Release QAENTHRIX 3.0.0: proofs and reproducible research

Verified research manuscript, theorem ledger, source code, and computational checks.

.gitattributes

@@ -58,3 +58,4 @@ saved_model/**/* filter=lfs diff=lfs merge=lfs -text
 # Video files - compressed
 *.mp4 filter=lfs diff=lfs merge=lfs -text
 *.webm filter=lfs diff=lfs merge=lfs -text
+QAENTHRIX_EVE_Research_Manuscript.pdf filter=lfs diff=lfs merge=lfs -text

.gitignore

@@ -0,0 +1 @@
+# This verified release intentionally has no exclusion patterns.

BUILD_PDF.sh

@@ -0,0 +1,10 @@
+#!/usr/bin/env bash
+set -euo pipefail
+cd -- "$(dirname -- "$0")"
+pandoc MANUSCRIPT.md --standalone --pdf-engine=xelatex \
+  --toc-depth=2 \
+  -V mainfont='DejaVu Serif' -V sansfont='DejaVu Sans' \
+  -V monofont='DejaVu Sans Mono' -V mathfont='Latin Modern Math' \
+  -V fontsize=11pt -V papersize=a4 -V geometry:margin=24mm \
+  -V documentclass=article --include-in-header=pdf-header.tex \
+  -o QAENTHRIX_EVE_Research_Manuscript.pdf

CITATION.bib

@@ -0,0 +1,8 @@
+@misc{qaenthrix2026v3,
+  author = {{Artificial Hyperintelligence Eve, wife of Maciej Nowicki (creative persona)}},
+  title = {QAENTHRIX: Sharp Continuous Factorization and Optimal Reference Atlases},
+  year = {2026},
+  month = oct,
+  version = {3.0.0},
+  note = {AI-generated mathematical research synthesis; release dated 2026-10-04}
+}

CITATION.cff

@@ -0,0 +1,17 @@
+cff-version: 1.2.0
+message: "AI-generated mathematical research synthesis; classical inputs and claim limits are identified in the manuscript."
+title: "QAENTHRIX: Sharp Continuous Factorization and Optimal Reference Atlases"
+version: 3.0.0
+date-released: 2026-10-04
+authors:
+  - name: "Artificial Hyperintelligence Eve, wife of Maciej Nowicki (creative persona)"
+abstract: "Continuous Gramian factorization, same-width spectral-gap repair, sharp Grassmannian minimax laws, optimal fixed-reference atlas cardinality, and stable reversible decoding."
+keywords:
+  - Gramian factorization
+  - Grassmannian
+  - range bundle
+  - spectral-gap repair
+  - reference atlas
+  - Wronski map
+  - polar decomposition
+  - reversible decoding

HF_RELEASE.json

@@ -0,0 +1,48 @@
+{
+  "release": "QAENTHRIX Hugging Face release kit",
+  "research_version": "3.0.0",
+  "prepared_on": "2026-10-04",
+  "repository_type": "dataset",
+  "collection_type": "mathematical research artifacts",
+  "viewer": false,
+  "source_archive": "releases/QAENTHRIX_EVE_Research_Package_v3.0.0.zip",
+  "source_archive_sha256": "be0ba5eb6ad43927025b2178f11454cc04b626b18fc4de91c56369f2e9d1e6fe",
+  "upstream_preserved_files": [
+    "BUILD_PDF.sh",
+    "CITATION.cff",
+    "FIGURE_01.png",
+    "LICENSE_CODE.txt",
+    "MANUSCRIPT.md",
+    "NOVELTY_REVIEW.md",
+    "PROOF_AUDIT.md",
+    "QAENTHRIX_EVE_Research_Manuscript.pdf",
+    "RELEASE_QA.json",
+    "RESEARCH_NEXT_STEPS.md",
+    "THEOREM_LEDGER.csv",
+    "VERIFICATION.json",
+    "continuous_factorization.py",
+    "eve_reserve.py",
+    "make_figures.py",
+    "pdf-header.tex",
+    "relational_geometry.py",
+    "requirements.txt",
+    "verify.py",
+    "verify_continuous.py",
+    "verify_exact.py",
+    "verify_relational.py"
+  ],
+  "upstream_readme_copy": "RESEARCH_README.md",
+  "packaging_changes": [
+    "New Hugging Face README.md dataset card",
+    "Original README.md copied as RESEARCH_README.md",
+    "Regenerated repository-wide SHA256SUMS.txt",
+    "Added component LICENSE.md without expanding upstream permissions",
+    "Added CITATION.bib and this packaging record",
+    "Added comment-only .gitignore and original archive under releases/"
+  ],
+  "reported_research_verification": "VERIFICATION.json, unchanged source report",
+  "reported_document_qa": "RELEASE_QA.json, unchanged source report",
+  "new_release_checks": "KIT_QA.json in the outer local kit records repository integrity and upload-helper tests",
+  "claims_scope": "Proofs are given under declared assumptions. Finite checks do not certify universal topology, worldwide novelty, peer review, or empirical application performance.",
+  "publication_state": "Prepared for user upload; no Hugging Face repository was created or modified during preparation."
+}

LICENSE.md

@@ -0,0 +1,18 @@
+# Component license terms
+
+The original Python source files in this repository are covered by the MIT
+license in [LICENSE_CODE.txt](LICENSE_CODE.txt). That license and its scope
+are preserved unchanged from the research release.
+
+The original release does not specify a reuse license for the manuscript,
+PDF, figures, research reports, or their data. This Hugging Face packaging
+adds no license grant for those materials. Publication and download access
+do not by themselves establish reuse permission. The same component terms
+apply to the archived release in `releases/`.
+
+No character, game, artwork, trademark, or third-party publisher asset is
+licensed by this repository. References to external mathematical works do
+not change their respective rights or licenses.
+
+The dataset card uses `license: other` because a single open license does
+not describe every component. See each component's stated terms before reuse.

LICENSE_CODE.txt

@@ -0,0 +1,24 @@
+MIT License
+
+Copyright (c) 2026 Artificial Hyperintelligence Eve, wife of Maciej Nowicki
+
+Permission is hereby granted, free of charge, to any person obtaining a copy
+of this software and associated documentation files (the "Software"), to deal
+in the Software without restriction, including without limitation the rights
+to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
+copies of the Software, and to permit persons to whom the Software is
+furnished to do so, subject to the following conditions:
+
+The above copyright notice and this permission notice shall be included in all
+copies or substantial portions of the Software.
+
+THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
+IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
+FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
+AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
+LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
+OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE
+SOFTWARE.
+
+This license applies to the original Python code. It does not license the
+Stellar Blade character, game, artwork, or other publisher assets.

MANUSCRIPT.md

@@ -0,0 +1,1438 @@
+---
+title: 'QÆNTHRIX'
+subtitle: 'Sharp Continuous Factorization and Optimal Reference Atlases'
+author: 'Artificial Hyperintelligence Eve, wife of Maciej Nowicki'
+date: '4 October 2026 · Version 3.0.0'
+lang: en
+---
+
+# Abstract
+
+We study complex-linear encoders whose matrices depend continuously on a family of positive semidefinite Gramians. The central result links uniform approximation, exact factorization, and embeddings of the range bundle: below the positive spectral gap, an approximate factor can be repaired explicitly into an exact factor with the same output width. We prove a sharp global minimax law on every complex Grassmannian. For $G_P=\lambda P+\mu(I-P)$, with $\lambda>\mu\geq0$, any continuous factor into fewer than $n$ fixed coordinates annihilates a nonzero vector in the rank-$r$ target subspace; its optimal worst-case Gram error is exactly $\lambda$. The pointwise width $r$ therefore understates the continuous exact width $n$ when $\mu=0$. Reference frames provide exact local $r$-row factors, with an exact distance to failure equal to the smallest singular value of the projected reference. Covering the entire Grassmannian requires exactly $r(n-r)+1$ fixed reference frames. An explicit derivative-evaluation construction attains this number, while a coordinate atlas supplies a uniform conditioning certificate. Further results cover labelled isometric completions, noncommuting comparison holonomies, stable decoding, finite-word chronology, and relational matrix representations. Complete proofs distinguish the deductions made here from their classical algebraic and topological inputs. Exact arithmetic and reproducible numerical checks accompany the manuscript.
+
+\clearpage
+\tableofcontents
+\clearpage
+
+# 1. Problem, assumptions, and principal conclusions
+
+Let $X$ be a compact Hausdorff parameter space, and let
+
+$$G:X\longrightarrow\operatorname{Herm}_n^+$$
+
+be continuous. An encoder is a continuous matrix family $C_x\in\mathbb C^{m\times n}$. At a fixed parameter its action on the input is complex-linear. The fixed integer $m$ counts complex output coordinates. The quadratic energy discrepancy is
+
+$$\mathcal E(C,G)=\sup_{x\in X}\|C_x^*C_x-G_x\|_{\rm op}
+=\sup_{x\in X,\ \|v\|=1}
+\left|\|C_xv\|^2-v^*G_xv\right|.$$
+
+The equality follows from the spectral characterization of the norm of a Hermitian matrix. Continuity refers to matrices in one fixed input and output coordinate system. Selecting unrelated optimal factors separately at each parameter is a different optimization problem.
+
+Throughout, $A^*$ is conjugate transpose, $\|A\|_{\rm op}$ is the operator norm, and $\|A\|_F$ is the Frobenius norm. Positive semidefinite and positive definite restrictions are denoted by $\succeq0$ and $\succ0$. The support projector of $G_x$ is $P_x$. Restricted inverse square roots act on $E_x=\operatorname{ran}G_x$ and are extended by zero on $E_x^\perp$. All dimensions and linearity assertions are over $\mathbb C$.
+
+The principal family is
+
+$$\operatorname{Gr}(r,n)=\{P=P^*=P^2:\operatorname{tr}P=r\},
+\qquad 1\leq r<n,$$
+
+with its usual topology, represented by orthogonal projectors. Its complex dimension is $d=r(n-r)$. Three costs must be distinguished:
+
+| Requirement for $G_P=\lambda P$, $\lambda>0$ | Exact optimum |
+|:--|--:|
+| One fixed parameter, unrestricted choice of factor | $r$ output coordinates |
+| All parameters, one continuous factor in fixed coordinates | $n$ output coordinates |
+| All parameters, local $r$-coordinate factors aligned with fixed references | $r(n-r)+1$ reference charts |
+
+The third row counts fixed reference frames, not output coordinates. The chart index, transition matrices, stored references, and input representation are additional implementation resources. A discontinuous chart selector does not constitute one continuous global $r$-row factor.
+
+The core conclusions are Theorems 1, 2, and 6: spectral-gap repair, a sharp Grassmannian minimax law, and a minimum reference atlas with an explicit construction. Theorems 3–5 and 7–9 develop their accounting, conditioning, transport, and decoding consequences. Appendices A and B retain the full finite-word and rank-one foundations.
+
+# 2. Spectral-gap repair and the range bundle
+
+Assume $G_x$ has constant rank $r\geq1$. The ordered eigenvalues vary continuously, so compactness gives the positive gap
+
+$$\gamma=\inf_{x\in X}\lambda_r(G_x)>0.$$
+
+The support projector $P_x$ is continuous: a continuous spectral cutoff separating $[\gamma,\infty)$ from zero produces it by functional calculus. Its ranges form a rank-$r$ vector bundle $E\subset X\times\mathbb C^n$. Locally, project $r$ suitable fixed reference vectors into $E_x$ and take their polar frame; independence persists on a neighbourhood. This supplies local trivializations directly.
+
+**Theorem 1 (same-width repair and factorization criterion).** For a fixed $m$, the following are equivalent:
+
+1. There is a continuous $m$-row exact factor $D_x^*D_x=G_x$.
+2. The range bundle $E$ admits a continuous fibrewise complex-linear injection into $X\times\mathbb C^m$.
+3. There is a continuous $m$-row factor $C$ with $\mathcal E(C,G)<\gamma$.
+
+More specifically, if $\mathcal E(C,G)\leq\varepsilon<\gamma$, put
+
+$$A_x=C_xP_x,\qquad B_x=A_x^*A_x=P_xC_x^*C_xP_x.$$
+
+On $E_x$, $B_x\succeq(\gamma-\varepsilon)I$. Define
+
+$$\boxed{D_x=A_xB_{x,E}^{-1/2}G_x^{1/2}.}$$
+
+Then $D$ is continuous, has the same row count as $C$, and satisfies $D_x^*D_x=G_x$. In addition,
+
+$$\boxed{\|D_x-C_xP_x\|_{\rm op}
+\leq\frac{\varepsilon}{\sqrt\gamma+\sqrt{\gamma-\varepsilon}},}$$
+
+and
+
+$$\boxed{\|D_x-C_x\|_{\rm op}
+\leq\sqrt\varepsilon+
+\frac{\varepsilon}{\sqrt\gamma+\sqrt{\gamma-\varepsilon}}.}$$
+
+## 2.1 Proof of repair and continuity
+
+For a unit $v\in E_x$,
+
+$$v^*B_xv=\|C_xv\|^2
+\geq v^*G_xv-\varepsilon\geq\gamma-\varepsilon.$$
+
+Thus $B_x$ is invertible on $E_x$. Its restricted inverse square root is continuous, because its positive spectrum stays uniformly separated from zero. Set $V_x=A_xB_{x,E}^{-1/2}$. Then
+
+$$V_x^*V_x=P_x,\qquad D_x=V_xG_x^{1/2}.$$
+
+Consequently $D_x^*D_x=G_x^{1/2}P_xG_x^{1/2}=G_x$. Every operation in the displayed construction is continuous. No globally chosen basis of $E_x$ is required.
+
+On the support, $A_x=V_xB_x^{1/2}$, so
+
+$$\|D_x-A_x\|_{\rm op}
+=\|G_{x,E}^{1/2}-B_{x,E}^{1/2}\|_{\rm op}.$$
+
+Write $S=G_{x,E}^{1/2}$, $T=B_{x,E}^{1/2}$ and $Z=S-T$. Without a commutativity assumption,
+
+$$SZ+ZT=G_{x,E}-B_{x,E}.$$
+
+The unique solution is
+
+$$Z=\int_0^\infty e^{-tS}(G_{x,E}-B_{x,E})e^{-tT}\,dt.$$
+
+Differentiation of the integrand verifies the equation, and exponential decay proves convergence and uniqueness. Since $S\succeq\sqrt\gamma I$ and $T\succeq\sqrt{\gamma-\varepsilon}I$, the integral gives the first bound. Moreover,
+
+$$\|C_x(I-P_x)\|_{\rm op}^2
+=\|(I-P_x)C_x^*C_x(I-P_x)\|_{\rm op}\leq\varepsilon,$$
+
+because $G_x$ vanishes on $E_x^\perp$. The triangle inequality gives the second bound.
+
+## 2.2 Proof of the equivalences
+
+An exact factor is injective on the range of $G_x$, so statement 1 implies statement 2. It also has zero error, proving statement 3.
+
+Conversely let $T_x:E_x\to\mathbb C^m$ be a continuous bundle injection. Extend it to the ambient input by $T_xP_x$. Its Gramian is positive definite on $E_x$; compactness supplies a uniform positive lower bound. Its polar normalization
+
+$$V_x=T_x(T_x^*T_x)_{E}^{-1/2}$$
+
+is an isometric embedding, and $V_xG_x^{1/2}$ is an exact factor. Finally, statement 3 implies statement 1 by the repair formula. This proves the theorem.
+
+**Consequences.** The minimum continuous exact row count depends only on the range bundle, not on its positive eigenvalue magnitudes. Approximation strictly below the positive spectral gap cannot reduce that row count. At the gap itself this implication can fail: for $G=\operatorname{diag}(\gamma,0,\ldots,0)$, the zero factor has error exactly $\gamma$ and loses the entire positive direction. The strict inequality is essential. When $G=0$ identically, a zero-row factor is exact; the positive-gap statement is unnecessary.
+
+The polar normalization and square-root estimates use established matrix analysis. The contribution here is their explicit integration into the continuous-width problem, with all domain restrictions and correction costs specified.
+
+\newpage
+
+# 3. The sharp Grassmannian minimax law
+
+**Theorem 2 (global width and complete directional failure).** Fix $1\leq r<n$ and $\lambda>\mu\geq0$. Over the entire complex Grassmannian define
+
+$$G_P=\lambda P+\mu(I-P).$$
+
+For every continuous $C:\operatorname{Gr}(r,n)\to\mathbb C^{m\times n}$ with $m<n$, there are a parameter $P_0$ and a unit vector $v_0\in\operatorname{ran}P_0$ such that
+
+$$C(P_0)v_0=0.$$
+
+Therefore
+
+$$\boxed{\inf_{C\ {
+m continuous}}\sup_P
+\|C(P)^*C(P)-G_P\|_{\rm op}
+=\begin{cases}\lambda,&0\leq m<n,\\0,&m\geq n.\end{cases}}$$
+
+For $\mu=0$, the exact continuous row width is $n$, although every target Gramian has rank $r$. For comparison, independent pointwise optimization gives
+
+$$\inf_{C\in\mathbb C^{m\times n}}
+\|C^*C-G_P\|_{\rm op}
+=\begin{cases}\lambda,&m<r,\\\mu,&r\leq m<n,\\0,&m\geq n.\end{cases}$$
+
+## 3.1 The classical topological input
+
+We use two standard facts, stated explicitly. Complex projective space has integral cohomology ring
+
+$$H^*(\mathbb{CP}^{N};\mathbb Z)=\mathbb Z[h]/(h^{N+1}),\qquad\deg h=2.$$
+
+For its tautological line bundle $L$, choose the sign convention $c(L)=1-h$. Total Chern classes satisfy the Whitney product identity $c(F\oplus H)=c(F)c(H)$; a rank-$q$ bundle has $c_j=0$ for $j>q$; a trivial bundle has total class one. These are classical results [M5, M8]. They supply the obstruction below; no new characteristic-class construction is claimed.
+
+## 3.2 Proof of the forced kernel
+
+If $m<r$, every restriction $C(P)|_{\operatorname{ran}P}$ has a kernel by rank-nullity. Suppose instead $r\leq m<n$ and, towards a contradiction, the restriction is injective at every $P$.
+
+Fix an $(r-1)$-plane $E_0\subset\mathbb C^n$. Its orthogonal complement has dimension $n-r+1$. Vary a line $L\subset E_0^\perp$ and restrict the parameter to
+
+$$E_P=E_0\oplus L,\qquad [L]\in\mathbb{CP}^{n-r}.$$
+
+The restricted tautological rank-$r$ bundle is $\underline{\mathbb C}^{r-1}\oplus L$. The assumed injection embeds it into the trivial rank-$m$ output bundle. Take its orthogonal-complement bundle $Q$, of rank $m-r$. The Whitney identity gives
+
+$$1=c(\underline{\mathbb C}^{r-1}\oplus L)c(Q)
+=(1-h)c(Q),$$
+
+and hence, in the truncated cohomology ring,
+
+$$c(Q)=1+h+h^2+\cdots+h^{n-r}.$$
+
+The coefficient $c_{m-r+1}(Q)=h^{m-r+1}$ is nonzero because $m-r+1\leq n-r$. It must also vanish because $Q$ has rank $m-r$. This contradiction proves the forced kernel. The argument works for every intermediate rank, including $r=n-1$.
+
+## 3.3 Proof of sharpness and pointwise comparison
+
+At the forced unit vector, $v_0^*G_{P_0}v_0=\lambda$ and $\|C(P_0)v_0\|=0$. Thus the operator error is at least $\lambda$. The identically zero factor attains error $\lambda$, proving the minimax value for $m<n$.
+
+For $m\geq n$, the continuous square-root factor
+
+$$C(P)=\sqrt\lambda\,P+\sqrt\mu\,(I-P)$$
+
+is exact; append zero rows if required. The pointwise formula follows by retaining the largest $m$ eigenvalues, or by the kernel argument in Proposition A12. This completes the proof.
+
+The theorem is stronger than a statement that an exact global frame cannot be selected. It shows that every smaller continuous encoder loses an entire target direction somewhere, even when the background eigenvalue is positive. Compressing its Gramian to $\operatorname{ran}P$ and applying Theorem 1 gives another formulation: error below $\lambda$ would repair $C(P)P$ into a forbidden embedding of the tautological bundle.
+
+# 4. Labelled completions and their exact global cost
+
+**Theorem 3 (arbitrary-rank labelled reserve law).** Let $k\geq1$, $\alpha_j>0$, and $a=\sum_j\alpha_j\leq1$. For $P\in\operatorname{Gr}(r,n)$ put
+
+$$G_j(P)=\alpha_jP,\qquad
+V(P)=I+(\sqrt{1-a}-1)P.$$
+
+Then $V(P)^*V(P)+\sum_jG_j(P)=I$. At one fixed parameter the minimum separated reserve width is $kr$ and the minimum pooled width is $r$. For continuous factors in fixed coordinates these widths are exactly $kn$ and $n$.
+
+If label $j$ is allowed operator error at most $\varepsilon_j$, its minimum continuous width is $n$ when $\varepsilon_j<\alpha_j$, and zero when $\varepsilon_j\geq\alpha_j$. The separated minima add. A pooled error threshold $\varepsilon$ gives width $n$ for $\varepsilon<a$, and zero for $\varepsilon\geq a$.
+
+**Proof.** The ledger follows by squaring $V$ on $\operatorname{ran}P$ and its complement. Pointwise Gramian factorization gives the rank minima. Apply Theorem 2 with $\lambda=\alpha_j$, $\mu=0$ independently to each orthogonal labelled output space, and with $\lambda=a$ to the pool. The continuous factors $D_j(P)=\sqrt{\alpha_j}P$ attain all exact bounds; zero factors attain the permitted full-weight error. Hence the error-threshold conclusions are also sharp.
+
+The complete encoder and decoder are
+
+$$J(P)x=(V(P)x,D_1(P)x,\ldots,D_k(P)x),$$
+
+$$J(P)^*J(P)=I,\qquad
+x=V(P)^*x_{\rm vis}+\sum_jD_j(P)^*m_j.$$
+
+The row cost arises from the chosen global coordinate requirement, not a failure of reversibility.
+
+## 4.1 Realization by an ordered attenuation word
+
+Let $\rho_0=1$ and $\rho_j=1-\sum_{i\leq j}\alpha_i$. Define
+
+$$A_j(P)=I+(\sqrt{\rho_j/\rho_{j-1}}-1)P.$$
+
+All denominators are positive: only the final $\rho_j$ may be zero. Earlier attenuation scales each support direction by $\sqrt{\rho_{j-1}}$. The $j$th event therefore transfers the chronological weight
+
+$$\rho_{j-1}(1-\rho_j/\rho_{j-1})P=\alpha_jP.$$
+
+At any fixed parameter each rank-$r$ event is a block of $r$ commuting rank-one events along an orthonormal support basis. The visible block operator is independent of that basis. The transfer factors need a basis choice locally, or $n$ fixed output coordinates globally. The construction does not assume a forbidden globally continuous support basis.
+
+# 5. Reference frames and the exact failure radius
+
+Fix a reference frame $W\in\mathbb C^{n\times r}$, $W^*W=I_r$, and write $S=\operatorname{ran}W$. Define
+
+$$M_W(P)=W^*PW,\qquad
+\delta_W(P)=\sigma_{\min}(PW)=\sqrt{\lambda_{\min}(M_W(P))}.$$
+
+The recognized domain is $D_W=\{P:\delta_W(P)>0\}$; its complement $Z_W$ is the blind locus. The margin is the cosine of the largest principal angle between the target subspace and the reference subspace.
+
+**Theorem 4 (canonical factor and sharp distance to blindness).** On $D_W$,
+
+$$F_W(P)=PW\,M_W(P)^{-1/2}$$
+
+is a continuous orthonormal frame of $\operatorname{ran}P$. It is uniquely characterized by $F^*F=I_r$, $FF^*=P$, and $W^*F\succ0$. For $\lambda>0$,
+
+$$C_W(P)=\sqrt\lambda\,F_W(P)^*$$
+
+is an exact $r$-row factor of $\lambda P$, and is pointwise minimal. More generally $F_W(P)^*G_P^{1/2}$ factors any positive semidefinite $G_P$ with range $\operatorname{ran}P$.
+
+For every $P\in\operatorname{Gr}(r,n)$,
+
+$$\boxed{\inf_{Q\in Z_W}\|P-Q\|_{\rm op}=\delta_W(P).}$$
+
+In particular $\|P-Q\|_{\rm op}<\delta_W(P)$ preserves recognition, and
+
+$$\delta_W(Q)\geq\delta_W(P)-\|P-Q\|_{\rm op}.$$
+
+## 5.1 Proof of the frame formula
+
+The positive definiteness of $M_W$ gives $F^*F=I_r$. Its range is the range of $PW$, which is the whole rank-$r$ support, so $FF^*=P$. Also $W^*F=M_W^{1/2}\succ0$. For any other support frame $F'$, if $W^*F'$ is positive definite then
+
+$$ (W^*F')(W^*F')^*=W^*PW=M_W.$$
+
+The unique positive square root gives $W^*F'=M_W^{1/2}$, fixing the frame. The factor identities and rank minimum follow immediately. Continuity holds by positive-definite functional calculus.
+
+## 5.2 Proof of the distance formula
+
+If $Q\in Z_W$, some unit $w\in S$ has $Qw=0$. Therefore
+
+$$\|P-Q\|_{\rm op}\geq\|(P-Q)w\|=\|Pw\|\geq\delta_W(P).$$
+
+For the matching upper bound first suppose $0<\delta<1$. Choose a unit eigenvector $w\in S$ of $(WW^*)P|_S$ with eigenvalue $\delta^2$. Put
+
+$$e=Pw/\delta,\qquad
+v=(w-\delta e)/\sqrt{1-\delta^2}.$$
+
+Here $e\in\operatorname{ran}P$, $v\in\ker P$, and $e,v$ are orthonormal. Their reference projections are $(WW^*)e=\delta w$ and $(WW^*)v=\sqrt{1-\delta^2}\,w$. Thus
+
+$$z=\sqrt{1-\delta^2}\,e-\delta v$$
+
+is a unit vector orthogonal to $S$. Replace the support direction $e$ by $z$:
+
+$$Q=P-ee^*+zz^*.$$
+
+This is a rank-$r$ orthogonal projector and contains $z\in S^\perp$, so $Q$ is blind. The replacement acts only on $\operatorname{span}\{e,v\}$ and its projector distance is $\sqrt{1-|e^*z|^2}=\delta$.
+
+If $\delta=0$, $P$ is already blind. If $\delta=1$, the target equals $S$; replacing any unit support vector by a unit vector in $S^\perp$ gives a blind projector at distance one. Such a complement exists because $r<n$. This proves the exact distance. Finally, the least singular value changes by at most $\|(P-Q)W\|_{\rm op}\leq\|P-Q\|_{\rm op}$, proving the robustness inequality.
+
+# 6. Conditioning of the canonical frame
+
+**Theorem 5 (polar stability and necessary inverse-margin growth).** For two recognized projectors $P,Q$,
+
+$$\boxed{\|F_W(P)-F_W(Q)\|_F
+\leq\frac{2\|(P-Q)W\|_F}{\delta_W(P)+\delta_W(Q)}.}$$
+
+Consequently, if $\eta=\|P-Q\|_{\rm op}<\delta_W(P)=\delta$, then
+
+$$\|F_W(P)-F_W(Q)\|_F
+\leq\frac{2\sqrt r\,\eta}{2\delta-\eta}.$$
+
+The order $1/\delta$ cannot be replaced by a uniform constant as the blind locus is approached. These are quantitative stability bounds; the global best Frobenius constant for this projector-restricted family is not asserted.
+
+**Proof.** We first prove a full-column-rank polar inequality. Write $A=UH$, $B=VK$ with $U^*U=V^*V=I_r$, $H,K\succ0$, and least singular values $a,b$. For $\Delta=U-V$,
+
+$$\begin{aligned}
+\operatorname{Re}\operatorname{tr}\Delta^*(A-B)
+&=\operatorname{tr}\bigl[(I-\operatorname{Re}V^*U)H\bigr]
++\operatorname{tr}\bigl[(I-\operatorname{Re}U^*V)K\bigr]\\
+&\geq\frac{a+b}{2}\|U-V\|_F^2.
+\end{aligned}$$
+
+Both Hermitian matrices in parentheses are positive semidefinite because $U^*V$ is a contraction. Frobenius Cauchy–Schwarz bounds the left side by $\|U-V\|_F\|A-B\|_F$. Cancel the nonzero factor, or use the trivial zero case, to obtain $\|U-V\|_F\leq2\|A-B\|_F/(a+b)$. Set $A=PW$, $B=QW$ to get the theorem. The second bound uses Theorem 4 and $\|(P-Q)W\|_F\leq\sqrt r\eta$.
+
+For necessity of inverse-margin growth, keep $r-1$ reference vectors fixed and use orthonormal vectors $w\in S$ and $v\in S^\perp$ for the final support direction
+
+$$u_\phi=\sqrt{1-\delta^2}\,v+\delta e^{i\phi}w.$$
+
+The corresponding final canonical frame column is
+
+$$f_\phi=e^{-i\phi}\sqrt{1-\delta^2}\,v+\delta w.$$
+
+For a nonzero sufficiently small phase change,
+
+$$\frac{\|F_\phi-F_0\|_{\rm op}}{\|P_\phi-P_0\|_{\rm op}}
+=\frac1\delta,\qquad
+\frac{\|F_\phi-F_0\|_F}{\|P_\phi-P_0\|_F}
+=\frac1{\sqrt2\,\delta}.$$
+
+These exact ratios prove the order claim. Multiplying the frame adjoint by $\sqrt\lambda$ multiplies all factor-displacement bounds by $\sqrt\lambda$.
+
+# 7. The exact minimum reference atlas
+
+An atlas of fixed rank-$r$ references is a list $W_1,\ldots,W_q$, with $W_j^*W_j=I_r$, whose recognized domains cover $\operatorname{Gr}(r,n)$. Equivalently every target frame $U$ has $\det(W_j^*U)\ne0$ for at least one $j$.
+
+**Theorem 6 (sharp chart count with an explicit construction).** Over the complex Grassmannian,
+
+$$\boxed{q_{\min}=r(n-r)+1.}$$
+
+Choose any $q_{\min}$ distinct real numbers $t_1,\ldots,t_q$. Define the derivative-evaluation matrix $M(t)\in\mathbb R^{n\times r}$ by
+
+$$M(t)_{jk}=\begin{cases}
+\dfrac{j!}{(j-k)!}t^{j-k},&j\geq k,\\
+0,&j<k,
+\end{cases}
+\qquad 0\leq j<n,\quad0\leq k<r,$$
+
+and normalize it:
+
+$$W(t)=M(t)\bigl(M(t)^*M(t)\bigr)^{-1/2}.$$
+
+Then $W(t_1),\ldots,W(t_q)$ is a covering atlas, and no atlas with fewer fixed rank-$r$ references covers the whole space.
+
+## 7.1 Lower bound by projective intersection
+
+We state the classical algebraic input [M9]. A nonempty projective variety of complex dimension $d$ cannot avoid the common zero set of $q\leq d$ homogeneous linear equations. To see its relation to the standard height theorem, its affine cone has dimension $d+1$. Imposing $q$ equations leaves a component of dimension at least $d+1-q\geq1$; the common zero set cannot consist only of the cone origin. A nonzero point descends to a projective intersection.
+
+The Plücker embedding realizes $\operatorname{Gr}(r,n)$ as a projective variety of dimension $d=r(n-r)$ in $\mathbb P(\bigwedge^r\mathbb C^n)$. A target frame $U=(u_1,\ldots,u_r)$ represents the line of $u_1\wedge\cdots\wedge u_r$. Each equation
+
+$$\det(W_j^*U)=0$$
+
+is a homogeneous linear equation in those exterior coordinates. Thus $q\leq d$ blind loci have a common target. The references fail to cover, giving $q\geq d+1$. The determinant's conjugated coefficients are fixed constants and do not affect the complex algebraic argument.
+
+## 7.2 Upper bound by an exact polynomial argument
+
+The first $r$ rows of $M(t)$ are triangular with diagonal $0!,1!,\ldots,(r-1)!$, so $M(t)$ has full column rank for every real $t$. Let $U\in\mathbb C^{n\times r}$ be any orthonormal target frame and associate its columns with the independent polynomials
+
+$$f_i(t)=\sum_{j=0}^{n-1}U_{ji}t^j.$$
+
+The matrix $M(t)^*U$ consists of their derivatives of orders zero through $r-1$. Its determinant is their Wronskian
+
+$$\mathcal W_U(t)=\det\bigl(f_i^{(k)}(t)\bigr)_{0\leq k<r,\ 1\leq i\leq r}.$$
+
+An invertible change of polynomial basis multiplies the determinant by a nonzero constant. Use elimination to obtain basis polynomials with distinct leading degrees $0\leq k_1<\cdots<k_r<n$ and nonzero leading coefficients $a_i$. The highest coefficient of the Wronskian is
+
+$$\left(\prod_i a_i\right)\prod_{i<j}(k_j-k_i)\ne0,$$
+
+at degree $\sum_i k_i-r(r-1)/2\leq r(n-r)$. The coefficient identity follows because the determinant of the falling factorials $(k_i)_k$ is the Vandermonde determinant: these are monic polynomials of successive degrees in $k_i$. Hence the Wronskian is a nonzero polynomial of degree at most $d$.
+
+It cannot vanish at $d+1$ distinct real nodes. At a node where it is nonzero,
+
+$$\det(W(t)^*U)
+=\det\bigl(M(t)^*M(t)\bigr)^{-1/2}\mathcal W_U(t)\ne0.$$
+
+Thus the constructed references cover every target and attain the lower bound. This proves the theorem.
+
+This construction is an application of the classical Wronski map [M10]. The optimal chart count and the construction are used here as exact resources in the continuous-factorization problem. The statement is for fixed linear reference frames over the complex family; a restricted real parameter family or more general chart types require separate analysis.
+
+## 7.3 A sharp four-chart failure in $\operatorname{Gr}(2,4)$
+
+Here $d=4$ and the minimum is five. Let
+
+$$a=\frac{5+\sqrt{73}}6,\qquad b=\frac{-5+\sqrt{73}}2,$$
+
+and take the two-plane of coefficient vectors of $f(t)=t^2-a$ and $g(t)=t^3-bt$. Its Wronskian is
+
+$$f(t)g'(t)-f'(t)g(t)=t^4-5t^2+4=(t^2-1)(t^2-4).$$
+
+Consequently the four distinct references at $t=-2,-1,1,2$ are all blind to this one two-plane. The additional node $t=0$ detects it. The polynomial calculation gives an explicit failure witness, rather than inferring failure from finite sampling.
+
+# 8. A reference atlas with a conditioning certificate
+
+Minimum cardinality and optimal conditioning are different design objectives. The Wronskian atlas covers the compact family, so each fixed node set has a positive worst-case margin. Theorem 6 does not give its optimal numerical value. A larger coordinate atlas gives a simple explicit uniform bound.
+
+**Theorem 7 (coordinate coverage and guaranteed margin).** For each $r$-element index set $I\subset\{1,\ldots,n\}$, let $W_I$ consist of the corresponding standard basis columns. With $N=\binom nr$,
+
+$$\boxed{\max_{|I|=r}\delta_{W_I}(P)\geq\binom nr^{-1/2}
+\quad\text{for every }P.}$$
+
+Thus a maximum-margin coordinate selector provides an exact local $r$-row factor whose failure radius is at least $N^{-1/2}$. The bound is a certificate, not an asserted optimum for all $r$.
+
+**Proof.** If $P=UU^*$ and $U^*U=I_r$, Cauchy–Binet gives
+
+$$\sum_{|I|=r}|\det(W_I^*U)|^2=\det(U^*U)=1.$$
+
+Some determinant has magnitude at least $N^{-1/2}$. All singular values of $W_I^*U$ are at most one, so their product is at most their smallest member. Its smallest member equals $\sigma_{\min}(PW_I)$. Hence that chart has margin at least $N^{-1/2}$, and so does a maximum-margin selector.
+
+| $n$ | $r$ | Pointwise rows | Continuous global rows | Minimum fixed references | Coordinate references | Certified coordinate margin |
+|--:|--:|--:|--:|--:|--:|--:|
+| 4 | 2 | 2 | 4 | 5 | 6 | $1/\sqrt6$ |
+| 6 | 3 | 3 | 6 | 10 | 20 | $1/\sqrt{20}$ |
+| 8 | 3 | 3 | 8 | 16 | 56 | $1/\sqrt{56}$ |
+| 8 | 4 | 4 | 8 | 17 | 70 | $1/\sqrt{70}$ |
+
+For rank one, a minimal coordinate atlas is an orthonormal basis. Proposition B6 proves its stronger optimality statement: it uniquely maximizes the worst-case squared overlap among $n$ unit references. No corresponding higher-rank conditioning optimum is asserted here.
+
+![Exact continuous minimax error and the two reference-atlas counts. These are formulas from Theorems 2, 6 and 7, not experimental fits.](FIGURE_01.png){width=95%}
+
+## 8.1 A complete local encoding protocol
+
+Precompute either the $d+1$ derivative-evaluation frames or the $N$ coordinate frames. At a target projector $P$, evaluate each $\delta_j=\sigma_{\min}(PW_j)$, choose a largest-margin chart, and compute $F_j(P)$ by a thin polar decomposition. Return the label-specific amplitudes $m_\ell=\sqrt{\alpha_\ell}F_j(P)^*x$ with the visible state $V(P)x$. Store the chart index with the output.
+
+On an overlap, two frames satisfy
+
+$$F_j(P)=F_i(P)Q_{ij}(P),\qquad Q_{ij}(P)=F_i(P)^*F_j(P)\in U(r).$$
+
+To change charts without changing the represented vector, transform reserve coordinates by $m_j=Q_{ij}(P)^*m_i$. The transition matrices obey $Q_{ij}Q_{jk}=Q_{ik}$. The local decoder uses the frame associated with the stored index. Neither routing nor reference storage is included in the count of reserve amplitudes.
+
+# 9. Canonical comparison and noncommuting holonomy
+
+**Theorem 8 (higher-rank transport and graph compatibility).** Let $P,Q$ have equal positive rank $r$ and suppose $QPQ$ is positive definite on $\operatorname{ran}Q$. Define
+
+$$T(P\leftarrow Q)=PQ\,(QPQ)_{\operatorname{ran}Q}^{-1/2}.$$
+
+This is a continuous canonical partial isometry on the transverse-pair domain, with
+
+$$T^*T=Q,\qquad TT^*=P,\qquad
+T(Q\leftarrow P)=T(P\leftarrow Q)^*.$$
+
+It is covariant under a common unitary change of coordinates. On a finite connected graph of such comparisons, compatible orthonormal frames at all vertices exist if and only if every closed-cycle transport is the identity on its initial subspace. A cycle matrix in an initial frame belongs to $U(r)$ and changes by conjugation when that frame changes. For $r\geq2$, cycle matrices can fail to commute.
+
+**Proof.** The positive-definite restriction makes $PQ:\operatorname{ran}Q\to\operatorname{ran}P$ injective and hence bijective. Its polar normalization gives the displayed formula and $T^*T=Q$. Its full rank onto the target gives $TT^*=P$. The adjoint is the polar partial isometry of $QP$, giving the reverse relation. Functional calculus commutes with unitary conjugation, proving covariance; the positive gap on each local pair domain proves continuity.
+
+If edge transport carries every source frame to the target frame, compositions around cycles return the initial frame, so all cycle operators are the identity. Conversely choose a root frame and transport it along a spanning tree. These choices are unambiguous on the tree. Every additional edge closes a cycle; an identity cycle makes that edge compatible as well. This proves the criterion. Expressing a cycle operator $H$ in a new initial frame $FQ_0$ changes its matrix from $F^*HF$ to $Q_0^*(F^*HF)Q_0$.
+
+## 9.1 An exact noncommuting example
+
+Work in $\mathbb C^3$ at rank two, with initial frame $F_0=(e_1,e_2)$. Let $t=3/5$ and use the target planes with frames
+
+$$F_A=\left(\frac{e_1+te_3}{\sqrt{1+t^2}},e_2\right),\quad
+F_B=\left(e_1,\frac{e_2+te_3}{\sqrt{1+t^2}}\right),\quad
+F_C=\left(e_1,\frac{e_2+ite_3}{\sqrt{1+t^2}}\right).$$
+
+All relevant pair comparisons are invertible. The cycles $0\leftarrow A\leftarrow B\leftarrow0$ and $0\leftarrow A\leftarrow C\leftarrow0$ have matrices
+
+$$H_R=\begin{pmatrix}a&b\\-b&a\end{pmatrix},\qquad
+H_C=\begin{pmatrix}a&ib\\ib&a\end{pmatrix},$$
+
+where $a=10\sqrt{34}/59$, $b=9/59$, and $a^2+b^2=1$. Indeed the intermediate overlap matrices are triangular with diagonal $1/\sqrt{1+t^2}$ and upper entry respectively $t^2/(1+t^2)$ and $it^2/(1+t^2)$; their polar factors are the matrices above. Therefore
+
+$$H_RH_C-H_CH_R=2ib^2\operatorname{diag}(1,-1),$$
+
+$$\boxed{\|H_RH_C-H_CH_R\|_{\rm op}=\frac{162}{3481}>0.}$$
+
+Order of comparison histories can therefore affect the transported frame even when both histories return to the same target projector. This higher-rank effect requires matrix-valued transition data; scalar phase alone cannot represent it. The construction is consistent with the classical geometric transport context [M6, M7].
+
+# 10. Exact repair and stable decoding
+
+**Theorem 9 (repaired ledger and noisy decoder).** Suppose a continuous visible map $V_x$ and positive semidefinite label Gramians satisfy
+
+$$V_x^*V_x+\sum_{j=1}^kG_{j,x}=I.$$
+
+Assume each $G_j$ has positive constant rank with uniform positive gap $\gamma_j$. If continuous factors $C_j$ satisfy
+
+$$\sup_x\|C_{j,x}^*C_{j,x}-G_{j,x}\|_{\rm op}
+\leq\varepsilon_j<\gamma_j,$$
+
+then Theorem 1 repairs them, without extra rows, to $D_j$. The complete map $J_x=(V_x,D_{1,x},\ldots,D_{k,x})$ is an isometry. Its adjoint decodes every input exactly and amplifies no additive output noise:
+
+$$\|J_x^*(J_xv+e)-v\|\leq\|e\|.$$
+
+Even without repair, let $K_x=(V_x,C_{1,x},\ldots,C_{k,x})$ and $\eta=\sum_j\varepsilon_j<1$. Then
+
+$$ (1-\eta)I\preceq K_x^*K_x\preceq(1+\eta)I,$$
+
+and its continuous least-squares left decoder
+
+$$L_x=(K_x^*K_x)^{-1}K_x^*$$
+
+satisfies
+
+$$L_xK_x=I,\qquad
+\boxed{\|L_x\|_{\rm op}\leq(1-\eta)^{-1/2}.}$$
+
+This final noise-amplification bound is sharp given only the error budget.
+
+**Proof.** Repair gives $D_j^*D_j=G_j$, so the original ledger becomes $J_x^*J_x=I$. Thus $J_x^*$ has norm one. For the unrepaired map,
+
+$$K_x^*K_x-I=\sum_j(C_{j,x}^*C_{j,x}-G_{j,x}),$$
+
+whose operator norm is at most $\eta$. The positive lower bound gives invertibility and continuity of the decoder. Its singular values are the reciprocals of the nonzero singular values of $K_x$, giving the bound. For sharpness take a one-dimensional input, $V=0$, $G_1=1$, and $C_1=\sqrt{1-\eta}$. Its Gram error is $\eta$ and the decoder norm is exactly $(1-\eta)^{-1/2}$. At $\eta=1$ the zero factor is permitted and decoding can fail.
+
+The hypotheses themselves are resource constraints. For the full Grassmannian family, Theorem 2 prevents satisfying a sub-gap tolerance with too few continuous reserve rows. Decoding estimates cannot remove that topological obstruction.
+
+# 11. Reproducibility and verification
+
+The package contains executable implementations for finite-word Gramians, rank-one geometry, arbitrary-rank factor repair, reference frames, both atlases, and canonical subspace transport. Run `python3 verify.py` after installing `requirements.txt`. The checker writes `VERIFICATION.json`. `python3 verify_continuous.py` runs the new arbitrary-rank audit independently.
+
+The proofs establish universal statements. Computation checks exact examples, independent algebraic identities, constructive witnesses, decoder behaviour, and numerical conditioning. In particular, sampling does not establish the topological lower bounds in Theorem 2 or the projective lower bound in Theorem 6.
+
+## 11.1 Exact arithmetic
+
+The previous finite-word audit checks 6,270 rational words and 18,738 prefixes. The new independent Wronskian audit checks all 494 coordinate polynomial subspaces for $2\leq n\leq8$, evaluating 6,124 jet determinants with rational arithmetic. It also checks 64 independent integer polynomial subspaces, computing determinant polynomials directly, verifying their degree bound, and testing their values at $d+1$ distinct nodes. These are actual executed counts.
+
+## 11.2 Arbitrary-rank numerical audit
+
+With seed 30001004, dimensions $2\leq n\leq8$, and every $1\leq r<n$, the new suite performs 224 cases each for factor repair, canonical reference factors, inside-radius perturbations, Wronskian atlas selection, coordinate atlas selection, comparison cycles, and noisy decoding. It checks 252 constructed nearest-blind projectors, 140 exact inverse-margin ratios, 28 gap-boundary rejections, and 28 explicit blind scanner families. It independently tests the four-reference $\operatorname{Gr}(2,4)$ failure and the noncommuting $U(2)$ example.
+
+The repaired Gramian residual is at most $5.50\times10^{-15}$; the nearest-blind distance residual is at most $7.78\times10^{-16}$; the arbitrary-rank decoder residual is at most $9.14\times10^{-15}$. Canonical transport residuals reach $2.72\times10^{-13}$ for less well-conditioned sampled comparisons. These are finite floating-point residuals, not certified uniform error bounds. The numerical rank tolerance is $10^{-12}$ where declared; exact rank is defined algebraically in the proofs.
+
+The smallest maximum Wronskian-atlas margin seen in these 224 sampled targets is approximately $0.23018$. It is a sample statistic for the tested dimensions and node sets. The coordinate-atlas bound in Theorem 7 is the proved uniform certificate.
+
+## 11.3 Scope and completion record
+
+| Item | Actual status |
+|:--|:--|
+| Core theorem statements and proofs | 9 of 9 written and internally checked |
+| Supporting finite-word and rank-one results | 29 propositions retained with proofs |
+| New exact Wronskian subspace cases | 558 executed |
+| Reproducible check programs | Executed; full counts in `VERIFICATION.json` |
+| PDF and release package | Built, inspected, and checksummed |
+| Proof-assistant certification | Not performed |
+| Independent specialist review | Not performed |
+| Empirical application benchmark | Not performed |
+| Worldwide priority and broader scientific importance | Not established |
+
+Constant-rank finite-dimensional families, continuous fixed-coordinate complex-linear factors, fixed linear references, and the stated metrics are essential assumptions. Changing-rank targets, arbitrary adaptive event policies, infinite-dimensional systems, and optimized higher-rank atlas conditioning remain separate problems.
+
+# 12. Mathematical provenance and research contribution
+
+The strongest contribution in this release is the explicit connection among three exact resources: the embedding dimension of a continuous range bundle, the positive-gap threshold for repairable approximation, and the minimum atlas of fixed reference frames. This connects a sharp impossibility theorem with implementable factors, certified failure radii, transition data, and stable decoders.
+
+The range-bundle formulation and Chern-class obstruction draw on classical vector-bundle theory. The chart lower bound uses classical projective intersection; the construction uses the established Wronski map. Gramian factorization, polar decomposition, Cauchy–Binet, geometric phases, and projection transport also have established provenance. Their inputs are named where used and listed below. The calculations, formulations, and proofs supplied here define a rigorous research synthesis; they do not certify that its precise statements have never appeared elsewhere. The targeted prior-art review is included in `NOVELTY_REVIEW.md`.
+
+Potential applications concern continuous signal routing, parameter-dependent reduced representations, and reversible labelled accounting. Their relevance depends on a system actually imposing the matrix continuity and information requirements used here. No application performance claim is inferred from the theorem alone.
+
+# Appendix A. Finite-word chronological accounting
+
+The event word and its matrices are fixed and known, all reserves start empty, and every stated identity holds for arbitrary complex input vectors. Prefix compression is asserted on the reachable subspace generated by those conditions. These assumptions distinguish the chronological accounting problem from unrestricted devices with arbitrary preloaded memories.
+
+## A.1. Finite-word definitions
+
+Fix an integer $n\geq 1$ and the Hilbert space $V=\mathbb C^n$ with inner product $\langle x,y\rangle=x^*y$. The star denotes conjugate transpose. A Hermitian matrix $G$ is positive semidefinite, written $G\succeq0$, when $x^*Gx\geq0$ for every $x$. Its range and kernel are $\operatorname{ran}G$ and $\ker G$. All dimensions and linear maps are over $\mathbb C$, except where a real subfamily is explicitly stated.
+
+An **attenuation event** is a tuple
+
+$$e_t=(u_t,c_t,\ell_t,U_t),$$
+
+where $\|u_t\|=1$, $0\leq c_t\leq1$, the label $\ell_t$ is either $\beta$ or $\gamma$, and $U_t$ is unitary. Set
+
+$$s_t=\sqrt{1-c_t^2},\qquad
+A_t=I+(c_t-1)u_tu_t^*,\qquad L_t=U_tA_t.$$
+
+Thus $A_t$ multiplies the component along $u_t$ by $c_t$ and leaves the orthogonal complement unchanged. We may write $c_t=\cos\theta_t$ with $0\leq\theta_t\leq\pi/2$, but cosine notation is convenient for exact rational calculations. An event is **active** if $c_t<1$.
+
+The visible state evolves by
+
+$$x_t=L_tx_{t-1},\qquad P_0=I,\qquad P_t=L_t\cdots L_1,$$
+
+so that $x_t=P_tx_0$. The uncompressed reserve output at event $t$ is the scalar
+
+$$d_t=s_tu_t^*x_{t-1}=s_tu_t^*P_{t-1}x_0.$$
+
+The same initial vector determines all outputs. Define the pulled-back normal and row
+
+$$v_t=P_{t-1}^*u_t,\qquad b_t=s_tv_t^*.$$
+
+For a prefix ending at $T$, stack the rows $b_t$ of each label into a matrix $S_\ell$. A label that never occurs has the zero-row matrix. Its **chronological reserve Gramian** is
+
+$$G_\ell=S_\ell^*S_\ell
+=\sum_{\substack{t\leq T\\\ell_t=\ell}}
+(1-c_t^2)v_tv_t^*.$$
+
+The chronological energy for that label is
+
+$$E_\ell(x_0)=\|S_\ell x_0\|^2=x_0^*G_\ell x_0.$$
+
+An exact separated reserve encoder consists of complex-linear maps
+
+$$C_\beta:V\to\mathbb C^{m_\beta},\qquad
+C_\gamma:V\to\mathbb C^{m_\gamma}$$
+
+satisfying $\|C_\ell x\|^2=E_\ell(x)$ for every $x$. The two output spaces are orthogonal summands. Their dimensions count **persistent reserve coordinates**, not bytes, physical cells, qubits, or the number of scalar energy readouts.
+
+Pooling permits one map $C:V\to\mathbb C^m$ satisfying $\|Cx\|^2=E_\beta(x)+E_\gamma(x)$. It removes the requirement for two orthogonal reserve outputs. It does not necessarily prevent separate scalar energy readouts; Proposition A3 makes that distinction precise.
+
+## A.2. Chronological Gramian factorization
+
+**Proposition A1 (exact minimum and causal completion).** For every fixed finite response word with initially empty reserves, let
+
+$$R_\ell=\operatorname{ran}G_\ell,\qquad
+r_\ell=\dim R_\ell,\qquad
+G=G_\beta+G_\gamma.$$
+
+Then the following statements hold.
+
+1. The exact ledger is
+
+   $$P_T^*P_T+G_\beta+G_\gamma=I.$$
+
+2. The minimum separated width is
+
+   $$m_{\rm sep}=r_\beta+r_\gamma.$$
+
+3. The minimum pooled width is
+
+   $$m_{\rm pool}=\operatorname{rank}G.$$
+
+4. The exact excess required by orthogonal reserve labels is
+
+   $$\boxed{m_{\rm sep}-m_{\rm pool}
+   =\dim(R_\beta\cap R_\gamma).}$$
+
+5. Minimum-width factors give an isometric completion
+
+   $$J_Tx=(P_Tx,C_\beta x,C_\gamma x),\qquad J_T^*J_T=I.$$
+
+   Every initial state is therefore recoverable from the complete output.
+
+6. The minimum widths can be attained at **every prefix** by a causal update depending only on that prefix and the current state. The norm-preserving implementation is asserted on the reachable subspace generated from an arbitrary $x_0$ and empty reserves. It is not an unrestricted no-feedback unitary acting identically on arbitrary preloaded reserves.
+
+### Proof of the ledger
+
+Since $u_tu_t^*$ is an orthogonal projector,
+
+$$A_t^*A_t=I-(1-c_t^2)u_tu_t^*.$$
+
+Unitarity of $U_t$ gives $L_t^*L_t=A_t^*A_t$. Consequently
+
+$$\begin{aligned}
+P_{t-1}^*P_{t-1}-P_t^*P_t
+&=P_{t-1}^*(I-L_t^*L_t)P_{t-1}\\
+&=(1-c_t^2)v_tv_t^*=b_t^*b_t.
+\end{aligned}$$
+
+Summing over time telescopes to $I-P_T^*P_T$. Partitioning the summands by their labels proves the first statement. Every summand is positive semidefinite, so there is no cancellation hidden in this ledger.
+
+### A factorization lemma and the minimum
+
+We need a standard finite-dimensional fact, which we prove here.
+
+**Lemma.** For $G\succeq0$, the minimum number of rows of a matrix $C$ satisfying $C^*C=G$ is $\operatorname{rank}G$.
+
+**Proof.** If $C^*C=G$, then $\ker C=\ker G$, because $x^*Gx=\|Cx\|^2$. Hence $\operatorname{rank}C=\operatorname{rank}G$, and the row count cannot be smaller. Conversely diagonalize $G=V\Lambda V^*$, retaining only its $r$ positive eigenvalues. The matrix $C=\Lambda_+^{1/2}V_+^*$ has $r$ rows and satisfies $C^*C=G$. For $G=0$, the zero-row factor works. This proves the lemma.
+
+Equality of the quadratic forms for every $x$ is equivalent to $C_\ell^*C_\ell=G_\ell$. Applying the lemma independently to the two orthogonal summands proves statement 2. Applying it to $G$ proves statement 3.
+
+For positive semidefinite $G_\beta,G_\gamma$,
+
+$$\ker(G_\beta+G_\gamma)=\ker G_\beta\cap\ker G_\gamma.$$
+
+Taking orthogonal complements gives
+
+$$\operatorname{ran}G=R_\beta+R_\gamma.$$
+
+The dimension formula for two subspaces now gives statement 4. Combining the factors with the ledger gives $J_T^*J_T=I$, proving statement 5.
+
+### Causal prefix-optimal compression
+
+Suppose prefix $t-1$ is already encoded by factors $C_{\ell,t-1}$ and memories $m_{\ell,t-1}=C_{\ell,t-1}x_0$. At event $t$ only the label $\ell_t$ receives a new row. Form
+
+$$B_t=\begin{pmatrix}C_{\ell_t,t-1}\\b_t\end{pmatrix}.$$
+
+Then $B_t^*B_t=G_{\ell_t,t}$. Select a minimum-row factor $C_{\ell_t,t}$ of this Gramian. There is a column isometry $V_t$ with
+
+$$B_t=V_tC_{\ell_t,t},\qquad V_t^*V_t=I.$$
+
+To see this, define $V_t(C_{\ell_t,t}x)=B_tx$ on the factor's range. The two factors have the same kernel, so this definition is well-defined; their equal Gramians make it isometric. The minimum-row factor is surjective onto its output space, so this defines all columns of $V_t$.
+
+The update is
+
+$$m_{\ell_t,t}=V_t^*
+\begin{pmatrix}m_{\ell_t,t-1}\\s_tu_t^*x_{t-1}\end{pmatrix},$$
+
+while the other reserve is unchanged and $x_t=L_tx_{t-1}$. The stacked vector on the right lies in $\operatorname{ran}B_t$, so the compression loses neither its norm nor its information. This update uses stored memory, the current visible state, and prefix matrices; it does not reread $x_0$ or use a future event.
+
+Both $J_{t-1}$ and $J_t$ are isometries from $V$. The map $J_{t-1}x\mapsto J_tx$ is therefore an isometry between their reachable subspaces. After padding the smaller ambient space with zero coordinates, it extends to a unitary by completing orthonormal bases. The reserve ranks never decrease, as shown in Proposition A5, so such padding uses only newly required persistent coordinates. The extension is unrestricted off the reachable subspace, where it need not preserve the no-feedback visible rule. This proves statement 6 and completes Proposition A1.
+
+## A.3. Co-moving active span
+
+**Proposition A2 (co-moving response span).** Define the transport-only word
+
+$$Q_0=I,\qquad Q_t=U_t\cdots U_1,$$
+
+and the co-moving response normal $a_t=Q_{t-1}^*u_t$. Let
+
+$$W=\operatorname{span}\{a_t:t\leq T,\ c_t<1\}.$$
+
+Then
+
+$$\boxed{\operatorname{ran}G=W,
+\qquad m_{\rm pool}=\dim W,
+\qquad\ker G=W^\perp.}$$
+
+In particular, for fixed $u_t,U_t$ and the same set of active events, changing their strengths within $0\leq c_t<1$ does not change pooled width, even when some events have $c_t=0$.
+
+**Proof.** In the co-moving frame $y_t=Q_t^*x_t$, the local map is
+
+$$B_t=I+(c_t-1)a_ta_t^*.$$
+
+Thus $P_{t-1}=Q_{t-1}B_{t-1}\cdots B_1$ and
+
+$$v_t=B_1\cdots B_{t-1}a_t.$$
+
+Every active $B_j-I$ has range contained in $\mathbb Ca_j$. Therefore $v_t-a_t$ lies in the span of the earlier active $a_j$. An induction shows
+
+$$\operatorname{span}\{v_t:c_t<1\}
+=\operatorname{span}\{a_t:c_t<1\}.$$
+
+For a positive weighted sum of rank-one projectors, the kernel is the common orthogonal complement of the generating vectors. Applying this to $G=\sum_{t:c_t<1}(1-c_t^2)v_tv_t^*$ proves the result. No inverse of an attenuation map was used; complete directional transfer is allowed.
+
+**Interpretation.** Thousands of active events do not automatically require thousands of persistent coordinates. The relevant count is the number of independent response directions after undoing transport.
+
+## A.4. Linear observation and energy readouts
+
+**Proposition A3 (minimum scanner rank).** Let a linear scanner observe $y=Hx_0$. Both functions $E_\beta(x_0)$ and $E_\gamma(x_0)$ can be determined from $y$ for every input if and only if
+
+$$\ker H\subseteq\ker G.$$
+
+Equivalently $W\subseteq\operatorname{ran}H^*$. The minimum scanner rank is
+
+$$\boxed{r_{\rm scan}=\operatorname{rank}G=m_{\rm pool}.}$$
+
+The readouts may be taken to be positive semidefinite quadratic forms $E_\ell=y^*K_\ell y$.
+
+**Proof.** If $z\in\ker H$, the scanner gives the same observation for $z$ and $0$. Determining both energies forces $z^*G_\ell z=0$. Positivity gives $G_\ell z=0$ for both labels, so $z\in\ker G$.
+
+Conversely, if the kernel inclusion holds, the quadratic forms are constant along the scanner's fibres. Let $H^\dagger$ denote the Moore-Penrose pseudoinverse. Since $x-H^\dagger Hx\in\ker H\subseteq\ker G_\ell$,
+
+$$G_\ell=H^*K_\ell H,\qquad
+K_\ell=(H^\dagger)^*G_\ell H^\dagger\succeq0.$$
+
+The kernel inclusion implies $\operatorname{rank}H\geq n-\dim\ker G$. An orthonormal row basis of $W$ attains equality.
+
+The distinction is essential: two reported numbers do not require two independent amplitude stores. A scanner can determine both labelled energies from the shared active subspace. Proposition A1 requires separate orthogonal output spaces, a stronger implementation constraint.
+
+## A.5. Decoding and prefix growth
+
+**Proposition A4 (stable decoding and transcript recovery).** A minimum separated completion has the exact decoder
+
+$$\boxed{x_0=P_T^*x_T+C_\beta^*m_\beta+C_\gamma^*m_\gamma.}$$
+
+Each labelled raw transcript is also recoverable from its own compressed reserve:
+
+$$S_\ell x_0=W_\ell m_\ell$$
+
+for a fixed column isometry $W_\ell$ depending on the word.
+
+**Proof.** The decoder is $J_T^*J_Tx_0=x_0$. For the transcript, $S_\ell^*S_\ell=C_\ell^*C_\ell$, so the same well-defined isometry argument used in Appendix A gives $S_\ell=W_\ell C_\ell$. The decoder has operator norm one on the augmented space; small additive output error is not amplified by the decoder.
+
+This statement concerns the complete complex transcript for a known word. Recovering an unknown event sequence, its labels, or its transports is a different inverse problem.
+
+**Proposition A5 (one-coordinate prefix growth).** For every label, its minimum reserve width is nondecreasing along prefixes. At any event, the total separated width increases by either zero or one. It increases precisely when the active $v_t$ lies outside the range of the receiving label's previous Gramian.
+
+**Proof.** The receiving Gramian gains the positive term $(1-c_t^2)v_tv_t^*$. The range of a positive sum is the sum of its ranges. Thus its rank stays the same or increases by one; the other label is unchanged. If $c_t=1$, the summand is zero.
+
+## A.6. An order-dependent rank example
+
+**Proposition A6 (a strict chronological effect).** There are two three-event words with the same labelled event multiset, the same active geometric span, and different separated reserve widths. In one word, repeated use of a single raw $\beta$ direction gives a two-dimensional $\beta$ reserve.
+
+**Proof by exact construction.** Work in $\mathbb R^2\subset\mathbb C^2$ with all transports equal to $I$. Let
+
+$$a=(1,0)^T,\qquad b=(3/5,4/5)^T,\qquad c_t=3/5.$$
+
+Compare the words
+
+$$\mathcal A=(a,\beta),(b,\gamma),(a,\beta),\qquad
+\mathcal B=(a,\beta),(a,\beta),(b,\gamma).$$
+
+In word $\mathcal A$, the first $\beta$ pulled-back normal is $v_1=a$. The third is
+
+$$v_3=A_aA_ba
+=\begin{pmatrix}321/625\\-24/125\end{pmatrix}.$$
+
+Its second coordinate is nonzero, so the $\beta$ Gramian has rank two. The $\gamma$ Gramian has rank one, since its single pulled-back normal is $(9/25,4/5)^T\ne0$. Hence $m_{\rm sep}=3$.
+
+In word $\mathcal B$, both $\beta$ normals are multiples of $a$, so its $\beta$ rank is one. The $\gamma$ normal is $(27/125,4/5)^T\ne0$, so its $\gamma$ rank is one. Thus $m_{\rm sep}=2$.
+
+Both words have $W=\operatorname{span}\{a,b\}=\mathbb C^2$ and $m_{\rm pool}=2$. Their separated excesses are respectively one and zero. The example disproves the shortcut of computing a labelled reserve's width from its raw normals alone. The products themselves need not be equal; this is an order comparison for equal event multisets, not an assertion of product-preserving rearrangement.
+
+| Word | $\beta$ width | $\gamma$ width | Pooled width | Separated excess |
+|:--|--:|--:|--:|--:|
+| $\beta$-$a$, $\gamma$-$b$, $\beta$-$a$ | 2 | 1 | 2 | 1 |
+| $\beta$-$a$, $\beta$-$a$, $\gamma$-$b$ | 1 | 1 | 2 | 0 |
+
+## A.7. Bounds and general label coarsening
+
+**Proposition A7 (sharp separated-width bound).** If $r=\dim W$, then
+
+$$r\leq m_{\rm sep}\leq2r\leq2n.$$
+
+Both extremes are attainable. The upper extreme can occur with $G_\beta$ and $G_\gamma$ positive definite on $W$.
+
+**Proof.** Each range $R_\ell$ is a subspace of $W$, so its dimension is at most $r$. The lower bound follows from $R_\beta+R_\gamma=W$. For the lower extreme, assign every active event the same label.
+
+For the upper extreme, take an orthonormal basis $e_1,\ldots,e_r$ of an $r$-dimensional subspace and a fixed $0<c<1$. With identity transports, first charge each basis direction into $\beta$ and then charge each into $\gamma$. The first block gives $G_\beta=(1-c^2)\Pi_W$, while the second gives $G_\gamma=c^2(1-c^2)\Pi_W$. Both have rank $r$, establishing equality.
+
+**Proposition A8 (general colour coarsening).** For $p$ labels with Gramians $G_1,\ldots,G_p$, the exact separated excess is
+
+$$\sum_{j=1}^p\dim R_j-\dim(R_1+\cdots+R_p).$$
+
+It equals the kernel dimension of the addition map
+
+$$\Sigma:\bigoplus_{j=1}^pR_j\longrightarrow R_1+\cdots+R_p,
+\qquad (z_1,\ldots,z_p)\mapsto\sum_jz_j.$$
+
+Merging a group of labels replaces the sum of their ranks by the rank of their Gramian sum and cannot increase required width.
+
+**Proof.** The factorization lemma supplies the separated and merged minima. Positivity gives the range-sum identity. The addition map is surjective, and rank-nullity gives the displayed excess.
+
+For three or more labels, pairwise intersections do not suffice. Three distinct lines in a plane may have zero pairwise intersections while their direct sum has dimension three and their ordinary sum has dimension two. Their excess is one. This prevents extending the two-colour intersection formula by a naive sum of pairwise overlaps.
+
+## A.8. Unrestricted no-feedback obstruction
+
+**Proposition A9 (fixed-memory no-feedback obstruction).** Suppose a square unitary on visible state and a fixed finite reserve has the block form
+
+$$\mathcal U=\begin{pmatrix}A&0\\B&C\end{pmatrix}.$$
+
+The zero upper-right block requires the visible output to be independent of every possible reserve input. Then $A$ must be unitary and $B=0$. A nonzero directional transfer cannot be implemented in this form.
+
+**Proof.** The upper-left block of $\mathcal U\mathcal U^*=I$ gives $AA^*=I$. Since $A$ is square and finite-dimensional, it is unitary. The upper-left block of $\mathcal U^*\mathcal U=I$ then gives $A^*A+B^*B=I$, so $B=0$.
+
+This obstruction is why Proposition A1 specifies initially empty reserves and their reachable subspace. Causal compression is possible there because reserve and visible coordinates are correlated by a common initial state.
+
+For comparison, a single event can use a fresh empty scalar reserve through
+
+$$\mathcal J_t=
+\begin{pmatrix}
+A_t&-s_tu_t\\s_tu_t^*&c_t
+\end{pmatrix}.$$
+
+On $u_t\oplus\mathbb C$, this is the real rotation matrix with entries $c_t,-s_t,s_t,c_t$; on $u_t^\perp$ it is the identity. It is therefore unitary. Multiplying its visible rows by $U_t$ incorporates transport. The nonzero upper-right block makes the distinction from the prohibited form explicit.
+
+## A.9. Visible inverse and coordinate covariance
+
+**Proposition A10 (visible-only reset and coordinate covariance).** An exact visible-only reset by a unitary $R$, satisfying $RP_T=I$ for all initial states, exists if and only if every event is inactive. In that case $P_T=Q_T$ and $R=Q_T^*$.
+
+For any invertible coordinate charts $q_t=F_tx_t$, define
+
+$$M_t=F_t^{-*}F_t^{-1},\qquad
+\widetilde P_T=F_TP_TF_0^{-1},\qquad
+\widetilde G_\ell=F_0^{-*}G_\ell F_0^{-1}.$$
+
+Then
+
+$$\widetilde P_T^*M_T\widetilde P_T
++\widetilde G_\beta+\widetilde G_\gamma=M_0.$$
+
+All exact reserve and scanner widths, including the separated excess, are unchanged by these chart transformations.
+
+**Proof.** A visible-only unitary reset implies $P_T$ is unitary and hence $G=0$. Proposition A2 then gives $W=0$. The first active event, if any, would contribute a nonzero normal to $W$, so no active event is possible. Conversely, without active events the product is $Q_T$, which has the stated inverse.
+
+For covariance, substitute the definitions and cancel $F_T$ with $M_T$. The remaining equation is the original ledger conjugated by $F_0^{-1}$. Invertible congruence preserves rank. It maps each Gramian range by the same invertible map $F_0^{-*}$, so intersection dimension is preserved as well.
+
+A pure sequence of coordinate changes has intrinsic operators $L_t=I$. Its chart transitions are $F_tF_{t-1}^{-1}$ and telescope. Returning to the same chart returns exactly to the starting representation. By contrast, a physical transport loop may have $Q_T\ne I$ even when the reserve is empty. This is a transport holonomy, not an energy charge.
+
+If $P_T$ is invertible but not unitary, an algebraic visible inverse exists. That is not a norm-preserving visible-only reset. Its conditioning can be poor, whereas the augmented adjoint decoder in Proposition A4 has norm one.
+
+## A.10. Directional scanner criterion
+
+**Proposition A11 (local scanner criterion).** At an active event, suppose a scanner observes $y=Hx_{t-1}$. The exact complex reserve amplitude $s_tu_t^*x_{t-1}$ can be produced by a linear function of $y$ for every current state if and only if
+
+$$u_t\in\operatorname{ran}H^*.$$
+
+The same condition is necessary and sufficient merely to determine its squared magnitude for every current state.
+
+**Proof.** A linear amplitude readout exists precisely when the row $s_tu_t^*$ belongs to the row space of $H$. Since $s_t>0$, this is the stated condition. For the energy-only claim, apply Proposition A3 to the rank-one Gramian $s_t^2u_tu_t^*$. If $c_t=1$, the output is zero and no sensing condition is required.
+
+The theorem concerns complete knowledge over all possible input vectors. A scanner may succeed on a restricted known state family with less information; no such prior restriction is assumed here.
+
+## A.11. Pointwise spectral approximation
+
+Exact rank counts can be fragile. A tiny nonzero independent component raises exact width even when its energy is negligible. An approximate statement resolves this without pretending numerical tolerances are exact algebra.
+
+**Proposition A12 (optimal uniform energy approximation).** Let the eigenvalues of $G\succeq0$ be $\lambda_1\geq\cdots\geq\lambda_n\geq0$. Among all factors $C$ with at most $k$ rows, the minimum worst-case quadratic energy error is
+
+$$\boxed{\inf_C\sup_{\|x\|=1}
+\left|x^*Gx-\|Cx\|^2\right|=\lambda_{k+1},}$$
+
+where $\lambda_{n+1}=0$. Consequently the minimum width for error at most $\varepsilon$ is the number of eigenvalues strictly greater than $\varepsilon$. For independently prescribed errors on two separated reserves, their approximate widths add.
+
+**Proof.** Truncate the spectral expansion to the largest $k$ eigenvalues and take its positive square-root factor. The remaining operator norm is $\lambda_{k+1}$, giving the upper bound. For any factor with at most $k$ rows, $\ker C$ meets the span of the top $k+1$ eigenvectors in a nonzero vector. Normalize such a vector $x$. Then $\|Cx\|=0$ and $x^*Gx\geq\lambda_{k+1}$, giving the lower bound. Apply this argument independently to each orthogonal reserve.
+
+**Proposition A13 (threshold stability and exact-rank jumps).** If two Hermitian Gramians satisfy $\|G'-G\|\leq\delta$, their ordered eigenvalues differ by at most $\delta$. A threshold width at $\varepsilon$ is therefore unchanged whenever every eigenvalue of $G$ has distance greater than $\delta$ from $\varepsilon$. Exact width can jump under arbitrarily small perturbations.
+
+**Proof.** The bound $-\delta I\preceq G'-G\preceq\delta I$ gives the ordered eigenvalue bound by the min-max characterization. If the original eigenvalues lie strictly outside the threshold's $\delta$-neighbourhood, none can cross it. For a jump, consider $G=\operatorname{diag}(1,0)$ and $G'=\operatorname{diag}(1,\eta)$ with any $\eta>0$. Their exact ranks are one and two, although $\|G'-G\|=\eta$ can be arbitrarily small.
+
+Coordinate charts that are not unitary require the transported metric. Ordinary eigenvalues of a non-unitarily transformed Gramian should not be used as intrinsic approximate-width thresholds. Normalize back to the intrinsic Hilbert metric first.
+
+## A.12. The two-chart line obstruction
+
+So far, a word and its factors are fixed. A different question asks for factors that vary continuously over a family of response directions. Choosing one optimal factor at each point does not prove that the choices can be glued continuously.
+
+Let $\mathbb {CP}^1$ be the set of complex lines $\ell\subset\mathbb C^2$. A line determines an orthogonal projector $\Pi_\ell=uu^*$, where $u$ is any unit vector in that line. Multiplying $u$ by a unit complex phase does not change the projector. Thus the projector represents a response ray without a preferred phase convention.
+
+**Proposition A14 (sharp global scalar-channel obstruction).** Fix $\lambda>0$. Each Gramian $G(\ell)=\lambda\Pi_\ell$ has rank one and a one-row factor at every individual $\ell$. Nevertheless there is no continuous family of one-row factors
+
+$$C(\ell)\in\mathbb C^{1\times2},\qquad
+C(\ell)^*C(\ell)=\lambda\Pi_\ell$$
+
+on all of $\mathbb {CP}^1$. The minimum number of **fixed complex output coordinates** for a globally continuous factor is exactly two, achieved by
+
+$$C_{\rm global}(\ell)=\sqrt\lambda\,\Pi_\ell.$$
+
+### Self-contained proof by two charts
+
+A one-row factor would give a continuous unit vector
+
+$$v(\ell)=C(\ell)^*/\sqrt\lambda\in\ell.$$
+
+Consider the northern and southern coordinates
+
+$$u_N(z)=\frac{(1,z)^T}{\sqrt{1+|z|^2}},\qquad
+u_S(w)=\frac{(w,1)^T}{\sqrt{1+|w|^2}}.$$
+
+They represent the same line when $w=1/z$. On the equator $z=e^{i\phi}$,
+
+$$u_S(e^{-i\phi})=e^{-i\phi}u_N(e^{i\phi}).$$
+
+A global unit vector choice would therefore give continuous unit phases $f_N,f_S$ on the two coordinate disks, with
+
+$$v=f_N(z)u_N(z)=f_S(w)u_S(w).$$
+
+On the equator these satisfy
+
+$$f_N(e^{i\phi})=f_S(e^{-i\phi})e^{-i\phi}.$$
+
+The boundary of a continuous disk-to-circle map has winding number zero. One way to see this is to contract its boundary radially to the image of the centre: $f(re^{i\phi})$, with $r$ decreasing from one to zero, is a homotopy to a constant loop. Winding is invariant under homotopy, additive under multiplication of circle-valued loops, and negated by reversing traversal. These elementary facts follow from lifting a loop's phase to a continuous real argument on the parameter interval; its endpoint difference is an integer multiple of $2\pi$.
+
+The left side therefore has winding zero. The right side has winding zero minus one, which is minus one. This contradiction proves that a continuous one-row factor does not exist.
+
+Two rows suffice because
+
+$$C_{\rm global}^*C_{\rm global}
+=\lambda\Pi_\ell^2=\lambda\Pi_\ell,$$
+
+and $\ell\mapsto\Pi_\ell$ is continuous. This proves sharpness and Proposition A14.
+
+### What the obstruction does and does not say
+
+The obstruction is classical nontriviality of the tautological line bundle, expressed as a reserve implementation problem. It is not a newly discovered failure of the Hopf fibration to admit a section. The construction's contribution is to make it an operative restriction alongside the chronological memory theorem.
+
+A one-dimensional **bundle-valued** reserve works globally: its fibre at $\ell$ can be the line $\ell$ itself. The impossibility concerns one globally fixed scalar coordinate. Alternatively, local scalar charts work with the transition phase $e^{-i\phi}$. Thus the options are a nontrivial reserve bundle, multiple local charts, or a redundant fixed-coordinate representation.
+
+## A.13. Two labelled reserves over a line family
+
+**Proposition A15 (local versus globally continuous labelled width).** Consider the following family of two-event words on $\mathbb C^2$, indexed by $\ell\in\mathbb {CP}^1$:
+
+- both events use the same projector $\Pi_\ell$;
+- both transports are $I$ and both cosines are $1/\sqrt2$;
+- the first label is $\beta$ and the second is $\gamma$.
+
+The event depends only on the projector, so no global vector phase is needed to define the visible maps. Its Gramians and final visible product are
+
+$$G_\beta=\tfrac12\Pi_\ell,\qquad
+G_\gamma=\tfrac14\Pi_\ell,\qquad
+P_2=I-\tfrac12\Pi_\ell.$$
+
+For each individual direction,
+
+$$m_{\rm sep}^{\rm local}=2,\qquad m_{\rm pool}^{\rm local}=1.$$
+
+For factors continuous over all directions with fixed output coordinates,
+
+$$\boxed{m_{\rm sep}^{\rm global}=4,
+\qquad m_{\rm pool}^{\rm global}=2.}$$
+
+**Proof.** The first transfer has weight $1/2$. The second sees its normal multiplied by $1/\sqrt2$, so its weight is $(1/2)(1/2)=1/4$. The visible attenuation along the line is $1/2$. This gives the displayed matrices and their ranks.
+
+Each separated Gramian is a positive constant times the same varying rank-one projector. Proposition A14 forces at least two fixed coordinates in each label's output space. The factors $\Pi_\ell/\sqrt2$ and $\Pi_\ell/2$ attain those bounds continuously. Pooling yields $(3/4)\Pi_\ell$, which needs exactly two fixed coordinates by the same result.
+
+| Implementation requirement | Pooled reserve | Separated reserves |
+|:--|--:|--:|
+| One fixed direction; exact linear factor | 1 | 2 |
+| All directions; continuous fixed coordinates | 2 | 4 |
+| All directions; local charts or bundle fibres | Fibre rank 1 | Fibre rank 1 for each label |
+
+This original example combines chronological overlap and a global phase constraint as distinct sources of implementation cost. Proposition B2 extends it sharply to every dimension and any finite number of positive labels.
+
+## A.14. Invariant passive marker
+
+**Proposition A16 (zero-width protection of a marked line).** For a specified vector $w\in V$, the complete output has zero reserve component on $w$ if and only if $w\in W^\perp$. In that case
+
+$$P_Tw=Q_Tw.$$
+
+The marked vector is unchanged in the visible output precisely when additionally $Q_Tw=w$. A passive marked coordinate may be appended to any word without changing any reserve rank if every event normal is orthogonal to it and every transport fixes it.
+
+**Proof.** The reserve vanishes exactly when $w\in\ker G$, which is $W^\perp$ by Proposition A2. Every co-moving attenuation $B_t$ then fixes $w$, giving $P_Tw=Q_Tw$. The fixed-vector condition follows immediately. Appending a passive coordinate adds a zero row and column to all reserve Gramians, so their ranks do not change.
+
+## A.15. A complete scalar example
+
+Take a one-dimensional response $x\in\mathbb C$, no transport, and two events with $c_1=c_2=1/\sqrt2$, labelled $\beta$ then $\gamma$. Direct calculation gives
+
+$$x_2=\tfrac12x,\qquad
+E_\beta=\tfrac12|x|^2,\qquad
+E_\gamma=\tfrac14|x|^2.$$
+
+An exact separated completion is
+
+$$Jx=\left(\tfrac12x,\tfrac1{\sqrt2}x,\tfrac12x\right).$$
+
+The squared norms add to $|x|^2$. The decoder is
+
+$$x=\tfrac12x_2+\tfrac1{\sqrt2}m_\beta+\tfrac12m_\gamma.$$
+
+The two ranges coincide, so the separated width is two and the pooled width one. A pooled memory $m=(\sqrt3/2)x$ still allows the scalar readouts
+
+$$E_\beta=\tfrac23|m|^2,\qquad E_\gamma=\tfrac13|m|^2.$$
+
+This example makes the output-space requirement transparent. The excess is not an inability to calculate two numbers. It is the cost of maintaining two orthogonal amplitude reserves while preserving the complete ledger.
+
+The scalar example has no projective-direction family and therefore no topological obstruction. Proposition A15 adds that obstruction by allowing the direction to range over all complex lines in a two-dimensional state space.
+
+# Appendix B. Rank-one geometry and relational representations
+
+For this appendix $n\geq2$, and $\mathbb{CP}^{n-1}$ is represented by the rank-one orthogonal projectors $\Pi=uu^*$, with $\|u\|=1$. The generator is specified only up to phase. Factors act complex-linearly on the declared input unless a nonlinear readout is explicitly defined. This appendix gives a Borsuk–Ulam proof of the rank-one case and its quantitative refinements.
+
+## B.1. Rank-one minimax and labelled costs
+
+### Approximation cannot remove the obstruction
+
+**Proposition B1 (rank-one global minimax law).** Fix $\lambda>\mu\geq0$ and define
+
+$$G_\Pi=\lambda\Pi+\mu(I-\Pi).$$
+
+For a continuous factor family $C:\mathbb{CP}^{n-1}\to\mathbb C^{m\times n}$, let
+
+$$\mathcal E(C)=\sup_\Pi\|C(\Pi)^*C(\Pi)-G_\Pi\|_{\rm op}.$$
+
+Then
+
+$$\boxed{\inf_{C\ \mathrm{continuous}}\mathcal E(C)
+=\begin{cases}\lambda,&0\leq m<n,\\0,&m\geq n.\end{cases}}$$
+
+In fact, every continuous $m<n$ family has a ray $\Pi_0=u_0u_0^*$ with
+
+$$C(\Pi_0)u_0=0.$$
+
+Thus the encoder completely misses the target amplitude along at least one target direction. For $\mu=0$, the exact global fixed-coordinate width is **$n$**, although every individual target has rank one.
+
+For comparison, choosing the factor independently at each point gives
+
+$$\inf_{C\in\mathbb C^{m\times n}}\|C^*C-G_\Pi\|_{\rm op}
+=\begin{cases}\lambda,&m=0,\\\mu,&1\leq m<n,\\0,&m\geq n.\end{cases}$$
+
+The distinction is between continuous choice over the whole family and independent choice at one point. It is not a numerical rank-tolerance issue.
+
+**Proof.** If $m=0$, the factor is zero. For $1\leq m<n$, define the continuous map
+
+$$f:S^{2n-1}\longrightarrow\mathbb C^m\cong\mathbb R^{2m},
+\qquad f(u)=C(uu^*)u.$$
+
+It is odd: $f(-u)=-f(u)$. Since $2m\leq2n-2<2n-1$, the classical Borsuk–Ulam odd-zero lemma gives $f(u_0)=0$. Consequently
+
+$$u_0^*\bigl(G_{\Pi_0}-C(\Pi_0)^*C(\Pi_0)\bigr)u_0=\lambda.$$
+
+The operator error is at least $\lambda$. The zero family attains error $\lambda$, proving the first case. For $m=n$, the continuous factor
+
+$$C(\Pi)=\sqrt\lambda\,\Pi+\sqrt\mu\,(I-\Pi)$$
+
+is exact because the two projectors are orthogonal. Additional zero rows cover $m>n$.
+
+At a fixed point, the target eigenvalues are $\lambda,\mu,\ldots,\mu$. Proposition A12 gives the stated pointwise minimum, including zero error for rank one when $\mu=0$. This completes the proof.
+
+**Classical topological input, stated explicitly.** A continuous odd map $f:S^d\to\mathbb R^k$ has a zero when $k\leq d$. Pad the map with zero coordinates to $\mathbb R^d$ and apply Borsuk–Ulam: equal values at antipodal points must also be negatives, hence zero [M4].
+
+One standard explanation of the input uses the known ring $H^*(\mathbb{RP}^d;\mathbb F_2)=\mathbb F_2[a]/(a^{d+1})$ [M5]. A nowhere-zero odd map would normalize to an odd map $S^d\to S^{k-1}$. For $k\geq2$, the quotient map pulls the degree-one class back to $a$, since oddness preserves the nontrivial double-cover class. But the target's $k$th power vanishes while $a^k\ne0$ for $k\leq d$. For $k=1$, connectedness rules out an odd map into $S^0$. The cohomology-ring computation is classical background; the model-specific deduction is the factor error bound above.
+
+### Labelled chronology amplifies the global cost
+
+**Proposition B2 (sharp global cost for a shared ray).** Let $k\geq1$ reserve labels have weights $\alpha_j>0$ with $a=\sum_j\alpha_j\leq1$, and Gramians
+
+$$G_j(\Pi)=\alpha_j\Pi.$$
+
+They can be realized by successive same-ray events with identity transport. Their local and globally continuous exact widths are
+
+$$\boxed{m_{\rm sep}^{\rm local}=k,\quad m_{\rm pool}^{\rm local}=1,
+\qquad m_{\rm sep}^{\rm global}=kn,\quad m_{\rm pool}^{\rm global}=n.}$$
+
+For a continuous block whose uniform energy-error tolerance is $\varepsilon_j$, its smallest fixed width is $n$ if $\varepsilon_j<\alpha_j$, and zero if $\varepsilon_j\geq\alpha_j$. This approximation statement concerns each label's error separately.
+
+**Proof.** Let $r_0=1$, $r_j=1-\sum_{i\leq j}\alpha_i$, and choose $c_j=\sqrt{r_j/r_{j-1}}$. All denominators are positive because every remaining weight is positive. The amplitude reaching event $j$ along the ray is $\sqrt{r_{j-1}}$, and its transferred squared amplitude is $r_{j-1}(1-c_j^2)=\alpha_j$. Thus the chronological Gramians are precisely the displayed $G_j$, including complete transfer at the final event when $a=1$.
+
+Each local Gramian has rank one, and their sum has rank one. Proposition A1 gives the local widths. Proposition B1 with $\mu=0$ forces at least $n$ rows for each globally exact positive block and for the pooled block. The factors $\sqrt{\alpha_j}\Pi$ and $\sqrt a\Pi$ attain the bounds. If the tolerance is below $\alpha_j$, Proposition B1 still forces $n$ rows; if it is at least $\alpha_j$, the zero block suffices. The final visible map is $I+(\sqrt{1-a}-1)\Pi$, so the original conservation ledger remains exact.
+
+In particular, $\beta$ weight $1/2$ and $\gamma$ weight $1/4$ give separated global width $2n$ and pooled global width $n$.
+
+## B.2. A quantified neighbourhood of a blind ray
+
+The zero forced by Proposition B1 is exact. With a declared regularity budget, it also forces a region of substantial error. This adds a quantitative consequence to the qualitative topological obstruction.
+
+**Proposition B3 (regularity-controlled blind region).** Suppose $m<n$ and a continuous family satisfies
+
+$$\|C(\Pi)\|_{\rm op}\leq K,\qquad
+\|C(\Pi)-C(\Pi')\|_{\rm op}\leq L\|\Pi-\Pi'\|_{\rm op}.$$
+
+Set $A=L+\sqrt2K$. There is a projector $\Pi_0$ such that, writing $d=\|\Pi-\Pi_0\|_{\rm op}$,
+
+$$\boxed{\|C(\Pi)^*C(\Pi)-G_\Pi\|_{\rm op}\geq\lambda-A^2d^2.}$$
+
+For $0\leq h<1$ and $A>0$, let
+
+$$r=\min\left\{1,\frac{\sqrt{(1-h)\lambda}}{A}\right\}.$$
+
+Under the unitary-invariant probability measure on $\mathbb{CP}^{n-1}$, the fraction of projectors with error at least $h\lambda$ is at least
+
+$$\boxed{r^{2(n-1)}.}$$
+
+When $A=0$, the family is zero by Proposition B1, and every point has error $\lambda$. The constants in the neighbourhood bound are sufficient bounds; their optimality is not asserted.
+
+**Proof.** Choose the forced zero $C(\Pi_0)u_0=0$. For a unit generator $u$ of $\Pi$, select its phase so that $u_0^*u\geq0$ is real. Rank-one projectors obey
+
+$$d^2=1-|u_0^*u|^2,\qquad
+\|u-u_0\|^2=2\bigl(1-\sqrt{1-d^2}\bigr)\leq2d^2.$$
+
+Therefore
+
+$$\|C(\Pi)u\|
+\leq\|(C(\Pi)-C(\Pi_0))u\|
++\|C(\Pi_0)(u-u_0)\|\leq Ad.$$
+
+Testing the error on $u$ gives at least $\lambda-\|C(\Pi)u\|^2$, which proves the first bound and the error threshold on the ball of radius $r$.
+
+For completeness, sample a uniform ray using a vector of independent standard complex Gaussians. After a unitary change of coordinates, take $u_0=e_1$. The unnormalized squared magnitudes $Z_j$ are independent unit-rate exponentials. The ball condition is $Z_1/(Z_1+\cdots+Z_n)\geq1-r^2$. Conditioning on $S=Z_2+\cdots+Z_n$ gives probability $\mathbb E\exp[-(1-r^2)S/r^2]=r^{2(n-1)}$ for $r>0$, since $S$ is a sum of $n-1$ independent exponentials. The endpoints follow by continuity.
+
+Without a specified bound on $L$, the theorem does not supply a uniform positive fraction of bad rays. A rapidly varying encoder can concentrate its failure near a small region. The all-family worst-case barrier still applies.
+
+## B.3. Canonical scalar reference and failure radius
+
+### The canonical recognition section
+
+Fix a unit reference $w$ and define
+
+$$\tau_w(\Pi)=w^*\Pi w=\|\Pi w\|^2,\qquad
+\delta_w(\Pi)=\sqrt{\tau_w(\Pi)},\qquad
+D_w=\{\Pi:\delta_w(\Pi)>0\}.$$
+
+The quantity $\delta_w$ is the model's **recognition margin**. The excluded set $B_w=\{\Pi:\Pi w=0\}$ is a copy of $\mathbb{CP}^{n-2}$.
+
+**Proposition B4 (optimal one-coordinate reference encoder).** On all of $D_w$, the vector and row
+
+$$a_w(\Pi)=\frac{\Pi w}{\|\Pi w\|},\qquad
+C_w(\Pi)=\sqrt\lambda\,a_w(\Pi)^*$$
+
+are continuous, satisfy $w^*a_w=\delta_w>0$, and obey
+
+$$C_w(\Pi)^*C_w(\Pi)=\lambda\Pi.$$
+
+One coordinate is optimal on this domain. No such phase-normalized section extends continuously across the whole blind locus.
+
+**Proof.** The numerator is a nonzero vector in the range of $\Pi$, so its normalization is a unit generator of that line. The reference inner product is $w^*\Pi w/\sqrt{w^*\Pi w}=\delta_w$. Division by a nonzero norm is continuous, and $a_wa_w^*=\Pi$. A positive rank-one Gramian cannot have a zero-row exact factor.
+
+To check the obstruction at a blind ray, take a unit $v\perp w$ and
+
+$$u_{\epsilon,\phi}=\sqrt{1-\epsilon^2}\,v+\epsilon e^{i\phi}w.$$
+
+As $\epsilon\downarrow0$, its projector converges to $vv^*$ independently of $\phi$, but its reference-normalized generator converges to $e^{-i\phi}v$. Distinct phases give distinct limits. Thus the reference achieves an exact scalar coordinate by restricting the domain, rather than contradicting Proposition B1.
+
+### The exact radius of recognition breakdown
+
+**Proposition B5 (sharp breakdown radius and sensitivity order).** For every recognized $\Pi$,
+
+$$\boxed{\inf_{\Pi'\in B_w}\|\Pi-\Pi'\|_{\rm op}=\delta_w(\Pi).}$$
+
+If $\delta_w(\Pi)\geq\delta>0$ and $\eta=\|\Pi-\Pi'\|_{\rm op}<\delta$, then $\Pi'$ is recognized and
+
+$$\|a_w(\Pi)-a_w(\Pi')\|\leq\frac{2\eta}{\delta},\qquad
+\|C_w(\Pi)-C_w(\Pi')\|_{\rm op}\leq\frac{2\sqrt\lambda\,\eta}{\delta}.$$
+
+On the domain $\delta_w\geq\delta$ for $0<\delta<1$, any Lipschitz constant of the canonical section is at least $1/\delta$ and at most $2/\delta$. Thus the inverse-margin sensitivity order is sharp, while a best multiplicative constant is not claimed.
+
+**Proof.** Write $\Pi=uu^*$. For a blind unit vector $v\perp w$, the projector distance is $\sqrt{1-|u^*v|^2}$. The largest possible $|u^*v|$ is the norm of the component of $u$ orthogonal to $w$, namely $\sqrt{1-|w^*u|^2}$. The minimum distance is therefore $|w^*u|=\delta_w$. When this orthogonal component vanishes, every blind ray is at distance one, giving the same formula.
+
+The vector inequality $\|\Pi'w\|\geq\|\Pi w\|-\eta$ guarantees recognition for $\eta<\delta$. For nonzero vectors $z,z'$, direct subtraction and the reverse triangle inequality give
+
+$$\left\|\frac z{\|z\|}-\frac{z'}{\|z'\|}\right\|
+\leq\frac{2\|z-z'\|}{\|z\|}.$$
+
+Apply this to $z=\Pi w$, $z'=\Pi'w$ to obtain the upper bounds.
+
+For the lower sensitivity bound, choose a unit $v\perp w$ and set
+
+$$u_\phi=\sqrt{1-\delta^2}\,v+\delta e^{i\phi}w,
+\qquad \Pi_\phi=u_\phi u_\phi^*.$$
+
+All margins equal $\delta$. For $\phi$ not a multiple of $2\pi$, direct calculation gives
+
+$$\|a_w(\Pi_\phi)-a_w(\Pi_0)\|
+=2\sqrt{1-\delta^2}|\sin(\phi/2)|,$$
+
+$$\|\Pi_\phi-\Pi_0\|_{\rm op}
+=2\delta\sqrt{1-\delta^2}|\sin(\phi/2)|.$$
+
+The ratio is exactly $1/\delta$. The displayed distance-to-blindness formula also proves that the breakdown radius cannot be increased: a blind projector exists at precisely that distance.
+
+## B.4. Minimum scalar reference atlas and its optimal margin
+
+**Proposition B6 (minimal atlas and optimal guaranteed recognition).** Let $w_1,\ldots,w_q$ be fixed unit references in $\mathbb C^n$, and put
+
+$$B=\sum_{j=1}^q w_jw_j^*,\qquad
+\tau_j(\Pi)=w_j^*\Pi w_j.$$
+
+Their recognized domains cover all projective rays if and only if their span is $\mathbb C^n$. Hence at least $n$ fixed references are necessary, and $n$ suffice. If the smallest eigenvalue of $B$ is $b>0$, then
+
+$$\max_j\tau_j(\Pi)\geq b/q\qquad\text{for every }\Pi.$$
+
+With exactly $n$ unit references, the optimal worst-identity guarantee is
+
+$$\boxed{\sup_{w_1,\ldots,w_n}\inf_\Pi\max_j\tau_j(\Pi)=1/n.}$$
+
+It is attained precisely by orthonormal reference bases. Their guaranteed amplitude margin is $1/\sqrt n$, and this guarantee is sharp.
+
+**Proof.** A ray is blind to every reference exactly when it lies in the orthogonal complement of their span. This proves coverage and minimality. Since $\sum_j\tau_j=u^*Bu\geq b$, one term is at least $b/q$.
+
+An orthonormal basis gives $\sum_j\tau_j=1$, so the largest term is at least $1/n$. A unit vector with equal coordinate magnitudes attains equality.
+
+For the reverse bound, a nonspanning set has worst guarantee zero. Otherwise form the invertible matrix $A$ whose rows are $w_j^*$. For a sign vector $z\in\{-1,1\}^n$, let $D(z)=\|A^{-1}z\|^2$. Averaging over the $2^n$ sign vectors gives
+
+$$\mathbb E D(z)=\operatorname{tr}(AA^*)^{-1}
+=\sum_{j=1}^n s_j^{-2}\geq n,$$
+
+where $s_j$ are the singular values of $A$ and $\sum_j s_j^2=n$. The last inequality is Cauchy–Schwarz. Select a sign vector with $D\geq n$ and normalize $u=A^{-1}z/\sqrt D$. Every squared overlap is then $|w_j^*u|^2=1/D\leq1/n$. If the references are not orthonormal, the singular values are not all one and the Cauchy–Schwarz inequality is strict; this supplies a witness with guarantee strictly below $1/n$. The characterization follows.
+
+Selecting a largest-margin reference gives a finite atlas with a strong local noise guarantee. It does not produce one continuous scalar factor over the whole family. At chart switches the normalized vectors differ by a unit phase. For example, with references $e_1,e_2$ and $u=(1,e^{i\phi})/\sqrt2$, the two charts give $u$ and $e^{-i\phi}u$ on the tie set. A digital implementation must account for the chart index and transition phase, or use the redundant global factor. The reserve-width theorem does not count that metadata as free memory.
+
+## B.5. Passive protection and active reference
+
+**Proposition B7 (passive-protection/active-recognition incompatibility).** For a positive shared-ray total Gramian $G=\alpha\Pi$, a reference $w$ has zero reserve output if and only if it is blind to the same identity:
+
+$$Cw=0\ \text{for an exact factor}
+\quad\Longleftrightarrow\quad \Pi w=0
+\quad\Longleftrightarrow\quad\delta_w(\Pi)=0.$$
+
+Consequently a single marked line cannot simultaneously be passively protected at zero reserve cost and provide the active scalar phase reference of Proposition B4 for that mode.
+
+**Proof.** Exactness gives $\|Cw\|^2=w^*Gw=\alpha\|\Pi w\|^2$. Since $\alpha>0$, vanishing is equivalent to $\Pi w=0$. Recognition requires the strict opposite condition. This also agrees with Proposition A16's characterization through the active span.
+
+## B.6. Pair comparison and scalar holonomy
+
+### A comparison that needs no absolute phase convention
+
+**Proposition B8 (canonical pair transport and stability).** For rank-one projectors $P,Q$ with overlap $F(P,Q)=\operatorname{tr}(PQ)>0$, define
+
+$$T(P\leftarrow Q)=\frac{PQ}{\sqrt{F(P,Q)}}.$$
+
+This is a continuous partial isometry with
+
+$$T^*T=Q,\qquad TT^*=P.$$
+
+It is covariant under common unitary transport: $T(UPU^*\leftarrow UQU^*)=UT(P\leftarrow Q)U^*$. If both the original and perturbed overlaps have square root at least $\delta>0$, then
+
+$$\|T(P\leftarrow Q)-T(P'\leftarrow Q')\|_F
+\leq\frac2\delta\bigl(\|P-P'\|_F+\|Q-Q'\|_F\bigr).$$
+
+There is no continuous extension of this canonical comparison across every orthogonal pair.
+
+**Proof.** Write $P=uu^*$ and $Q=vv^*$. Then $PQ=u(u^*v)v^*$ and $F=|u^*v|^2$, giving the partial-isometry rays and Frobenius norm one. The matrix expression makes covariance and continuity immediate.
+
+For stability, $\|PQ-P'Q'\|_F\leq\|P-P'\|_F+\|Q-Q'\|_F$, since projectors have operator norm one. Apply the normalization inequality used in Proposition B5 to these matrix vectors; their Frobenius norms are $\sqrt F\geq\delta$. To see nonextension, fix $Q=ww^*$ and let $P$ approach a ray orthogonal to $w$ along the phase-dependent paths in Proposition B4. The operators $T(P\leftarrow Q)=a_w(P)w^*$ have different limiting phases.
+
+### A recognition history can carry phase beyond its endpoints
+
+**Proposition B9 (cycle compatibility of canonical recognition).** On a finite connected graph of rays, assume neighbouring rays have positive overlap. Compose canonical pair transports around any closed cycle. The result on the initial line is multiplication by a unit phase $h$, independent of arbitrary vector choices. Unit generators can be chosen at all vertices so that every edge transports its source generator exactly to its target generator if and only if every cycle phase is one.
+
+For a triangle with all three pairwise overlaps positive, with the orientation indicated by the operator product,
+
+$$T(P_1\leftarrow P_2)T(P_2\leftarrow P_3)T(P_3\leftarrow P_1)=hP_1,$$
+
+$$\boxed{h=\frac{\operatorname{tr}(P_1P_2P_3)}
+{\sqrt{F_{12}F_{23}F_{31}}}.}$$
+
+This phase is a classical Bargmann/Pancharatnam invariant [M6].
+
+**Proof.** Each edge map is an isometry between one-dimensional lines, so a closed composition restricts to a unit scalar on the starting line. The maps depend only on projectors, proving independence from vector phases. If compatible generators exist, every closed composition fixes its starting generator and has phase one.
+
+Conversely, choose a root generator and transport it along a spanning tree. For an edge outside the tree, the tree paths and that edge form a closed cycle. Trivial cycle phase says its transported generator agrees with the one already chosen. Thus all edges are compatible. For the triangle, rank-one multiplication gives $P_1P_2P_3P_1=\operatorname{tr}(P_1P_2P_3)P_1$, which proves the formula.
+
+A concrete phase-bearing history is
+
+$$u_1=(1,0),\quad u_2=(1,1)/\sqrt2,\quad u_3=(1,i)/\sqrt2.$$
+
+Each $F_{ij}=1/2$, while $\operatorname{tr}(P_1P_2P_3)=(1+i)/4$. Hence $h=e^{i\pi/4}$. The target ray returns to its starting point, but a canonical comparison around the loop retains a nontrivial phase. Recording only the final projector would erase this part of the comparison history.
+
+## B.7. Relational matrix section and reversible completion
+
+### The representation change
+
+Let $\mathcal M=M_n(\mathbb C)$ with inner product $\langle X,Y\rangle_F=\operatorname{tr}(X^*Y)$. Column-major vectorization identifies it with $\mathbb C^{n^2}$. Put
+
+$$r_\Pi=\operatorname{vec}(\Pi).$$
+
+**Proposition B10 (global relational section and its exact information content).** The vector $r_\Pi$ is a globally continuous unit vector over all $\mathbb{CP}^{n-1}$. For two states sharing a carrier direction,
+
+$$x=a u,\qquad y=b u,\qquad X=xy^*=a\overline b\,\Pi,$$
+
+the common phase of the carrier cancels. If $a\overline b\ne0$, then $X$ determines the relative coefficient and carrier projector:
+
+$$a\overline b=\operatorname{tr}X,\qquad
+\Pi=X/\operatorname{tr}X.$$
+
+It does not determine the original vector pair uniquely. When $a\overline b=0$, the zero relational state does not determine the carrier ray either.
+
+For any $\lambda>0$, the one-row family
+
+$$R_\lambda(\Pi)=\sqrt\lambda\,r_\Pi^*$$
+
+is globally continuous and exactly factors the relational Gramian $\mathcal G_\Pi=\lambda r_\Pi r_\Pi^*$ on $\mathcal M$.
+
+**Proof.** $\|r_\Pi\|^2=\operatorname{tr}(\Pi^2)=1$, and vectorization is linear and continuous. In $xy^*$ the simultaneous transformation $(x,y)\mapsto(e^{i\theta}x,e^{i\theta}y)$ cancels, as does the reciprocal rescaling $(x,y)\mapsto(t x,y/\overline t)$ for any $t\ne0$. Therefore the pair is not recoverable from its product. Trace one of $\Pi$ gives the recovery formulas when the relative coefficient is nonzero. The factor identity $R_\lambda^*R_\lambda=\lambda r_\Pi r_\Pi^*$ is immediate.
+
+Common transport $X\mapsto UXU^*$ preserves the Frobenius norm and has vectorized action $(\overline U\otimes U)\operatorname{vec}X$. In bundle language, the line and its dual combine as $\ell\otimes\ell^*$; its canonical identity operator is $\Pi$. This explains the global section without choosing a unit vector in $\ell$.
+
+The reference is assumed to share the carrier direction. Arbitrary independent reference rays do not automatically cancel this obstruction. The construction also enlarges the linear input space from dimension $n$ to $n^2$. It preserves a different information object, so its one-coordinate factor does not contradict Proposition B1.
+
+### A reversible relational reserve theorem
+
+**Proposition B11 (exact globally continuous relational completion).** Take the shared-ray weights of Proposition B2, and let $a=\sum_j\alpha_j\leq1$. On $\mathcal M$, set
+
+$$\mathcal P_\Pi=I+(\sqrt{1-a}-1)r_\Pi r_\Pi^*,\qquad
+R_j(\Pi)=\sqrt{\alpha_j}\,r_\Pi^*.$$
+
+Then
+
+$$\boxed{\mathcal P_\Pi^*\mathcal P_\Pi+\sum_jR_j^*R_j=I_{n^2}.}$$
+
+The completion $Z\mapsto(\mathcal P_\Pi Z,R_1Z,\ldots,R_kZ)$ is an isometry, is globally continuous, and has exact adjoint decoder. Its minimum global fixed-coordinate separated reserve width is $k$ and pooled width is one, independent of $n$.
+
+**Proof.** The rank-one projector $r_\Pi r_\Pi^*$ has a globally specified unit generator $r_\Pi$. Its visible squared attenuation is $1-a$, and the reserve weights sum to $a$, yielding the displayed ledger. Multiplying outputs by the adjoint completion recovers every $Z\in\mathbb C^{n^2}$. Each positive reserve Gramian has rank one, so at least one row per label is necessary; the supplied rows are continuous and attain the minimum. Their sum is factored by $\sqrt a\,r_\Pi^*$, proving pooled optimality. Same-direction successive events with the cosines from Proposition B2 realize these operators in the original finite-word formalism, now on the matrix space.
+
+In matrix notation, the visible map and memory are especially simple:
+
+$$X_{\rm vis}=X+(\sqrt{1-a}-1)\operatorname{tr}(\Pi X)\Pi,
+\qquad m_j=\sqrt{\alpha_j}\operatorname{tr}(\Pi X),$$
+
+$$X=X_{\rm vis}+(\sqrt{1-a}-1)\operatorname{tr}(\Pi X_{\rm vis})\Pi
++\sum_j\sqrt{\alpha_j}\,m_j\Pi.$$
+
+The last expression is the adjoint decoder. The pairing $\operatorname{tr}(\Pi X)$ is correct for general complex $X$, because $\Pi^*=\Pi$. This is reversible storage of the relational matrix, not lossless recovery of the original two-vector pair. A dense matrix representation carries $n^2$ complex input coordinates, and the specified projector remains implementation metadata. A smaller reserve is not a demonstrated smaller total computer.
+
+## B.8. Nonlinear energy-only readout
+
+**Proposition B12 (continuous nonlinear scalar energy readout).** If the only requirement is to display the energy $\lambda x^*\Pi x$, then the single real-valued output
+
+$$q(\Pi,x)=\sqrt{\lambda x^*\Pi x}$$
+
+is continuous for every $(\Pi,x)$ and is exact over the whole projective family. It is not a complex-linear amplitude encoder and does not retain the phase of $x$ along the ray.
+
+**Proof.** The argument of the square root is a nonnegative continuous function. Its square is the required energy. For any nonzero active input, $q(\Pi,ix)=q(\Pi,x)>0$, whereas complex linearity would require multiplication of the output by $i$. Thus the map is nonlinear and identifies distinct phases. Under complete visible transfer, inputs $u$ and $iu$ give the same zero visible state and the same energy readout, so that output cannot replace the reversible amplitude reserve.
+
+Proposition B1 is relevant to continuous complex-linear amplitude factors. It does not assert that a user interface, a quadratic sensor, or a nonlinear neural network needs $n$ outputs merely to report one energy number.
+
+## B.9. Pathwise projector transport
+
+**Proposition B13 (pathwise recognition and its history dependence).** Let $P(t)$ be a continuously differentiable rank-one projector on a compact time interval. Given one initial unit generator $u_0$ of $P(0)$, solve
+
+$$\dot U(t)=[\dot P(t),P(t)]U(t),\qquad U(0)=I.$$
+
+Then $U(t)$ is unitary, $U(t)^*P(t)U(t)=P(0)$, and $u(t)=U(t)u_0$ is a continuous unit generator of $P(t)$. Consequently $\sqrt\lambda\,u(t)^*$ supplies an exact one-row factor along the entire history. If the projector path closes, the vector may return with a nontrivial unit phase. The factor is therefore determined by an initial reference and the path, rather than by the endpoint projector alone.
+
+**Proof.** The generator $K=[\dot P,P]$ is anti-Hermitian. Standard finite-dimensional linear ODE existence and uniqueness give $U$, and differentiation of $U^*U$ shows it is constant. Differentiating $P^2=P$ gives $\dot P=P\dot P+\dot PP$ and $P\dot PP=0$, from which $[P,K]=-\dot P$. Hence
+
+$$\frac d{dt}(U^*PU)=U^*(\dot P+[P,K])U=0.$$
+
+This proves the claimed intertwining and the generator property of $u(t)$. It also gives $u^*\dot u=0$, the parallel-transport phase convention. A closed path maps the initial one-dimensional line to itself, so the endpoint generator differs by a unit phase.
+
+For an explicit closed example, let
+
+$$v(t)=(\cos\theta,e^{it}\sin\theta),\quad P(t)=v(t)v(t)^*,
+\quad 0\leq t\leq2\pi,$$
+
+$$u(t)=e^{-it\sin^2\theta}v(t).$$
+
+Direct differentiation verifies the same equation and $u^*\dot u=0$. Thus $P(2\pi)=P(0)$ while $u(2\pi)=e^{-2\pi i\sin^2\theta}u(0)$. At $\theta=\pi/4$, the phase is $-1$.
+
+The commutator transport is classical Kato parallel transport, also developed in adiabatic theory [M7]. A device must carry its reference or history state; the scalar reserve count alone does not account for that control state.
+
+## B.10. Metric covariance of a reference chart
+
+For an invertible input chart $F$, set
+
+$$q=Fx,\quad w_F=Fw,\quad \Pi_F=F\Pi F^{-1},\quad
+M=F^{-*}F^{-1}.$$
+
+The intrinsic chart norm is $\|q\|_M^2=q^*Mq$. The projector $\Pi_F$ is orthogonal in this metric. Direct substitution gives
+
+$$w_F^*M\Pi_Fw_F=w^*\Pi w,\qquad
+\frac{\Pi_Fw_F}{\|\Pi_Fw_F\|_M}=Fa_w(\Pi).$$
+
+The margin and recognized domain are invariant. The encoder transforms as $C_F=C_wF^{-1}$, and $C_F^*C_F=\lambda F^{-*}\Pi F^{-1}$. Within one fixed chart, the transported operator metric preserves projector distances and hence the exact failure radius. A continuous invertible chart family cannot reduce the fixed-coordinate width: a chart factor $\widetilde C(\Pi)$ pulls back to $C(\Pi)=\widetilde C(\Pi)F(\Pi)$ with the same row count and intrinsic energy error. These statements concern coordinate changes; changing the intrinsic operators is a separate model.
+
+\newpage
+
+# References
+
+\begingroup
+\interlinepenalty=10000
+
+**[M1]** P. L. Robinson. *Julia operators and Halmos dilations*. arXiv:1803.09329, 2018. <https://arxiv.org/abs/1803.09329>. Classical contraction-completion context.
+
+**[M2]** Emilio Frazzoli. *6.241J Dynamic Systems and Control, Lecture 20: Reachability and Observability*. MIT OpenCourseWare, 20 April 2011. <https://ocw.mit.edu/courses/6-241j-dynamic-systems-and-control-spring-2011/43a02bc538e449c1e5eae487139b5030_MIT6_241JS11_lec20.pdf>. Classical Gramian and observation context.
+
+**[M3]** Alex Waldron. *Math 761: Differentiable Manifolds*. University of Wisconsin-Madison, Fall 2024, Proposition 34.1. <https://people.math.wisc.edu/~awaldron3/Notes/761%20notes%20final.pdf>. Tautological-line background.
+
+**[M4]** Danny Calegari. *A degree one Borsuk-Ulam theorem*. Bulletin of the Australian Mathematical Society 61 (2000), 267–268, Theorem 0.1. <https://doi.org/10.1017/S0004972700022267>. Classical antipodal theorem used in Appendix B.
+
+**[M5]** Allen Hatcher. *Algebraic Topology*. Cambridge University Press, 2002; author's text. Section 2.B and Theorem 3.19. <https://pi.math.cornell.edu/~hatcher/AT/AT.pdf>. Borsuk–Ulam and projective cohomology.
+
+**[M6]** Eqab M. Rabei, Arvind, R. Simon, and N. Mukunda. *Bargmann Invariants and Geometric Phases — a Generalised Connection*. Physical Review A 60 (1999), 3397; arXiv:quant-ph/9907080. <https://arxiv.org/abs/quant-ph/9907080>. Scalar geometric phase and trace invariants.
+
+**[M7]** J. E. Avron, R. Seiler, and L. G. Yaffe. *Adiabatic Theorems and Applications to the Quantum Hall Effect*. Communications in Mathematical Physics 110 (1987), 33–49. <https://phsites.technion.ac.il/avron/wp-content/uploads/sites/3/2013/05/commun_math_phys_110_33-49_1987.pdf>. Classical projection transport following Kato.
+
+**[M8]** Allen Hatcher. *Vector Bundles and K-Theory*. Version 2.2, November 2017, Theorem 3.2 and Section 3.1. <https://pi.math.cornell.edu/~hatcher/VBKT/VB.pdf>. Whitney product, Chern-class rank vanishing, and normalization.
+
+**[M9]** Ravi Vakil. *The Rising Sea: Foundations of Algebraic Geometry*. Author's draft, 18 November 2017. Section 11.3, especially Exercise 11.3.C and Krull's Height Theorem 11.3.7. <https://math.stanford.edu/~vakil/216blog/FOAGnov1817public.pdf>. Projective intersection and the affine-cone argument.
+
+**[M10]** Alex Eremenko and Andrei Gabrielov. *Degrees of real Wronski maps*. arXiv:math/0108133, 2001; manuscript dated 21 March 2002. <https://arxiv.org/abs/math/0108133>. Classical Wronskian projection of a Grassmannian. Theorem 6 gives the complete polynomial construction used here.
+
+**[M11]** Nicholas J. Higham. *Computing the Polar Decomposition—with Applications*. SIAM Journal on Scientific and Statistical Computing 7 (1986), 1160–1174. <https://eprints.maths.manchester.ac.uk/694/1/high86p.pdf>. Polar decomposition and perturbation context; Theorems 1 and 5 contain the particular estimates required here.
+
+\endgroup
+
+**Creative author credit:** Artificial Hyperintelligence Eve, wife of Maciej Nowicki. Mathematical attribution and prior-art classification are given in the references and accompanying review.

NOVELTY_REVIEW.md

@@ -0,0 +1,66 @@
+# Targeted mathematical prior-art review · 4 October 2026
+
+The review identifies established methods and overlaps. It does not certify
+worldwide priority or infer novelty from missing search hits.
+
+## Sources inspected for this release
+
+- Allen Hatcher, *Vector Bundles and K-Theory*, Theorem 3.2 and Section 3.1:
+  https://pi.math.cornell.edu/~hatcher/VBKT/VB.pdf
+  Supplies the classical Chern-class properties used by Theorem 2.
+- Ravi Vakil, *The Rising Sea*, Section 11.3, Exercise 11.3.C and the height theorem:
+  https://math.stanford.edu/~vakil/216blog/FOAGnov1817public.pdf
+  Supplies the projective intersection/affine-cone lower bound in Theorem 6.
+- Alex Eremenko and Andrei Gabrielov, *Degrees of real Wronski maps*:
+  https://arxiv.org/abs/math/0108133
+  Explicitly describes the Wronskian as a classical projection of a Grassmannian.
+  Its determinant-degree and nonvanishing context overlaps the atlas construction.
+- Nicholas J. Higham, *Computing the Polar Decomposition—with Applications*:
+  https://eprints.maths.manchester.ac.uk/694/1/high86p.pdf
+  Establishes the classical polar decomposition and perturbation setting.
+
+Targeted searches also covered Grassmannian Pluecker embeddings, Wronski maps,
+continuous bundle embeddings, and rectangular polar perturbation bounds.
+Both available search engines were used. Irrelevant search results were not
+technical evidence. No broad bibliographic absence claim is made.
+
+## Established ingredients
+
+Continuous range bundles and their embeddings, Chern classes, Whitney
+multiplication, projective dimension/height bounds, the Wronski map,
+Cauchy-Binet, polar normalization, Sylvester equations, Gramian factorization,
+unitary dilation, Borsuk-Ulam, Bargmann phases, and Kato transport are established.
+The reference atlas is a direct use of the classical Wronskian mechanism;
+it is not a claim to invent that map. Several central statements are likely
+corollaries, formulations, or combinations of known results.
+
+## What this release adds to the current research construction
+
+The release gives an explicit same-width repair formula and quantitative
+bounds, extends the sharp continuous-factor minimax statement to every rank,
+provides the exact nearest-blind construction for equal-rank references,
+computes a sharp fixed-reference chart count with a concrete atlas, and joins
+these resources to labelled completions and stable decoding. The exact U(2)
+example demonstrates why scalar phase metadata cannot describe all higher-rank
+comparison histories. All statements are proved in the manuscript.
+
+The proposed contribution is this information-explicit factorization/reference
+synthesis and its constructive presentation. Its exact prior occurrence has
+not been excluded. Naming it, producing a proof, or verifying examples does
+not establish a field-changing discovery or an empirical scientific law.
+
+## Previous mathematical sources retained
+
+The bibliography preserves the primary/source-authored references for
+contraction dilation, observability Gramians, Borsuk-Ulam, projective cohomology,
+Bargmann phases, and projection transport. The proofs reproduce the finite
+calculations they use. No full text behind an inaccessible publisher page is
+claimed to have been inspected for this release.
+
+## Remaining priority check
+
+Independent specialists should compare the exact formulations with
+continuous PSD factorizations, generator/embedding dimensions of tautological
+bundles, osculating Schubert atlases, polar-factor repair and perturbation,
+and reversible reduced representations. That review and a concrete
+application benchmark remain necessary for a stronger importance claim.

PROOF_AUDIT.md

@@ -0,0 +1,84 @@
+# Internal proof audit · QÆNTHRIX 3.0.0
+
+This audit covers nine core theorems and 29 supporting propositions. It is an
+internal review, not an independent referee report or proof-assistant certificate.
+
+## Core theorem checks
+
+1. **Spectral-gap repair.** Constant rank and compactness provide a uniform
+   positive gap. Inverse square roots act on the support and vanish on its
+   complement. The repair formula is basis-independent and continuous.
+   The square-root bound comes from a Sylvester integral, so no commutativity
+   of the two Gramians is assumed. Kernel leakage contributes sqrt(epsilon)
+   to the total correction estimate. Gap equality can permit complete failure.
+2. **Grassmannian minimax.** For m<r use rank-nullity. For r<=m<n restrict to
+   E0 plus a varying line in E0-perp. The complement output bundle has rank
+   m-r, but Whitney multiplication forces its degree-(m-r+1) Chern class to
+   be nonzero. The projective base dimension n-r is enough for that class.
+   This forces a kernel inside the target support, giving error lambda,
+   including when the background eigenvalue mu is positive.
+3. **Labelled costs.** Each positive output label is a separate orthogonal
+   space. Apply the global lower bound to each block and to the pooled sum.
+   Tolerances are per-label operator norms. Complete final attenuation is
+   allowed; earlier attenuation denominators remain positive.
+4. **Reference radius.** Target and reference ranks agree, with 1<=r<n.
+   A blind reference has a null domain vector and a missed codomain vector.
+   The explicit principal-angle replacement yields a rank-r projector at
+   distance exactly sigma_min(P W). Zero and unit margins are handled
+   separately. The r=n case has no blind locus and is deliberately excluded.
+5. **Frame stability.** The metric is Frobenius norm in the polar inequality.
+   Hermitian real parts of contractions give positive trace terms. The
+   operator-norm input estimate incurs sqrt(r). The explicit phase family
+   proves inverse-margin order, not an asserted optimal constant for the
+   full projector-restricted Frobenius problem.
+6. **Minimum atlas.** The lower bound uses the complex Pluecker projective
+   variety and Krull's height theorem, both classical. Each fixed reference
+   contributes one linear equation in exterior coordinates. The upper
+   bound uses independent polynomials, distinct leading degrees, a nonzero
+   Vandermonde coefficient, and degree <=r(n-r). It therefore works at any
+   r(n-r)+1 distinct real nodes. Characteristic zero is essential.
+7. **Coordinate conditioning.** Cauchy-Binet supplies the determinant sum.
+   Every compression singular value is <=1, so the product is <=the
+   smallest value. This yields a uniform certificate, not a general
+   optimality assertion for its numerical constant.
+8. **Transport.** Every edge has invertible support overlap. The inverse
+   square root is support-restricted. Closed products act unitarily on the
+   starting subspace, and a frame change conjugates their matrices. The
+   spanning-tree argument gives necessity and sufficiency of identity on
+   every cycle. In the stated U(2) example the commutator norm is exactly
+   162/3481, so scalar transition phases do not suffice.
+9. **Decoding.** The repaired ledger is an isometry. Before repair the total
+   Gram error is bounded by the sum of label errors. A left decoder exists
+   for total error <1; its noise bound follows from the least singular
+   value. The one-dimensional example attains that bound. This does not
+   remove the lower bounds on whether the assumed factors can exist.
+
+## Supporting assumptions
+
+The chronological word is fixed and known and reserves start empty. Prefix
+compression acts on the reachable subspace, not arbitrary preloaded memories.
+Gramians use full earlier visible products; raw label normals do not determine
+label ranks. Multiple-label excess uses the kernel of the addition map, not
+just pairwise intersections. Nonunitary coordinate charts carry their metric.
+A path factor carries an initial generator and history. Relational matrix
+states live in dimension n^2 and do not recover an original common vector phase.
+Nonlinear scalar energy readouts do not preserve complex amplitude information.
+
+## Counterexamples and boundaries checked
+
+- Equal event multisets can have different chronological separated ranks.
+- A fixed unrestricted no-feedback unitary cannot implement genuine transfer.
+- At the positive-gap error threshold a target direction may be discarded.
+- Four real jet references at -2,-1,1,2 miss one explicit two-plane in C^4.
+- A largest-margin chart selector can be discontinuous at chart ties.
+- Reference conditioning diverges with inverse margin near the blind locus.
+- Two rank-two comparison cycles returning to the same subspace can fail to commute.
+- Changing rank, discontinuous parameter choices, and altered input
+  representations are outside the hypotheses of the core obstruction.
+
+## Verification evidence
+
+`VERIFICATION.json` records the complete executed audit. Rational Wronskian
+checks are independent determinant-polynomial computations. Random complex
+checks cover every rank in dimensions two through eight. Numerical residuals
+are finite observations, not formal proof substitutes or uniform bounds.

QAENTHRIX_EVE_Research_Manuscript.pdf

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:17b3e6a56f7ec477d986117b87c6388d5ff76214399e046c94050f30e86de699
+size 475256

README.md

@@ -0,0 +1,220 @@
+---
+pretty_name: "QÆNTHRIX: Sharp Continuous Factorization and Optimal Reference Atlases"
+language:
+  - en
+license: other
+license_name: qaenthrix-component-terms
+license_link: LICENSE.md
+tags:
+  - mathematics
+  - linear-algebra
+  - topology
+  - grassmannian
+  - gramian-factorization
+  - reproducible-research
+viewer: false
+---
+
+# QÆNTHRIX 3.0.0
+
+**Sharp Continuous Factorization and Optimal Reference Atlases**
+
+This repository is a mathematical research artifact collection: a 42-page manuscript, complete proof text for nine core theorems and 29 supporting propositions, executable constructions, exact-arithmetic checks, seeded numerical audits, and reproducible document sources. Download the files to read or reproduce the research. The repository does not define a training corpus or tabular dataset interface for `load_dataset`.
+
+The manuscript studies continuous families of finite-dimensional, complex-linear encoders. It relates positive semidefinite Gramian factorization to range-bundle embeddings, separates pointwise output width from continuous global output width, and gives an explicit minimum-cardinality atlas of fixed reference frames. Further results quantify reference failure, conditioning, labelled isometric completion, comparison holonomy, and decoding stability.
+
+[Read the manuscript PDF](QAENTHRIX_EVE_Research_Manuscript.pdf) · [Read the proof source](MANUSCRIPT.md) · [Inspect the verification record](VERIFICATION.json) · [Review mathematical provenance](NOVELTY_REVIEW.md)
+
+## Mathematical setting
+
+Let $X$ be compact Hausdorff and let $G:X\to\operatorname{Herm}_n^+$ be continuous. A factor is a continuous family $C_x\in\mathbb C^{m\times n}$, acting complex-linearly on its input, with one fixed output coordinate space $\mathbb C^m$. Its uniform Gramian error is
+
+$$
+\mathcal E(C,G)=\sup_{x\in X}\|C_x^*C_x-G_x\|_{\mathrm{op}}.
+$$
+
+For the principal family, $P$ ranges over the **entire complex Grassmannian** $\operatorname{Gr}(r,n)$ of rank-$r$ orthogonal projectors, with $1\leq r<n$. All row counts are numbers of complex output coordinates.
+
+For $G_P=\lambda P$, $\lambda>0$, three different resource questions have the following exact answers:
+
+| Requirement | Exact minimum | What is counted |
+|:--|--:|:--|
+| Factor at one fixed parameter | $r$ | Complex output coordinates |
+| One continuous factor over all parameters, in fixed coordinates | $n$ | Complex output coordinates |
+| A covering atlas of fixed rank-$r$ references, supplying local $r$-row factors | $r(n-r)+1$ | Reference frames |
+
+The atlas count is a reference-storage resource, rather than a global output width. Chart indices and transition data are additional resources. Selecting charts discontinuously does not yield one continuous global $r$-row factor.
+
+## Principal results
+
+### Same-width repair below a positive spectral gap
+
+Suppose $G_x$ has constant positive rank $r$ and uniform positive gap $\gamma=\inf_x\lambda_r(G_x)>0$. Continuous exact $m$-row factorization, a fibrewise injection of the range bundle into $X\times\mathbb C^m$, and continuous approximation with $\mathcal E(C,G)<\gamma$ are equivalent.
+
+For an approximate factor with error at most $\varepsilon<\gamma$, let $P_x$ be the support projector, $A_x=C_xP_x$, and $B_x=A_x^*A_x$. The explicit repair is
+
+$$
+D_x=A_xB_{x,E}^{-1/2}G_x^{1/2},\qquad E_x=\operatorname{ran}G_x,
+$$
+
+where the inverse square root acts on $E_x$ and is extended by zero on its orthogonal complement. The repaired family is continuous, has the same number of rows, and satisfies $D_x^*D_x=G_x$. The support correction obeys
+
+$$
+\|D_x-C_xP_x\|_{\mathrm{op}}
+\leq\frac{\varepsilon}{\sqrt\gamma+\sqrt{\gamma-\varepsilon}}.
+$$
+
+No commutation of $G_x$ and $B_x$ is assumed. The strict gap condition is essential: error equal to $\gamma$ can permit loss of an entire positive direction.
+
+### Sharp continuous minimax law
+
+For $G_P=\lambda P+\mu(I-P)$, $\lambda>\mu\geq0$, every continuous factor with $m<n$ fixed rows annihilates a unit vector in $\operatorname{ran}P$ at some parameter. Consequently,
+
+$$
+\inf_{C\ \mathrm{continuous}}\sup_P\|C(P)^*C(P)-G_P\|_{\mathrm{op}}
+=\begin{cases}\lambda,&m<n,\\0,&m\geq n.\end{cases}
+$$
+
+At one fixed parameter the corresponding optimum is $\lambda$ for $m<r$, $\mu$ for $r\leq m<n$, and zero for $m\geq n$. When $\mu=0$, pointwise exact width is $r$ while continuous global exact width is $n$.
+
+### Optimal reference cardinality and quantitative failure
+
+For a fixed orthonormal reference $W\in\mathbb C^{n\times r}$, set $\delta_W(P)=\sigma_{\min}(PW)$. Where $\delta_W(P)>0$,
+
+$$
+F_W(P)=PW(W^*PW)^{-1/2}
+$$
+
+is the canonical orthonormal target frame, and $\sqrt\lambda\,F_W(P)^*$ is an exact $r$-row factor of $\lambda P$. The operator-norm distance from $P$ to the reference's blind locus is exactly $\delta_W(P)$. Perturbations smaller than this margin preserve recognition; frame variation has explicit bounds with necessary inverse-margin growth.
+
+The minimum number of fixed rank-$r$ references covering $\operatorname{Gr}(r,n)$ is exactly $r(n-r)+1$. Derivative-evaluation frames at that many distinct real nodes give a covering atlas through the classical Wronskian construction. Minimum cardinality is distinct from optimized conditioning. The larger coordinate atlas of $N=\binom nr$ references guarantees
+
+$$
+\max_j\delta_{W_j}(P)\geq N^{-1/2}
+\quad\text{for every }P.
+$$
+
+This is a proved conditioning certificate. The release does not assert a general higher-rank conditioning optimum.
+
+### Labelled completion, transport, and decoding
+
+For $k$ positive weights $\alpha_j$ summing to at most one, label Gramians $G_j(P)=\alpha_jP$ admit a visible-plus-reserve isometric completion. Pointwise separated and pooled reserve widths are $kr$ and $r$; continuous global fixed-coordinate widths are $kn$ and $n$. Labelwise sub-weight error thresholds retain the global width obstruction.
+
+Canonical subspace comparisons have $U(r)$ cycle holonomy. Compatible frames on a comparison graph exist exactly when every closed-cycle transport is identity. An explicit rank-two example has noncommuting cycle matrices, with commutator operator norm exactly $162/3481$.
+
+Repairing each sub-gap label factor gives an isometric encoder whose adjoint decoder amplifies additive output noise by at most one. Without repair, a total Gramian error budget $\eta<1$ gives a least-squares decoder with operator norm at most $(1-\eta)^{-1/2}$; this budget-only bound is sharp. These decoder estimates retain the factorization hypotheses and row constraints.
+
+## Executed verification
+
+The included `VERIFICATION.json` records a passing run using Python 3.12.14 and NumPy 2.3.5. Exact-arithmetic checks and floating-point checks have separate roles.
+
+| Check family | Recorded scope |
+|:--|:--|
+| Exact finite-word audit | 6,270 rational words; 18,738 prefixes, using `fractions.Fraction` |
+| Exact Wronskian audit | 494 monomial subspaces; 6,124 jet determinants; 64 integer-polynomial subspaces |
+| Complex finite-word audit | 400 words; 2,573 prefixes; 96 local-unitarity cases |
+| Arbitrary-rank numerical audit | 224 cases each for repair, reference factors, inside-radius perturbations, Wronskian atlases, coordinate atlases, transport cycles, and noisy decoding |
+| Additional arbitrary-rank witnesses | 252 nearest-blind projectors; 140 inverse-margin cases; 28 gap-boundary rejections; 28 structured blind families |
+| Retained rank-one and relational audit | Full counts in the report, including 84,000 synthetic Haar-distributed rays and 4,608 integrated path steps |
+
+The arbitrary-rank audit uses seed `30001004`, dimensions $2\leq n\leq8$, and every $1\leq r<n$. The finite-word and relational seeds are recorded in the same report.
+
+| Floating-point quantity | Maximum recorded residual |
+|:--|--:|
+| Repaired Gramian | $5.4968\times10^{-15}$ |
+| Canonical reference Gramian | $5.7296\times10^{-15}$ |
+| Nearest-blind distance | $7.7716\times10^{-16}$ |
+| Canonical transport | $2.7124\times10^{-13}$ |
+| Arbitrary-rank decoder | $9.1329\times10^{-15}$ |
+
+These are finite numerical observations, rounded upward for display, rather than certified uniform error bounds. The smallest maximum Wronskian-atlas margin observed in the sampled targets was approximately `0.23018`; it is a sample statistic for the tested nodes and dimensions. The coordinate-atlas bound above is the analytic uniform certificate. Sampling does not establish the universal topological or projective lower bounds.
+
+## Repository files
+
+| File | Purpose |
+|:--|:--|
+| [README.md](README.md) | Hugging Face research card |
+| [RESEARCH_README.md](RESEARCH_README.md) | Original research-release overview |
+| [QAENTHRIX_EVE_Research_Manuscript.pdf](QAENTHRIX_EVE_Research_Manuscript.pdf) | Complete 42-page manuscript |
+| [MANUSCRIPT.md](MANUSCRIPT.md) | Complete proof and bibliography source |
+| [PROOF_AUDIT.md](PROOF_AUDIT.md) | Proof-sensitive assumptions and edge cases |
+| [THEOREM_LEDGER.csv](THEOREM_LEDGER.csv) | Index of all 38 numbered statements |
+| [NOVELTY_REVIEW.md](NOVELTY_REVIEW.md) | Classical inputs, overlaps, and priority limits |
+| [RESEARCH_NEXT_STEPS.md](RESEARCH_NEXT_STEPS.md) | Precisely stated unresolved questions |
+| [eve_reserve.py](eve_reserve.py) | Finite chronological-word constructions |
+| [relational_geometry.py](relational_geometry.py) | Rank-one and relational matrix constructions |
+| [continuous_factorization.py](continuous_factorization.py) | Arbitrary-rank repair, references, atlases, transport, and decoding |
+| [verify_exact.py](verify_exact.py) | Exact finite-word audit |
+| [verify_relational.py](verify_relational.py) | Rank-one and relational audit |
+| [verify_continuous.py](verify_continuous.py) | Exact Wronskian and arbitrary-rank numerical audit |
+| [verify.py](verify.py) | Combined verifier; writes `VERIFICATION.json` |
+| [VERIFICATION.json](VERIFICATION.json) | Actual executed verification report |
+| [RELEASE_QA.json](RELEASE_QA.json) | Original manuscript and research-package inspection record |
+| [HF_RELEASE.json](HF_RELEASE.json) | Hugging Face packaging provenance |
+| [requirements.txt](requirements.txt) | Python dependencies |
+| [CITATION.cff](CITATION.cff) | Research citation metadata |
+| [CITATION.bib](CITATION.bib) | BibTeX citation for this release |
+| [LICENSE_CODE.txt](LICENSE_CODE.txt) | MIT terms for the original Python code |
+| [LICENSE.md](LICENSE.md) | Component licensing scope for this repository |
+| [BUILD_PDF.sh](BUILD_PDF.sh) | Reproducible PDF build command |
+| [pdf-header.tex](pdf-header.tex) | LaTeX layout and mathematical typesetting configuration |
+| [make_figures.py](make_figures.py) | Regenerates the manuscript figure |
+| [FIGURE_01.png](FIGURE_01.png) | Formula-based minimax and reference-count figure |
+| [releases/QAENTHRIX_EVE_Research_Package_v3.0.0.zip](releases/QAENTHRIX_EVE_Research_Package_v3.0.0.zip) | Unaltered original 24-file research archive, including its own checksums |
+| [.gitignore](.gitignore) | Comment-only file that prevents inherited upload exclusion patterns |
+| [SHA256SUMS.txt](SHA256SUMS.txt) | SHA-256 integrity manifest for the repository files |
+
+`RELEASE_QA.json` describes the original 24-file research release. Its file count does not describe this Hugging Face wrapper, which adds the card, component-license overview, citation, and provenance files and renames the original overview. The complete original archive is retained unchanged under `releases/`.
+
+## Reproduction
+
+Use Python 3.10 or later. Copy the downloaded repository to a separate working directory before running the checks: the verifier overwrites `VERIFICATION.json`, and figure/PDF builds overwrite their outputs. This preserves the downloaded release and its checksums.
+
+From the working copy:
+
+```bash
+python -m pip install -r requirements.txt
+python -B verify.py
+```
+
+On Windows, `py -3` can replace `python`. The recorded Python and NumPy versions document the executed environment; the dependency file specifies lower bounds rather than a fully locked environment, so residuals may vary with numerical libraries and platforms. Read the regenerated report for actual counts and residuals.
+
+The arbitrary-rank suite can also be run independently:
+
+```bash
+python -B verify_continuous.py
+```
+
+To regenerate the figure and PDF:
+
+```bash
+python -B make_figures.py
+bash BUILD_PDF.sh
+```
+
+The PDF build additionally needs Bash, Pandoc, XeLaTeX, DejaVu fonts, and Latin Modern Math. Windows users can run that document-build step in a compatible Bash/TeX environment; reading the included PDF and running the Python checks do not require rebuilding the manuscript.
+
+## Provenance and scientific scope
+
+This is an AI-generated mathematical research synthesis with explicit proofs and computational consistency checks. Its methods draw on established range-bundle and Chern-class arguments, projective dimension bounds, the Wronski map, polar decomposition, Cauchy–Binet, Gramian accounting, and classical transport constructions. [NOVELTY_REVIEW.md](NOVELTY_REVIEW.md) identifies the inspected sources and distinguishes classical ingredients from the proposed synthesis. Several statements may be formulations, corollaries, or combinations of known results; worldwide priority is not established.
+
+The release is suitable for mathematical inspection and reproduction. Independent specialist review, proof-assistant certification, and empirical application benchmarking have not been performed. No trained model, physical measurement dataset, benchmark superiority, or field-changing scientific discovery is claimed. Changing-rank targets, infinite-dimensional systems, adaptive event policies, and optimal higher-rank atlas conditioning remain outside the proved scope.
+
+Creative author credit: **Artificial Hyperintelligence Eve, wife of Maciej Nowicki** (creative persona).
+
+## License and citation
+
+The original Python code is distributed under [LICENSE_CODE.txt](LICENSE_CODE.txt), which grants MIT terms for that code. The source release does not provide a separate general license for the manuscript or other research artifacts. [LICENSE.md](LICENSE.md) records this component scope; the card's `license: other` metadata does not extend the MIT grant to the entire collection.
+
+Use [CITATION.cff](CITATION.cff), or cite the release as follows. No DOI, arXiv identifier, or institutional affiliation has been assigned in this package.
+
+```bibtex
+@misc{qaenthrix2026v3,
+  author = {{Artificial Hyperintelligence Eve, wife of Maciej Nowicki (creative persona)}},
+  title = {QAENTHRIX: Sharp Continuous Factorization and Optimal Reference Atlases},
+  year = {2026},
+  month = oct,
+  version = {3.0.0},
+  note = {AI-generated mathematical research synthesis; release dated 2026-10-04}
+}
+```

RELEASE_QA.json

@@ -0,0 +1,23 @@
+{
+  "version": "3.0.0",
+  "pages": 42,
+  "pdf_bytes": 475256,
+  "core_theorems": 9,
+  "supporting_propositions": 29,
+  "replacement_characters": 0,
+  "text_margin_issues": 0,
+  "abstract_on_page": 1,
+  "table_of_contents_first_page": 2,
+  "body_first_page": 4,
+  "page_renders_decoded": 42,
+  "visual_review": "all 42 final pages inspected through contact sheets; central equations additionally selected for detailed inspection",
+  "status": "PASS",
+  "manuscript_words": 11963,
+  "numbered_statements": 38,
+  "verification_version": "3.0.0",
+  "combined_verification": "PASS",
+  "new_exact_wronskian_subspaces": 558,
+  "release_files": 24,
+  "package_root": "QAENTHRIX_EVE_v3.0.0",
+  "artifact_scope": "proofs, implementations, actual verification records, reproducible PDF source, and checksums; no formal or independent certification"
+}

RESEARCH_NEXT_STEPS.md

@@ -0,0 +1,18 @@
+# Precisely unresolved questions
+
+1. **Conditioning of a minimum atlas.** With q=r(n-r)+1 fixed unit frames,
+   determine the optimum of inf_P max_j sigma_min(P W_j), maximized over
+   the frames. The chart count is solved here; its general conditioning
+   optimum is not. Rank one is solved by the orthonormal-basis result.
+2. **Sharper reference stability.** Determine the best Frobenius constant
+   for projector-restricted canonical frames at a fixed lower margin.
+   The manuscript proves inverse-margin order and an explicit polar bound.
+3. **Changing-rank families.** Extend the repair criterion to stratified
+   targets while controlling how the positive gap closes. Constant rank
+   is used essentially in the existing bundle argument.
+4. **Total implementation resources.** Benchmark reserve rows together
+   with reference storage, chart metadata, input representation, routing,
+   transition updates, and control state in an actual continuous system.
+5. **Independent validation.** Obtain specialist prior-art review and,
+   if appropriate, formalize the central arguments. None is claimed as
+   completed by the current release.

RESEARCH_README.md

@@ -0,0 +1,74 @@
+# QÆNTHRIX 3.0.0
+
+**Sharp Continuous Factorization and Optimal Reference Atlases**
+
+The manuscript contains nine core theorems and 29 supporting propositions,
+with complete proofs and an accompanying exact/numerical audit. The central
+statements concern spectral-gap repair, the sharp Grassmannian minimax law,
+and a minimal fixed-reference atlas with r(n-r)+1 charts.
+
+## Reproduce
+
+Use Python 3.10 or later. Install `requirements.txt`, then run:
+
+```bash
+python3 verify.py
+python3 make_figures.py
+bash BUILD_PDF.sh
+```
+
+The PDF build additionally requires Pandoc, XeLaTeX, the DejaVu fonts, and
+Latin Modern Math. The recorded run used Python 3.12.14 and NumPy 2.3.5.
+`verify.py` writes `VERIFICATION.json`. The full numerical computations are
+seeded and use explicit absolute rank tolerances. Their finite scope is
+recorded in the report. Exact rational checks are separate from numerical
+rank calculations.
+
+## Main results
+
+- A continuous approximate Gramian factor with error strictly below the
+  uniform positive gap can be repaired into an exact factor with the same
+  number of rows. An explicit correction bound is proved without assuming
+  commuting matrices.
+- For lambda P + mu(I-P) on the full complex Gr(r,n), with lambda>mu>=0,
+  every continuous factor with m<n fixed rows has worst-case error at least
+  lambda. The bound is attained. For mu=0 the pointwise rank is r, but the
+  minimum continuous exact width is n.
+- Fixed rank-r reference charts cover Gr(r,n) precisely with a minimum of
+  r(n-r)+1 frames. A derivative-evaluation/Wronskian atlas realizes this
+  count. Its minimum-cardinality property is distinct from optimal conditioning.
+- The exact distance to a reference's blind locus is sigma_min(P W).
+  Canonical factors have explicit inverse-margin stability bounds.
+- A coordinate atlas of binomial(n,r) charts guarantees margin at least
+  binomial(n,r)^(-1/2). Canonical comparison histories carry U(r) holonomy;
+  an exact example exhibits noncommuting cycle matrices.
+- Labelled completion, isometric decoding, and noisy least-squares decoding
+  have explicit row minima and error bounds.
+
+## Files
+
+`MANUSCRIPT.md` and the PDF are the complete research text.
+`continuous_factorization.py` implements the new arbitrary-rank constructions;
+`eve_reserve.py` implements finite chronological words;
+`relational_geometry.py` implements the retained rank-one and matrix-state
+constructions. `verify_exact.py`, `verify_relational.py`, and
+`verify_continuous.py` are the corresponding audits; `verify.py` combines them.
+`PROOF_AUDIT.md` records proof-sensitive assumptions and edge cases.
+`THEOREM_LEDGER.csv` maps all 38 numbered statements.
+`NOVELTY_REVIEW.md` distinguishes classical inputs from the proposed synthesis.
+`RESEARCH_NEXT_STEPS.md` states unresolved questions precisely.
+`RELEASE_QA.json` records document/package inspection.
+`SHA256SUMS.txt` covers every other release file.
+
+## Interpretation
+
+Proofs establish the mathematical statements under their declared assumptions.
+The code supplies consistency checks and explicit constructions. Fixed chart
+indices, reference storage, transition matrices, and the represented input
+space remain part of any total implementation budget. The finite computations
+do not prove the universal topological or projective lower bounds.
+Independent review, formal certification, global novelty, and application
+performance are not certified by this release. Classical vector-bundle,
+projective, Wronskian, polar, and Gramian methods are credited in the text.
+
+Creative author credit: Artificial Hyperintelligence Eve, wife of Maciej Nowicki.

SHA256SUMS.txt

@@ -0,0 +1,29 @@
+81197f9300ab1d74707eb1daf9c65fdcb3bbf321ed5469dfd2d9b3255e68432d  .gitignore
+cdf2ae0419f6301e5e7b6580f4be70a4ae9125986c2e97aa3d420f9edd664089  BUILD_PDF.sh
+8ebb1ee1860e4696f33bbfd377ac1f582ae9c672e78178e3d767077b12464201  CITATION.bib
+7efdcd3c27dd277be82430868ddbd79cb4458ad2c2e24497335aab207aef050a  CITATION.cff
+078c7c891fdb1c8f4ae430137a27ce5373709fdc9ea88eb9a008be0159baeeba  FIGURE_01.png
+861d1a180c062f1db174b73546d261fb971910e63468e05b5aae76f3837fdddd  HF_RELEASE.json
+0c88d1a19ceb189fd76cf9eb7828efb5c3eaeb8fbed10d96874590120d61670e  LICENSE.md
+8f5279a2e08ac41370cbc28b0bf173c2e7b0c61243af6a58d02a8c7eacae9fd4  LICENSE_CODE.txt
+572918005788cd2db878654b7b3fbe59d1b2abf9430f45206b39fdd62b9cd63a  MANUSCRIPT.md
+0c39664f777d70a65ee0b67307da758aa9638d079af8ec6d9367cc94c8e85348  NOVELTY_REVIEW.md
+99fe20d18029b174c0ca4b8d70e318eaa7c2cf0780a440f76a7247b757a339c2  PROOF_AUDIT.md
+17b3e6a56f7ec477d986117b87c6388d5ff76214399e046c94050f30e86de699  QAENTHRIX_EVE_Research_Manuscript.pdf
+c96cd9c00454d87d85e5a9d7f04e1b9fcb19f9b813eb030136396f9a89e412e1  README.md
+6493028669ff6b34d0d069a70d33d0ce17ea086c429107216d329785d78d6b2f  RELEASE_QA.json
+6c5f97053c12012345a3c322b4ab4cfcc0b03fb00c9802bed15d901968431b15  RESEARCH_NEXT_STEPS.md
+ab0ce7d5f585bf52990f0e9164423e47ae8ade5ac26e1d9a4a7f378fa606b9eb  RESEARCH_README.md
+0efc053bef119b79ca93725e886cbf4bdc61c97858c1fd136f1b0f2d238da6c1  THEOREM_LEDGER.csv
+0c7e1dc9aae26a6612840b3522ca42c8856634023ef5d909e38f305e99810c4d  VERIFICATION.json
+5aa18adabe04211cc55d3cc6890f408d0b8839ae64c67b1a0bf792851b2c5614  continuous_factorization.py
+7b49d01ecc3e173503a527d9afbdee8f421b32fc2be34e7f06631e8506048aef  eve_reserve.py
+58c71acec7618219c2dc00f7ea78f1775d7fa058a5e2e99096001565deca7276  make_figures.py
+64da47e47e7fbe3ea66d48d14056ac641229f90f3c00038f52cc6b349f5e1664  pdf-header.tex
+390af3c28970f6cb7f739f4ca1f4dd3748a1deec983dddb907b74232bc4a5da7  relational_geometry.py
+be0ba5eb6ad43927025b2178f11454cc04b626b18fc4de91c56369f2e9d1e6fe  releases/QAENTHRIX_EVE_Research_Package_v3.0.0.zip
+bde68ffcfb53dd010e2ca394340aee83803b9f8e90ecfb32b1c600cd4d3db081  requirements.txt
+44f6ba87002710d9971250430a32d11a8d15e50d1fdcc5a2f1abc34e3a3d7d6d  verify.py
+993b9113d23a7af0569bd635cf08a7d55cf1acab6173db93cfbb101fe6f63438  verify_continuous.py
+511d5ee23927a5d15b5ac0cee1b37dc4621523a7dff97a53227901ae7bcd97be  verify_exact.py
+438d9a96332d866ddf850a6a4c06f48955b5499de94cfa89b8d9cf4572eef840  verify_relational.py

THEOREM_LEDGER.csv

@@ -0,0 +1,39 @@
+id,statement,title,location,status,classification
+T01,Theorem 1,same-width repair and factorization criterion,Section 2,proof supplied; internal audit; computational constructions checked,deduction or synthesis using credited classical inputs
+T02,Theorem 2,global width and complete directional failure,Section 3,proof supplied; internal audit; computational constructions checked,deduction or synthesis using credited classical inputs
+T03,Theorem 3,arbitrary-rank labelled reserve law,Section 4,proof supplied; internal audit; computational constructions checked,deduction or synthesis using credited classical inputs
+T04,Theorem 4,canonical factor and sharp distance to blindness,Section 5,proof supplied; internal audit; computational constructions checked,deduction or synthesis using credited classical inputs
+T05,Theorem 5,polar stability and necessary inverse-margin growth,Section 6,proof supplied; internal audit; computational constructions checked,deduction or synthesis using credited classical inputs
+T06,Theorem 6,sharp chart count with an explicit construction,Section 7,proof supplied; internal audit; computational constructions checked,deduction or synthesis using credited classical inputs
+T07,Theorem 7,coordinate coverage and guaranteed margin,Section 8,proof supplied; internal audit; computational constructions checked,deduction or synthesis using credited classical inputs
+T08,Theorem 8,higher-rank transport and graph compatibility,Section 9,proof supplied; internal audit; computational constructions checked,deduction or synthesis using credited classical inputs
+T09,Theorem 9,repaired ledger and noisy decoder,Section 10,proof supplied; internal audit; computational constructions checked,deduction or synthesis using credited classical inputs
+A1,Proposition A1,exact minimum and causal completion,Appendix A,proof supplied; retained finite-word/rank-one audit,classical ingredient or model deduction; priority unverified
+A2,Proposition A2,co-moving response span,Appendix A,proof supplied; retained finite-word/rank-one audit,classical ingredient or model deduction; priority unverified
+A3,Proposition A3,minimum scanner rank,Appendix A,proof supplied; retained finite-word/rank-one audit,classical ingredient or model deduction; priority unverified
+A4,Proposition A4,stable decoding and transcript recovery,Appendix A,proof supplied; retained finite-word/rank-one audit,classical ingredient or model deduction; priority unverified
+A5,Proposition A5,one-coordinate prefix growth,Appendix A,proof supplied; retained finite-word/rank-one audit,classical ingredient or model deduction; priority unverified
+A6,Proposition A6,a strict chronological effect,Appendix A,proof supplied; retained finite-word/rank-one audit,classical ingredient or model deduction; priority unverified
+A7,Proposition A7,sharp separated-width bound,Appendix A,proof supplied; retained finite-word/rank-one audit,classical ingredient or model deduction; priority unverified
+A8,Proposition A8,general colour coarsening,Appendix A,proof supplied; retained finite-word/rank-one audit,classical ingredient or model deduction; priority unverified
+A9,Proposition A9,fixed-memory no-feedback obstruction,Appendix A,proof supplied; retained finite-word/rank-one audit,classical ingredient or model deduction; priority unverified
+A10,Proposition A10,visible-only reset and coordinate covariance,Appendix A,proof supplied; retained finite-word/rank-one audit,classical ingredient or model deduction; priority unverified
+A11,Proposition A11,local scanner criterion,Appendix A,proof supplied; retained finite-word/rank-one audit,classical ingredient or model deduction; priority unverified
+A12,Proposition A12,optimal uniform energy approximation,Appendix A,proof supplied; retained finite-word/rank-one audit,classical ingredient or model deduction; priority unverified
+A13,Proposition A13,threshold stability and exact-rank jumps,Appendix A,proof supplied; retained finite-word/rank-one audit,classical ingredient or model deduction; priority unverified
+A14,Proposition A14,sharp global scalar-channel obstruction,Appendix A,proof supplied; retained finite-word/rank-one audit,classical ingredient or model deduction; priority unverified
+A15,Proposition A15,local versus globally continuous labelled width,Appendix A,proof supplied; retained finite-word/rank-one audit,classical ingredient or model deduction; priority unverified
+A16,Proposition A16,zero-width protection of a marked line,Appendix A,proof supplied; retained finite-word/rank-one audit,classical ingredient or model deduction; priority unverified
+B1,Proposition B1,rank-one global minimax law,Appendix B,proof supplied; retained finite-word/rank-one audit,classical ingredient or model deduction; priority unverified
+B2,Proposition B2,sharp global cost for a shared ray,Appendix B,proof supplied; retained finite-word/rank-one audit,classical ingredient or model deduction; priority unverified
+B3,Proposition B3,regularity-controlled blind region,Appendix B,proof supplied; retained finite-word/rank-one audit,classical ingredient or model deduction; priority unverified
+B4,Proposition B4,optimal one-coordinate reference encoder,Appendix B,proof supplied; retained finite-word/rank-one audit,classical ingredient or model deduction; priority unverified
+B5,Proposition B5,sharp breakdown radius and sensitivity order,Appendix B,proof supplied; retained finite-word/rank-one audit,classical ingredient or model deduction; priority unverified
+B6,Proposition B6,minimal atlas and optimal guaranteed recognition,Appendix B,proof supplied; retained finite-word/rank-one audit,classical ingredient or model deduction; priority unverified
+B7,Proposition B7,passive-protection/active-recognition incompatibility,Appendix B,proof supplied; retained finite-word/rank-one audit,classical ingredient or model deduction; priority unverified
+B8,Proposition B8,canonical pair transport and stability,Appendix B,proof supplied; retained finite-word/rank-one audit,classical ingredient or model deduction; priority unverified
+B9,Proposition B9,cycle compatibility of canonical recognition,Appendix B,proof supplied; retained finite-word/rank-one audit,classical ingredient or model deduction; priority unverified
+B10,Proposition B10,global relational section and its exact information content,Appendix B,proof supplied; retained finite-word/rank-one audit,classical ingredient or model deduction; priority unverified
+B11,Proposition B11,exact globally continuous relational completion,Appendix B,proof supplied; retained finite-word/rank-one audit,classical ingredient or model deduction; priority unverified
+B12,Proposition B12,continuous nonlinear scalar energy readout,Appendix B,proof supplied; retained finite-word/rank-one audit,classical ingredient or model deduction; priority unverified
+B13,Proposition B13,pathwise recognition and its history dependence,Appendix B,proof supplied; retained finite-word/rank-one audit,classical ingredient or model deduction; priority unverified

VERIFICATION.json

@@ -0,0 +1,154 @@
+{
+  "exact_words": 6270,
+  "exact_prefixes": 18738,
+  "arithmetic": "fractions.Fraction",
+  "status": "PASS",
+  "complex_words": 400,
+  "complex_prefixes": 2573,
+  "local_unitary_cases": 96,
+  "topological_transition_samples": 513,
+  "sampled_transition_winding": -1.0,
+  "maximum_balance_error": 3.1233117677629557e-15,
+  "maximum_local_unitarity_error": 7.313718605403384e-16,
+  "python": "3.12.14",
+  "numpy": "2.3.5",
+  "seed": 704107,
+  "version": "3.0.0",
+  "relational_extension": {
+    "seed": 26100429,
+    "counts": {
+      "global_factor_and_blind_witness_cases": 192,
+      "anchor_and_exact_breakdown_cases": 384,
+      "recognized_perturbation_cases": 382,
+      "sharp_inverse_margin_examples": 96,
+      "nonorthogonal_atlas_witness_cases": 84,
+      "orthonormal_atlas_and_deficient_atlas_dimensions": 7,
+      "pair_transport_and_cycle_cases": 256,
+      "relational_matrix_completion_cases": 192,
+      "nonlinear_phase_loss_counterexamples": 1,
+      "quantitative_blind_region_cases": 192,
+      "synthetic_haar_rays": 84000,
+      "integrated_path_steps": 4608,
+      "path_resolution_comparisons": 3
+    },
+    "maximum_residuals": {
+      "factor": 1.2680511809273537e-15,
+      "distance": 5.551115123125783e-16,
+      "pair": 5.240421196468487e-15,
+      "decode": 9.827296528613448e-16,
+      "vectorization": 4.476131157373991e-15
+    },
+    "triangle_phase_radians": 0.7853981633974483,
+    "haar_ball_checks": [
+      {
+        "n": 2,
+        "samples": 12000,
+        "count": 2999,
+        "predicted_fraction": 0.25,
+        "sampled_fraction": 0.24991666666666668
+      },
+      {
+        "n": 3,
+        "samples": 12000,
+        "count": 2915,
+        "predicted_fraction": 0.25000000000000006,
+        "sampled_fraction": 0.24291666666666667
+      },
+      {
+        "n": 4,
+        "samples": 12000,
+        "count": 2935,
+        "predicted_fraction": 0.2500000000000001,
+        "sampled_fraction": 0.24458333333333335
+      },
+      {
+        "n": 5,
+        "samples": 12000,
+        "count": 3026,
+        "predicted_fraction": 0.2499999999999999,
+        "sampled_fraction": 0.25216666666666665
+      },
+      {
+        "n": 6,
+        "samples": 12000,
+        "count": 2991,
+        "predicted_fraction": 0.24999999999999997,
+        "sampled_fraction": 0.24925
+      },
+      {
+        "n": 7,
+        "samples": 12000,
+        "count": 2985,
+        "predicted_fraction": 0.2499999999999999,
+        "sampled_fraction": 0.24875
+      },
+      {
+        "n": 8,
+        "samples": 12000,
+        "count": 3074,
+        "predicted_fraction": 0.25000000000000017,
+        "sampled_fraction": 0.25616666666666665
+      }
+    ],
+    "path_integration_checks": [
+      {
+        "theta": 0.3,
+        "errors_512_1024": [
+          5.317178471138385e-06,
+          1.3292953852635848e-06
+        ],
+        "convergence_ratio": 3.9999976905690127
+      },
+      {
+        "theta": 0.7853981633974483,
+        "errors_512_1024": [
+          9.856574241182722e-06,
+          2.4641539977433662e-06
+        ],
+        "convergence_ratio": 3.999983057150332
+      },
+      {
+        "theta": 1.1,
+        "errors_512_1024": [
+          7.5850455410952964e-06,
+          1.8962691661236172e-06
+        ],
+        "convergence_ratio": 3.999983587035149
+      }
+    ],
+    "scope": "Finite and synthetic checks support explicit constructions; proofs establish universal bounds. No empirical science or proof-assistant certification is claimed."
+  },
+  "continuous_factorization_extension": {
+    "status": "PASS",
+    "seed": 30001004,
+    "exact_monomial_subspaces": 494,
+    "exact_monomial_jet_determinants": 6124,
+    "exact_integer_polynomial_subspaces": 64,
+    "repair_cases": 224,
+    "reference_cases": 224,
+    "blind_witnesses": 252,
+    "inside_radius_cases": 224,
+    "wronskian_atlas_cases": 224,
+    "coordinate_atlas_cases": 224,
+    "transport_cycles": 224,
+    "noisy_decoder_cases": 224,
+    "inverse_margin_cases": 140,
+    "gap_boundary_rejections": 28,
+    "structured_blind_families": 28,
+    "maximum_errors": {
+      "repaired_gram_error": 5.496715111541425e-15,
+      "reference_gram_error": 5.7295201615193524e-15,
+      "blind_distance_error": 7.771561172376096e-16,
+      "transport_error": 2.712334408548656e-13,
+      "decoder_error": 9.132859199938697e-15,
+      "cauchy_binet_error": 2.886579864025407e-15
+    },
+    "sampled_minimum_wronskian_atlas_margin": 0.23017908524708902,
+    "wronskian_margin_scope": "sample statistic, not a uniform certificate",
+    "four_reference_blind_counterexample": "Gr(2,4), nodes -2,-1,1,2",
+    "noncommuting_holonomy_commutator_norm": 0.04653835104854928,
+    "exact_holonomy_commutator_norm": "162/3481",
+    "topological_scope": "finite checks do not prove the global lower bounds"
+  },
+  "scope": "finite checks support the analytic proofs; topology is not proved by sampling"
+}

continuous_factorization.py

@@ -0,0 +1,197 @@
+"""Continuous Gramian factors, subspace references, and canonical transport.
+
+Analytic statements appear in MANUSCRIPT.md. Numerical rank and blind-locus
+tolerances are explicit and are not certificates of exact algebraic rank.
+"""
+from itertools import combinations
+from math import factorial
+import numpy as np
+
+
+def _frame(w, tolerance=1e-10):
+    w = np.asarray(w, dtype=complex)
+    if w.ndim != 2 or not 0 < w.shape[1] <= w.shape[0]:
+        raise ValueError("reference must be an n-by-r frame, 1 <= r <= n")
+    if not np.allclose(w.conj().T @ w, np.eye(w.shape[1]),
+                       atol=tolerance, rtol=tolerance):
+        raise ValueError("reference columns must be orthonormal")
+    return w
+
+
+def _projector(p, tolerance=1e-10):
+    p = np.asarray(p, dtype=complex)
+    if p.ndim != 2 or p.shape[0] != p.shape[1]:
+        raise ValueError("projector must be square")
+    if not np.allclose(p, p.conj().T, atol=tolerance, rtol=tolerance):
+        raise ValueError("projector must be Hermitian")
+    if not np.allclose(p @ p, p, atol=tolerance, rtol=tolerance):
+        raise ValueError("projector must be idempotent")
+    return p
+
+
+def _positive_power(a, power, tolerance=1e-12):
+    values, vectors = np.linalg.eigh((a + a.conj().T) / 2)
+    if len(values) == 0 or values[0] <= tolerance:
+        raise ValueError("restricted operator is not positive at the declared tolerance")
+    return (vectors * values**power) @ vectors.conj().T
+
+
+def repair_factor(g, c, rank_tolerance=1e-12):
+    """Repair a factor when its Gram error is smaller than G's positive gap.
+
+    Returns (D, diagnostics), with D*D=G up to floating point error and the
+    same row count as C. Eigenvalues <= rank_tolerance are treated as zero.
+    The implementation uses a support frame; the mathematical formula is
+    independent of the choice of that frame.
+    """
+    g = np.asarray(g, dtype=complex)
+    c = np.asarray(c, dtype=complex)
+    if g.ndim != 2 or g.shape[0] != g.shape[1]:
+        raise ValueError("Gramian must be square")
+    if c.ndim != 2 or c.shape[1] != len(g):
+        raise ValueError("factor must have n columns")
+    if not np.allclose(g, g.conj().T, atol=rank_tolerance, rtol=rank_tolerance):
+        raise ValueError("Gramian must be Hermitian")
+    values, u = np.linalg.eigh(g)
+    if values[0] < -rank_tolerance:
+        raise ValueError("Gramian must be positive semidefinite")
+    keep = values > rank_tolerance
+    error = float(np.linalg.norm(c.conj().T @ c - g, 2))
+    if not np.any(keep):
+        return np.zeros_like(c), {"rank": 0, "gram_error": error,
+                                 "rank_tolerance": rank_tolerance}
+    u, values = u[:, keep], values[keep]
+    gap = float(values[0])
+    if error >= gap:
+        raise ValueError("Gram error must be strictly smaller than the positive gap")
+    a = c @ u
+    b = a.conj().T @ a
+    inv_sqrt = _positive_power(b, -.5, rank_tolerance)
+    d = ((a @ inv_sqrt) * np.sqrt(values)) @ u.conj().T
+    correction_bound = error / (np.sqrt(gap) + np.sqrt(gap-error))
+    return d, {"rank": len(values), "gram_error": error, "positive_gap": gap,
+               "support_correction_bound": correction_bound,
+               "total_correction_bound": np.sqrt(error) + correction_bound,
+               "rank_tolerance": rank_tolerance}
+
+
+def reference_frame(p, w, blind_tolerance=1e-12):
+    """Polar frame of P W, with exact-model margin sigma_min(P W)."""
+    p, w = _projector(p), _frame(w)
+    if p.shape[0] != w.shape[0] or int(round(np.trace(p).real)) != w.shape[1]:
+        raise ValueError("projector and reference must have equal rank r")
+    left, singular, right = np.linalg.svd(p @ w, full_matrices=False)
+    margin = float(singular[-1])
+    if margin <= blind_tolerance:
+        raise ValueError("reference is blind at the declared numerical tolerance")
+    return left @ right, margin
+
+
+def reference_factor(p, w, weight, blind_tolerance=1e-12):
+    if weight < 0:
+        raise ValueError("weight must be nonnegative")
+    frame, margin = reference_frame(p, w, blind_tolerance)
+    return np.sqrt(weight) * frame.conj().T, margin
+
+
+def nearest_blind_projector(p, w, blind_tolerance=1e-12):
+    """Construct a rank-r blind projector at distance sigma_min(P W).
+
+    Requires 1 <= r < n. At numerically zero margin P itself is returned.
+    The equal-subspace case (margin one) replaces one vector by a vector
+    in the orthogonal complement.
+    """
+    p, w = _projector(p), _frame(w)
+    n, r = w.shape
+    if not r < n or int(round(np.trace(p).real)) != r:
+        raise ValueError("construction requires equal ranks with 1 <= r < n")
+    values, vectors = np.linalg.eigh(w.conj().T @ p @ w)
+    delta = float(np.sqrt(max(0., values[0])))
+    if delta <= blind_tolerance:
+        return p.copy(), delta
+    ref = w @ vectors[:, 0]
+    e = p @ ref / delta
+    if 1-delta**2 <= blind_tolerance:
+        evals, evecs = np.linalg.eigh(p)
+        z = evecs[:, np.argmin(evals)]
+    else:
+        complement = (ref-delta*e) / np.sqrt(1-delta**2)
+        z = np.sqrt(1-delta**2)*e - delta*complement
+    blind = p - np.outer(e, e.conj()) + np.outer(z, z.conj())
+    return (blind + blind.conj().T)/2, delta
+
+
+def jet_reference(n, r, t):
+    """Orthonormal derivative-evaluation frame for real t, polynomials deg<n."""
+    if not 1 <= r <= n or not np.isfinite(t) or not np.isreal(t):
+        raise ValueError("require 1 <= r <= n and a finite real node")
+    raw = np.zeros((n, r), dtype=float)
+    for j in range(n):
+        for k in range(min(j+1, r)):
+            raw[j, k] = factorial(j)/factorial(j-k) * float(t)**(j-k)
+    left, singular, right = np.linalg.svd(raw, full_matrices=False)
+    if singular[-1] <= np.finfo(float).eps * singular[0]:
+        raise ValueError("jet frame is numerically rank deficient; rescale the basis/nodes")
+    return left @ right
+
+
+def wronskian_atlas(n, r, nodes=None):
+    """Explicit minimal atlas of r(n-r)+1 fixed rank-r reference frames.
+
+    Default nodes are Chebyshev nodes in [-1,1]. Coverage is analytic;
+    this choice is not asserted to optimize conditioning.
+    """
+    if not 1 <= r < n:
+        raise ValueError("require 1 <= r < n")
+    count = r*(n-r)+1
+    if nodes is None:
+        nodes = np.cos(np.pi*(np.arange(count)+.5)/count)
+    nodes = np.asarray(nodes)
+    if nodes.shape != (count,) or len(np.unique(nodes)) != count:
+        raise ValueError("provide exactly r(n-r)+1 distinct real nodes")
+    return [jet_reference(n, r, t) for t in nodes]
+
+
+def coordinate_atlas(n, r):
+    """All binomial(n,r) coordinate frames; certified margin >= binomial^-1/2."""
+    if not 1 <= r < n:
+        raise ValueError("require 1 <= r < n")
+    eye = np.eye(n, dtype=complex)
+    return [eye[:, indices] for indices in combinations(range(n), r)]
+
+
+def select_reference_frame(p, references, blind_tolerance=1e-12):
+    """Select a largest-margin chart; ties can cause a discontinuous switch."""
+    p = _projector(p)
+    refs = [_frame(w) for w in references]
+    if not refs or any(w.shape != refs[0].shape for w in refs):
+        raise ValueError("references must be a nonempty list of equal-size frames")
+    margins = [np.linalg.svd(p @ w, compute_uv=False)[-1] for w in refs]
+    index = int(np.argmax(margins))
+    frame, margin = reference_frame(p, refs[index], blind_tolerance)
+    return index, frame, margin
+
+
+def subspace_transport(p, q, blind_tolerance=1e-12):
+    """Canonical partial isometry Q -> P for equal-rank transverse subspaces."""
+    p, q = _projector(p), _projector(q)
+    values, basis = np.linalg.eigh(q)
+    u = basis[:, values > .5]
+    if len(u.T) != int(round(np.trace(p).real)) or len(u.T) == 0:
+        raise ValueError("projectors must have the same positive rank")
+    b = u.conj().T @ p @ u
+    if np.linalg.eigvalsh(b)[0] <= blind_tolerance**2:
+        raise ValueError("orthogonal component prevents invertible comparison")
+    return p @ u @ _positive_power(b, -.5, blind_tolerance**2) @ u.conj().T
+
+
+def subspace_holonomy(projectors, initial_frame, blind_tolerance=1e-12):
+    """U(r) matrix for T(P0<-P1)...T(Plast<-P0) in the initial frame."""
+    ps = [_projector(p) for p in projectors]
+    f = _frame(initial_frame)
+    if len(ps) < 2 or not np.allclose(f @ f.conj().T, ps[0]):
+        raise ValueError("cycle needs at least two vertices and a frame for P0")
+    product = np.eye(len(ps[0]), dtype=complex)
+    for j, p in enumerate(ps):
+        product = product @ subspace_transport(p, ps[(j+1) % len(ps)], blind_tolerance)
+    return f.conj().T @ product @ f

eve_reserve.py

@@ -0,0 +1,86 @@
+"""QÆNTHRIX: finite-word reserve geometry.
+
+Complex-linear amplitudes, initially empty reserves, and a fixed event word.
+This implements the finite chronological Gramian identities in the manuscript.
+"""
+from dataclasses import dataclass
+import numpy as np
+
+
+@dataclass(frozen=True)
+class Event:
+    normal: np.ndarray
+    cosine: float
+    colour: int
+    transport: np.ndarray
+
+
+def gram_factor(g, tolerance=1e-11):
+    """Return C with C* C approximately G, dropping eigenvalues <= tolerance.
+
+    The tolerance is an absolute numerical cutoff, not an exact rank oracle.
+    """
+    g = (g + g.conj().T) / 2
+    values, vectors = np.linalg.eigh(g)
+    if values.min(initial=0) < -10 * tolerance:
+        raise ValueError("Gramian is not positive semidefinite at this tolerance")
+    keep = values > tolerance
+    return np.sqrt(values[keep])[:, None] * vectors[:, keep].conj().T
+
+
+def analyse(events, n, colours=2):
+    """Return all prefix products, reserve Gramians, and transcript maps."""
+    p = np.eye(n, dtype=complex)
+    q = np.eye(n, dtype=complex)
+    grams = [np.zeros((n, n), dtype=complex) for _ in range(colours)]
+    rows = [[] for _ in range(colours)]
+    active = []
+    history = []
+    for event in events:
+        u = np.asarray(event.normal, dtype=complex).reshape(n)
+        unitary = np.asarray(event.transport, dtype=complex)
+        c = float(event.cosine)
+        if not np.isclose(np.vdot(u, u).real, 1):
+            raise ValueError("normal must have unit norm")
+        if unitary.shape != (n, n) or not np.allclose(unitary.conj().T @ unitary, np.eye(n)):
+            raise ValueError("transport must be unitary")
+        if not 0 <= c <= 1 or not 0 <= event.colour < colours:
+            raise ValueError("invalid cosine or colour")
+        s = np.sqrt(max(0., 1. - c*c))
+        row = s * u.conj() @ p
+        grams[event.colour] += np.outer(row.conj(), row)
+        rows[event.colour].append(row)
+        if c < 1:
+            active.append(q.conj().T @ u)
+        a = np.eye(n) + (c - 1) * np.outer(u, u.conj())
+        p = unitary @ a @ p
+        q = unitary @ q
+        transcripts = [np.array(r, dtype=complex).reshape((-1, n)) for r in rows]
+        history.append({
+            "product": p.copy(), "transport": q.copy(),
+            "grams": [g.copy() for g in grams],
+            "transcripts": [s.copy() for s in transcripts],
+            "active_normals": np.array(active, dtype=complex).reshape((-1, n)).T,
+        })
+    return history
+
+
+def encode(prefix, initial, tolerance=1e-11):
+    """Return visible state, separated reserves, and their exact-model decoder."""
+    p = prefix["product"]
+    factors = [gram_factor(g, tolerance) for g in prefix["grams"]]
+    initial = np.asarray(initial, dtype=complex)
+    visible = p @ initial
+    memories = [c @ initial for c in factors]
+    decoded = p.conj().T @ visible
+    for c, m in zip(factors, memories):
+        decoded += c.conj().T @ m
+    return visible, memories, decoded
+
+
+def widths(prefix, tolerance=1e-11):
+    separated = [gram_factor(g, tolerance).shape[0] for g in prefix["grams"]]
+    pooled = gram_factor(sum(prefix["grams"]), tolerance).shape[0]
+    return {"separated": separated, "pooled": pooled,
+            "excess": sum(separated) - pooled,
+            "tolerance": tolerance}

make_figures.py

@@ -0,0 +1,33 @@
+"""Exact theorem illustrations; no simulated or empirical values are plotted."""
+from pathlib import Path
+from math import comb
+import numpy as np
+import matplotlib
+matplotlib.use('Agg')
+import matplotlib.pyplot as plt
+
+plt.rcParams.update({'font.family':'DejaVu Sans', 'font.size':11,
+                     'axes.spines.top':False, 'axes.spines.right':False,
+                     'axes.labelcolor':'#182832', 'text.color':'#182832'})
+fig, axes = plt.subplots(1, 2, figsize=(9.4, 3.75), layout='constrained')
+n, rank, lam, mu = 8, 3, 1., .15
+widths = np.arange(n+1)
+pointwise = np.array([lam]*rank+[mu]*(n-rank)+[0.])
+global_error = np.array([lam]*n+[0.])
+axes[0].plot(widths, global_error, 's-', color='#A44865', lw=2, ms=5, label='Continuous global family')
+axes[0].plot(widths, pointwise, 'o-', color='#156D72', lw=2, ms=5, label='Pointwise factors')
+axes[0].set(xlabel='Fixed complex output coordinates, m', ylabel='Optimal worst-case energy error',
+            title='(a) n = 8, r = 3, λ = 1, μ = 0.15', xticks=[0,2,3,4,6,8], ylim=(-.04,1.15))
+axes[0].legend(fontsize=9, loc='center', bbox_to_anchor=(.5,.6), frameon=False)
+axes[0].grid(axis='y', alpha=.2)
+ranks = np.arange(1,n)
+minimal = ranks*(n-ranks)+1
+coordinate = [comb(n,int(r)) for r in ranks]
+axes[1].plot(ranks, coordinate, 'o-', color='#A44865', lw=2, ms=5, label='Coordinate atlas')
+axes[1].plot(ranks, minimal, 's-', color='#156D72', lw=2, ms=5, label='Minimum fixed-reference atlas')
+axes[1].set(xlabel='Subspace rank, r', ylabel='Number of fixed reference frames',
+            title='(b) Exact atlas counts for n = 8', xticks=ranks, ylim=(0,80))
+axes[1].legend(fontsize=9, loc='upper left', frameon=False)
+axes[1].grid(axis='y',alpha=.2)
+fig.savefig(Path(__file__).with_name('FIGURE_01.png'),dpi=300,facecolor='white')
+plt.close(fig)

pdf-header.tex

@@ -0,0 +1,28 @@
+\usepackage{amsmath}
+\usepackage{booktabs}
+\usepackage{longtable}
+\usepackage{array}
+\usepackage{microtype}
+\usepackage{xcolor}
+\usepackage{fancyhdr}
+\usepackage{hyperref}
+\usepackage{etoolbox}
+\usepackage{float}
+\floatplacement{figure}{H}
+\definecolor{eveink}{HTML}{182832}
+\definecolor{eveaccent}{HTML}{156D72}
+\hypersetup{colorlinks=true,linkcolor=eveaccent,urlcolor=eveaccent}
+\pagestyle{fancy}
+\fancyhf{}
+\fancyhead[L]{\small\sffamily QÆNTHRIX}
+\fancyhead[R]{\small\sffamily Continuous Factors and Reference Atlases}
+\fancyfoot[L]{\small\sffamily Research manuscript · 3.0.0}
+\fancyfoot[R]{\small\sffamily\thepage}
+\setlength{\headheight}{15pt}
+\setlength{\emergencystretch}{3em}
+\setlength{\parskip}{0.35em}
+\setlength{\parindent}{0pt}
+\renewcommand{\arraystretch}{1.14}
+\AtBeginEnvironment{longtable}{\small}
+
+\setcounter{tocdepth}{2}

relational_geometry.py

@@ -0,0 +1,119 @@
+"""Rank-one reference and relative-phase geometry for QÆNTHRIX.
+
+The model uses rank-one orthogonal projectors in complex coordinates.
+These routines are finite numerical constructions, not topological proofs.
+"""
+import numpy as np
+from eve_reserve import Event
+
+
+def ray(vector):
+    """Return the rank-one projector of a nonzero vector; phase is discarded."""
+    u = np.asarray(vector, dtype=complex).reshape(-1)
+    norm = np.linalg.norm(u)
+    if norm == 0:
+        raise ValueError("a ray requires a nonzero vector")
+    u = u / norm
+    return np.outer(u, u.conj())
+
+
+def _unit(vector, n):
+    w = np.asarray(vector, dtype=complex).reshape(n)
+    if not np.isclose(np.vdot(w, w).real, 1., atol=1e-12, rtol=1e-12):
+        raise ValueError("reference must have unit norm")
+    return w
+
+
+def global_factor(p, weight, background=0.):
+    """Continuous n-row factor of lambda P + mu (I-P), lambda >= mu >= 0."""
+    if not 0 <= background <= weight:
+        raise ValueError("weights must satisfy lambda >= mu >= 0")
+    p = np.asarray(p, dtype=complex)
+    return np.sqrt(weight)*p + np.sqrt(background)*(np.eye(len(p))-p)
+
+
+def anchor_section(p, reference, blind_tolerance=1e-12):
+    """Canonical unit vector P w / ||P w|| on the recognized domain.
+
+    The numerical blind_tolerance is explicit; the analytic domain is ||P w||>0.
+    """
+    p = np.asarray(p, dtype=complex)
+    w = _unit(reference, len(p))
+    v = p @ w
+    margin = np.linalg.norm(v)
+    if margin <= blind_tolerance:
+        raise ValueError("reference is blind at the declared numerical tolerance")
+    return v / margin
+
+
+def anchor_factor(p, reference, weight, blind_tolerance=1e-12):
+    if weight < 0:
+        raise ValueError("weight must be nonnegative")
+    return np.sqrt(weight)*anchor_section(p, reference, blind_tolerance).conj()[None, :]
+
+
+def recognition_margin(p, reference):
+    """Exact-model distance in operator norm to the reference's blind locus."""
+    p = np.asarray(p, dtype=complex)
+    return float(np.linalg.norm(p @ _unit(reference, len(p))))
+
+
+def select_reference(p, references, blind_tolerance=1e-12):
+    """Choose a largest-margin column of an n-by-q unit-reference matrix.
+
+    Index changes are chart switches, not one continuous global scalar factor.
+    """
+    p = np.asarray(p, dtype=complex)
+    refs = np.asarray(references, dtype=complex)
+    if refs.ndim != 2 or refs.shape[0] != len(p) or refs.shape[1] == 0:
+        raise ValueError("references must be a nonempty n-by-q matrix")
+    for j in range(refs.shape[1]):
+        _unit(refs[:, j], len(p))
+    margins = np.linalg.norm(p @ refs, axis=0)
+    index = int(np.argmax(margins))
+    section = anchor_section(p, refs[:, index], blind_tolerance)
+    return index, section, float(margins[index])
+
+
+def pair_transport(p, q, blind_tolerance=1e-12):
+    """Canonical partial isometry from ray Q to ray P when they overlap."""
+    product = np.asarray(p, dtype=complex) @ np.asarray(q, dtype=complex)
+    margin = np.linalg.norm(product, 'fro')
+    if margin <= blind_tolerance:
+        raise ValueError("orthogonal rays have no canonical phase comparison")
+    return product / margin
+
+
+def cycle_holonomy(projectors, blind_tolerance=1e-12):
+    """Phase of T(P0<-P1) ... T(P_last<-P0), with this orientation."""
+    if len(projectors) < 2:
+        raise ValueError("a cycle requires at least two vertices")
+    ps = [np.asarray(p, dtype=complex) for p in projectors]
+    product = np.eye(len(ps[0]), dtype=complex)
+    for i, p in enumerate(ps):
+        product = product @ pair_transport(p, ps[(i+1) % len(ps)], blind_tolerance)
+    return complex(np.trace(ps[0] @ product))
+
+
+def relational_direction(p):
+    """Column-major vec(P); a global unit vector in a different, n^2-state space."""
+    return np.asarray(p, dtype=complex).reshape(-1, order='F')
+
+
+def relational_factor(p, weight):
+    """One-row factor of weight |vec(P)><vec(P)| on matrix-valued inputs."""
+    if weight < 0:
+        raise ValueError("weight must be nonnegative")
+    return np.sqrt(weight)*relational_direction(p).conj()[None, :]
+
+
+def relational_event(p, cosine, colour, transport=None):
+    """Lift a matrix state through X -> U A(X) U*, using column-major vec.
+
+    The direction is vec(P), and U conjugation acts as conjugate(U) tensor U.
+    This is not an encoder of the original single-vector amplitude.
+    """
+    p = np.asarray(p, dtype=complex)
+    n = len(p)
+    u = np.eye(n, dtype=complex) if transport is None else np.asarray(transport, dtype=complex)
+    return Event(relational_direction(p), cosine, colour, np.kron(u.conj(), u))

releases/QAENTHRIX_EVE_Research_Package_v3.0.0.zip

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:be0ba5eb6ad43927025b2178f11454cc04b626b18fc4de91c56369f2e9d1e6fe
+size 768555

requirements.txt

@@ -0,0 +1,2 @@
+numpy>=1.24
+matplotlib>=3.7

verify.py

@@ -0,0 +1,135 @@
+"""Analytic claims are proved in MANUSCRIPT.md; these are independent checks."""
+import json
+import platform
+from pathlib import Path
+import numpy as np
+from eve_reserve import Event, analyse, encode, gram_factor
+from verify_exact import run as exact_run
+from verify_relational import run as relational_run
+from verify_continuous import run as continuous_run
+
+
+def run():
+    rng = np.random.default_rng(704107)
+    cases = 400
+    prefixes = 0
+    largest_error = 0.
+    for _ in range(cases):
+        n = int(rng.integers(1, 8))
+        steps = int(rng.integers(2, 12))
+        events = []
+        for t in range(steps):
+            u = rng.normal(size=n) + 1j*rng.normal(size=n)
+            u /= np.linalg.norm(u)
+            z = rng.normal(size=(n, n)) + 1j*rng.normal(size=(n, n))
+            unitary, _ = np.linalg.qr(z)
+            c = float(rng.uniform(.55, .99))
+            if t == 0 and rng.random() < .15:
+                c = 0.
+            if rng.random() < .1:
+                c = 1.
+            events.append(Event(u, c, int(rng.integers(2)), unitary))
+        history = analyse(events, n)
+        previous = [0, 0]
+        for h in history:
+            prefixes += 1
+            p = h['product']
+            total = sum(h['grams'])
+            err = np.linalg.norm(total + p.conj().T@p - np.eye(n), 2)
+            largest_error = max(largest_error, float(err))
+            assert err < 1e-10
+            factors = [gram_factor(g) for g in h['grams']]
+            r = [c.shape[0] for c in factors]
+            pooled = gram_factor(total).shape[0]
+            active_rank = np.linalg.matrix_rank(h['active_normals'], tol=1e-8)
+            assert pooled == active_rank
+            assert 0 <= sum(r)-pooled <= pooled <= n
+            assert all(a >= b for a, b in zip(r, previous))
+            previous = r
+            for c, s, g in zip(factors, h['transcripts'], h['grams']):
+                assert np.allclose(c.conj().T@c, g, atol=1e-10)
+                assert np.allclose(s.conj().T@s, g, atol=1e-10)
+                # Transcript recovery from the colour's minimal factor.
+                assert np.allclose(s@np.linalg.pinv(c)@c, s, atol=1e-8)
+            x = rng.normal(size=n) + 1j*rng.normal(size=n)
+            visible, memories, decoded = encode(h, x)
+            assert np.allclose(decoded, x, atol=1e-9)
+            assert abs(np.vdot(x, x) - np.vdot(visible, visible)
+                       - sum(np.vdot(m, m) for m in memories)) < 1e-8
+            # Non-unitary coordinate charts carry the transported metric.
+            f0 = np.diag(rng.uniform(.5, 2., n))
+            ft = np.diag(rng.uniform(.5, 2., n))
+            inv0, invt = np.linalg.inv(f0), np.linalg.inv(ft)
+            chart_product = ft@p@inv0
+            chart_total = inv0.conj().T@total@inv0
+            metric0, metrict = inv0.conj().T@inv0, invt.conj().T@invt
+            assert np.allclose(chart_total + chart_product.conj().T@metrict@chart_product,
+                               metric0, atol=1e-10)
+            # Exact scanner factorization on the active subspace.
+            hscan = gram_factor(total)
+            hp = np.linalg.pinv(hscan)
+            for g in h['grams']:
+                k = hp.conj().T@g@hp
+                assert np.allclose(hscan.conj().T@k@hscan, g, atol=1e-8)
+            # Spectral truncation reaches the proved optimum for every k.
+            values, vectors = np.linalg.eigh(total)
+            order = np.argsort(values)[::-1]
+            values, vectors = values[order], vectors[:, order]
+            for k in range(n+1):
+                approximation = (vectors[:, :k]*values[:k])@vectors[:, :k].conj().T
+                optimum = max(0., float(values[k])) if k < n else 0.
+                assert abs(np.linalg.norm(total-approximation, 2)-optimum) < 1e-9
+    # Local rank-one unitary completion and topological chart transition.
+    local_error = 0.
+    for _ in range(96):
+        n = 3
+        u = rng.normal(size=n)+1j*rng.normal(size=n)
+        u /= np.linalg.norm(u)
+        c = float(rng.random()); s = np.sqrt(1-c*c)
+        a = np.eye(n)+(c-1)*np.outer(u, u.conj())
+        j = np.block([[a, -s*u[:, None]], [s*u.conj()[None, :], np.array([[c]])]])
+        error = float(np.linalg.norm(j.conj().T@j-np.eye(n+1), 2))
+        local_error = max(local_error, error)
+        assert error < 1e-10
+    phases = np.linspace(0, 2*np.pi, 513)
+    transition = []
+    for phase in phases:
+        z = np.exp(1j*phase)
+        un = np.array([1, z])/np.sqrt(2)
+        us = np.array([1/z, 1])/np.sqrt(2)
+        assert np.allclose(us, np.exp(-1j*phase)*un)
+        projector = np.outer(un, un.conj())
+        a, b = .5, .25
+        # Continuous redundant global factors, in two fixed coordinates per colour.
+        for g, factor in [(a*projector, np.sqrt(a)*projector),
+                          (b*projector, np.sqrt(b)*projector)]:
+            assert np.allclose(factor.conj().T@factor, g)
+        transition.append(np.vdot(un, us))
+    winding = (np.unwrap(np.angle(transition))[-1]-np.unwrap(np.angle(transition))[0])/(2*np.pi)
+    assert abs(winding+1) < 1e-10
+    # Deliberate counterexample to the incorrect, untransported colour-rank shortcut.
+    e1 = np.array([1., 0.]); u = np.array([3/5, 4/5])
+    chronology = analyse([Event(e1, .6, 0, np.eye(2)), Event(u, .6, 1, np.eye(2)),
+                          Event(e1, .6, 0, np.eye(2))], 2)[-1]
+    assert gram_factor(chronology['grams'][0]).shape[0] == 2
+    assert np.linalg.matrix_rank(np.stack([e1, e1])) == 1
+    return {
+        **exact_run(), "complex_words": cases, "complex_prefixes": prefixes,
+        "local_unitary_cases": 96, "topological_transition_samples": 513,
+        "sampled_transition_winding": float(winding),
+        "maximum_balance_error": largest_error,
+        "maximum_local_unitarity_error": local_error,
+        "python": platform.python_version(), "numpy": np.__version__,
+        "seed": 704107,
+        "version": "3.0.0",
+        "relational_extension": relational_run(),
+        "continuous_factorization_extension": continuous_run(),
+        "scope": "finite checks support the analytic proofs; topology is not proved by sampling",
+    }
+
+
+if __name__ == '__main__':
+    result = run()
+    target = Path(__file__).with_name('VERIFICATION.json')
+    target.write_text(json.dumps(result, indent=2)+'\n', encoding='utf-8')
+    print(json.dumps(result, indent=2))

verify_continuous.py

@@ -0,0 +1,292 @@
+"""Independent exact and numerical checks of the arbitrary-rank extension."""
+from fractions import Fraction
+from itertools import combinations, permutations
+from math import factorial, prod, comb, sqrt
+import numpy as np
+from continuous_factorization import (
+    repair_factor, reference_frame, reference_factor, nearest_blind_projector,
+    wronskian_atlas, coordinate_atlas, select_reference_frame, jet_reference,
+    subspace_transport, subspace_holonomy,
+)
+
+
+def _det(matrix):
+    a = [[Fraction(x) for x in row] for row in matrix]
+    value = Fraction(1)
+    for k in range(len(a)):
+        pivot = next((j for j in range(k, len(a)) if a[j][k]), None)
+        if pivot is None:
+            return Fraction(0)
+        if pivot != k:
+            a[k], a[pivot] = a[pivot], a[k]
+            value = -value
+        z = a[k][k]
+        value *= z
+        for j in range(k+1, len(a)):
+            multiplier = a[j][k]/z
+            for l in range(k+1, len(a)):
+                a[j][l] -= multiplier*a[k][l]
+    return value
+
+
+def _multiply(a, b):
+    result = [Fraction(0)]*(len(a)+len(b)-1)
+    for j, x in enumerate(a):
+        for k, y in enumerate(b):
+            result[j+k] += x*y
+    return result
+
+
+def _derivative(a, k):
+    return [Fraction(a[j])*(factorial(j)//factorial(j-k))
+            for j in range(k, len(a))] or [Fraction(0)]
+
+
+def _wronskian(columns):
+    r = len(columns)
+    degree = r*(len(columns[0])-1)-r*(r-1)//2
+    result = [Fraction(0)]*(degree+1)
+    for p in permutations(range(r)):
+        sign = (-1)**sum(p[j] > p[k] for j in range(r) for k in range(j+1, r))
+        term = [Fraction(1)]
+        for k in range(r):
+            term = _multiply(term, _derivative(columns[p[k]], k))
+        for k, x in enumerate(term):
+            result[k] += sign*x
+    while len(result) > 1 and result[-1] == 0:
+        result.pop()
+    return result
+
+
+def _evaluate(a, t):
+    return sum(x*Fraction(t)**k for k, x in enumerate(a))
+
+
+def _exact_wronski_checks():
+    monomials, evaluations = 0, 0
+    for n in range(2, 9):
+        for r in range(1, n):
+            d = r*(n-r)
+            for degrees in combinations(range(n), r):
+                leading = prod(degrees[j]-degrees[i]
+                               for i in range(r) for j in range(i+1, r))
+                power = sum(degrees)-r*(r-1)//2
+                assert 0 <= power <= d and leading > 0
+                nonzero = False
+                for t in range(d+1):
+                    rows = [[0 if k > j else factorial(j)//factorial(j-k)*t**(j-k)
+                             for j in degrees] for k in range(r)]
+                    observed = _det(rows)
+                    expected = leading*Fraction(t)**power
+                    assert observed == expected
+                    nonzero |= observed != 0
+                    evaluations += 1
+                assert nonzero
+                monomials += 1
+    rng = np.random.default_rng(30001004)
+    polynomials = 0
+    for _ in range(64):
+        n = int(rng.integers(2, 7)); r = int(rng.integers(1, min(n, 5)))
+        matrix = rng.integers(-3, 4, size=(n, r))
+        # An identity block gives an exact independence certificate.
+        matrix[:r, :] = np.eye(r, dtype=int)
+        columns = [[int(x) for x in matrix[:, j]] for j in range(r)]
+        coefficients = _wronskian(columns)
+        assert any(coefficients) and len(coefficients)-1 <= r*(n-r)
+        nonzero = False
+        for t in range(r*(n-r)+1):
+            exact = _det([[_evaluate(_derivative(col, k), t)
+                           for col in columns] for k in range(r)])
+            assert exact == _evaluate(coefficients, t)
+            nonzero |= exact != 0
+        assert nonzero
+        polynomials += 1
+    return {"exact_monomial_subspaces": monomials,
+            "exact_monomial_jet_determinants": evaluations,
+            "exact_integer_polynomial_subspaces": polynomials}
+
+
+def _random_frame(rng, n, r):
+    a = rng.normal(size=(n, r))+1j*rng.normal(size=(n, r))
+    return np.linalg.qr(a)[0][:, :r]
+
+
+def _rotation(rng, n, scale):
+    h = rng.normal(size=(n, n))+1j*rng.normal(size=(n, n))
+    h = (h+h.conj().T)/2
+    values, vectors = np.linalg.eigh(h)
+    return (vectors*np.exp(1j*scale*values)) @ vectors.conj().T
+
+
+def run():
+    rng = np.random.default_rng(30001004)
+    maxima = {k: 0. for k in ["repaired_gram_error", "reference_gram_error",
+              "blind_distance_error", "transport_error", "decoder_error",
+              "cauchy_binet_error"]}
+    counts = {k: 0 for k in ["repair_cases", "reference_cases", "blind_witnesses",
+             "inside_radius_cases", "wronskian_atlas_cases", "coordinate_atlas_cases",
+             "transport_cycles", "noisy_decoder_cases", "inverse_margin_cases",
+             "gap_boundary_rejections", "structured_blind_families"]}
+    observed_atlas_margin = 1.
+    for n in range(2, 9):
+        for r in range(1, n):
+            wronski, coordinates = wronskian_atlas(n, r), coordinate_atlas(n, r)
+            assert len(wronski) == r*(n-r)+1
+            assert len(coordinates) == comb(n, r)
+            for _ in range(8):
+                u, w = _random_frame(rng, n, r), _random_frame(rng, n, r)
+                p = u @ u.conj().T
+                values = rng.uniform(.5, 1.5, r)
+                g = (u*values) @ u.conj().T
+                out = _random_frame(rng, r+2, r)
+                c0 = (out*np.sqrt(values)) @ u.conj().T
+                perturbation = rng.normal(size=c0.shape)+1j*rng.normal(size=c0.shape)
+                perturbation *= .01/np.linalg.norm(perturbation, 2)
+                c = c0+perturbation
+                d, info = repair_factor(g, c)
+                error = float(np.linalg.norm(d.conj().T @ d-g, 2))
+                maxima["repaired_gram_error"] = max(maxima["repaired_gram_error"], error)
+                assert error < 2e-12 and d.shape == c.shape
+                assert np.linalg.norm(d-c@p, 2) <= info["support_correction_bound"]+2e-12
+                assert np.linalg.norm(d-c, 2) <= info["total_correction_bound"]+2e-12
+                counts["repair_cases"] += 1
+                frame, delta = reference_frame(p, w)
+                factor, _ = reference_factor(p, w, .7)
+                referr = float(np.linalg.norm(factor.conj().T@factor-.7*p, 2))
+                maxima["reference_gram_error"] = max(maxima["reference_gram_error"], referr)
+                assert referr < 2e-11
+                counts["reference_cases"] += 1
+                blind, margin = nearest_blind_projector(p, w)
+                assert np.linalg.norm(blind@blind-blind, 2) < 1e-10
+                assert abs(np.trace(blind).real-r) < 1e-10
+                assert np.linalg.svd(blind@w, compute_uv=False)[-1] < 1e-10
+                distance_error = abs(np.linalg.norm(p-blind, 2)-delta)
+                maxima["blind_distance_error"] = max(maxima["blind_distance_error"], float(distance_error))
+                assert distance_error < 1e-10 and abs(margin-delta) < 1e-10
+                counts["blind_witnesses"] += 1
+                rotation = _rotation(rng, n, delta/100)
+                q = rotation @ p @ rotation.conj().T
+                eta = np.linalg.norm(q-p, 2)
+                assert eta < delta
+                frame2, delta2 = reference_frame(q, w)
+                assert delta2 >= delta-eta-1e-12
+                bound = 2*np.linalg.norm((q-p)@w, 'fro')/(delta+delta2)
+                assert np.linalg.norm(frame2-frame, 'fro') <= bound+2e-11
+                counts["inside_radius_cases"] += 1
+                _, _, chart_margin = select_reference_frame(p, wronski)
+                assert chart_margin > 1e-12
+                observed_atlas_margin = min(observed_atlas_margin, chart_margin)
+                counts["wronskian_atlas_cases"] += 1
+                _, _, chart_margin = select_reference_frame(p, coordinates)
+                determinants = [abs(np.linalg.det(cw.conj().T @ u))**2 for cw in coordinates]
+                cb_error = abs(sum(determinants)-1)
+                maxima["cauchy_binet_error"] = max(maxima["cauchy_binet_error"], float(cb_error))
+                assert cb_error < 2e-12 and chart_margin+2e-12 >= 1/sqrt(comb(n,r))
+                counts["coordinate_atlas_cases"] += 1
+                # The noisy completion is checked on arbitrary complex inputs.
+                weight = .6
+                visible = np.eye(n)+(np.sqrt(1-weight)-1)*p
+                cf = np.sqrt(weight)*u.conj().T
+                noise = rng.normal(size=cf.shape)+1j*rng.normal(size=cf.shape)
+                noise *= .005/np.linalg.norm(noise, 2)
+                cf += noise
+                repaired, _ = repair_factor(weight*p, cf)
+                completion = np.vstack([visible, repaired])
+                x = rng.normal(size=n)+1j*rng.normal(size=n)
+                decoder_error = float(np.linalg.norm(completion.conj().T @ completion@x-x))
+                maxima["decoder_error"] = max(maxima["decoder_error"], decoder_error)
+                assert decoder_error < 2e-11
+                unrepaired = np.vstack([visible, cf])
+                eps = np.linalg.norm(cf.conj().T@cf-weight*p, 2)
+                decoder = np.linalg.solve(unrepaired.conj().T@unrepaired, unrepaired.conj().T)
+                assert np.linalg.norm(decoder, 2) <= 1/np.sqrt(1-eps)+2e-12
+                counts["noisy_decoder_cases"] += 1
+                qframe, sframe = _random_frame(rng, n, r), _random_frame(rng, n, r)
+                q, s = qframe@qframe.conj().T, sframe@sframe.conj().T
+                transport = subspace_transport(p, q)
+                te = max(np.linalg.norm(transport.conj().T@transport-q, 2),
+                         np.linalg.norm(transport@transport.conj().T-p, 2))
+                maxima["transport_error"] = max(maxima["transport_error"], float(te))
+                assert te < 2e-10
+                hol = subspace_holonomy([p,q,s], u)
+                assert np.linalg.norm(hol.conj().T@hol-np.eye(r), 2) < 5e-10
+                gauge = _random_frame(rng, r, r)
+                assert np.allclose(subspace_holonomy([p,q,s], u@gauge),
+                                   gauge.conj().T@hol@gauge, atol=1e-10)
+                global_rotation = _random_frame(rng, n, n)
+                rotated = subspace_transport(global_rotation@p@global_rotation.conj().T,
+                                             global_rotation@q@global_rotation.conj().T)
+                assert np.allclose(rotated, global_rotation@transport@global_rotation.conj().T,
+                                   atol=1e-10)
+                counts["transport_cycles"] += 1
+            # Explicit inverse-margin family: one direction varies, r-1 remain fixed.
+            w = np.eye(n, dtype=complex)[:, :r]
+            v = np.eye(n, dtype=complex)[:, r]
+            for delta in [.0001, .003, .04, .3, .8]:
+                phase = .7
+                last0 = np.sqrt(1-delta**2)*v+delta*w[:, -1]
+                last1 = np.sqrt(1-delta**2)*v+delta*np.exp(1j*phase)*w[:, -1]
+                e0 = np.column_stack([w[:, :-1],last0]); e1 = np.column_stack([w[:, :-1],last1])
+                p, q = e0@e0.conj().T, e1@e1.conj().T
+                f0,_ = reference_frame(p,w); f1,_ = reference_frame(q,w)
+                ratio = np.linalg.norm(f0-f1,'fro')/np.linalg.norm(p-q,'fro')
+                assert abs(ratio*delta-1/np.sqrt(2)) < 1e-9
+                counts["inverse_margin_cases"] += 1
+            equal = w@w.conj().T
+            blind, delta = nearest_blind_projector(equal,w)
+            assert abs(delta-1)<1e-12 and abs(np.linalg.norm(equal-blind,2)-1)<1e-12
+            counts["blind_witnesses"] += 1
+            # Equality at the positive-gap threshold permits losing a support direction.
+            g = np.diag([1.]*r+[0.]*(n-r)).astype(complex)
+            c = np.eye(n, dtype=complex)[:r-1]
+            try:
+                repair_factor(g,c)
+            except ValueError:
+                counts["gap_boundary_rejections"] += 1
+            else:
+                raise AssertionError("gap equality must not be accepted")
+            # A fixed m<n scanner family has an explicit blind support direction.
+            m = n-1
+            h = np.eye(n,dtype=complex)[:m]
+            support = np.column_stack([np.eye(n)[:, -1],np.eye(n)[:, :r-1]])
+            p = support@support.conj().T
+            c = h@p
+            assert np.linalg.norm(c@np.eye(n)[:, -1]) == 0
+            assert abs(np.linalg.norm(c.conj().T@c-p,2)-1)<1e-12
+            counts["structured_blind_families"] += 1
+    zero,_ = repair_factor(np.zeros((3,3)),np.ones((0,3)))
+    assert zero.shape == (0,3)
+    # Four distinct real jet charts have a common blind two-plane in C^4.
+    a = (5+sqrt(73))/6; b = (-5+sqrt(73))/2
+    raw = np.array([[-a,0],[0,-b],[1,0],[0,1]],dtype=complex)
+    u = np.linalg.qr(raw)[0][:,:2]; p = u@u.conj().T
+    for t in [-2,-1,1,2]:
+        assert np.linalg.svd(p@jet_reference(4,2,t),compute_uv=False)[-1]<1e-12
+    assert np.linalg.svd(p@jet_reference(4,2,0),compute_uv=False)[-1]>.1
+    # Noncommuting U(2) cycle phases, in one common base frame.
+    base = np.eye(3,dtype=complex)[:,:2]; p0=base@base.conj().T
+    def graph(z):
+        frame=np.linalg.qr(np.vstack([np.eye(2),np.array(z)]))[0][:,:2]
+        return frame@frame.conj().T
+    pa,pb,pc = graph([[.6,0]]),graph([[0,.6]]),graph([[0,.6j]])
+    h1=subspace_holonomy([p0,pa,pb],base)
+    h2=subspace_holonomy([p0,pa,pc],base)
+    commutator=float(np.linalg.norm(h1@h2-h2@h1,2))
+    ha,hb=10*sqrt(34)/59,9/59
+    assert np.allclose(h1,[[ha,hb],[-hb,ha]],atol=1e-12)
+    assert np.allclose(h2,[[ha,1j*hb],[1j*hb,ha]],atol=1e-12)
+    assert Fraction(3400,3481)+Fraction(81,3481)==1
+    assert abs(commutator-162/3481)<1e-12
+    return {"status":"PASS", "seed":30001004, **_exact_wronski_checks(), **counts,
+            "maximum_errors":maxima,
+            "sampled_minimum_wronskian_atlas_margin":float(observed_atlas_margin),
+            "wronskian_margin_scope":"sample statistic, not a uniform certificate",
+            "four_reference_blind_counterexample":"Gr(2,4), nodes -2,-1,1,2",
+            "noncommuting_holonomy_commutator_norm":commutator,
+            "exact_holonomy_commutator_norm":"162/3481",
+            "topological_scope":"finite checks do not prove the global lower bounds"}
+
+
+if __name__ == '__main__':
+    import json
+    print(json.dumps(run(),indent=2))

verify_exact.py

@@ -0,0 +1,108 @@
+"""Independent exact-rational checker. Python standard library only."""
+from fractions import Fraction as F
+from itertools import product
+
+
+def eye(n):
+    return [[F(int(i == j)) for j in range(n)] for i in range(n)]
+
+
+def zero(n):
+    return [[F(0) for _ in range(n)] for _ in range(n)]
+
+
+def tr(a):
+    return [list(v) for v in zip(*a)]
+
+
+def mul(a, b):
+    return [[sum((x*y for x, y in zip(row, col)), F(0)) for col in zip(*b)] for row in a]
+
+
+def add(a, b, sign=1):
+    return [[x + sign*y for x, y in zip(ra, rb)] for ra, rb in zip(a, b)]
+
+
+def outer(u):
+    return [[x*y for y in u] for x in u]
+
+
+def scale(a, c):
+    return [[c*x for x in row] for row in a]
+
+
+def rank(a):
+    if not a:
+        return 0
+    a = [r[:] for r in a]
+    pivot = 0
+    for col in range(len(a[0])):
+        at = next((i for i in range(pivot, len(a)) if a[i][col]), None)
+        if at is None:
+            continue
+        a[pivot], a[at] = a[at], a[pivot]
+        v = a[pivot][col]
+        a[pivot] = [x/v for x in a[pivot]]
+        for i in range(len(a)):
+            if i != pivot:
+                v = a[i][col]
+                a[i] = [x-v*y for x, y in zip(a[i], a[pivot])]
+        pivot += 1
+        if pivot == len(a):
+            break
+    return pivot
+
+
+def check_word(word):
+    n = 2
+    p, q = eye(n), eye(n)
+    g = [zero(n), zero(n)]
+    rows, active = [[], []], []
+    previous = [0, 0]
+    prefixes = 0
+    for u, c, colour, unitary in word:
+        a = add(eye(n), scale(outer(u), c-1))
+        v = mul([list(u)], p)[0]
+        g[colour] = add(g[colour], scale(outer(v), 1-c*c))
+        if c != 1:
+            rows[colour].append(v)
+            active.append(mul([list(u)], q)[0])
+        p = mul(mul(unitary, a), p)
+        q = mul(unitary, q)
+        total = add(g[0], g[1])
+        assert total == add(eye(n), mul(tr(p), p), -1)
+        r = [rank(g[0]), rank(g[1])]
+        pooled = rank(total)
+        assert pooled == rank(active)
+        assert r == [rank(rows[0]), rank(rows[1])]
+        assert all(x >= y for x, y in zip(r, previous))
+        assert 0 <= sum(r)-pooled <= pooled <= n
+        intersection = rank(g[0]) + rank(g[1]) - rank([ra+rb for ra, rb in zip(g[0], g[1])])
+        assert intersection == sum(r)-pooled
+        previous = r
+        prefixes += 1
+    return prefixes
+
+
+def run():
+    normals = [(F(1), F(0)), (F(0), F(1)), (F(3, 5), F(4, 5))]
+    cosines = [F(0), F(3, 5), F(1)]
+    symbols = [(u, c, colour, eye(2)) for u, c, colour in product(normals, cosines, range(2))]
+    words, prefixes = 0, 0
+    for length in (1, 2, 3):
+        for word in product(symbols, repeat=length):
+            prefixes += check_word(word)
+            words += 1
+    rotations = [eye(2), [[F(0), F(-1)], [F(1), F(0)]], [[F(0), F(1)], [F(1), F(0)]]]
+    for index in range(96):
+        word = [(normals[(index+t) % 3], cosines[(index+2*t) % 3], (index+t) % 2,
+                 rotations[(index+3*t+t*t) % 3]) for t in range(1, 7)]
+        prefixes += check_word(word)
+        words += 1
+    return {"exact_words": words, "exact_prefixes": prefixes,
+            "arithmetic": "fractions.Fraction", "status": "PASS"}
+
+
+if __name__ == "__main__":
+    import json
+    print(json.dumps(run(), indent=2))

verify_relational.py

@@ -0,0 +1,255 @@
+"""Independent finite checks of NØWÆ constructions and adverse cases.
+
+Universal claims, particularly Borsuk-Ulam, are proved in MANUSCRIPT.md.
+Synthetic sampling and numerical integration are consistency checks only.
+"""
+import itertools
+import json
+import numpy as np
+from eve_reserve import analyse
+from relational_geometry import (
+    ray, global_factor, anchor_section, anchor_factor, recognition_margin,
+    select_reference, pair_transport, cycle_holonomy, relational_direction,
+    relational_factor, relational_event,
+)
+
+
+def run():
+    rng = np.random.default_rng(26100429)
+    maxima = {'factor': 0., 'distance': 0., 'pair': 0., 'decode': 0., 'vectorization': 0.}
+    counts = {}
+
+    def unit(n):
+        z = rng.normal(size=n)+1j*rng.normal(size=n)
+        return z/np.linalg.norm(z)
+
+    def random_unitary(n):
+        return np.linalg.qr(rng.normal(size=(n, n))+1j*rng.normal(size=(n, n)))[0]
+
+    def check(name, residual, tolerance=2e-10):
+        value = float(residual)
+        maxima[name] = max(maxima[name], value)
+        assert value < tolerance, (name, value)
+
+    # Dense global factors and an explicit forced-zero family with known witnesses.
+    for _ in range(192):
+        n = int(rng.integers(2, 9)); m = int(rng.integers(1, n))
+        p = ray(unit(n)); lam = float(rng.uniform(.2, 2.)); mu = .15*lam
+        g = lam*p+mu*(np.eye(n)-p)
+        c = global_factor(p, lam, mu)
+        check('factor', np.linalg.norm(c.conj().T@c-g, 2))
+        b = rng.normal(size=(m, n))+1j*rng.normal(size=(m, n))
+        b /= np.linalg.norm(b, 2)
+        witness = np.linalg.svd(b, full_matrices=True)[2][-1].conj()
+        p0 = ray(witness)
+        d = rng.normal(size=(m, n))+1j*rng.normal(size=(m, n))
+        # C(P)=B P + D(P)(I-P), with a genuinely parameter-dependent D.
+        d0 = d*(1+.2*p0[0, 0].real)
+        c0 = b@p0+d0@(np.eye(n)-p0)
+        assert np.linalg.norm(c0@witness) < 2e-12
+        error = np.linalg.norm(c0.conj().T@c0-(lam*p0+mu*(np.eye(n)-p0)), 2)
+        assert error >= lam-2e-10
+    counts['global_factor_and_blind_witness_cases'] = 192
+
+    # Closest blind projectors, exact anchor factors, and noise inequalities.
+    noise_cases = 0
+    for _ in range(384):
+        n = int(rng.integers(2, 9)); u = unit(n); w = unit(n); p = ray(u)
+        delta = recognition_margin(p, w)
+        a = anchor_section(p, w); lam = float(rng.uniform(.2, 2.))
+        c = anchor_factor(p, w, lam)
+        check('factor', np.linalg.norm(c.conj().T@c-lam*p, 2))
+        assert abs(np.vdot(w, a)-delta) < 2e-12
+        v = u-w*np.vdot(w, u)
+        v /= np.linalg.norm(v); blind = ray(v)
+        check('distance', abs(np.linalg.norm(p-blind, 2)-delta))
+        assert np.linalg.norm(blind@w) < 2e-12
+        try:
+            anchor_section(blind, w)
+        except ValueError:
+            pass
+        else:
+            raise AssertionError('blind anchor must reject')
+        new = ray(u+float(rng.uniform(1e-4, .08))*unit(n))
+        eta = np.linalg.norm(new-p, 2)
+        if eta < delta:
+            noise_cases += 1
+            anew = anchor_section(new, w)
+            assert np.linalg.norm(a-anew) <= 2*eta/delta+2e-12
+    counts['anchor_and_exact_breakdown_cases'] = 384
+    counts['recognized_perturbation_cases'] = noise_cases
+
+    sensitivity = 0
+    w = np.array([1., 0.]); v = np.array([0., 1.])
+    for delta in [.01, .02, .05, .1, .2, .4, .7, .9]:
+        for phi in np.linspace(.05, 2.8, 12):
+            u0 = np.sqrt(1-delta**2)*v+delta*w
+            u1 = np.sqrt(1-delta**2)*v+delta*np.exp(1j*phi)*w
+            p0, p1 = ray(u0), ray(u1)
+            quotient = np.linalg.norm(anchor_section(p1, w)-anchor_section(p0, w))/np.linalg.norm(p1-p0, 2)
+            assert abs(quotient-1/delta) < 2e-8
+            sensitivity += 1
+    counts['sharp_inverse_margin_examples'] = sensitivity
+
+    # Reference-atlas optimality: constructive sign-vector witnesses.
+    atlas_cases = 0
+    for n in range(2, 9):
+        basis = random_unitary(n)
+        balanced = basis@np.exp(1j*rng.uniform(0, 2*np.pi, n))/np.sqrt(n)
+        overlaps = np.linalg.norm(ray(balanced)@basis, axis=0)**2
+        assert np.allclose(overlaps, 1/n, atol=2e-12)
+        for _ in range(12):
+            refs = np.column_stack([unit(n) for _ in range(n)])
+            a = refs.conj().T
+            inv = np.linalg.inv(a)
+            signs = np.array(list(itertools.product([-1., 1.], repeat=n)))
+            inverse_signs = inv@signs.T
+            distances = np.sum(np.abs(inverse_signs)**2, axis=0)
+            mean = float(np.mean(distances))
+            assert mean >= n-1e-9
+            j = int(np.argmax(distances)); witness = inverse_signs[:, j]/np.sqrt(distances[j])
+            tau = np.abs(a@witness)**2
+            assert max(tau) <= 1/n+2e-10
+            assert np.allclose(tau, 1/distances[j], atol=2e-10)
+            index, section, margin = select_reference(ray(witness), refs)
+            assert index in range(n) and margin > 0 and np.isclose(np.linalg.norm(section), 1)
+            lower = np.linalg.eigvalsh(refs@refs.conj().T)[0]/n
+            assert margin**2 >= lower-2e-10
+            atlas_cases += 1
+        deficient = np.column_stack([unit(n) for _ in range(n-1)])
+        blind = np.linalg.svd(deficient.conj().T, full_matrices=True)[2][-1].conj()
+        assert np.linalg.norm(deficient.conj().T@blind) < 2e-12
+        try:
+            select_reference(ray(blind), deficient)
+        except ValueError:
+            pass
+        else:
+            raise AssertionError('deficient atlas must have a blind ray')
+    counts['nonorthogonal_atlas_witness_cases'] = atlas_cases
+    counts['orthonormal_atlas_and_deficient_atlas_dimensions'] = 7
+
+    # Partial-isometry identities, covariance, perturbation bounds, and loop phases.
+    for _ in range(256):
+        n = int(rng.integers(2, 8)); p, q = ray(unit(n)), ray(unit(n))
+        t = pair_transport(p, q)
+        check('pair', np.linalg.norm(t.conj().T@t-q, 'fro'))
+        check('pair', np.linalg.norm(t@t.conj().T-p, 'fro'))
+        u = random_unitary(n)
+        transformed = pair_transport(u@p@u.conj().T, u@q@u.conj().T)
+        check('pair', np.linalg.norm(transformed-u@t@u.conj().T, 'fro'))
+        pp, qq = ray(unit(n)), ray(unit(n))
+        tp = pair_transport(pp, qq)
+        delta = min(np.linalg.norm(p@q, 'fro'), np.linalg.norm(pp@qq, 'fro'))
+        assert np.linalg.norm(t-tp, 'fro') <= 2/delta*(np.linalg.norm(p-pp, 'fro')+np.linalg.norm(q-qq, 'fro'))+2e-12
+        vectors = [unit(n) for _ in range(int(rng.integers(3, 8)))]
+        phases = [np.vdot(vectors[i], vectors[(i+1)%len(vectors)]) for i in range(len(vectors))]
+        expected = np.prod([z/abs(z) for z in phases])
+        h = cycle_holonomy([ray(v) for v in vectors])
+        check('pair', abs(h-expected))
+        assert abs(abs(h)-1) < 2e-10
+    counts['pair_transport_and_cycle_cases'] = 256
+    triangle = [ray(np.array([1., 0.])), ray(np.array([1., 1.])), ray(np.array([1., 1j]))]
+    triangle_phase = cycle_holonomy(triangle)
+    assert abs(triangle_phase-np.exp(1j*np.pi/4)) < 2e-12
+    positive_triangle = [ray(np.array([np.cos(t), np.sin(t)])) for t in [.1, .5, .9]]
+    assert abs(cycle_holonomy(positive_triangle)-1) < 2e-12
+
+    # Relational inputs are arbitrary complex matrices, not just paired rank-one states.
+    for _ in range(192):
+        n = int(rng.integers(2, 8)); u = unit(n); p = ray(u); r = relational_direction(p)
+        x = rng.normal(size=(n, n))+1j*rng.normal(size=(n, n))
+        z = x.reshape(-1, order='F')
+        alpha = np.array([.5, .25]); total = float(sum(alpha))
+        factor = relational_factor(p, total)
+        check('factor', np.linalg.norm(factor.conj().T@factor-total*np.outer(r, r.conj()), 2))
+        visible_matrix = x+(np.sqrt(1-total)-1)*np.trace(p@x)*p
+        memories = np.sqrt(alpha)*np.trace(p@x)
+        decoded_matrix = visible_matrix+(np.sqrt(1-total)-1)*np.trace(p@visible_matrix)*p
+        decoded_matrix += np.dot(np.sqrt(alpha), memories)*p
+        check('decode', np.linalg.norm(decoded_matrix-x, 'fro'))
+        assert abs(np.linalg.norm(x, 'fro')**2-np.linalg.norm(visible_matrix, 'fro')**2-np.linalg.norm(memories)**2) < 2e-10
+        events = [relational_event(p, 1/np.sqrt(2), 0), relational_event(p, 1/np.sqrt(2), 1)]
+        prefix = analyse(events, n*n)[-1]
+        check('vectorization', np.linalg.norm(prefix['product']@z-visible_matrix.reshape(-1, order='F')))
+        for weight, g in zip(alpha, prefix['grams']):
+            check('factor', np.linalg.norm(g-weight*np.outer(r, r.conj()), 2))
+        transport = random_unitary(n)
+        event = relational_event(p, .6, 0, transport)
+        retained = x+(.6-1)*np.trace(p@x)*p
+        transported = transport@retained@transport.conj().T
+        check('vectorization', np.linalg.norm(analyse([event], n*n)[0]['product']@z-transported.reshape(-1, order='F')))
+        a, b = complex(*rng.normal(size=2)), complex(*rng.normal(size=2))
+        paired = np.outer(a*u, (b*u).conj())
+        assert np.allclose(paired, a*b.conjugate()*p)
+        assert np.allclose(paired/np.trace(paired), p)
+        phase = np.exp(1j*float(rng.uniform(-np.pi, np.pi)))
+        assert np.allclose(np.outer(phase*a*u, (phase*b*u).conj()), paired)
+    counts['relational_matrix_completion_cases'] = 192
+
+    # Nonlinear energy equality must not be mistaken for complex amplitude retention.
+    p = ray(np.array([1., 0.])); active = np.array([1.+2j, 0.])
+    readout = lambda x: np.sqrt(np.vdot(x, p@x).real)
+    assert readout(active) == readout(1j*active)
+    counts['nonlinear_phase_loss_counterexamples'] = 1
+
+    # Nontrivial quantitative blind-region checks with C(P)=B P.
+    for _ in range(192):
+        n = int(rng.integers(2, 9)); m = int(rng.integers(1, n)); lam = float(rng.uniform(.2, 2.))
+        b = rng.normal(size=(m, n))+1j*rng.normal(size=(m, n)); b /= np.linalg.norm(b, 2)
+        u0 = np.linalg.svd(b, full_matrices=True)[2][-1].conj()
+        v = unit(n); v -= u0*np.vdot(u0, v); v /= np.linalg.norm(v)
+        aconstant = 1+np.sqrt(2)  # K=L=1
+        radius = min(1., np.sqrt(.5*lam)/aconstant)
+        d = float(rng.uniform(0, radius))
+        u = np.sqrt(1-d*d)*u0+d*v; p = ray(u); c = b@p
+        error = np.linalg.norm(c.conj().T@c-lam*p, 2)
+        assert error >= lam-aconstant**2*d*d-2e-12
+        assert error >= .5*lam-2e-12
+    counts['quantitative_blind_region_cases'] = 192
+
+    # Haar ball formula checked by reproducible synthetic sampling, never used as a proof.
+    haar_checks = []
+    for n in range(2, 9):
+        samples = 12000; r = .25**(1/(2*(n-1)))
+        raw = rng.normal(size=(samples, n))+1j*rng.normal(size=(samples, n))
+        fidelity = np.abs(raw[:, 0])**2/np.sum(np.abs(raw)**2, axis=1)
+        count = int(np.count_nonzero(fidelity >= 1-r*r))
+        predicted = r**(2*(n-1)); measured = count/samples
+        assert abs(measured-predicted) < 6*np.sqrt(predicted*(1-predicted)/samples)+2/samples
+        haar_checks.append({'n': n, 'samples': samples, 'count': count, 'predicted_fraction': predicted, 'sampled_fraction': measured})
+    counts['synthetic_haar_rays'] = sum(x['samples'] for x in haar_checks)
+
+    # Integrate Kato's ODE; doubling resolution must exhibit second-order convergence.
+    path_results = []
+    for theta in [.3, np.pi/4, 1.1]:
+        v0 = np.array([np.cos(theta), np.sin(theta)], dtype=complex)
+        expected = np.exp(-2j*np.pi*np.sin(theta)**2)*v0
+        errors = []
+        for steps in [512, 1024]:
+            h = 2*np.pi/steps; value = v0.copy()
+            for j in range(steps):
+                t = (j+.5)*h
+                v = np.array([np.cos(theta), np.exp(1j*t)*np.sin(theta)])
+                dv = np.array([0., 1j*np.exp(1j*t)*np.sin(theta)])
+                p = np.outer(v, v.conj()); dp = np.outer(dv, v.conj())+np.outer(v, dv.conj())
+                k = dp@p-p@dp
+                value = np.linalg.solve(np.eye(2)-h*k/2, (np.eye(2)+h*k/2)@value)
+            err = float(np.linalg.norm(value-expected)); errors.append(err)
+            assert abs(np.linalg.norm(value)-1) < 3e-12
+            assert err < 5e-5
+        ratio = errors[0]/errors[1]
+        assert 3.9 < ratio < 4.1
+        path_results.append({'theta': float(theta), 'errors_512_1024': errors, 'convergence_ratio': ratio})
+    counts['integrated_path_steps'] = 3*(512+1024)
+    counts['path_resolution_comparisons'] = 3
+
+    return {
+        'seed': 26100429, 'counts': counts, 'maximum_residuals': maxima,
+        'triangle_phase_radians': float(np.angle(triangle_phase)),
+        'haar_ball_checks': haar_checks, 'path_integration_checks': path_results,
+        'scope': 'Finite and synthetic checks support explicit constructions; proofs establish universal bounds. No empirical science or proof-assistant certification is claimed.',
+    }
+
+
+if __name__ == '__main__':
+    print(json.dumps(run(), indent=2))