Verify finite profiles and formalize multiplication transfer interfaces

[rfu08]

Change and proof scope

Publish the completed verification and formalization checkpoints in research/machine-transfer-verification/, alongside the original rank-first package. The update makes the finite certificates, actual-real characteristic proofs, physical-interface lemmas, and concrete tape primitives reviewable under one portable source inventory.

The additions include:

• Actual real-characteristic Lean proofs for the pinned PR64, PR68 and PR73 profiles, and a floor-child recurrence/state-cost theorem with explicit physical hypotheses.
• Literal fixed-alphabet stack push/pop and a six-state binary counter with restored heads, first-completion semantics and a fixed-direction amortized bound.
• Constructive all-positive-width row supply, padding below factor two, descendant descriptor bounds, rational localization and eligible-prime-power transport.
• Tensor/gauge identities and complete externally checked ambient schedule/profile correspondence, including reverse execution, copied centers, exterior transitions and endpoint correction.
• Source-pinned PR71/73 reproduction and PR76 finite-certificate audit. PR76's supplied words, fresh CRT profiles, dirty-basis replay and exact arithmetic pass at conditional κ = 5.1738676045141e-5; heuristic discovery regeneration was not rerun. The latest literal Lean profile remains PR73 (κ = 5.1414646104194e-5).
• A bounded combination experiment for PR69 balanced coarse sums and PR71 anchored splits, with both changed axes checked and finite conditional κ = 5.1544825802029e-5, above PR73 and below PR76. No literal Lean profile is claimed for this experiment.

The integrated Lean inventory contains 27 unique modules and 298 named theorem/lemma audits, including 13 earlier DirtyWrapper declarations and 12 helpers omitted from the historical selected prints. These counts are audit coverage, not independent new results. Duplicate source copies are byte-checked and compiled once.

This remains a draft research contribution. Actual complete digit/row routing, selected table and child-plan instantiation, fixed-tape recursive scheduler composition, and analytic resampling/precision/exact-recovery/setup integration remain required. The counter's primitive bound does not include controller dispatch, payload emission or template reset. No unconditional multiplication theorem, global optimum or measured practical speedup is claimed.

Reproduction and validation

• make machine-transfer-check: publication hashes; exact PR64/68/73 Lean literal binding; independent rational characteristic and 47 inequalities/seven margins for all five profiles; exact negative control showing PR73 does not dominate PR68 at every power.
• make formal-machine-transfer-verify: PASS all 298 audits: fresh isolated compilation and complete axiom audit using the existing Lean 4.21.0 / pinned Mathlib project. Reports go to ignored build/.
• The offline target joins verify-certificates; the formal target joins existing historical formal CI. Source-pinned finite/CRT replay commands and earlier receipts are retained separately; offline checks do not pretend to rerun them.
• The original ranked-pair-check remains passing. The broad unchanged repository suite previously passed at 790a14e; this update uses fresh focused checks and the full new Lean inventory rather than relabeling that earlier run as current.

The package README is the current review map; older checkpoint documents retain their dated scope. The manuscript is the completed full-transfer snapshot. Unfinished routing drafts, raw GitHub dumps, compiler binaries and build logs are excluded.

Attribution

The public constructions remain credited to their authors: Avi Eisenberg (#62), eumemic (#57/#69), Chafik Boukhalfa (#60/#71), Rohan Arun (#65/#67), Dominik Scholz (#63/#68/#73/#76), Alejandro Zarzuelo Urdiales (#70), Thomas DiFiore (#74), and all inherited contributors and source-specific notices. The original #63/#64 identical-construction comparison is preserved. This verification, formalization and bounded combination work used substantial OpenAI Codex assistance, including separate agent checks. Apache-2.0 and inherited notices are retained.