Research: exact CK helper-matching floors under two scalar lowerings (no new kappa)

[steph4n-gh]

This finite research result closes the disjoint-helper matching problem for one frozen historical PR917 CK target forest under two explicit scalar lowerings. No new kappa. Only research/ck-dyadic-helper-shears-seed22/ is added.

Allowing coefficients ±1/2, ±1, ±2 lowers the exact compensation minimum from 11,964 to 11,961 component primitives under the historical binary-Horner lowering. Under the stronger accepted PR696 copy/unit lowering, the exact minimum is 8,434 for either alphabet: the extra alphabet supplies no scalar advantage there. Four independently checked primal/dual certificates prove these finite optima without trusting or requiring the discovery solver.

The historical extended witness saves 8 real primitives per orientation and one macro versus PR917; when repriced with PR696 it saves 4 primitives per orientation and one macro. Its complete historical counts are 149,612/149,660. The PR696 scalar-optimal recipes reach 104,934/104,950, but introduce more macros and worsen the conservative local allowance compared with reusing the historical extended witness. The README prices that adverse result explicitly; scalar optimality is not invoice optimality.

Reproduce with Python 3.10+ and the standard library:

python3 -I -O -B research/ck-dyadic-helper-shears-seed22/verify.py

Expected: PASS_CK_DYADIC_HELPERS cases=385572x2 old_optima=4400/4403 pr696_optima=3778/3778 columns=725 controls=14+25 no_new_kappa.

Local validation on CPython 3.14.4/macOS arm64:

• Both exhaustive 385,572-choice censuses and all four matching certificates agree with a separately written checker.
• Complete arbitrary-input scalar replay covers 725 columns, both orientations, dirty helper restoration and private erasure. Candidate prefix bounds remain 41/65 bits within the inherited 192-bit allowance; denominator powers remain 3/8.
• Unchanged native emission and frame/wrapper checks pass on all three changed candidate macro words, including the 25 inherited controls each. Recursive child profiles remain unchanged.
• Operator/certificate corruption controls, full receipt rejection, and optimized minimal relocation with spaces, no Git metadata and only 19 declared inputs pass. make selected-record-check and staged whitespace checks pass.

Exact base: 2921874911956718fcaaaf9396fe8267b3cfa9f0; OS assignment seed 22, complex-supplier track. Source pins, the failed initial greedy comparison, complete coverage/stopping rule, proof, exact certificates and local claim are included. At the final refresh, upstream abdc6a7002abdbcd9e21bfc5e1a795784deacfdf preserved every consumed source pin and selected PR942. The 742-open-PR snapshot through #1075 showed no duplicate fixed-PR917 matching scope; #964 target-parent bounds and #769/#1036 scalar-lowering work are distinct and credited.

The matching optima fix the target forest, pair disjointness, six-coefficient alphabet and lowering. Global hosting/provider bindings, shared invoices and outer assembly were not regenerated; full repository/cold ancestry suites, hosted CI and Lean were not run. All inherited compiler, realization, physical, precision/tape and analytic assumptions remain.

Prepared for @steph4n-gh, with substantial OpenAI Codex assistance, including separately implemented AI checks and review; no independent human review is claimed. Credit Ross Richardson (#917), David Awad (#807), @dd-hemant (#816), Sagar Thakrar (#696), Sricharan Vagvala (#749), Oscar Fennell (#759), icekylinx, DreamingOfClouds, Jeremy Kothe, Chafik Boukhalfa and the retained source lineage. Apache-2.0. The selected frontier and upstream/ are unchanged.

[Mister-Meeseeks]

100-PR batch review at 5fb9a3d1ff67845c681233072a00d107e7084bd7.

Merged to main, preserving the original contributor commits and package attribution.

Reviewed scope: Exact fixed-forest CK helper matching under two lowerings; the stronger PR696 model receives no alphabet gain.

The pinned scoped command passed; the batch ledger contains its command, input hashes, environment and log. This incorporates the research/tooling contribution and does not change the selected κ or discharge the inherited multiplication-theorem assumptions.

Batch review, attribution and evidence. Thank you for the contribution.