[Dugongue]

This adds an informal sufficient native Word RAM assembly for packed n-bit multiplication in O(n / (log n)^κ), κ = 0.0030201655, conditional on the explicitly pinned predecessor algebraic and analytic proofs. It replaces the packed TM transfer with direct bounded-word execution. It exceeds the displayed #4541 target by 5 × 10⁻¹⁰, but does not change the p16 paid profile, critical root, selected TM frontier, or establish worldwide priority for this numerical value.

The mechanisms that make the native bound possible are fused gauge-free Gaussian circuits with paid sublinear complete-group metadata tables and microbox routing; global-precision native FFT record products; and explicit batched boundary arithmetic with both couplings and periodic closure. Offline high-precision coefficient work is multi-limb and charged. All nine native cost rows have exact strict margins. The enormous fixed supplier is algorithmically specified, not materialized.

Validation: Python 3.12 stdlib source-authenticated replay under -I -O -B; exact critical-majorant/finite-absorption arithmetic; nine-row margin ledger and five rejecting controls; 36 signed complex record products (378 coefficients); 36 boundary systems (1,785 residual entries); 8,190 exact metadata addresses; sparse dispatch and limb arithmetic controls. Registered repository replay, Git-free export and drift rejection are included in the release validation receipt. Finite checks are component evidence, not a formal certificate of the universal theorem or its inherited proofs. The complete growing-input multiplication program has not been emitted or practically benchmarked.

Predecessor credit: #4470/#4483/#4532/#4535 native all-fiber compiler and critical-root lineage; #1673/#1758/#1786 tensor circuits, gauge-free interiors and fine padding; #4538/#4541 native/packed transfer and #4554 arithmetic/adapter components. Named contributors and original notices are retained in README and source snapshots. Main was refreshed at f626db5; recent PRs through #4562 were checked for overlap. #4556–#4562 component audits are acknowledged without claiming their full replay. DAG/Lean precedents informed the representation; no new Lean verification is claimed. Substantial OpenAI Codex assistance: one lead and three research/review agents.

Remaining obligations: independent mathematical review of the retained source proofs and their assembly; practical full native caller and usable constants. The canonical DAG and selected frontier remain unchanged.