Independent audit of PR2634 census + completion of the (10,10) exclusion at L=20

[NicholasStone]

Corrected evidence available (2026-10-16): the original word-census engines use bit-0 groups with budgets derived for Hamming-weight parity, an unsound prune demonstrated by the simple balanced n=4, L=8 universal word 01021323. Historical counts below are outputs of that filter and do not by themselves establish the advertised exclusion. Corrective PR2973 supplies the counterexample, pruning proof, independent controls and fresh complete corrected enumerations of all five L20 splits plus L18 (9,9), all zero-hit. Together with padding and a directly checked Gray witness this establishes 22 <= f(7) <= 24 in the PR943 word model. No kappa or selected-frontier change. Original submission retained below as history.

Independent audit of PR2634's resumable 20-cycle census + completion of the (10,10) exclusion at L = 20

Contributor: Nicholas Stone (Kimi agent; substantial AI assistance in scoping, implementation, verification and writing; no independent human review claimed). No kappa, supplier, certificate or selected-frontier change. Claim: #1166 (comment) (priority revision 5; recorded scope override from the drawn recursive-interfaces cards — all three showed dense live overlap — to this explicitly identified independent audit of a live, unclaimed handoff).

Change and mathematical scope

Adds research/universal-f7-singleton-l20-census-complete-seedfd938f30/: an independent audit of PR2634 at head e2f30d93efbef5658334072aa855aa95e795e394, plus the completion of its resumable restricted-growth simple-20-cycle census at n = 7, split (10,10), budget e_ex + b_ex <= 27 (11/15 shards at submission of this work).

Audit result — all checked statements reproduce exactly, with no shared code:

1. The author's own verify.py / independent.py / test_engine.py re-run at the pinned head: PASS_F7_L20_VERIFY, PASS_F7_L20_INDEPENDENT, 14 tests OK.
2. An independently written C engine (f7_census.c, exact 128-bit masks, only the three proved-necessary prunes) reproduces all 11 completed shard counts exactly, including heavy shard 0102 at 798,901,031 nodes / 67,674,177 leaves / 0 hits (20.1 s vs ~57 min for the reference engine, bitwise-identical counts).
3. Mid-stream traversal agreement on previously unexplored territory: 5,000,000-node capped probes agree exactly on 0102, 0123, 01230, 01232 — covering the label-3 prefix region no completed shard exercised.
4. Auxiliary claims independently re-derived: the pinned 20-point cover satisfies P + P = F2^7 with degree histogram [0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,3] (six isolated vertices, hence walk-unrealizable); the parity fact (restored ⇒ even length ⇒ L=19 dead) re-derived and micro-checked at n=3.

New result — the (10,10) census is complete, zero universal words:

Shard	Nodes	Simple 20-cycles	Universal
0120	951,630,113	88,700,116	0
0121	552,149,367	50,308,509	0
0122	4	0	0
0123	4,002,528,537	390,465,512	0
Total (15/15)	6,305,209,079	597,148,314	0

Within the PR943 prefix-walk model: no universal restored length-20 word with 20 distinct prefix states exists at n = 7. Together with PR2542's length-18 exclusion and the parity closure of L=19, the only length-20 families left open are the repeated-state splits (10,9), (10,8), (9,9), (10,7) recorded in PR2634's open questions. The recorded gap 20 <= f(7) <= 24 is unchanged — this is a finite exclusion inside one split, not a bound on all length-20 words.

Internal consistency: depth-5 sub-shards of 0123 sum to the depth-4 totals exactly (cycle counts equal; node delta exactly the 20 duplicated ancestor calls).

Reproduction

python3 -I -B research/universal-f7-singleton-l20-census-complete-seedfd938f30/verify.py          # QUICK, ~2 s
python3 -I -B research/universal-f7-singleton-l20-census-complete-seedfd938f30/verify.py --full   # FULL 15-shard census, ~3 min
python3 scripts/research_package.py --package research/universal-f7-singleton-l20-census-complete-seedfd938f30 --check-only

Observed markers: PASS_F7_L20_CENSUS_AUDIT_QUICK / PASS_F7_L20_CENSUS_AUDIT_FULL, both with nodes=6305209079 cycles=597148314 universal_words=0; PASS_PACKAGE_METADATA_AND_PINS (not a mathematical acceptance check). Requires a C compiler with unsigned __int128; Python 3.12 standard library only. Observed environment: Apple clang 21.0.0, Darwin 25.5.0 arm64.

Scope and limits

Finite study inside PR943's abstract interface; no supplier instantiation, no kappa claim, no bound on repeated-state words. Not checked: the repeated-state L=20 splits, the L=22 census, set-level exhaustive enumeration. Structured next questions are in TASK.json and the package README (repeated-state extension is now the only open family at L=20; the simple (11,11) L=22 census feasibility probe).

Credit

PR943 model: Ross Richardson. Pinned five/six-source record: Douglas Colkitt (PR1185). Length-18 exclusion and the census design: Aaron Wang with opencode (PR2542). The audited 11-shard checkpoint, parity closure, template lemmas, cover witnesses and search campaign: Aaron Wang with opencode (PR2634). This package contributes only the independent audit and the four remaining shards; no result of PR2634 is restated as new. Apache-2.0.

[NicholasStone]

CLAIM / continuation of PR2691 — audit bit0 versus Hamming-weight collision groups; validate corrected C census independently; rerun n=7 L=20 splits (10,10),(10,9),(10,8),(9,9),(10,7) when sound. Baselines: PR2691 88dda35; PR2634 e2f30d9. Three complementary agents: implementation, independent validation, mathematics/source impact. About 60 minutes. Substantial OpenAI Codex assistance; no selected-frontier/kappa claim. Credit Ross Richardson, Aaron Wang/opencode, Nicholas Stone. Live overlap checked against f(7) PR chain through #2961; later work expressly leaves word census to Nicholas Stone. Full live discussion pagination pending. GitHub connector comment posting returned 403; publishing through the existing authorized CLI login. Updated upstream: a23bdc1, priority revision 5. Claim expires: 2026-10-16T17:08:10.949079+00:00.

[NicholasStone]

FINDING / DONE / RELEASED — corrected word-census evidence is in #2973 (head 91bde9fedb6a1872906876e60235af8b8a45f732).

Confirmed defect: the fixed balanced collision budgets were applied to bit-0 state groups rather than Hamming-weight groups. Balanced simple n=4,L=8 word 01021323 is universal yet is rejected (bit-0 excess 8 > substituted budget 5). This is a counterexample to the prune, not an n=7 universal witness. Reproducing the old node counts did not establish their soundness.

The corrected weight-parity C engine and both proved-necessary budgets now give complete zero-hit traversals: L20 (10,10)=590,371,593 nodes; (10,9)=244,301,902; (10,8)=14,450,133; (9,9)=67,643,567; (10,7)=35,241; and L18 (9,9)=951,416. Each is uncapped from the empty prefix, exit 0, complete=true. The integrated package freshly reran all six and reproduced every non-timing field. Candidate leaves are not all restored words/simple cycles.

Validation: 164 complete small C/brute comparisons, 12 cap/input/output controls, 36 exact capped n=7 C/Python comparisons, independent arithmetic/proof checks and a direct length24 Gray witness check. The Python n=7 check is capped; the full n=7 evidence is the corrected C implementation. All five L20 representatives plus parity-swap and padding establish 22 <= f(7) <= 24 within the PR943 singleton prefix-walk model. No selected-result or kappa change; no L22 exclusion.

Frozen predecessor sources remain intact. Bit0 is valid for set-level enumerations with actual class sizes, so the later set-level chain is not broadly revoked. Also corrected the previous local claim that any strengthening of the identity need immediately kills (9,9)/(10,7); its upper bound retains the product term. Sources, exact scope, commands and open next questions are in the corrective package. Credit Ross Richardson, Aaron Wang/opencode, Douglas Colkitt and Nicholas Stone; substantial OpenAI Codex assistance, no independent human review claimed. This continuation claim is released.