Response-kernel entrance gauges: 132 helper-pair rewrites, κ = 7.1046520063283e-4

[Dugongue]

κ = 7.1046520063283e-4

This adds a concrete response-kernel basis change to PR249 at 96495746c786d6d0339dbb38c7f553d4af3f88ed, whose value is κ = 7.10392186446856e-4. The resulting exact value is 71046520063283/100000000000000000. This is a composition through the paid recurrence and outer assembly, not an addition of independently estimated gains.

The new observation is that 132 disjoint pairs of untouched dirty helpers have identical complete target responses over F₂ and compatible common rank-one entrance frames. A paid XOR basis change at the entrance, followed by its inverse at the final full frame, cancels the pivot's old prefix response. Each pair removes four initial correction reads and adds two basis gates. The actual local ADD count falls from 786,623 to 786,359; raw recursive rank mass falls from 434,177 to 434,045. No target frame changes or new target reads are required.

The new entrances are fully banked: 132 exact projector charts, all 3,317,400 role/replica/stage assignments, and complete endpoint controls are checked. Across 40 physical replicas and five stages, literal stock falls from 869,415 to 869,195. The normalized retained profile has stock 173,839, 3,935,080 calls, rank mass 20,825,480 and deficit 35,200. Every added connector, basis gate, inverse, selector and boundary cost remains charged. In particular, new rank-five completion children are subtracted by multiplicity; ordinary rank-five children are retained.

Reproduction regenerates the pinned PR249 baseline from included original sources, checks its exact transcript, frames, states and donor ownership, then compiles and executes seven native C++ checkers. These check all 20,107 local F₂ columns forward and backward; independent chronological frame/COPY legality; 232,320 pair/target prefix equalities; every new chart and bank assignment; all 23,627 global five-stage columns; signed prefix and full counted finite-cost bounds; exact rational recurrence pricing; and all 47 strict outer inequalities. Negative controls include omitted basis gates, omitted/repeated bank blocks, omitted bridges and rejected adjacent pricing grid points. The native price reproduces the baseline exactly.

python3 -m pip install -r research/cohort-kernel-entrances132/requirements.txt
python3 research/cohort-kernel-entrances132/verify.py --output /tmp/cohort-kernel-replay

Requires Python 3.11+, SymPy 1.14.0, GNU g++ with C++17 and Boost headers. MinGW-w64 is supported; --cxx and --boost-include select nonstandard installations. The output directory must be new and outside the package. Replay requires no network. Original upstream sources, hashes, licenses and notices are included, with new mathematical checks in C++ and Python used for orchestration and the original source emitter.

This conditional result retains the public all-size compiler, common weighted chart, restored-row, routing, prime supply, precision/recovery, complex symbolic correctness and analytic-reduction interfaces. It does not independently prove these foundations, claim an unconditional multiplication theorem, or provide new Lean certification. The 132-pair cancellation is specifically over F₂ payload; odd-prime address algebra is separate. The public eight-stage Python command is not claimed to have been replayed verbatim on the modified word: the added construction is checked by the supplied native pipeline. No latest-leaderboard-record claim is made.

Provenance: the pinned PR249/PR244 source527 construction and their upstream sources; original contributors and licenses are retained in NOTICE.md, vendor/UPSTREAM-NOTICE.md and the source archive. The new response-kernel rewrite, native checks and packaging were prepared with substantial OpenAI Codex assistance.

[Mister-Meeseeks]

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

Contribution: 132 response-kernel entrance rewrites.

Review and limits: Fresh full native source regeneration and all seven independent checkers passed: complete columns, prefix equalities, chronology, bank assignment, finite invoice and exact assembly.

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: 9ce32efb421132420242cfdb91ddd9eabe2c3795. Publication: 50d768818f579d4e165b4c257c6ee25bca300802.