Parallel-effort entry: Lean proof of #130's frame lemma and check of #193's complex word, Fourier side (no κ claim)

[jacobalansussman]

Change and mathematical scope

One bullet in CONTRIBUTORS.md, under "Additional public parallel efforts", linking a separate Lean 4 development on the Fourier side: jacobalansussman/wht-power-saving-lean, pinned at fdfb781. Documentation only; no κ is claimed.

What it adds for this project

1. The general frame lemma of Three-stage Cayley covers for conditional kappa = 3.146011e-4 > 2^-12 #130, proved in Lean. notes/general-clifford-frames.tex states it for arbitrary binary subspaces, degenerate ones included, and pull-request texts here list "General Clifford/tensor" among the assumptions. Work/GFrame (frame_lemma) proves it for the RAM model of OpenAI's family 130, with the cost of the two adapter steps: each is one linear pass.
2. A complete machine check of the complex word of Pair donors to avoid parity erasure on the source-assisted v4 word: conditional κ = 6.647872e-4 #193, rebuilt from the published data of Annealed modules, terminal sinks and physical bit words on paired cubes: conditional κ = 6.558894e-4 #168 and Pair donors to avoid parity erasure on the source-assisted v4 word: conditional κ = 6.647872e-4 #193 (43,036 gates, 9,412 helper arrays). The Lean kernel replays every addition in exact arithmetic and checks every frame and block count, inside my layout and statement. Pair donors to avoid parity erasure on the source-assisted v4 word: conditional κ = 6.647872e-4 #193 states its own scope as "no globally renumbered scalar transcript of the new complex word and no full Clifford/router replay".
3. A five-stage "bridged" layout. Every pair has a twin on the same line, so two pairs are exchanged with five invocations in place of six. The word of Pair donors to avoid parity erasure on the source-assisted v4 word: conditional κ = 6.647872e-4 #193 gives 7.009184e-4 in its three-stage layout and 7.474547e-4 in this one. As far as my scans found (pull requests 1 to 207, by title and body), it is in no pull request here; if it is, please say so and I will credit it.

Credit. The circuit is this community's in every part: paired cubes, icekylinx (#144, crediting an664's #128 and eumemic's #117); modules, carrier links, frames and physical layer, eumemic (#168); source-parity local word and frame flow, icekylinx (#184); its move to the v4 modules and the donor pairing, ikeboy (#191, #193). The frame lemma (#130) and whole-residual batching (#10) are icekylinx's. Method, RAM model and recursion engine are OpenAI's (family 130). Mine are the Lean proof of the lemma for that model, the engine and checker built on it, the bridged layout, and the machine check. Full list: RELATED-WORK.md in the linked repository.

Scope and limits. The theorem is a Walsh-Hadamard statement, O(n (log n)^(1 - δ)) with δ = 7.474547e-4, in family 130's RAM model, in whole-block accounting. It is not this project's multiplication theorem and not a κ: the bit side and the analytic, routing and fixed-tape interfaces are outside it, the frame assumption stands as it is in this project's framework, and the bridged layout is untested there. That the rebuilt list is the word of #193 rests on a Python conversion and on equal counts, not on Lean. The Walsh-Hadamard statement (about 60 lines) was written for the linked project and is the part to read. The proofs have not yet had a line-by-line human review, and I would welcome one.

Built by me with a team of AI agents (Claude) that I directed; the proofs are theirs and are machine-checked.

Please edit, move or close this as you see fit. If a short note in the tree or a pinned replay would serve better, I am glad to prepare it.

Validation

This change. One added line in one file; git diff --check is clean. make verify was not run: no generator, certificate, patch, note or test is touched.

The linked repository (its own report; nothing in it has been replayed here).

• Proof arguments: Lean theorems for the frame lemma, the generalised engine, and the soundness of the certificate checker.
• Finite checks: one certificate, evaluated by the Lean kernel in 16 segments plus 63 block counts.
• The Lean comparator accepted the proof three times against a 331-line statement importing only Mathlib (axioms propext, Classical.choice, Quot.sound) and rejects a saving one unit larger.
• Limits: macOS with a stand-in for the comparator's Linux sandbox; second kernel (nanoda) off; no build from nothing yet. Commands: VERIFY.md. Re-runs are welcome, most of all on Linux.

🤖 Generated with Claude Code