Conditional κ = 0.001673626873024029 by composing six cycles on PR3354

[hexapode]

Change and mathematical scope

Compose six bin-disjoint three-helper cycles with the latest PR #3354 map:

• Before: κ = 0.001673620586419375.
• After: κ = 0.001673626873024029.
• Exact gain: 3143302327/500000000000000000 = 6.286604654e-9.

PR3354 appeared while we independently found a similar cycle. Instead of publishing a duplicate, this preserves its exact cycle and adds six compatible cycles from our search. Eighteen additional helpers relocate; forty-four rows change including offsets. The complete paid profile changes by -6 H1 -6 H2 +6 H3, with rank mass 1,041,408 and deficit 5,120 unchanged. Strict concavity proves the moment gain. Complex still binds.

This is a conditional finite-composition packing improvement, not an optimum or complete multiplication theorem. The selected word, helper widths, 144-call order, suppliers and entire physical invoice are preserved. Selected-result.json is unchanged.

Task and overlap

Package: research/t16-composed-cycles. Canonical base: 6f100d4. Commit: 3bdc260.

Exact latest predecessor: #3354 head 2007d0d, reconstructed from canonical PR2169 data plus an authenticated 176-row delta. Its original construction, delta, source manifest, result and proof are retained. The six new cycles are (1813,2979,650), (1814,2981,861), (1815,2986,869), (1859,2993,1009), (1861,2995,231), (1862,3002,260).

Card selected-packing, revision6, refined to this explicit composition. Public PR listings were refreshed during the round and before submission; #3354 overtook our first local witness and became the baseline. Coverage does not establish global novelty. Higher numerical claims in the #3190/#3251 chain have the separate operator objection in #3296 and are not treated as admitted baselines.

Validation

Standard-library CPython3.12.1, macOS arm64:

• Complete unchanged eleven-stage strict admission passes on all69,120 roles,144 calls and10,008 physical charts; controls69/12/6/15 plus11 independent owner controls.
• Both exact supplier-pricing engines, native/conservative bootstraps and all47 strict outer inequalities per path pass.
• Full typed physical invoice equality passes, allowing only three freshly bound source hashes. Prefix397 remains paid at512; row coefficient16,469.
• Eleven behavioral groups pass normally and optimized, including incomplete/overlapping cycles, wrong physical family, changed width, corrupt predecessor and invoice value/type/key faults.
• A separate private derivation reconstructs every candidate row from PR3354 and independently checks disjoint bins and the exact paid-profile delta.
• Committed-byte preflight with source history and 61-file Git-free export pass.

Reproduce:

python3.12 -Werror -B scripts/research_package.py --package research/t16-composed-cycles

No hosted CI, fresh full cold native rebuild, formal-kernel proof or independent human review 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 (#3354/#3338), Gonzalo Mier (#3308), Philippe Van Velzen (#2169), George Lydakis, Rohan Arun, Jeremy Kothe, Josie Elliston, Hao Yan and the retained compiler/native/bit/moment lineage. Original notices and byte-identical admission code are preserved. New work by Pierre (hexapode), with substantial OpenAI Codex assistance from four concurrent agents. Apache-2.0.

Open next questions

Can a neutral rearrangement unlock another legal exchange on this exact combined map? Preserve all prior gains, nominate one explicit cycle and replay its full invoice. This construction does not exhaust global packing or interacting exchanges. Private search helpers are excluded from the contribution.