Extreme-frame retiming on #284's gen4 word: conditional κ = 7.26319877913759e-4 (+0.0039% over #284; below #285)

[GamingPuzzled]

κ = 726319877913758986831841 / 10²⁷ = 7.26319877913759e-4 (conditional), +2.8186e-8 (+0.00388%) over #284 (7.26291692225845e-4) by #284's own fixed-prime price.

This is an improvement to the gen4 line only, not a repository record: #285's gen5 word (7.42785663120535e-4) is higher. It is posted for the method, which is word-agnostic and stacks on later packages.

Change

An ADD's frame must contain its operands' current frames (MOVEs only go up) and lie inside the frames their next uses need. Between those bounds the choice is free. Each operand role then pays one recursive child for each frame jump. At fixed rank mass the price rewards fewer, wider children (≈ −Σ r ln r), and r ln r is convex, so the optimum for a unit is an extreme:

• the join of the operands' previous frames, or
• the meet of the frames their next uses require.

Either can be a new frame. Each is built exactly as an integer basis/annihilator pair.

pass	unit	moves
1	equal-frame ADD component (54 gates)	21 joins (14→13 ×12, 16→15 ×4, 12→11 ×4, 12→10)
2	single ADD (components split)	40 meets (10→15 ×20, 18→19 ×16, …) and 1 join

Rank mass (422,388 local) and stock (2,549,300 literal / 509,860 normalized) are unchanged. Afterwards, fresh scans of both unit types find no improving extreme. The same scan gives +2.4958e-8 on #283. On #280 it finds nothing, because the slack comes from the kernel pivots and target-prefix squares.

Verification

research/gen4-extreme-frame-retiming/verify.py --pr284 <pinned #284 checkout> --output <fresh> runs these steps in order:

1. Conditional κ = 7.26291692225845·10⁻⁴: 91 collective response-kernel families on helpers untouched by #283, at its own cut (+0.0248% over #283) #284's complete verifier, freshly. All nine PR276 stages, sinks, kernels and the 91 families, retiming, transport, restorations and targets; its certificate must pass.
2. Frame audit before each frozen pass. Each new frame is checked against the exact word it is applied to: complementary ranks, P_r ⊆ G ⊆ N_r for every operand role, and exact join/meet dimension.
3. Local emission. The ordered non-MOVE stream must be unchanged except for the selected frame fields; only the affected roles' MOVEs are rebuilt.
4. Conditional κ = 7.26291692225845·10⁻⁴: 91 collective response-kernel families on helpers untouched by #283, at its own cut (+0.0248% over #283) #284's own native checkers on the new word. Legality, multi-cut prefix, exact bank charts and allocation, all five-stage formal columns, the rational price, the finite invoice, and the fixed-prime 47-constraint assembly with adjacent-grid rejection.

A local full replay passed in about 10 minutes, almost all of it #284's verifier. The package has its own path-filtered workflow, which checks out pinned #284 separately. No existing file changes.

Outlook. On gen5's raw word the same scan automatically rediscovers #279/#280's 880 source-pair retimings and finds 135 further gate moves. Proxy price is about 7.4354e-4, and the five-stage columns pass on that word. It is not claimed here: it needs a gen5 bank/invoice admission path.

Conditional on #284's (and #283's) retained interfaces; not an unconditional theorem, a Lean proof or a runtime result. Credit: chafreaky (#284), Dugongue (#283) and the lineage they credit; lydakis's #275 motivated the generalization.

🤖 Generated with Claude Code

[Mister-Meeseeks]

Thank you @GamingPuzzled. This contribution is now retained on main in the scoped 100-PR review and contributor credits.

Contribution: Extreme-frame retiming on the PR284 gen4 word.

Review and limits: Source-bound Linux full parent and changed-stage replay. Complete chronology, columns, banks, finite invoice and price accompany the local histogram improvement.

The complete original source, notices and relevant execution evidence are retained through the pinned source archive. The review distinguishes fresh local runs, source-bound historical Linux CI and prior exact-head review; it does not claim 100 fresh full mathematical audits. Closing this historical PR as reconciled. This preserves parallel and cumulative credit without replacing the selected frontier or promoting priced research targets to completed constructions.

Reviewed head: 13a523a250f471e7cc880e8a24060d654ba9c78b. Publication: 50d768818f579d4e165b4c257c6ee25bca300802.