne k\) from \(\{j,k\}\) unchanged, changes \(f\) + by at most \(2\). +\end{enumerate} +The set \(A_*\), the function \(F\), and its formula are fixed from +an independent set of the entire graph before the fresh seeds above +are drawn. A seed enters \(f\) only through its endpoint questions. +\end{lemma} + +\begin{proof} +Take \(A_*\) to be the set of sum vectors of the stipulated high-norm +vertices in the independent set. The number of those vertices gives +the probability bound even if several vertices represent the same +vector. For distinct \(x,y\in A_*\), their representing vertices +are distinct and nonadjacent, so \Cref{graph:edge} gives +\(\norm{x+y}>2t\). If \(x=y\), the same inequality follows from +\(\norm{2x}=2\norm{x}>2t\). + +Distance to a nonempty finite set is the minimum of the distances to +its points. The triangle inequality makes each distance function +\(1\)-Lipschitz; the half-difference in +\eqref{ana:distance-function} is therefore \(1\)-Lipschitz. +Symmetry of the norm gives +\(\dist_*(-z,-A_*)=\dist_*(z,A_*)\), proving oddness. For +\(z\in A_*\), its distance to \(A_*\) is zero, while +\[ + \dist_*(z,-A_*)=\min_{x\in A_*}\norm{z+x}>2t. +\] +Finiteness preserves the strict inequality, so \(F(z)>t\). + +For fixed seeds, negating all weights preserves their joint law and +negates \(f\). This proves the first assertion in +\eqref{ana:variance}. Also \(F(0)=0\), and the increment bound +\(\norm{z(i,s)}\leq\|s\|_\infty\) implies \(|f|\leq d\) +on the sampling space. + +Round every continuous entry to the midpoint of its grid interval. +The resulting entries are independent uniform elements of +\(\mathcal T\); interval-boundary choices have probability zero. +The change in one increment has norm at most \(1/p\), by linearity +in its weights and the increment bound. Thus rounding changes the +argument of \(F\), and hence its value, by at most \(d/p4m\). For the restriction +conditions \eqref{src:restriction-budget}, we have \(h