Research: exact seven-source singleton minimum f(7) = 24

[conorholds]

Change and mathematical scope

Scoped finite computational proof; no new κ. Adds research/connector-covering-code/ and proves that the minimum universal restored seven-source singleton word has 24 steps in Ross Richardson’s PR943 interface: immutable sources, an arbitrary dirty connector, singleton source additions, paired readouts and eventual restoration.

This answers open question 1 of Nicholas Stone’s PR2973, which establishes 22 <= f(7) <= 24. The existing length-24 construction is Douglas Colkitt’s PR1185; the new result excludes every shorter word.

The proof combines published covering-code and saturating-set results, a component obstruction for minimum covers, two independent support searches excluding the 21-prefix case, and complete path searches excluding both simple 22-cycle multiplicity patterns.

Reusable corollary: every difference cover P+P = F2^7 has at least twenty points. This excludes all 17-, 18- and 19-point covers directly from the literature and settles the remaining set-level questions in the PR2773/2870/2911/2933/2961/3023/3034 line. The covering bound and component argument also give f(7) >= 22 without any word census. The full f(7)=24 proof still uses freshly replayed five-/six-source multiplicity censuses.

Limits: the theorem concerns the original singleton alphabet. No unrestricted-query, DUO/TRIO, physical implementation, supplier admission or multiplication-exponent improvement is claimed. Only the nine-file research package is added; selected records and frozen predecessors are unchanged.

Task and overlap

• Original task: connector-alphabet-change; scope changed from an alphabet-extension experiment to the exact seven-source singleton minimum.
• Contribution base: 95ff4e2dd396c505932a43ff7c1cdfdd94b8c867.
• Frozen scientific source pin: 9ee73fecdbae9f892a6496dbdfae872217cf4601.
• Guidance checkpoint: d8e0f83f00191df62174b641e9fff055c357f641, priority revision 6.

This closes a named interface question rather than meeting the original alphabet-extension gate. The census-free lower bound is also relevant to the revision-6 census-pruning-audit card; it does not validate the disputed historical prunes or improve a recursive budget.

Nearest overlaps:

• PR2973 corrects the PR2542/2634/2691 word-census pruning and leaves length 22 open. PR2984 and PR3005 replay that lower bound. None is a proof dependency here.
• Aaron Wang’s PR3023 excludes one 18-point-cover class; the subsequent PR3034, head a05791d428f5d06802fb4c96b33732e6347c64eb, leaves the remaining classes undecided. The published twenty-point cover bound resolves that set-level remainder directly. The KB3i handoff releases its scope.
• PR2396 changes the source-call alphabet; its DUO result is outside this singleton optimum.

SOURCES.json and TASK.json retain scientific source pins, predecessor heads, scope history and earlier overlap checks. A final read-only refresh at 2026-10-16 20:47 UTC searched 293 open PRs, 2,440 closed/merged PRs and 1,788 Discussion/comment/reply records, with complete pagination. No competing exact f(7)=24 result or complete length-22 exclusion was found. This metadata search is not an exhaustive novelty certificate. No upstream coordination message has been posted.

Validation

Python 3.12 and a C++17 compiler, from the repository root:

python3.12 -I -B scripts/research_package.py --package research/connector-covering-code --check-only
python3.12 -I -B scripts/research_package.py --package research/connector-covering-code

Recorded environment: macOS arm64, CPython 3.12.13, Apple Clang 21.0.0, C++17 -O3, with a task-local macOS 15.2 SDK override. Registered execution is serial and requires no Python solver dependency.

All five registered commands passed in 87.415 seconds. The complete native exclusion covers five A and 77 B orbits, 22,594,820 gap vectors and 16,055,378 search nodes, with zero witnesses. Both Python checkers run normally and under -O, agreeing on all 234 surviving character cases and 2,808 structural branches.

Controls rediscover universal 24-step fixtures through full gap enumeration and recursive branching, and reject wrong joins, wrong counts and repeated prefixes. Four altered control receipts were rejected; a deliberately incomplete run failed without claiming exhaustion. Source pins, committed bytes, attribution, patch round-trip and diff checks passed.

Both published premises were read directly during the AI audit, and all 815 transcribed coordinates matched the paper. Pattern A has independent implementation and solver corroboration. Pattern B has one complete native implementation: its independent Z3 checks return three UNSAT and 74 inconclusive timeouts. A prior complete replay without conditional pruning and sanitizer checks of the controls and B orbit 77 provide additional coverage of the same search core, not a second implementation.

Not run: the original published classification searches, a complete B exhaustion by a second implementation, proof-assistant verification, independent human review, hosted CI or the selected supplier’s global cold replay. Finite replays check the enumerations and controls; the theorem also relies on the written completeness reductions and published premises.

Credit

Ross Richardson/PR943 supplies the connector model and five-source result; Douglas Colkitt/PR1185 supplies the six-source result and seven-source upper construction. Struik’s Bound 3.13 credits Ytrehus; Davydov, Marcugini and Pambianco supply the published classification and all 41 transcribed representatives.

Credit Aaron Wang’s set-level work, Nicholas Stone’s census and correction, Tekalign’s PR2984 replay and Nick Petalas’s PR3005 independent replay. Detailed predecessor roles, exact pins and assistance disclosures remain in the package README and metadata. New work is Apache-2.0.

Open next questions

The README and TASK.json record the mathematical next questions and blockers. Two useful follow-ups are:

1. Independently re-exhaust all 77 B orbits with a separate implementation, starting from the documented path constraints and positive controls. This would address the principal remaining implementation-independence limitation; the current solver timeouts do not resolve it.
2. Replace the finite support/path searches or the published classification dependency with a shorter invariant-based proof. This would reduce the verification burden; no such replacement is established here.