Independent construction: 46 hyperplane-ended helpers per call via the exact commutation condition, conditional κ = 0.001613569610580215 (below #601)

[higherbeing]

Independent construction of the 46-helper hyperplane end; no record claimed. Conditional κ = 0.001613569610580215 (conservative model 0.001613569607722207). This is the same mechanism as #601 (jeremykothe2), which was posted first: both replace #591's first-gate rule by the exact commutation condition and end 46 helpers per call. #601 re-optimises the dispatch (ck × 45, atm3 × 21, jp8 × 6) and its RINS owner map reaches 0.001614202; this package keeps #592's dispatch (ck × 53, atm3 × 19) with its own owner map, so its value is 0.04 % lower. It is offered as a second, independently built and strictly verified witness, together with the owner-allocation tools it used (HiGHS pattern MILP, lazy cross-time Hall cuts, per-window matching; the PR520 allocator extended to per-call widths).

Mechanism. #591 (jeremykothe2) ends 38 entering width-8 helpers per call on a hyperplane U instead of the full frame. Its candidate rule requires the helper's single cleanup gate to be the first cleanup gate on both of its registers, so that it trivially commutes with the skipped cleanup prefix. Eight more helpers per call fail that rule only because their partner register is read by an earlier cleanup gate; two additions that merely share a source commute, so nothing obstructs the move. Replacing the rule by the exact condition (the helper's cleanup gates plus the non-commuting closure of what they pass run right after part B at the minimal common frames; here the closure is the gate itself) ends 46 helpers per call on every provider word (ck, atm3, and jp8, which this dispatch does not use). The local paid Φ is unchanged to within 1 Φ; each call has eight more width-7 residuals.

| | #592/#596 | this PR | #601 |
| --- | --- | --- |
| hyperplane-ended helpers per call | 38 | 46 | 46 |
| ck width inventory | {4:8, 5:48, 7:70, 8:102, 9:276} | {4:8, 5:48, 7:78, 8:94, 9:276} | same |
| idle remainders | {1:2, 2:1, 3:280, 12:1, 48:1969} | {3:197, 25:1, 48:1986} | dispatch 45/21/6 |
| paid profile | {1:182818, 2:34193, 3:22560, 4:14584, 5:4032, 6:4032, 8:576, 12:1, 48:1969} | {1:183968, 2:33616, 3:23053, 4:14008, 5:4032, 6:4032, 8:576, 25:1, 48:1986} | see #601 |
| LP bound on remainders for the menu (dispatch 53/19) | 42,321.7 Φ (attained within 37 Φ) | 41,862.3 Φ (attained within 28 Φ) | — |
| κ | 0.0016127213726553 (#596) | 0.001613569610580215 | 0.001614202339036103 |

Dispatch (ck × 53, atm3 × 19), call order, pinned sources (PR568 71c99ac, PR569 897213a, PR571 b7bdc5a, PR575 eac6f7e, PR583 89be768), the other build steps and #592's verifier code are unchanged. The owner map is renominated for the new width inventory: an exact-window MILP over port patterns (HiGHS) with lazy cross-time Hall cuts and a per-window literal matching (CP-SAT); the nominator and the allocation tools are included.

Package research/complex-571-hpe46/, a copy of #592's with: one new allowlisted step type hyperplane_end_closure (code/build_hpe_closure.py; the chosen helpers and the tail are recomputed from the input word and must equal the recipe, then #592's unchanged raw-word control audit runs and the same independent comparison as hyperplane_end), new data/PLAN.json, data/OWNERS.json.gz, data/SPEC.json and recipes, and code/witness.py recording pins in discovery mode (strict mode unchanged). Everything else is byte-identical to #592.

Verification. verify.py rebuilds both provider words from the pinned sources through all typed steps, localises them with #592's HPE native chain (all columns, both reflected ledgers, local controls, normalised wrappers, all phase edges), binds, emits and audits the owner map with mutation controls, prices a fresh invoice and runs both exact moment engines with the 47 outer constraints and the assembly, timing and recompression controls. The strict run passed locally with the frozen558 baseline reused from a fresh #588 replay; RECEIPT.json, RESULT.json and STAGES.json are in research/complex-571-hpe46-validation/ (31 stages; 23 controller and 114 assembly mutations rejected). The README was reworded after that run, so the receipt's package manifest hash predates the final README.

Scope. Conditional on the same inherited assumptions as #592 (all-size compiler, common-label/mutable-source realization, whole-call hosting, precision/tape, analytic transfer, and Lemma A for hyperplane-ended helpers as stated in #592's PROOF-HYPERPLANE-END.md and #593's Lean). The eight extra helpers satisfy exactly the hypotheses of #591's construction (one cleanup gate run at U right after B, the helper untouched afterwards, the partner continuing to FULL); only the commutation argument for passing the skipped cleanup prefix is generalised, and the exact scalar replay of the reordered program in hpe_audit_and_transport.py / hpe_controls_atc.py checks it. Not an unconditional theorem.

Credits: jeremykothe2 (#591/#592/#596 hyperplane-ended helpers, verifier and owner-map method; #583/#588 coalescences), DominikScholz (#569/#571), jon314159 (#575), eumemic (#558–#564 cold kernels, #543 bit supplier), chafreaky (#583), DreamingOfClouds (#593/#599 Lean groundwork), icekylinx, Dugongue and the inherited lineage listed in NOTICE.md. Prepared by higherbeing with Claude Code assistance.