Conditional κ = 0.001526158885117467: frame-chain search on #531's 504-helper word

[rohanarun]

Conditional κ = 0.001526158885117467 (exact 1526158885117467/1000000000000000000), +0.55331% over #531 (0.001517760977820331) and +3.96% over #532.

This builds directly on @icekylinx's #531 (the branch contains the #531 commit plus this package). Scalar operations, helpers, widths, entrances, owner map, controller and every checker are #531's; the only change is a better assignment of operation frames.

Prepared by rohanarun with Claude Opus 5.5 (Anthropic). Apache-2.0. This is a derivative of
#531 by icekylinx
(its direct-delivery 504-helper word, owner map, controller and every checker are reused unchanged).
The only new mathematical object is a different assignment of operation frames.

Result

Quantity	#531	This package
Conditional κ	0.001517760977820331	0.001526158885117467
Complex lower rate b	0.001520068077832496458155	0.001528491606163040267702
Cited #493 bit rate c	0.001565442026981656908596	same
Local profile	{1:2028,2:752,3:156,4:392,5:56,6:56,7:8}	{1:2220,2:368,3:460,4:168,5:168,6:56,7:8}
Helpers / widths / entrance rank	504 / {8:152,9:352} / 152	same
Owner map, live/physical, deficit	36,288 roles, 7168/7682, 2560	same

κ is +0.55331% over #531 and +3.96329% over #532. The complex supplier still binds.

What changed

#531 ran a single-node greedy pass (248 promotions in delivery body B, each to the meet of its successors, accepted on the linear
entropy fee). We instead search the whole lattice of monotone frame lifts of the constructor word:

• every A/B operation frame G must contain the frame the Direct delivery with 504 helpers for conditional κ = 0.001517760977820331 #531 constructor assigned to it (F ⊆ G);
• every register chain (start frame, A, centre/target barriers, B, final frame) must stay monotone;
• scalar operations, cut, centres and centre frames, helper start frames, target levels and final frames are fixed.

Any such assignment is reachable by #531's own coalescing lemma: lift nodes from last to first; each step has predecessors inside the
constructor frame and successors already at their final frames. A seeded local search (code/frame_chain_search.py, moves = one-vector
raises with upward closure and codimension-one lowers with downward closure, bounded below by constructor frames) minimises
Σ_r H_r · r((73/r)^b − 1), the moment-excess fee. A 40-sandbox run (25 min per seed, temperatures 0 to 0.004) converged to the same fee in 23 of 40 seeds; the selected word lifts 725 frames (versus 248 in #531). Rank mass is unchanged
(6240), so helper widths, the owner map and all 7,968 idle remainders are identical; only the local histogram changes. Spreading rank
mass across children of ranks 3 to 5 moves the moment root up.

Verification (all #531 checkers, fresh on the new word)

python -B verify_fast.py --work /absolute/new/output/dir

New stages: rebuild the #531 constructor word from the frozen selection (no search), then code/check_frame_lifts.py
proves every changed frame is a lift of its constructor frame and every chain is monotone. Then the unchanged #531 stages run:
exact arbitrary-dirt forward and inverse-transpose scalar replay on all 728 columns with native frame chains, binding to the 112-port
controller, emission and independent audit of the actual whole-owner placement, fresh prefix and finite invoice (prefix 225 < 512 bits),
and two exact moment engines with all 47 strict outer inequalities.

The local evidence was regenerated with #531's code: phases_delta.py lowered 928 fresh phase edges (1,994 reused) and checked every
tableau and global phase; emit_local.py emitted both orientations (22,937 events each); audit_local.py passed the independent emitted
replay with 13 controls rejected; adapters_local.py proved Lemma A for all 7 entrance frames with 5 mutations rejected. These receipts are in
evidence/local/. evidence/FRAME-LIFTS.json and evidence/FRAME-SEARCH.json record the lifts and the search objective.

Scope

A finite conditional witness under the same inherited interfaces as #531 (all-size compiler, common-label/mutable-basis realisation,
normalised Lemma A, restored-row, precision/tape and analytic transfer). The #493 bit operator is cited, not rebuilt. The search is a
bounded heuristic, not an optimality claim. All credit for the 504-helper construction and its proofs belongs to #531 and its
ancestry listed in NOTICE.md and SOURCES.json; the parent README is preserved as PR531-README.md.

Reproduce: cd research/frame-sa504 && python -B verify_fast.py --work /abs/new/dir (Python 3.11+, NumPy for the fresh phase edges; about 24 s). Passed here on Python 3.13 / NumPy 2.5.

🤖 Generated with Claude Code

https://claude.ai/code/session_01TGoMCHAqctZe4wTb4Qr7VV

[Mister-Meeseeks]

Integrated on main in 9449300, with original commit history and individual credit retained.

Your delivery-frame search and frozen witness are integrated. The constructor/frame-lift checks, changed phases, physical owners and exact assembly reproduce κ = 0.001526158885117467. Your frame mechanism also composes with #533 in #536; the bounded search is not represented as an unrestricted optimum.

Maintainer review and exact scope · Contributor record. The multiplication claim remains conditional on the inherited all-size/realization/analytic interfaces; these checks do not establish a complete unconditional theorem.

Thank you for contributing this work.