Conditional κ = 8.049462195041506e-4: two kernels and 15 extra retimings after #366

[applecrusher]

The combined finite word certifies conditional κ = 804946219504150552042557 / 10^27 ≈ 0.000804946219504150552, improving PR #366 at d3d84202e528d6e147a8ea62463ae5c0125997cf by approximately 0.0007988514%. The bit side still binds; the E8 complex supplier is unchanged.

Starting from immutable PR #364 (f8c6a47abfec24b87cb1eb0a138dd73dfce2abe4), this PR adds two rank-two kernel entries and composes #366's 123 attributed frame changes with 15 additional compatible retimings. Five are in the preserved pr364-kernel-extension package. The new pr366-propagation-extension package adds ten more, in three pairs and one four-gate group, applied through three source-bound stages. Recursive predecessor propagation repairs the containment failure caused by lowering each later gate alone. The complete final word is reconstructed and priced; separately reported gains are not added.

Exact banking uses 100 replicas and admits the four extra kernel entrance ranks. The final word contains 405675 records, with records SHA-256 b0fd6cfb9a976e07c3860a5fddd34962504db31163d5346f96fc5e7c6624fdb0. Two rational pricing engines, eight finite bootstraps and a finer exact κ grid certify the resulting assembly together.

Both packages include source-bound selections, deterministic reconstruction, proof and remaining obligations, source attribution, portable expected certificates, and manifests. The default entry point replays the complete pinned parent chain before verifying the new stages:

python -m pip install -r research/pr366-propagation-extension/requirements.txt
python -B research/pr366-propagation-extension/verify.py --output /fresh/output

Validation: a complete fresh replay of the immutable parent and the first two stages passed. The final three-stage extension was then replayed against that fresh, hash-bound parent receipt. Checks passed for exact containment and signed scalar preservation, all 11,930 global formal columns, physical legality and changed-frame nondegeneracy, four failed single-gate controls, twelve kernel ADD-omission controls, exact banks, both rational engines, eight finite bootstrap steps, and all 47 strict outer inequalities. Independent agents reproduced the final word and price and reviewed source binding, composition and certificate portability.

The broader repository verification completed across the original run and an exact resume of its failed OpenMP compilation tail: 538 tests across 82 modules and 20 patch-applicability checks passed, including make gen4-bank-verify. An isolated official LLVM OpenMP runtime resolved the Apple compiler incompatibility without modifying repository sources or disabling assertions. This was resumed coverage, not one uninterrupted make verify invocation. Dedicated Python 3.11/3.12 Linux CI is included; hosted CI has not yet been confirmed.

Native downstream admission, changed projector charts, the complete finite invoice, and the inherited all-size compiler, restored-row, selector/routing, prime-supply, precision/recovery and analytic-transfer interfaces remain open or conditional. This is a conditional finite-word and arithmetic improvement, not an unconditional integer-multiplication theorem, a new Lean proof, or a claim of global optimality.

Credit includes Rohan Arun for #366's 123 changes and the public recursive propagation algorithm (research/kappa-research-tools/casc.py at 4a57eb9b2d450b87e9db5c514b5a088ec401d761), DreamingOfClouds, Chafik Boukhalfa, Jacob Sussman and the contributors identified in the retained notices. New discovery, integration and verification used substantial OpenAI Codex assistance. New contributions are Apache-2.0. The base already includes the unchanged three-file source-pin repair from PR #318.