Translated partial-swap frames for a conditional 5499/10^9 > 2^-18 bound

[jacklightChen]

This PR gives the conditional final multiplication saving

κ = 5499/10^9 = 5.499 × 10^-6 > 2^-18

in T(n) = O(n (log n)^(1−κ)) under the retained analytic and fixed finite-alphabet multitape hypotheses. It passes both the 2^-19 and 2^-18 milestones; this is the final multiplication saving, not just a component exponent.

Construction

• Translate the dedicated stage-two auxiliary source/sink frames in Partial-swap networks for a conditional kappa = 1.884586e-6 bound #18's partial-swap bit network. The same scalar producers, role counts and common rational basis now support recursive widths 23 and 32729 instead of 575, 31625, 23 and 529. Arbitrary dirty scratch is restored, all other physical edges remain accounted for, and the exact bit saving is a_b = 11/10^6.
• Independently translate the complex phase sources in the retained Ternary five-subset networks for a conditional 373/10^11 > 2^-28 bound #7/Nested controlled bases for a conditional 9799/10^10 > 2^-20 bound #16 physical schedule. Operator-level identities, signed orthonormal directions and phase polarization give a_c = 18/10^6. This does not combine Partial-swap networks for a conditional kappa = 1.884586e-6 bound #18's separate complex data-entrance adjustment with an incompatible schedule.
• Prove the precision guard for the changed complex circuit: path rank budget 22876, maximum child rank 21924, rho = 2, C1 = 19991/10000. Exact assembly has a final absorption margin greater than 9 × 10^-10.

Base and review entry points

The target is main, pinned at 6e564879f51ae16f23d392e9e196c605f36d90df. Because #10 and #18 are not merged, their necessary sources are included. #18's text sources are imported unchanged from f2ab41aebad47861caf6316282c1793e5513845e, retaining all recorded source hashes and attribution. The new work is concentrated in:

• notes/translated-partial-{bit,complex,assembly}.tex and the complete 19-page artifacts/translated-partial-18-note.pdf;
• research/translated-partial/: reproducible finite controls, the exact witness and producer report;
• tests/test_translated_partial.py.

Relative to the inspected main and PRs through #18, this is a higher stated final bound. It is a conditional mathematical construction with written proofs and executable certificates, not formal verification or completed external peer review.

Validation completed

• Regenerated all Partial-swap networks for a conditional kappa = 1.884586e-6 bound #18 finite producers at dimensions 25, 23 and 57 from portable source once, including scalar maps, carrier dependencies, positive frames and complete physical histograms; reconstructed the retained h28 complex histogram.
• Verified the new dirty-auxiliary endpoint identities, prescribed common-basis corners, both physical orientations, stage boundaries, exact complex phases and the revised path guard.
• Exact rational moment certificates: bit gap > 1.59 × 10^-8 and complex gap > 6.66 × 10^-9; 29 strict assembly constraints and 7 strict final margins pass.
• 172 unit tests, 18 retained patch-application checks, and manuscript compilation pass. Negative controls reject unsupported bit saving 12 × 10^-6 and final κ = 2^-17.

Reproduction commands and conditional dependencies are documented in research/translated-partial/README.md.

Contributions

This is Zhihao Chen (jacklightChen)'s first submission of this translated partial-swap/phase-frame extension and assembled witness, developed with substantial OpenAI Codex assistance. The earlier GPT-6 Astra assistance attribution is preserved as recorded; no runtime-model identity or worldwide-first claim is inferred.

The construction builds directly on icekylinx's #18 partial-swap network, selected producers and common basis; eumemic's #13 source-frame idea (with Claude assistance); Zhihao Chen's #7 and #16 network/nested-basis work; and icekylinx's #10 batching and tape interfaces. Rohan Arun's #14 supplies concurrent overlapping data-corner refinements. All preceding authors and licenses remain credited in NOTICE and SOURCES.json.

Future research using these contributions should explicitly acknowledge Zhihao Chen (jacklightChen) and cite the relevant work, while also crediting the other dependencies it uses. This attribution request adds no license condition.

[Mister-Meeseeks]

Thank you @jacklightChen. This contribution is now recorded on main in the scoped reconciliation review and contributor credits.

Contribution: Translated second-stage partial swaps and corrected complex phase gauge.

Review and limits: Fresh translated-partial tests; phase and coefficient accounting remain part of the contract.

The complete original source, proof/research notes, notices and relevant replay/CI evidence are retained through the pinned source archive. The review distinguishes fresh runs, exact-head historical CI and prior integration evidence. Closing this historical PR as reconciled; its obsolete headline does not replace the current selected bound, and draft/retracted claims retain their explicit qualifications.

Reviewed head: 5ba6cf0bfb68f2be8d15610e7972207c50254d6a. Publication: b84ba67156a01f9c2cb34fe1e3cb1dcdeab00d9c.