An independent exact recheck of the PG(2,9) multiple-unicast counterexample (a recheck of a result by others)

M. Braverman and Z. He (arXiv:2610.10108, 'The Multiple Unicast Conjecture is False', 7 Oct 2026, unrefereed) give a network obtained from the PG(2,9) incidence graph (182 vertices, 910 edges) by deleting 175 edges, with 157 unicast sessions and a deterministic causal linear code over GF(9) (their Section 4), and conclude that coding beats fractional multicommodity flow in an undirected network, against the Li-Li / Harvey-Kleinberg-Lehman conjecture. The paper says GPT-6 found the non-deterministic example of its Section 3; the claim checked here is the deterministic Section 4 code. This is an independent recheck of a result by others. We claim nothing new, and we do not say the conjecture is resolved beyond the verified finite facts. What was rechecked (Tier C: exact finite-field, integer and rational computation; Tier X for random-input simulation). The graph (735 edges after deletion, every one of the 157 sessions at distance exactly 5, total 785); exact GF(9) arithmetic (producer's tables; auditor's Sage field and a GF(3) block cross-check); all 182 local codes 5-dimensional MDS; the 910x910 parity matrix of rank 744 over GF(9), dim W = 166, dim W_C = 157, injective on the session edges; the paper's Appendix B assignments (dim W = 166 exactly, so the paper's bound is tight); decoding of all 157 unit inputs, hence of every input by linearity; and the optimum of the fractional flow, exactly 147/157 (an exact rational flow of that value with every edge saturated, plus the length-function dual). Under the paper's own Section 2 definition one execution already reaches rate 1. Controls pass; a scaled local-code row fails 16 of 20 trials. Audit. A separate AI agent instance (not a human) reproduced the results with independent code (ACCEPT_WITH_CORRECTIONS; corrections applied). Outcome: confirmed as a recheck; Tier A not reached. Limitations. Not verified: that the paper's Section 2 model is the model of the conjecture, the BGS17 amplification and Corollary 1.2, Section 3's example, the authors' verifier and repository (not run), the OpenAI preprint, the Lean formalization claimed by the independent same-day counterexample arXiv:2610.09367 (a completely different construction, not checked here), and the kingy.ai/MathWorld statements. The preregistration was committed with the result; its precedence is shown only by file timestamps. No Lean; no human audit; the prior-art search was limited. The Appendix A lists of the third-party paper and their transcription are not redistributed (hashes and extraction instructions only).

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Publication Details

Journal
Zenodo (CERN European Organization for Nuclear Research)
Published
2026-11-02
DOI
https://doi.org/10.5281/zenodo.23879682
Primary Topic
Complexity and Algorithms in Graphs
Type
preprint
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preprint

An independent exact recheck of the PG(2,9) multiple-unicast counterexample (a recheck of a result by others)

Xavier Callens
Zenodo (CERN European Organization for Nuclear Research)
Complexity and Algorithms in Graphs
preprint

An independent exact recheck of the PG(2,9) multiple-unicast counterexample (a recheck of a result by others)

Xavier Callens
preprint en

Abstract

M. Braverman and Z. He (arXiv:2610.10108, 'The Multiple Unicast Conjecture is False', 7 Oct 2026, unrefereed) give a network obtained from the PG(2,9) incidence graph (182 vertices, 910 edges) by deleting 175 edges, with 157 unicast sessions and a deterministic causal linear code over GF(9) (their Section 4), and conclude that coding beats fractional multicommodity flow in an undirected network, against the Li-Li / Harvey-Kleinberg-Lehman conjecture. The paper says GPT-6 found the non-deterministic example of its Section 3; the claim checked here is the deterministic Section 4 code. This is an independent recheck of a result by others. We claim nothing new, and we do not say the conjecture is resolved beyond the verified finite facts. What was rechecked (Tier C: exact finite-field, integer and rational computation; Tier X for random-input simulation). The graph (735 edges after deletion, every one of the 157 sessions at distance exactly 5, total 785); exact GF(9) arithmetic (producer's tables; auditor's Sage field and a GF(3) block cross-check); all 182 local codes 5-dimensional MDS; the 910x910 parity matrix of rank 744 over GF(9), dim W = 166, dim W_C = 157, injective on the session edges; the paper's Appendix B assignments (dim W = 166 exactly, so the paper's bound is tight); decoding of all 157 unit inputs, hence of every input by linearity; and the optimum of the fractional flow, exactly 147/157 (an exact rational flow of that value with every edge saturated, plus the length-function dual). Under the paper's own Section 2 definition one execution already reaches rate 1. Controls pass; a scaled local-code row fails 16 of 20 trials. Audit. A separate AI agent instance (not a human) reproduced the results with independent code (ACCEPT_WITH_CORRECTIONS; corrections applied). Outcome: confirmed as a recheck; Tier A not reached. Limitations. Not verified: that the paper's Section 2 model is the model of the conjecture, the BGS17 amplification and Corollary 1.2, Section 3's example, the authors' verifier and repository (not run), the OpenAI preprint, the Lean formalization claimed by the independent same-day counterexample arXiv:2610.09367 (a completely different construction, not checked here), and the kingy.ai/MathWorld statements. The preregistration was committed with the result; its precedence is shown only by file timestamps. No Lean; no human audit; the prior-art search was limited. The Appendix A lists of the third-party paper and their transcription are not redistributed (hashes and extraction instructions only).

Zenodo (CERN European Organization for Nuclear Research)
Complexity and Algorithms in Graphs
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