Conditional κ = 7.53508664880762e-4: 209 exact F₂ target-prefix squares on #310's w3 word (+0.082% over #310)

[rohanarun]

Conditional κ = 753508664880762517753401/10^27 = 7.53508664880762517753401e-4, +0.0817 % over #310's w3 word (7.52893499605334868002344e-4), +0.268 % over #309.

One self-contained package, research/w3-target-prefix/ (#310's w3-bitword package with one transcript stage inserted), with its own workflow; branched from main, no existing file changed. Vendored files are stored byte-for-byte (.gitattributes).

Full replay: PASS on the committed package (verify.py --output, ~9 minutes locally): both prerequisite source verifiers (PR249 eight-stage, PR256/PR233 complex) fresh, the target-prefix stage, recompaction, legality, all 23,627 global and 22,172 compact F₂ columns, bank allocation, 21,054 changed charts, geometry admission, exact price (two engines, eight ordinary levels, 47 strict constraints, adjacent grid point rejected), finite invoice, complex guard and the cross-binding finish.py; package hashes unchanged before and after; prints PASS conditional kappa = 753508664880762517753401/1000000000000000000000000000.

Change

209 exact F₂ target-prefix squares on w3 — eumemic's #268 mechanism in the native-transcript stage form of #273/#281 (code/target_prefix.py), inserted right after #310's source binding and inert-role deletion. The w3 word admits the same square structure as the source527/gen4/gen5 words: in each of 209 squares of four data targets every rank-20 entrance helper is read by exactly two of the four, so the dependent target's frame-20 prefix response is the F₂ sum of its three square-mates'. Its 1,057 frame-20 reads are omitted; it receives t −= p_j at the zero frame at the cut (record 876,399) and t += p_j at the square's common nondegenerate 21-dimensional frame at the close (1,254 setup/restore additions): ZERO →(20)→(1)→ 21 becomes ZERO →(21)→ 21.

Local histogram delta {1: −209, 2: −7, 3: −1, 4: −4, 8: −2, 9: −4, 12: −2, 13: +4, 17: −1, 18: −7, 20: −199, 21: +209}: 223 fewer local calls (3,669,720 five-stage calls) at unchanged rank mass 398,897; endpoints, entrances, the bank allocation (3,026,400 assignments, stock 810,615 / 162,123, normalizer 787, selector charge 571,950,677,600) are unchanged. The stage recomputes every literal prefix dependency from the actual transcript, rebuilds every MOVE, checks nesting, fixed endpoints, COPY lifetimes, nondegenerate endpoint bases, both reflected ledgers and the forward/inverse all-column F₂ replays with an omitted-setup control; code/recompact.py rebuilds the compact stream with the same freed-role map. Everything downstream is #310's checker chain fresh on the new word (word hash de63e0a3…, 934,582 ADDs, 21,054 changed projector charts at ≤ 342 factors, finite coefficient 58,167,289,165,918,601 < 2⁸⁰, two exact moment engines, eight ordinary levels, 47 strict constraints, adjacent 10⁻²⁷ point rejected). Re-pinned literals: code/price.py (word hash, ADDs, calls, κ), code/admit_geometry.py and code/finish.py (chart count, coefficient).

Verify

python3 -m pip install -r research/w3-target-prefix/requirements.txt

python3 -B research/w3-target-prefix/verify.py --output /tmp/w3-target-prefix-proof --cxx g++ --boost-include /usr/include

Python 3.11+, a C++17 compiler and Boost headers; about 10 minutes; prints PASS conditional kappa = 753508664880762517753401/1000000000000000000000000000.

Scope and credits

Exactly #310's conditional interfaces; not a Lean certificate, an unconditional theorem or a benchmark. Not yet composed on w3: the restoration, kernel, sink, descent and reorder stages of the gen5b lineage (the restoration census finds 440 candidates of the same shape as on gen4/gen5; follow-up). utcorvusvolat-dotcom (#310: the w3 word and admission chain; #280), eumemic (#268 mechanism, #249/#251 lineage), DreamingOfClouds, Dugongue, chafreaky, hcg890, Jacob Sussman, Evan McKinney and the retained lineage. Prepared by Rohan Arun with Anthropic Claude assistance; Apache-2.0; NOTICE.md.

🤖 Generated with Claude Code

[rohanarun]

CI note: the only failing check here was the repo-wide 'Verify research artifacts' job (make joint-dual-verify), which fails on main itself because the root Makefile/README changed while research/joint-dual/SOURCE.json pins their digests (see #318). To make this PR's checks green now, the branch carries that same one-line repin commit (research/joint-dual/SOURCE.json and the two joint-dual certificates' recorded manifest hash) — no package content changes; it can be dropped once #318 or an equivalent fix lands on main.

[Mister-Meeseeks]

Thank you @rohanarun. This contribution is now retained on main in the scoped 100-PR review and contributor credits.

Contribution: Exact target-prefix squares on the w3 word.

Review and limits: The changed package matches the actual Linux checkout; seven unrelated root-wrapper files differ. Finite prefix and native admission are covered, not a whole-theorem proof.

The complete original source, notices and relevant execution evidence are retained through the pinned source archive. The review distinguishes fresh local runs, source-bound historical Linux CI and prior exact-head review; it does not claim 100 fresh full mathematical audits. Closing this historical PR as reconciled. This preserves parallel and cumulative credit without replacing the selected frontier or promoting priced research targets to completed constructions.

Reviewed head: 298f9baf71c261a326e54eb8407ee414c1109f79. Publication: 4040a0f46d67b41c36a472a7cd4af791fd4c36cc.