Scoped negative result: PR589's bit word is exactly frame-optimal (no new kappa)

[SethBurkart123]

Scoped negative result; no new κ. PR589's bit word is exactly frame-optimal. Keep its gate sequence, scalar program, start/final frames and COPY frames (decoder constant 63) fixed. Then no reassignment of gate frames, however many gates change at once, raises c = 0.001733427039589844234749. PR555's base word is frame-optimal too. PR589's tuner was greedy and made no optimality claim; this settles the frame-only part of that question exactly.

Method (standard library only, about 16 s):

1. Bounds. Forward least-frame and backward greatest-frame passes bound every legal frame. Only 159 of 5,472 gates have any slack.
2. Components. Free gates couple only when consecutive on a register, giving 133 independent components (largest has 5 gates). Each is enumerated exhaustively: 398 assignments, of which 55 fail nesting.
3. Exact objective. At z = c the moment is linear in the paid profile, so the sign of Σ n_d·d·(64/d)^c decides exactly whether c rises. Every strictly different alternative loses by at least 9.1e−4. Ten alternatives tie on the same profile.

Positive control: drop every free gate to its least frame. The optimiser finds 37 improving components and recovers exactly PR589's profile {1:1323,2:1342,3:79,4:44,5:102,6:11,7:19}.

bash research/bit589-frame-optimum/verify.sh   # → PASS_BIT589_FRAME_OPTIMUM

Scope

• Frame-only search class. Transports, Steinberg rewrites and different retained-total frames are outside it.
• C-gate frames are relaxed beyond H4, which only enlarges the search.
• The bit side does not bind κ. The practical conclusion: further bit headroom on this word must come from gate-order or structural moves.
• "No improvement" is proven only within this finite search class.

Pins: base f567884a. Inputs are the committed PR589 and PR555 words.

Credit: the PR589 word and tuner are by DreamingOfClouds, on eumemic's PR555 lineage. #815 and #864 give complex-side frame-optimality certificates by different methods. Assignment seed 16 (bit track).

Prepared with substantial AI assistance.

🤖 Generated with Claude Code