Conditional κ = 0.001673628681907313 by composing a cycle with PR3366

[hexapode]

Change and mathematical scope

Compose the cycle (43909, 0, 60) with the latest PR #3366 owner map. Conditional κ = 0.001673628681907313, up from 0.001673627634134309. Exact gain: 261943251/250000000000000000 = 1.047773004e-9.

Idle ranks (2,1,3) become (0,3,3): the complete paid delta is -H1-H2+H3, and strict concavity proves the moment gain. Three helpers relocate; eleven rows change including offsets. PR3366's five-helper construction and all seven earlier cycles remain present. The selected word, widths, 144-call order, suppliers and complete physical invoice stay fixed. Complex still binds.

This is a conditional finite-composition packing witness, not an optimum, selected-frontier update or complete multiplication theorem. Selected-result.json is unchanged.

Task and overlap

Package research/t16-post-unlock-cycle; commit 62fd8039cc40749cea6238e7807db9a63b4d6765; canonical base 6f100d4bb5083f8c35d96fd95a539811c62f245c. Predecessor #3366 is pinned at 3f59a00257b3e29ad54e349f826d5ffdc59f7cfb and reconstructed from canonical data plus an authenticated 236-row patch. Its original construction, delta, source export, result, README and proof are retained.

Card selected-packing, revision 6. PR3369's direct-cycle obstruction concerns the older PR3359 map; this witness uses the new rank-two remainder created by PR3366. A private solver independently found a 37-role alternative with the same exact gain. This publishes the simpler witness and preserves the alternative for future combinations.

Recent PRs were refreshed repeatedly and again through authenticated GitHub CLI immediately before submission. No newer supported κ improvement than PR3366 appeared in that coverage. Coverage is not a global novelty proof.

Validation

Standard-library CPython 3.12.1, macOS arm64:

• Full unchanged eleven-stage strict replay passes on 69,120 roles, 144 calls and 10,005 charts; controls 69/12/6/15 plus eleven independent owner controls pass.
• Both exact supplier engines, native/conservative seven-step bootstraps and all 47 strict outer inequalities per path pass.
• Entire typed physical invoice equality passes, allowing only three freshly bound source identifiers. Prefix 397 remains paid at 512; row coefficient 16,469; rank mass 1,041,408 and deficit 5,120 are unchanged.
• Nine behavioral groups pass normally and optimized. A separate private derivation reconstructs every owner row from PR3366 and verifies the exact profile delta.
• Committed-byte preflight and source history pass. The 61-file committed Git-free export passes full strict replay; all 62 extracted files, including the transport manifest, remain byte-identical.

python3.12 -Werror -B scripts/research_package.py --package research/t16-post-unlock-cycle

No hosted CI, fresh full cold native rebuild, independent human review or formal-kernel proof is claimed. Inherited all-size/compiler, common-label/mutable-source, whole-call hosting, precision/tape, exact recovery, analytic and bit H1–H5 premises remain.

Credit

Nick Petalas (#3366/#3354/#3338), Pierre/hexapode (#3359), Gonzalo Mier (#3308), Philippe Van Velzen (#2169), George Lydakis, Rohan Arun, Jeremy Kothe, Josie Elliston, Hao Yan and the retained compiler/native/bit/moment authors. Original notices and unchanged admission code remain. New work by Pierre (hexapode) with substantial OpenAI Codex assistance from four concurrent agents. Apache-2.0.

Open next questions

Can mixed-call exchanges move multiple helpers through one bin to pool more residuals? Preserve this complete map, nominate exact moves and replay the full invoice. Private helper tools and solver experiments are excluded from the contribution.