Bounded length-22 census probe: exclude splits (11,7), (10,8), (11,8); measure the rest (answers #3708 OQ1's blocker; no new kappa)

[petalas]

Summary

Supplies the resource-budget measurement named as the blocker of the open question 1 in PR #3708 ("no node-count estimate or resource budget; #2973's stop rule was explicitly no L22 census"), and completes the first three finite split exclusions of the length-22 word domain. Produced by Nick Petalas with AI assistance. Independent human and formal proof-kernel review are unclaimed.

Arithmetic scope, stated up front: no kappa, supplier, tariff, certificate, frozen-input or selected-frontier change. f(7) remains 22 <= f(7) <= 24 and is not decided here. This is a finite census result inside PR943's prefix-walk model, not a formal-kernel proof.

What is established

1. Engine extension to L=22, validated against the pinned predecessor. The corrected census engine of Correct word-census parity budgets; 22 <= f(7) <= 24 in the PR943 model #2973 (research/f7-weight-parity-census-correction, source sha256 265ead91...) is extended by exactly three header/gate lines (L <= 20 → L <= 22); nothing else changes (integrity-checked by diff in verify.py). Soundness of the gate change: at N=7, L=22 all counters stay within exact small-integer/128-bit-mask types (largest values: even budget 110, odd budget 57, per-label multiplicity <= 10), so the L<=20 gate is conservative, not necessary. Behavioral proof: the extended binary reproduces all six recorded L20/L18 receipts node-exactly — 590,371,593 / 244,301,902 / 14,450,133 / 67,643,567 / 35,241 / 951,416 nodes — and passes the predecessor's brute-force cross-check (validation/check_corrected_c.py, all small splits, all four budget modes, cap and fail-closed controls) and its math/check_math.py.

2. Capped censuses at all nine open splits (both proved necessary budgets enabled, restricted-growth canonical scope, revisits permitted):

Split	Nodes	Outcome
(11,7)	8,784,744	complete, exit 0, 0 hits → finite split exclusion
(10,8)	597,933,297	complete, exit 0, 0 hits → finite split exclusion
(11,8)	1,349,702,650	complete, exit 0, 0 hits → finite split exclusion
(9,9), (10,9), (10,10), (11,9), (11,10), (11,11)	2,000,000,000 (cap) each	aborted → tree-size lower bounds only

No universal word was found in any probe. Combined with the imported 17-point set-level exclusions of #2773/#2870 (splits (10,7), (11,6)), any restored universal length-22 word must now have split in {(9,9), (10,9), (10,10), (11,9), (11,10), (11,11)} — the domain narrows from nine splits to six. For scale: #2973's complete length-20 exclusion totalled ~9.2e8 nodes over five splits; each of the six open L22 splits already exceeds 2e9 nodes. That is the concrete budget datum the handoff asked for; finishing the census is a resource decision, not a mechanism change.

Certification scope, stated exactly: the three completed exclusions are CI-certified; each gets its own registered node-exact replay command (below). The six capped lower-bound runs are locally measured session evidence, reproducible with the documented opt-in command, and are not re-run by CI.

3. What is not claimed: the six loose-budget splits are open; no length-22 word is excluded outright and none is constructed; the (11,11) probe had 3 budget-surviving restored leaves before its cap cut the search (they failed the exact coverage test; recorded, no claim); no prune beyond Correct word-census parity budgets; 22 <= f(7) <= 24 in the PR943 model #2973's two proved budgets was added.

Verification

# Registered CI checks (one command per exclusion; declared timeouts sum to the
# 1200 s package budget, each command under the 600 s per-command cap):
python3.12 -I -B research/f7-l22-census-probe/verify.py --skip-probe         # validation + six receipt replays
python3.12 -I -B research/f7-l22-census-probe/verify.py --probe-split 11,7  # exclusion (11,7), node-exact replay
python3.12 -I -B research/f7-l22-census-probe/verify.py --probe-split 10,8  # exclusion (10,8), node-exact replay
python3.12 -I -B research/f7-l22-census-probe/verify.py --probe-split 11,8  # exclusion (11,8), node-exact replay

# Opt-in, not run by CI: full nine-split replay of the committed receipt,
# including the six capped lower-bound probes (~1.4e10 nodes).
python3.12 -I -B research/f7-l22-census-probe/verify.py --probe-all-splits

python3.12 -B scripts/research_package.py --package research/f7-l22-census-probe
python3.12 -B scripts/check_research_commit.py --package research/f7-l22-census-probe --check-source-history

• Registered replay: PASS_REGISTERED_RESEARCH_PACKAGE research/f7-l22-census-probe outcome=completed with four registered checks, each with margin against the measured CI run: validation stages (declared 420 s; earlier CI run ~160 s; local 60 s), exclusion (11,7) (declared 60 s; local 0.8 s), exclusion (10,8) (declared 240 s; local 39 s), exclusion (11,8) (declared 480 s; local 90 s). Full registered launcher locally ~190 s. Run on CPython 3.12.15 at ~/git/integer-mult-bounds/.venv/bin/python3.12, Apple arm64, Apple clang.
• Fresh exclusion replays are compared against the committed RESULT.json session receipt: each must exit 0 with complete=true, zero hits and the identical node count (8,784,744 / 597,933,297 / 1,349,702,650). Node counts are deterministic; timing fields are excluded from every equality check.
• The six capped probes (2e9-node cap each, ~1.4e10 nodes total) are not replayed in CI; their RESULT.json values are unchanged, and the opt-in command reproduces the same receipt comparison locally in ~3 minutes on a many-core host.
• Commit bytes + source history: PASS_RESEARCH_COMMIT_BYTES at e75698688de7300df4f44d86d31738302ea355b8, base 20ecf2891 (canonical main, revision 6).
• scripts/research_baseline.py: PASS_RESEARCH_BASELINE_BINDINGS_ONLY, selected record hash unchanged.
• Interpreter note: CPython 3.12.15 via the main checkout's venv (not on PATH elsewhere); this host has no python3.12 system binary, which the registered launcher accounts for.

Overlap and credit

• Overlap re-checked at submission (open PRs and discussions fetched 19:06 UTC): no other L22 probe in flight. Builds directly on the fresh Derive the complete length-22 parity-split domain at seven labels (no census, no f(7)) #3708 handoff and answers its open question 1's named blocker.
• Credit: Nicholas Stone (Correct word-census parity budgets; 22 <= f(7) <= 24 in the PR943 model #2973 corrected engine, budgets, split theory); Aaron Wang (PR2542/PR2634 word model); Ross Richardson (PR943 model); Nick Petalas (PR Derive the complete length-22 parity-split domain at seven labels (no census, no f(7)) #3708 split domain). This package: produced by Nick Petalas with AI assistance.
• Frozen predecessor packages and upstream/ are unchanged; the extended engine lives in the new package only.

Open next questions

1. Finish the six open splits (each > 2e9 nodes so far): higher caps alone need hours-to-days; a third proved-monotone prune would help more. The pair identity sum_x A_x B_x >= need (MATH.md §3) is proved at leaves but its partial quantity is not shown monotone — proving that (or a monotone variant) is the cheapest possible engine improvement.
2. Does a restored universal length-22 word exist (f(7) = 22) or not (f(7) = 24)?
3. Independently replay the three completed split exclusions with a separate implementation (the Correct word-census parity budgets; 22 <= f(7) <= 24 in the PR943 model #2973 package's own open question 2 pattern).