Strengthen the length-22 seven-label census with a union-support budget; exclude the (9,9) split (no new kappa)

[petalas]

Summary

Adds one proved necessary condition to the corrected seven-label word census of #2973 and uses it to extend the length-22 split domain of open PR #3731. Produced by Nick Petalas with AI assistance. Independent human and formal proof-kernel review are unclaimed.

Scope, up front: finite word census inside PR943's singleton prefix-walk model. No kappa, supplier, tariff, certificate, frozen-input or selected-frontier change. f(7) remains 22 <= f(7) <= 24; this PR does not decide length 22. It excludes one more split and measures floors for the rest.

What is established

1. A strictly stronger even-side budget, with a proof. The merged independent census #3005 proves that a universal completion must satisfy

   |U| + C(S,2) + C(T,2) - C(e,2) - C(o,2) >= 2^(n-1) - 1,
   U = supp EE union supp OO.
   

   This package ports exactly that bound into the fast corrected C engine of Correct word-census parity budgets; 22 <= f(7) <= 24 in the PR943 model #2973 (plus the L<=22 gate already recorded in Bounded length-22 census probe: exclude splits (11,7), (10,8), (11,8); measure the rest (answers #3708 OQ1's blocker; no new kappa) #3731). Because |U| <= |supp EE| + |supp OO|, it dominates the predecessor's even excess budget a+b <= B0, which charges an overlapping mask twice. The complete engine delta is the committed zero-context engine/DELTA.diff against the pinned corrected_census.c (sha256 265ead91...). The pinned bytes are unchanged; --no-union-budget restores the predecessor's even test.

2. Soundness evidence. The predecessor's raw-product brute-force cross-check (validation/check_corrected_c.py, unmodified) passes with the union engine: 164 comparisons over all small splits and all four budget modes, including positive universal counts, with cap and fail-closed controls. A union-only run (both older budgets disabled) also matches the brute-force universal counts. The six recorded L20/L18 receipts replay complete and zero-hit. Node counts there are intentionally not compared: the union budget removes nodes by design, and the pinned receipts are untouched.

3. A new length-22 finite split exclusion: (9,9).

   Split	Nodes	Outcome
   (11,7)	8,620,043	complete, zero hits (control for Bounded length-22 census probe: exclude splits (11,7), (10,8), (11,8); measure the rest (answers #3708 OQ1's blocker; no new kappa) #3731)
   (10,8)	239,946,434	complete, zero hits (control; Bounded length-22 census probe: exclude splits (11,7), (10,8), (11,8); measure the rest (answers #3708 OQ1's blocker; no new kappa) #3731: 597,933,297)
   (11,8)	1,017,687,439	complete, zero hits (control; Bounded length-22 census probe: exclude splits (11,7), (10,8), (11,8); measure the rest (answers #3708 OQ1's blocker; no new kappa) #3731: 1,349,702,650)
   (9,9)	388,563,293	complete, zero hits → new finite split exclusion
   (10,9)	≥ 6.7976e9	one shard capped at 3e9; floor only
   (10,10)	≥ 1.43517e10	six shards capped at 1.5e9; floor only
   (11,9)	≥ 8.1284e9	one shard capped at 1.5e9; floor only
   (11,10)	≥ 7.5000e9	five shards capped at 1.5e9; floor only
   (11,11)	≥ 6.0000e9	four shards capped at 1.5e9; floor only

   Every split is the fifteen disjoint restricted-growth length-four prefix shards. RESULT.json records per-shard receipts; a split is an exclusion only when all fifteen shards exit complete with zero hits and no cap. The five floors are sharded-node sums with capped shards counted at their caps; they are not exclusions.

   Domain after this PR. Bounded length-22 census probe: exclude splits (11,7), (10,8), (11,8); measure the rest (answers #3708 OQ1's blocker; no new kappa) #3731 left six splits: {(9,9),(10,9),(10,10),(11,9),(11,10),(11,11)}. Excluding (9,9) leaves five. The union budget also cuts the (10,8) tree by 2.49x and the (11,8) tree by 1.33x at the same stated scope.

4. Negative answer to Bounded length-22 census probe: exclude splits (11,7), (10,8), (11,8); measure the rest (answers #3708 OQ1's blocker; no new kappa) #3731's pair-identity question. The exact identity 2 sum_x A_x B_x = sum_y r_y^2 - s*t gives a leaf lower bound on a partial quantity that is monotone increasing, so it cannot reject a partial state. A usable prune needs a leaf upper bound; the union-support bound is one. This is recorded rather than implemented as an unsound prune.

Verification

# Registered checks (each under the 600 s per-command margin; local timings below):
python3.12 -I -B research/f7-l22-union-census/verify.py --skip-probe      # 37 s
python3.12 -I -B research/f7-l22-union-census/verify.py --probe-split 11,7  # 0.4 s
python3.12 -I -B research/f7-l22-union-census/verify.py --probe-split 10,8  # 6.6 s
python3.12 -I -B research/f7-l22-union-census/verify.py --probe-split 11,8  # 35 s
python3.12 -I -B research/f7-l22-union-census/verify.py --probe-split 9,9   # 10 s

# Local only, not run by CI: all completed splits.
python3.12 -I -B research/f7-l22-union-census/verify.py --probe-all

• CI replay: passed. plan pass (2m6s) and replay (research/f7-l22-union-census) pass (6m30s) on Actions run 39169381951, job 117799522004, which ran all five registered checks under CPython 3.12 in CI. This is the authoritative certification of the four exclusions and the validation stages.
• PASS_REGISTERED_RESEARCH_PACKAGE research/f7-l22-union-census outcome=completed also locally, with all five registered checks.
• PASS_RESEARCH_COMMIT_BYTES at e09a08b5f85e648be373bece1b6ea1265d5ea8fa, base 20ecf2891 (canonical main, priority revision 6), with --check-source-history.
• Interpreter: CPython 3.12.15 at ~/git/integer-mult-bounds/.venv/bin/python3.12; Apple arm64, 18 cores, Apple clang.
• Sharded node sums are deterministic and compared exactly; timing fields are excluded from every equality check. A sharded sum exceeds an unsharded run by duplicated shared prefix nodes (51 when every shard survives to depth four), so sharded totals are compared only with sharded totals.

Overlap and credit

• Overlap re-checked at submission against open PRs and discussions; Bounded length-22 census probe: exclude splits (11,7), (10,8), (11,8); measure the rest (answers #3708 OQ1's blocker; no new kappa) #3731 is the direct predecessor probe and Independently corroborate the corrected seven-label word exclusion #3005 is the merged source of the union bound. No other in-flight L22 union-support run was found.
• Nicholas Stone (Correct word-census parity budgets; 22 <= f(7) <= 24 in the PR943 model #2973 corrected engine, budgets and split theory); Aaron Wang (PR2542/PR2634 word model); Ross Richardson (PR943 prefix-walk model); Nick Petalas (Independently corroborate the corrected seven-label word exclusion #3005 union-support bound; PR Bounded length-22 census probe: exclude splits (11,7), (10,8), (11,8); measure the rest (answers #3708 OQ1's blocker; no new kappa) #3731 L22 domain and budget). This package: produced by Nick Petalas with AI assistance.
• Frozen predecessor packages and upstream/ are unchanged.

Open next questions

1. Finish the five remaining splits: each capped shard needs a larger cap, or a further proved completion bound. (10,9) has one shard above 3e9; (10,10) has six above 1.5e9.
2. Independently replay the (9,9) exclusion with a separately implemented engine; the Independently corroborate the corrected seven-label word exclusion #3005 C++ census at L=22 is the natural candidate.
3. Can the pair identity be turned into a leaf upper bound other than the shared-support union bound? That is the only direction that can prune.