Partial-swap networks for a conditional kappa = 1.884586e-6 bound

[icekylinx]

Result and dependency

This PR extends #10 at 62691e395a0458ce089a1c7b5d89e74291e95e29 with one new commit. It includes #10 in its ancestry and should be reviewed and merged after that dependency.

Under the retained analytic and fixed-tape interfaces,

$$ T(n)=O!\left(n(\log n)^{1-\kappa}\right),\qquad \kappa=\frac{942293}{500000000000}=1.884586\times10^{-6}. $$

Partial-swap frames remove the source rank penalty, giving s = Wm - 2N + 2L. The binary retained-total triple producer uses dimensions (25,23,57), compatible carrier reuse and positive frames. A common rational basis batches the paired auxiliary boundary into widths 25,7,25,25,32611. Additional whole-residual batching improves the complex component.

The component savings are 188459/50000000000 and 417/100000000. Exact rational certificates give strict bit and complex moment gaps, a uniform precision guard, and final absorption margin greater than 4.4822e-13.

Artifacts and scope

• Add the standalone proof artifacts/partial-swap-note.pdf and its LaTeX sources.
• Regenerate selected producer graphs, carrier dependencies, positive labels, matchings and complete histograms from portable source. Large intermediate binaries are temporary.
• Add the exact recursive-width, basis-combinatorics, precision-guard and assembly certificate; integrate reproduction into make verify.
• Preserve Batch recursive networks for a conditional kappa > 2^-23 bound #10's Gaussian scaling correction, source pins, Apache-2.0 notices and contributor attribution. Record icekylinx's contribution and substantial OpenAI GPT-6 Astra and Codex assistance.

The standalone note is the current proof. patches/batched-23.patch remains the inherited #10 manuscript patch. Generic common-basis existence and the analytic/tape interfaces remain written proof obligations; finite certificates do not formally verify the complete multiplication theorem.

Validation

• Regenerated all three selected scalar DAGs, verified their exact supports, and matched the full positive-frame histograms and role counts.
• Reconstructed all 29 complex histogram entries from the retained paired producer.
• Certified both recursive moments, the uniform precision guard, 29 strict parameter constraints and seven assembly margins with exact rationals.
• Passed 168 unit tests and all 18 inherited manuscript patch checks.
• Compiled the new proof PDF and checked the staged diff for whitespace errors.