The Polak–Schrijver code has exactly eight private pairs (with the shannon repository, Stages 0–2W)

Abstract

The lower bound on the Shannon capacity Θ(C7) improved four times between July and August 2026, and every improvement is built on one five-dimensional gadget: the 367-word independent set of Polak and Schrijver (IPL 143 (2019) 37–40) together with eight private pairs — vertices outside the code with a single neighbour in it. The private-pair count is the parameter those recursions are most sensitive to: a ninth pair would be worth 1.736·10−4 over the current record, about seven times the spread between the published recursions. Theorem (proved). In the whole of Z75 exactly eight vertices have a single neighbour in the Polak–Schrijver code, and they are precisely the eight in use. Their confusability graph is a matching with three edges, hence bipartite, so all eight are simultaneously usable and t* = 8. There is no ninth private pair for this code, and the route to a better bound that consists of adding one is closed. The proof is a non-mixing lemma plus two finite checks over 16807 vertices; it does not depend on the code in this repository. Exhaustive computations around the theorem. All eight 367-word codes the Polak–Schrijver construction can produce, with private-pair counts 5, 6, 6, 6, 7, 7, 7, 8 — the printed one is the unique best; the arithmetic that closes smaller codes (366 words would need 13 pairs, 365 would need 16); and a sweep of 8,260,976,640 (code, colouring, automorphism) triples under Aut(C7⊠5), in which the forbidden intersection is never empty — which is why the published construction has to replace one auxiliary vector, and the single offender is exactly the word (2,4,6,3,5) that Itty et al. replace. Exact repair reaches o = 322, the Buys–Polak–Zuiddam value, and no further. Stated as weaker than the theorem, because it is. Of the sixteen proper 2-colourings of the eight pairs, a 367-word admissible auxiliary set was found for exactly one — the one the literature uses, which no paper remarks on. For the other fifteen this is 1500 s of search finding nothing, not a proof, and the note says so. Method. An exact verifier in C with no dependencies, calibrated on the cases where it can fail (1177 corruptions, maximality sweeps, a deliberately defective build that has to be rejected); preregistration sealed before the first search run; RESULTS.md generated from stored results and never edited by hand; 171 verbatim quotations in SOURCES.md machine-checked against the dumped arXiv sources; and every number printed in the note reconciled by script against the computation behind it, an unsourced number counting as a failed build. Files. The note (PDF) and the repository archive at tag v1.1.0: the verifier, the reproductions, the sets with their SHA-256, the scripts behind every number, the note's LaTeX source and the literature checks. Code under Apache-2.0; texts, tables and data under CC BY 4.0. Third-party papers and publisher pages are not redistributed: the arXiv e-prints that Rule 0 checks quotations against are not in the archive and not in the repository's git history — two of the six are under arXiv's perpetual non-exclusive licence, which grants no redistribution right (sources/LICENCES.md). Only each dump's SHA-256 is kept, and scripts/fetch_arxiv.sh re-fetches the papers, so a re-fetch can be compared byte for byte with what the quotations were checked against. SOURCES.md carries only short verbatim quotations with precise references. Changes in 1.1.0. No result, number or proof changed. The archive now also carries the six SHA-256 files the 1.0.0 archive had lost, so the re-fetch check is possible from the record alone; the submission package for Information Processing Letters is included (elsarticle, five pages in the journal's print layout, against its limit of nine, plus the cover letter); the note's text lives in one place, included by both the standalone and the submission build, and the number reconciliation now sweeps 68 ledger entries across both plus the cover letter; and the git history was rewritten so that no commit carries a third-party paper. Public history. The repository's git history was rewritten on 2026-09-26 to remove the dumped arXiv e-prints from every commit, because two of the six papers the quotations are checked against are under arXiv's perpetual non-exclusive licence, which grants no redistribution right. All commit hashes therefore changed: COMMIT-MAP.txt in the repository maps old to new, the archive of version 1.0.0 was built from a commit that no longer exists in the public history, and for each published version the tarball in its record is authoritative rather than a commit hash. scripts/w_archive.py re-checks each record's tarball against the tag it names: v1.1.0 is identical to tag v1.1.0, and v1.0.0 differs from tag v1.0.0 only by the six sources/ /SHA256 files its pruning removed by mistake.