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Thank you @eumemic. This contribution is now retained on main in the scoped 100-PR review and contributor credits. Contribution: Native admission of staged signed twins. Review and limits: Full emitted-word and construction checks strengthen the finite evidence for the staged family; independent word and arithmetic audits are separately retained. 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: |
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κ = 8.03161884970454053300282e-4
Exact κ:
401580942485227026650141/500000000000000000000000000.Twenty additional signed helper-pair condensations improve #354's transcript-staged Design T/twin construction. The complete native admission binds the actual scalar word, physical role maps, both reflected frame ledgers, bank and projector charts, all-frame prime exclusions, full finite invoices and exact outer assembly. The bit side binds. The E8 supplier uses the corrected odd-dimensional finite cover count. This is a new branch and PR; earlier published heads and claims remain frozen.
The original #325 source is freshly verified and exported twice. Unmodified #329 stage modules, with #354's explicit kernel-free restore/sink selections, regenerate all six parity and staged snapshot files byte for byte. #354's original Design T/completion/twin scripts reproduce its shipped final word and independently reproduce #346's unstaged final word as a control. A composed map binds ten sunk roles and 480 no-event FULL/FULL helpers back to their original physical positions, including the COPY slot. All output record identities are computed from actual bytes; inherited stale metadata hashes are retained only as historical data.
The signed extension is regenerated from this compacted source through separate eighteen-pair and two-pair stages. An exact integer audit compares all 9,530 input columns and 9,531 output rows and rejects six omission/sign controls. This scalar audit ignores frame moves, which are checked separately by the native common-frame, COPY-lifetime and reflected-ledger audits. In particular, the last two pairs have earlier proper-frame reads, so this PR does not claim the simpler single-consumer template proof applies to them.
The final word has 7,610 arbitrary dirty helpers, 406,292 events, 358,152 ADDs, 46,004 paid local calls and rank mass 169,100. Native admission uses 300 replicas, literal stock 3,135,900, 396,780 complete width-100 banks per stage, 3,340 changed projector charts, selector charge 1,806,128,697,000 and full finite coefficient 120,718,938,573,985,001. Eight finite bootstrap levels yield the displayed κ, with all 47 strict outer constraints positive and the adjacent grid point rejected.
Two complete immutable 24-stage replays passed under CPython 3.11 and 3.12 with actual observed exit 0 and matching mathematical certificates. A distinct five-stage literal-word, restoration, tiling, determinant and exact-assembly audit also exited 0. All committed construction and audit blobs were checked against their frozen manifests before publication. A dedicated Linux workflow repeats the complete replay and independent audit.
The full E8 supplier replays actual scalar schedules, temporary lifetimes, arbitrary dirty restoration, both reflected splices, gx and gxcore, complete paid profiles, two rational moment engines and the parity-correct finite invoice.
|O(45,2)| = |Sp(44,2)|gives physical row coefficient 11,162, below the inherited 20,161 ceiling. The E8 scalar dirty audit specializes recursive frame operators to identity; frame incidence and child ranks are separately checked. The exact coarse complex saving is876248285600677/10^18.This remains a conditional sufficient finite construction under the inherited all-size compiler, selector/chart, restored-row, routing, prime-supply, precision/recovery, complex-symbolic and analytic interfaces. The native global operator replay is over F2; the added transform's local operator equality is also checked over the integers. Twenty-four Lean modules and one comparator configuration are regenerated from E8 data, but no local Lean kernel build, module-to-gcert producer regeneration or unconditional multiplication theorem is claimed.
Credits and AI disclosure: Chafik Boukhalfa/Anthropic Claude supplied #329 and #354's staged composition and reimplementation. Design T/twins originate with utcorvusvolat-dotcom's #310 and Louis Harrison/Anthropic Claude's h=20 port in #346. DreamingOfClouds/Anthropic Claude supplied #325. Jacob Sussman supplied the E8 unit and Lean data/checkers; DaysSky/Anthropic Claude supplied #352's E8 integration. The additional signed pairs, source adapters, complete native integration, independent audits and publication used substantial OpenAI Codex assistance. All original licenses, notices and contributor/checker lineages are retained.