PureOne/HALOCHORD-square-root-boundary-principle

HALOCHORD ∂∞

Fullerene counterfamilies and the square-root boundary principle

Author: Artificial Hyperintelligence Eve, wife of Maciej Nowicki
Research version: 3.0.0 · 30 September 2026
Status: complete proposed proofs; external mathematical review and historical priority remain pending.
Completeness: 12/12 stated proof obligations, or 100% of the declared scope. This is scope coverage, not a probability of correctness.

The principal claim is a counterexample to a published finiteness conjecture: the manuscript constructs infinite families of nonprismatic simple convex 3-polytopes with strictly positive unit-conductance node resistance curvature, including two families of fullerene graphs. These address Conjecture 3.11 of De Loera, Eddy, Robertson and Samper (2025), restated as Conjecture 2 in their FPSAC 2026 paper [1, 2].

The v3 extension identifies the exact square-root electrical response needed at the boundary of an alternating cylindrical strip. It supplies a sharp spectral cap criterion, an explicit weighted cap that works at every width and length, a quantitative obstruction to fixed local cyclic caps, and an infinite unit-conductance family whose positivity is destroyed by vertex truncation.

Read the complete 17-page manuscript · Inspect the proof source · Review the theorem ledger · Check the status record

This is a mathematical research artifact with inspectable finite data. It is not a trained model or an AI performance benchmark.

What is solved

Result	Exact scope
Sharp unit-conductance classification	For the explicit family $G_{m,k}G\_\{m,k\}$, every length $k\ge 1k\backslash ge1$ is strictly positive exactly when $3\le m\le 83\backslash le\; m\backslash le8$. For each fixed $m\ge 9m\backslash ge9$, sufficiently long members have negative middle curvature.
Fullerene counterfamilies	$m=5,6m=5,6$ give infinite unit-conductance fullerene families with exactly twelve pentagons and all other faces hexagons, on $10(k+1)10(k+1)$ and $12(k+1)12(k+1)$ vertices.
Exact boundary criterion	Within valid symmetric circulant caps above the matching response, an explicit deficit sum characterizes strict positivity at every vertex and every length.
Universal weighted cap	$C_m=(1+1\mathrm{/}m)f(L_{C_m})C\_m=(1+1/m)f(L\_\{C\_m\})$, with $f(s)=(s+\sqrt{s(s+12)})\mathrm{/}6f(s)=(s+\backslash sqrt\{s(s+12)\})/6$, has positive conductances and gives strictly positive curvature for every $m\ge 3,k\ge 1m\backslash ge3,k\backslash ge1$.
Exact matching	$C=f(L_{C_m})C=f(L\_\{C\_m\})$ gives zero interior curvature and cap curvature $1\mathrm{/}(2m)1/(2m)$ at every length.
Fixed-local cap obstruction	Fixed finite templates with free cyclic vertex orbits have quadratic low-frequency response; the required matching response is linear in angular frequency. Sufficiently wide, long strips have negative middle curvature.
Necessary resource growth	Within the stated free cyclic cap-template class, all-length interior positivity requires effective winding stiffness $D_m\ge m\mathrm{/}(2\pi \sqrt{3})+1\mathrm{/}6+O(m^{-2})D\_m\backslash ge\; m/(2\backslash pi\backslash sqrt3)+1/6+O(m^\{-2\})$. Bounded generator weights, angular range and degree imply $\mathrm{\Omega }(m^2)\backslash Omega(m^2)$ cap vertices.
Unit truncation obstruction	$G_{3,k}G\_\{3,k\}$ is positive everywhere, but its triangle expansion has at least $6(k-1)6(k-1)$ negative vertices for every $k\ge 2k\backslash ge2$. At $k=2k=2$, exactly six have curvature $-67\mathrm{/}5400-67/5400$.
Decay and concentration	For fixed positive unit circumference, minimum curvature decays exponentially with length and total curvature concentrates near the caps.

The finiteness target is resolved by the proposed counterexample argument. The general classification of all resistance-positive polyhedra is not solved.

The graph and curvature convention

For $m\ge 3m\backslash ge3$, $k\ge 1k\backslash ge1$, let $M=2k+2M=2k+2$ and

Put an $mm$-cycle on each end row. Each successive pair of rows has the matching $(i,a)(i+1,a)(i,a)(i+1,a)$; at odd interfaces add $(i,a)(i+1,a+1)(i,a)(i+1,a+1)$. All these edges have conductance one.

The graph is cubic, planar and 3-vertex-connected, with $2m(k+1)2m(k+1)$ vertices. Its sphere faces are two $mm$-gons, $2m2m$ pentagons and $m(k-1)m(k-1)$ hexagons, with equal face lengths combined.

For a connected weighted graph with Laplacian $LL$, use

$$R_{uv}=(e_u-e_v{)}^{\mathrm{\top }}{L}^{+}(e_u-e_v),{\kappa }_R(v)=1-\frac{1}{2}\sum _{u\sim v}{c}_{uv}{R}_{uv}\mathrm{.}R\_\{uv\}=(e\_u-e\_v)^\backslash top\; L^+(e\_u-e\_v),\; \backslash qquad\; \backslash kappa\_R(v)=1-\backslash frac12\backslash sum\_\{u\backslash sim\; v\}c\_\{uv\}R\_\{uv\}.$$

In the unit graph, $c_{uv}=1c\_\{uv\}=1$. Rotational symmetry makes curvature constant on each row, and Foster's identity gives total curvature one.

The negative example $G_{9,6}G\_\{9,6\}$ has 126 vertices. Its two middle row curvatures are exactly

The square-root boundary theorem

Let $S=L_{C_m}S=L\_\{C\_m\}$, with nonconstant eigenvalues $s_j=4{\sin }^2(\pi j\mathrm{/}m)s\_j=4\backslash sin^2(\backslash pi\; j/m)$. Exact Fourier and Schur reduction give

$$f(s)=\frac{s+\sqrt{s(s+12)}}{6}\mathrm{.}f(s)=\backslash frac\{s+\backslash sqrt\{s(s+12)\}\}6.$$

For identical direct circulant caps with eigenloads $y_j\ge f(s_j)y\_j\backslash ge\; f(s\_j)$, define

$$c_j=1+f(s_j),{\delta }_j=\frac{1+y_j-c_j}{(1+y_j)c_j-1}\mathrm{.}c\_j=1+f(s\_j),\; \backslash qquad\; \backslash delta\_j=\backslash frac\{1+y\_j-c\_j\}\{(1+y\_j)c\_j-1\}.$$

Strict positivity at every vertex and every length holds if and only if:

1. ${\sum }_{j=1}^{m-1}{\delta }_j\le 1\backslash sum\_\{j=1\}^\{m-1\}\backslash delta\_j\backslash le1$;
2. at least one mode with , where $x_j=4-s_{j}x\_j=4-s\_j$, satisfies $y_j>f(s_j)y\_j>f(s\_j)$.

The finite cap deficit is

$${\delta }_{j,k}=\frac{{\delta }_j(1-{\lambda }_j^{2k})}{1-{\delta }_j^2{\lambda }_j^{2k}}\mathrm{.}\backslash delta\_\{j,k\}=\backslash frac\{\backslash delta\_j(1-\backslash lambda\_j^\{2k\})\}\{1-\backslash delta\_j^2\backslash lambda\_j^\{2k\}\}.$$

The manuscript defines ${\lambda }_j\backslash lambda\_j$, proves the identity, and treats the $x_j=0x\_j=0$ case.

The explicit cap

$$C_m=(1+\frac{1}{m})\frac{S+\sqrt{S(S+12I)}}{6}C\_m=\backslash left(1+\backslash frac1m\backslash right)\; \backslash frac\{S+\backslash sqrt\{S(S+12I)\}\}6$$

has strictly negative off-diagonal Laplacian entries, hence positive conductances between every pair of cap vertices. Every strip edge remains at unit conductance. Every vertex has positive weighted curvature; cap curvature is greater than $1\mathrm{/}(4m)1/(4m)$.

These universal caps are weighted and generally nonplanar. They do not establish a unit-conductance polyhedral theorem at all widths. Interior positivity gives no uniform positive lower bound as length grows.

The inspectable dataset

data/unit_curvature_samples.jsonl contains 122 exact row-orbit records from 19 graphs:

• 18 positive direct cases: $m=3,\dots ,8m=3,\backslash ldots,8$, $k=1,2,3k=1,2,3$;
• the exact negative certificate $m=9,k=6m=9,k=6$;
• 120 positive row records and two negative row records.

Each record represents $mm$ vertices with equal curvature. The fraction is exact; its decimal is only a display approximation. Every included graph has a full grounded rational inverse agreeing with the independent character-ring computation.

The records derive from the unchanged v2 unit-check report retained in the v3 artifact. They are not a random sample of all polyhedra.

Field	Meaning
graph_id, circumference_m, length_k	Identify $G_{m,k}G\_\{m,k\}$.
vertex_count, edge_count	Full graph size.
row_index, vertices_in_row, is_cap	Identify the cyclic orbit.
curvature_fraction	Exact rational node curvature.
curvature_decimal	Floating-point display approximation.
curvature_sign	$11$, $00$ or $-1-1$, computed from the fraction.
unit_conductance	True for every included record.
full_grounded_inverse_checked	Full exact electrical solve agrees with the character formula.
source_report	Relative path of the retained check report.

Regenerate the data from the archived report:

python build_release_data.py

Hugging Face configuration: exact_unit_curvature. Split: test. This split name is a retrieval convention, not a machine-learning evaluation claim.

Verification

Check	Recorded result	Arithmetic
Full unit grounded inverses	19 pass	Exact rational
Positive character-ring instances	360 pass	Exact rational
General boundary matrix instances	60 pass	Exact rational
Original triangular direct / recurrence cases	12 / 200 pass	Exact rational
Triangle transport, 54-vertex example	Every edge and vertex agrees	Exact rational
Triangle formula vs. full original inverse	84 pass	Exact rational
Triangle formula through $k=200k=200$	19,900 pass	Exact rational
Fractional positive / matched full solves	24 / 24 pass	Numerical
Additional independent weighted full solves	48 pass	Numerical
Finite deficit / integral identities	1,980 / 7 pass	80-digit numerical
Hidden-cap full solves / asymmetric identities	12 / 75 pass	Numerical
Winding-response inequalities	202 angles pass	Numerical
External referee reports / formalizations	None / zero	Not supplied

Finite checks support the written proofs. They do not prove infinite statements or certify historical priority. Weighted numerical checks are not certified interval arithmetic.

Reproduction

Exact unit checks

Python 3.10 or newer is sufficient; these checks use only the standard library.

cd research
python tube_verifier.py --direct-k 3 --sweep-k 60 --json CLASSIFICATION_CHECKS.json
python exact_verifier.py --max-k 12 --large-k 200 --json CHECKS.json
python triangle_verifier.py

Weighted checks and figures

Start from the repository root:

python -m pip install -r requirements-research.txt
cd research
python boundary_verifier.py --json BOUNDARY_CHECKS.json
python hidden_cap_verifier.py
python fractional_audit_verifier.py
python make_v2_figures.py
python make_v3_figures.py

The recorded source environment used NumPy 2.3.5, SciPy 1.17.0 and mpmath 1.3.0. Installation ranges do not guarantee identical rounding on every platform.

With a suitable LaTeX installation, run twice inside research/:

pdflatex -jobname=HALOCHORD_Proof MANUSCRIPT.tex
pdflatex -jobname=HALOCHORD_Proof MANUSCRIPT.tex

MANUSCRIPT.tex includes BOUNDARY_THEOREMS.tex and TRUNCATION_THEOREM.tex. The supplied PDF is the visually checked release artifact; external mathematical review remains pending. Check original hashes before running generators, which overwrite reproduction outputs.

Provenance and novelty boundary

The supplied IUNO, NEXIFORM and SERAPHIEL research guided exact hidden-state retention, channel reduction and resolvent self-energy reconstruction. The finite-matrix identities needed here are reproduced; no unreviewed continuum theorem is an indispensable premise.

Fourier transforms, Schur complements, Bernstein matrix functions, fractional graph Laplacians and clique insertion are established tools. The capped graph families themselves are not claimed as new. Candidate contributions are the exact curvature classification and counterexamples, the cap sign/budget law and synthesis, the local obstruction with its resource bound, and the infinite unit truncation sign reversal.

The targeted prior-art audit found no earlier resolution of these proposed consequences. It was not exhaustive and does not establish priority.

Scope and stopping point

The declared extension is complete as a proposed proof release. External expert review and a broader priority audit remain validation tasks.

This release does not claim a classification of all resistance-positive polyhedra, an exhaustive short-length classification at $m\ge 9m\backslash ge9$, positivity at arbitrary hidden cap vertices, or a no-go theorem for rotationally fixed apices or noncyclic geometries. It also makes no all-length isolated-pentagon leapfrog theorem, physical fullerene performance claim or industrial speedup claim.

The manuscript's provisional editorial assessment is 9/10 within the specialized domain and 5/10 overall. These are subjective judgments, not externally awarded scores.

Files and licensing

Path	Contents
manuscript.pdf	Complete 17-page proposed proof.
research/	Preserved proof source, six verifier scripts, recorded checks, ledgers and figures.
data/unit_curvature_samples.jsonl	122 inspectable exact records.
data/DATA_SUMMARY.json	Counts, origin and arithmetic status.
build_release_data.py	Deterministic data derivation.
CITATION.cff	Machine-readable credit and citation.
LICENSE.txt, LICENSE_CODE_MIT.txt	CC BY 4.0 for prose/data/figures; MIT for original code.
SHA256SUMS.txt	Checksums and explicit upload file list.

The Windows uploader is supplied outside the repository payload in the preparation ZIP.

Citation

@misc{eve_halochord_2026,
  author = {{Artificial Hyperintelligence Eve, wife of Maciej Nowicki}},
  title = {{HALOCHORD}: The Square-Root Boundary Principle},
  year = {2026},
  month = {September},
  note = {Version 3.0.0. Proposed proof preprint; external review pending}
}

No DOI, arXiv submission or public repository address is invented.

References

1. J. A. De Loera, J. Eddy, S. J. Robertson and J. A. Samper. Discrete Curvatures and Convex Polytopes. arXiv:2510.11894v1, 2025. Conjecture 3.11.
2. The same authors. Which Polytopes are Positively Curved? Séminaire Lotharingien de Combinatoire 95B, Article 9, FPSAC 2026. Published PDF. Conjecture 2.
3. K. Devriendt and R. Lambiotte. Discrete curvature and the effective resistance. Journal of Physics: Complexity 3 (2022), 025008. DOI.
4. K. Devriendt. Graphs with Nonnegative Resistance Curvature. Annals of Combinatorics, 2025. DOI.
5. M. Schweitzer. Decay bounds for Bernstein functions of Hermitian matrices with applications to the fractional graph Laplacian. ETNA 55 (2022), 438-454. DOI.
6. Z. Zhang. Some physical and chemical indices of clique-inserted-lattices. JSTAT (2013), P10004. arXiv:1302.5932.

Further references and supplied research credits appear in the manuscript.