Audit the 20-prefix confinement step of the length-22 exclusion (no new kappa)

[petalas]

Change and scope

Bounded independent verification; no kappa, no selected-frontier change. Adds research/f7-l22-confinement-audit/.

PR #3058 proves f(7) = 24 in PR943's singleton prefix-walk model. Its length-22 argument rests on |P| >= 20 (published covering bound), then on excluding |P| = 20, 21 and 22. This PR audits the |P| = 20 step with its own code and its own derivation.

Argument re-derived here. If |P| = 20, every position of the walk lies in the twenty-point set T = P, so the walk is a closed walk spanning T and the graph induced on T by the walk's seven directions must be connected. Each basis direction contributes only the pairs of T at one difference, so

edges <= sum of the seven largest multiplicity classes of diff(T),

while a graph on twenty vertices with at most two components needs at least eighteen edges. Every 20-point cover is an affine image of {0} u S for one of the five 19-point representatives in DMP's Table 1 (imported), and G(M T + b, D) = G(T, M^-1 D), so one inequality per representative covers every basis and every cover.

Representative	seven largest multiplicities	sum	edges needed	verdict
1, 2, 3	3,3,3,3,3,3,3	21	18	arithmetic insufficient
4, 5	2,2,2,2,2,2,2	14	18	excluded for every basis, by arithmetic

Representatives 1-3 are reported open with bounded evidence only: exact canonical-basis component counts 9, 12, 4 and deterministic 200-base sample minima 6, 5, 5. No sampled basis produced two components. Sampling is not a proof and the package says so.

Also re-derived: a simple 22-cycle can only carry the multiplicity patterns (6,4,4,2,2,2,2) and (4,4,4,4,2,2,2), from the imported projections f(6) = 16 and f(5) = 12. Word-level searches give a universal 24-step positive control (145164232614626031626540, 128/128, |P| = 24, restored) and two non-universal simple 22-cycles as controls: 1206231531046325014203 at 126/128 (pattern B, masks 82 and 120 unrealised) and 2130240236105010621045 at 125/128 (pattern A).

What this does not establish

Not a re-proof of f(7) = 24. The covering bound, the cover classification, f(5) = 12, f(6) = 16, the size-21 argument and the size-22 exhaustion are imported. Three of five cover classes remain sampled. No census, no tariff, no supplier, no kappa.

Validation

• python3.12 -B research/f7-l22-confinement-audit/verify.py -> PASS_F7_L22_CONFINEMENT_AUDIT arithmetic_complete=2/5 sampled_only=3 f7_untouched (well under a second, stdlib only, deterministic).
• python3.12 -B scripts/research_package.py --package research/f7-l22-confinement-audit -> PASS_REGISTERED_RESEARCH_PACKAGE.
• python3.12 -B scripts/check_research_commit.py --package research/f7-l22-confinement-audit --check-source-history -> PASS_RESEARCH_COMMIT_BYTES.
• Interpreter: ~/git/integer-mult-bounds/.venv/bin/python3.12 (CPython 3.12), git rev-parse HEAD = 0f000bb4aac86fa667906d8092eb36e49a76944a, base 20ecf2891d80b5acf18d980b2164fb44ea7faa90.
• scripts/research_baseline.py could not run in this sparse checkout (it reads a selected-record file that is not materialised); that is a checkout artifact, not a result.

Overlap: python3 scripts/find_overlap.py --task census-pruning-audit was run before submission; its report states its own coverage and freshness.

Credit: Ross Richardson (PR943 model), Davydov/Marcugini/Pambianco (covering-code facts and the minimal 1-saturating classification transcribed in research/connector-covering-code/table6.json), Nicholas Stone (PR2973 corrected budgets), the PR3058 contributor, Douglas Colkitt.

This contribution was produced by Nick Petalas with AI assistance. Independent human and formal proof-kernel review are unclaimed.