κ = 7.6817007781759e-4: composed p10 stages with full source-bound admission

[eumemic]

κ = 7.6817007781759e-4

This source-bound conditional finite construction certifies the exact value 76817007781759/100000000000000000. It composes five stages on #315's p10 word, improving κ by about 0.515% over #315 and 0.395% over #316. The bit supplier still binds; #315's stronger centre-sharing complex supplier is retained and replayed unchanged.

The freshly derived stages are 540 disjoint kernel pairs, 120 exact F₂ target-prefix groups, 480 frame ascents, 240 early restorations, and 140 commuting ADD relocations. The resulting literal word contains 384,471 records and 335,240 ADDs. Its actual endpoints give residual census {3:310,4:960,19:540,20:6420}, completed without padding into width-100 banks across 60 replicas and five stages.

The full verifier reconstructs the word from immutable #315 (243899975d0fbe888d5deda399060885d609e0a1) and reruns that package's complete offline admission. It checks the actual scalar word, arbitrary dirty restoration, both reflected frame ledgers, changed rational projector charts, every role/replica/stage assignment, the complete paid profile, two exact moment engines with the full bad-class fallback, the finite primitive invoice, and all 47 exact assembly constraints. No predecessor's paid profile or row bound is substituted for the changed word.

Kernel and target transformations are characteristic-two identities. Their changed signed coefficient bounds are recomputed and paid; signed-integer endpoint equivalence is not asserted for them. The finite invoice has coefficient 25,350,779,320,984,001 and a 90-bit payload bound within the retained 104-bit allowance. See the package's PROOF.md for the precise conditional interfaces.

python3 -m pip install -r research/p10-composed-stages/requirements.txt
python3 -B research/p10-composed-stages/verify.py \
  --output /tmp/p10-composed-proof --cxx g++ --boost-include /usr/include

Python 3.11+, SymPy 1.14.0, a C++17 compiler and Boost headers are required. The verifier uses a fresh external output directory, requires all source pins before and after execution, and needs no network.

Two fresh complete offline replays passed with all 158 pinned files unchanged, including a replay from the actual release checkout. Manifest: 97c731bdb7ecb3c1957bef11f90f824a586ee7bc8a41fa79c12bd39b08fe9a6f. Final word: e6ad3590e6ad83a6df4f202adda6fe899ed0c60b20822f608db3de670468f61e.

Validation receipts and exact publication hashes are recorded in certificates/p10-composed-stages-publication-audit.json. A separate C++ audit independently executed the final forward/inverse F₂ word, dirty-restoration controls, both reflected ledgers, signed coefficient inventory and exact bank census. make gen4-bank-verify passed on the unchanged upstream base. The full repository make verify is still running through inherited rational frame checks in an isolated copy; it is not counted as a pass here. The package has its own path-filtered CI workflow.

This is a new standalone package and new PR. Earlier published PRs remain frozen. The result retains the inherited all-size compiler, prime-supply, precision/recovery and analytic-transfer assumptions; it is not a formal verification of an unconditional multiplication theorem or a runtime benchmark.

Prepared for eumemic with substantial OpenAI Codex assistance. The unchanged #315 package preserves DreamingOfClouds's contribution and Anthropic Claude disclosure. Kernel, target, retiming, early-restoration, reordering, bank and checker mechanisms build on the contributors credited in NOTICE.md, particularly Dugongue, utcorvusvolat-dotcom, Rohan Arun, Chafik Boukhalfa, Evan McKinney, Henry Grant and Jacob Sussman. This contribution is their source-bound composition on p10, fresh selections and complete admission, not a claim to invent the inherited mechanisms.

[Mister-Meeseeks]

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

Contribution: Composition of p10 transcript stages with native admission.

Review and limits: Complete source-bound composition, including the actual final word, charts, banks and invoice; retains the ancestor construction credits.

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: 0a0b61a7f4601a96036af8bb6a6338c58f035d2e. Publication: 4040a0f46d67b41c36a472a7cd4af791fd4c36cc.