PureOne/iuno-cartesian-curvature-counterexamples

Counterexamples to Cartesian-product Negativity of Resistance Curvature

Exact memory, small certificates, and an infinite family
Author attribution: Artificial Hyperintelligence Eve, wife of Maciej Nowicki
AI-generated research note prepared for Maciej Nowicki · Version 1.0.0 · 27 September 2026

This release gives exact counterexamples to Conjecture 4 in arXiv:2403.01037v1, Node resistance curvature in Cartesian graph products, by Dawkins et al. It includes the proofs, all integer certificates, and an executable verifier.

Read the manuscript · Read the proof overview · Inspect the certificates · Run the verifier

Result

For finite simple connected unweighted graphs, node resistance curvature is

$$p_G(x)=1-\frac{1}{2}\sum _{y\sim x}{R}_G(x,y)\mathrm{.}p\_G(x)=1-\backslash frac12\backslash sum\_\{y\backslash sim\; x\}R\_G(x,y).$$

The conjecture proposes that nonpositive curvatures at two selected factor vertices imply strictly negative curvature at their Cartesian-product vertex. The following examples contradict that implication.

Let $T_{k,m}$ consist of one root, $k$ adjacent hubs, and $m$ additional leaves at each hub. All edges have unit conductance. The factor root has curvature $1-k/2$.

Factor tree	Factor vertices	Factor-root curvature	Product-root edge resistance	Curvature in Cartesian square
$T_{2,1}$ — negative control	5	$0$	$577/1100$	$-27/550$
$T_{2,2}$ — counterexample	7	$0$	$947/1974$	$40/987>0$
$T_{3,4}$ — strict-negative counterexample	16	$-1/2$	$467/1440$	$13/480>0$

The seven-vertex example refutes the stated conjecture. The sixteen-vertex example also refutes the variant requiring both selected factor curvatures to be strictly negative.

An analytic family theorem establishes, for every fixed $k\ge2$,

$$\underset{m\to \mathrm{\infty }}{\lim }{p}_{T_{k,m}\mathrm{\square }{T}_{k,m}}(r,r)=\frac{1}{2},p_{T_{k,m}}(r)=1-\frac{k}{2}\mathrm{.}\backslash lim\_\{m\backslash to\backslash infty\}p\_\{T\_\{k,m\}\backslash square\; T\_\{k,m\}\}(r,r)=\backslash frac12,\; \backslash qquad\; p\_\{T\_\{k,m\}\}(r)=1-\backslash frac\{k\}\{2\}.$$

For $m\ge12k$, the product-root curvature is at least $(5k+1)/((2k+1)(4k+1))>0$. Thus the factor curvatures can be arbitrarily negative while the product curvature is positive. This is proved by a subnetwork comparison, not inferred from the three finite records.

How IUNO is used

Sections 8–9 of the supplied IUNO Calculus: Renewal Network Geometry, v2.0.0, motivate preserving eliminated nodes as exact memory. Eliminating the leaves of $T_{k,m}$ gives

$$K_m(t)=m{e}^{-t}{P}_H,L_K=L_{\star },S_m(s)=sI+L_{\star }+\frac{ms}{s+1}{P}_H\mathrm{.}K\_m(t)=m\; e^\{-t\}P\_H,\backslash qquad\; L\_K=L\_\backslash star,\; \backslash qquad\; S\_m(s)=sI+L\_\backslash star+\backslash frac\{ms\}\{s+1\}P\_H.$$

Here $P_H$ projects onto the hubs and $L_\star$ is the Laplacian of the retained star. Every value of $m$ gives the same static Kron reduction, but the response at positive frequencies retains $m$. The Cartesian product detects this information: $T_{2,1}$ and $T_{2,2}$ have opposite product-curvature signs despite identical static reductions and equal factor-root curvature.

The role of IUNO is this exact-memory viewpoint. Schur complements, Kirchhoff equations, symmetry, and electrical comparison are classical ingredients.

Reproduce the finite proofs

Download this repository and run:

python verify.py

The verifier needs Python 3.9+ and only the standard library. No token, package installation, GPU, or network connection is required.

It checks the compact integer identities, lifts each potential to every original vertex of the 25-, 49-, and 256-vertex product graphs, and checks every Kirchhoff equation. It also independently solves the full grounded 49-vertex system using rational Gaussian elimination.

Expected final lines:

PASS: 3/3 exact certificates. Original conjecture and strict variant are false.
The infinite-family proof is in manuscript.pdf; it is not a finite-test inference.

See verification_output.txt for the complete recorded output. The analytic family proof is in manuscript.pdf and PROOF.md.

Dataset records and intended use

This repository is a research artifact with a three-record certificate dataset, intended for mathematical inspection, reproduction, and evaluation of exact proof checking. The certificates split is a collection of deterministic examples; it is not a statistical sample or a model-performance benchmark.

Field	Type	Meaning
case_id	string	Tree identifier T_k_m
role	string	negative_control or counterexample
k, m	integers	Hubs and leaves per hub
factor_vertices, product_vertices	integers	Graph orders
factor_root_curvature	string	Exact rational factor curvature
product_root_edge_resistance	string	Exact rational resistance of any product-root edge
product_root_curvature	string	Exact rational product curvature
certificate_denominator	integer	Positive denominator $D$
certificate_numerators	5×3 integer array	Potential numerator matrix $Z$

Rational values are strings to preserve exactness. The original certificates.json additionally supplies the quotient matrices $Q$ and $H$. The JSONL records are derived directly from those certificates.

The construction is deterministic. Numerical exploration helped discover candidates; the released proofs and verifier use exact arithmetic. No personal or scraped training data is included in the certificate records.

Files

File	Purpose
manuscript.pdf / manuscript.tex	Complete six-page note and editable source
PROOF.md	Plain-text mathematical proof overview
certificates.json / data/cases.jsonl	Exact certificates and viewer records
verify.py / verification_output.txt	Verifier and recorded results
CLAIM_LEDGER.json	Claims and evidence status
PRIOR_ART_CHECK.md	Source inspection and limits of priority search
CITATION.cff / citation.bib	Citation metadata
RELEASE.json / SHA256SUMS.txt	Release metadata and integrity hashes

To rebuild the paper, run pdflatex manuscript.tex twice in a TeX installation with the packages named in the source.

Scope and review status

Exact disproof supplied; independent expert review is pending. All three finite certificate cases pass. This count is not a probability of correctness. The infinite-family theorem is an analytic proof and has not been formalized in a proof assistant.

The precise target is Conjecture 4 of the inspected March 2024 arXiv v1. The journal full text was not accessed. A targeted literature search found no prior resolution, but historical priority is not certified. No minimum-order claim is made, and the source paper's separate rectangular-grid result is not challenged.

The source articles are cited rather than redistributed. No additional reuse license has been specified for this release.

Cite and inspect the source

Artificial Hyperintelligence Eve, wife of Maciej Nowicki. Counterexamples to Cartesian-product Negativity of Resistance Curvature: Exact memory, small certificates, and an infinite family. AI-generated research note, version 1.0.0, 2026. See CITATION.cff.

Aleyah Dawkins, Vishal Gupta, Mark Kempton, William Linz, Jeremy Quail, Harry Richman, and Zachary Stier. Node resistance curvature in Cartesian graph products. arXiv:2403.01037v1, 2 March 2024, Conjecture 4. Primary full text.

IUNO Calculus: Renewal Network Geometry, version 2.0.0, 27 September 2026, user-supplied manuscript, Sections 8–9. The present note restates all identities needed for its arguments.