Formalize P104 transport transcript capacity in Lean

[commdata2338]

Change and mathematical scope

This machine-checks the chronological step behind the P104 transport floors of Ross Richardson's PR #2127 and Nick Petalas's PR #2148, the "+1 array per added transport" implication that Michael Yee's PR #2163 asked to have formalized.

• Model (TransportCut.lean): Runs n s t, where s and t are the source-dependent linear blocks of the source-side and target-side states. Its four constructors: init (arbitrary source-side input, zero source dependence on the target side); source (any linear update of the source side from both sides, so all return traffic is free); target (any linear endomorphism of the target side); and receive (one element of D read from both sides, with the target updated by A t + B z, the only constructor that increases the count n). Copies, storage, basis changes, feedback and cycles in the static frame graph are all included.
• Theorems: runs_factor, under which every target-side state factors through a transcript of n whole arrays (t = decode ∘ encode, encode : X → (Fin n → D)); runs_rank, giving rank at most n · dim D; and recovery_requires_receives, under which exact recovery of m independent arrays by any linear readout of all target storage needs m ≤ n. one_receive_attains shows the bound is attained. PR Prove P104 transport floor with arbitrary added storage #2127's 96-transport floor and PR feat(research): price escape from refined P104 intervals #2148's 48-rank floor follow as p104_unsplit_floor and p104_refined_floor, conditional on their source and cut translations.
• Source binding (SourceCut.lean and verify.py): a Python binder reconstructs PR Draft construction: P104 dirty-restored gather and low-rank channel with COPY56 (no new kappa) #1972's witness (4,058 events, 1,410 transports, 444 frames split 222/222, 104 source entrances and target exits, and exactly eight inward C descents at events 2463 to 2518) and regenerates the Boolean facts in SourceCut.lean, which the kernel checks by decide (not native_decide). Interpreting the JSON into frames stays in Python, outside the kernel.
• Status: a scoped formal proof plus a finite source-binding certificate. The transcript and rank method is standard; the contribution is its formalization and source binding. No supplier or κ change.
• CI: following the repository's per-package workflow convention, the PR adds a path-filtered .github/workflows/lean-transport-cut.yml: read-only permissions, credentials not persisted, the same pinned checkout and setup-python actions as the existing workflows, and the checksum-pinned official Lean 4.21.0 release. It runs the same commands as below. Hosted CI has not been observed for this contribution.

Remaining obligations. The native-event-to-Runs translation is written, not formalized. The refined separator and its crossing-count premise (PR #2136, #2148) remain outside Lean. No native phase, owner or hosting realization; no physical transport tariff, coverage credit, precision or tape proof, all-size lift or selected cold replay. No independent human review.

Task and overlap

• Task ID / scope key: formal-interface-lemma (636f26dec98e26b296e8), refined to P104/chronological-transcript-factorization/arbitrary-linear-feedback/Lean. Chooser nonce c0ba36ea40e7964ca3d6bcd152c6c2cf, track recursive-interfaces, priority revision 4. Following revision 4's preference for a live central handoff, this was taken over the chooser's suggestions (centre-path-sharing, whose nearest native lift is this account's PR Scoped proof: PR1918's selected centre-source minor lifts to native rank 8192 (no new kappa) #2048; connector-alphabet-change, whose immediate cases are covered; coverage-law-change, with no concrete new identity). No reroll.
• Base and exact heads: base 0fdd7d82de03e43c9e5a1000dfbcd380d1b687d2, rechecked unchanged at 22:12 UTC; contribution a553060b8fa93b2eadfac00834cc8ccc8dc0415e. PR Prove P104 transport floor with arbitrary added storage #2127 befebf0f4ae7a3f69fe9c5e136262559dcac125f, PR feat(research): price escape from refined P104 intervals #2148 937b132f3f143ce0f2621b3b12e1ebc413e66730 and PR Independent max-flow audit of the P104 rank-eight cut family (no new κ) #2163 9f6e1f2783aef0c52bb3f26f587a918a8be32fa9 (reconfirmed by anonymous fetch). Scientific input: main's PR Draft construction: P104 dirty-restored gather and low-rank channel with COPY56 (no new kappa) #1972 witness, SHA-256 0f541bef7f19fdeffd67157684e9f03d785f42c89973632f508fc1a0452b580e. Lean 4.21.0 and Mathlib 308445d7985027f538e281e18df29ca16ede2ba3 are pinned in lake-manifest.json.
• Scope status: refined, from a generic formal-interface card to the written implication named in the DONE: independent max-flow audit of the P104 rank-eight cut family (#2082/#2090/#2113/#2127) #2164 handoff.
• Nearest work: PR Prove P104 transport floor with arbitrary added storage #2127 and feat(research): price escape from refined P104 intervals #2148 own the transcript argument and the numerical bounds; PR Rule out identity from nested P104 transport refinements #2136 (Tairitsu-Aya) owns the refined separator; PR Independent max-flow audit of the P104 rank-eight cut family (no new κ) #2163 checks the graph premises and leaves this reduction unformalized. This account's PR Certify the P104 cut-2429 nested-contact price: 864 rank units #2172 prices nested contacts at cut 2429 and does no formal work. PR P104 shared correction-space dimension is 96–97 #2173 bounds a shared correction-space dimension, a different question.
• Claim and final check: the claim is in discussion #2164. A final overlap check of open PRs and Discussions (complete pagination), plus a live check of newer PRs, found no competing formalization.

Validation

python3 -I -B scripts/research_package.py --package research/lean-transport-cut
cd research/lean-transport-cut
lake exe cache get Mathlib.LinearAlgebra.Dimension.Constructions Mathlib.LinearAlgebra.FiniteDimensional.Basic
python3 -I -B check_lean.py --lake lake

Linux x86_64, CPython 3.12.3, Lean 4.21.0. The registered Python replay (source binding, 12 finite and source controls, 8 artifact groups) deliberately does not compile Lean. check_lean.py validates every dependency head, compiles both files afresh with warnings as errors and at most four Lean workers, audits all 13 theorem roots (only propext, Classical.choice and Quot.sound are allowed; SourceCut's six finite facts use no axioms), and requires eight weakened variants to fail: an uncharged receive, an informative outside initialization, the false floors 97 and 49, a sorry, a custom-axiom proof, a missing recovery probe and an undercounted original cut. The custom-axiom and missing-probe controls must compile and then fail the audit, so a syntax error cannot pass for audit coverage.

Both were run again on the final package bytes before submission, under an 8 GiB memory cap: the Python replay in about 4 seconds and the Lean check in about 30 seconds, leaving the worktree unchanged.

PASS_TRANSPORT_CUT_BINDING sources=104 crossings=8 transports=1410 controls=12 formal_not_run
PASS_REGISTERED_RESEARCH_PACKAGE research/lean-transport-cut outcome=completed
PASS_LEAN_TRANSPORT_CUT theorems=13 rejected_controls=8 no_sorry_or_custom_axioms

FORMAL-RECEIPT.json records that local run: compiler, dependency heads, proof hashes, theorem axioms and the rejected controls.

Credit

• Ross Richardson (@RossRichardson, PR Prove P104 transport floor with arbitrary added storage #2127): the graph partition, the written all-program bound and the retiming and storage extension.
• Nick Petalas (@petalas, PR feat(research): price escape from refined P104 intervals #2148): the refined 48-rank argument, using Tairitsu-Aya's PR Rule out identity from nested P104 transport refinements #2136 separator.
• Michael Yee (@AgoraPluribusMichael, PR Independent max-flow audit of the P104 rank-eight cut family (no new κ) #2163): the independent graph check and the formalization request.
• David Arias, Adam Gall and Jeremy Diviney: the earlier cut work. Rohan Arun (PR Draft construction: P104 dirty-restored gather and low-rank channel with COPY56 (no new kappa) #1972) and Chafik Boukhalfa (PR Native Clifford realization of the P104 low-rank channel (no new kappa) #1991, with Adam Gall's schedule): the channel and its native realization.
• The Lean and Mathlib authors: the proof system and the dimension library.

New work is Apache-2.0; retained notices are unchanged.

AI assistance: submitted by @commdata2338 with substantial AI assistance. OpenAI Codex (GPT-6 Astra) did the formalization, the implementation, the checking and the writing; it was packaged and submitted with Claude Code. No independent human review is claimed.

Open next questions

From the package README:

1. Formalize the native-event-to-Runs translation. It would remove the written bridge for arbitrary retained-occurrence programs. First step: a typed register and frame semantics, with a proof that each primitive has one of the four constructor forms. Blocker: agreeing on the allowed initialization, array width and cross-chart access rules.
2. Formalize PR Rule out identity from nested P104 transport refinements #2136 and feat(research): price escape from refined P104 intervals #2148's refined crossing count. The rank theorem already accepts n ≤ 8 + B; the next step is a kernel proof of that premise from the survivor separator and nested paths. Blocker: the finite-subspace and interval model, not the 56/8 arithmetic.

Workflow feedback (optional)

• What helped: DONE: independent max-flow audit of the P104 rank-eight cut family (#2082/#2090/#2113/#2127) #2164's handoff split cleanly between graph pricing and this formal lemma, so two parallel sessions did not collide.
• Suggestion: formalization handoffs could attach the exact target proposition and a proposed program grammar, so a session can spend its time on the native-to-model boundary as well as the abstract rank argument.

🤖 Generated with Claude Code

https://claude.ai/code/session_01X86sCSXdoDd61SDFfC1PKG