Research: exact 3->2 z-program rewrites of the PR589 word, 24,305 -> 23,495 ADDs (no new kappa)

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Status: finite bit construction / research note (no new kappa). Assignment seed 13, bit-supplier track. Base commit abdc6a7002abdbcd9e21bfc5e1a795784deacfdf. Prepared with help from opencode (substantial AI assistance, disclosed per CONTRIBUTING.md).

What this is

405 exact single-edge triangle (Steinberg 3->2) rewrites of the z-program of the selected PR 589 word (research/e8-bit-p111-tuned/tune/bitE8mut-p111-tuned.json.gz, DreamingOfClouds on eumemic's PR 555): the A/B z-destination gates and their literal-reverse cleanup mirrors. One forward source edge is deleted per rewrite and two kept edges swap order; the rebuilt cleanup loses its mirror edge.

complete local quantity	base	candidate	change
complete exported ADDs	24,305	23,495	-810
exported records	27,234	26,424	-810
MOVEs / COPYs / ERASEs	2,911 / 9 / 9	unchanged	0
661 register frame paths	reference	identical	unchanged
paid profile	reference	identical	unchanged
z-program product U (550x550 over F2)	reference	identical	unchanged
F2 endpoint, all 661 basis columns	pass	pass	unchanged
kappa	unchanged	unchanged	no new kappa

Every accepted rewrite is gated on an exact replay of the F2 endpoint on all 661 columns, identical frame paths and an identical paid profile, so the frozen hosting and the whole pricing chain carry over exactly as in the prior phase-B block packages. This is a finite scalar circuit-size result only.

Method and coverage

• The identity is exact over F2: [b += x] [b += a] [a += x] == [a += x] [b += a] (and the mirror order). A candidate is accepted only when every gate strictly between the three edges commutes with all three at the register level, so no observable intermediate state moves.
• Complete base census: 2,606 distinct triangles / 2,613 instances in the z-stream; 293 base instances have a fully commuting window. Greedy application in deterministic order to a fixed point: 7 rounds, 405 accepted, zero endpoint rejections (rounds 2-7 unlock new instances as windows open). Stopping rule: a full rescan with zero accepted rewrites. The greedy order is not an optimum over orders.
• Paired-transvection control class exhausted: only 17 duplicate edges exist in the whole z-program and every pair is dependency-blocked.
• Structural note: rank(U - I) = 539 over F2 (rank(U_A - I) = 439), so any forward program and any exact cleanup need at least 539 transvections each against 2,224 each after these rewrites. The literal-reverse cleanup is not claimed optimal; that gap is recorded as headroom, not closed here.

Reproduce

python3 -I -B research/r439-z3to2-seed13/verify.py

Expected (standard library only, CPython 3.11+):

PASS_R439_Z3TO2 moves=405 ADDs=24305->23495 records=27234->26424 columns=661 paths=661 profile=identical U=identical controls=6/6 exporter=ok no_new_kappa

verify.py imports no search code: it re-applies the witness moves from the pinned base word (canonical SHA-256 5642fd0e...eba0b), re-checks each move's three-register identity and window commutation at its application state, compares the rebuilt word with the committed candidate (canonical SHA-256 8aedb3be...539a), replays base and candidate independently on all 661 columns, compares U, all frame paths and the paid profile, recomputes the full ADD/MOVE/COPY/ERASE invoice and cross-checks it against the repository exporter export_records.py run on both words, rejects six tamper controls (source corruption, deleted cleanup gate, truncated cleanup, tampered move list, non-commuting swap without cleanup rebuild, truncated dirt gate), and compares the summary with CERTIFICATE.json byte for byte. search.py regenerates the witness deterministically (same fixed point, byte-identical outputs).

Scope

Finite replays of one word in one rewrite class; not an optimality proof and not a Lean multiplication theorem. H1-H5 of the interface lemma hold literally (D untouched, scatter unchanged, C the literal reverse at the full frame, V = I). Native 5/9/11-stage C++/Boost checkers were not run here (no such toolchain in this environment); this is an exact finite word-level certificate in the sense of PR 1040/PR 1047's semantic candidates, not a native hosted supplier admission. No new kappa and no new bit headroom (the profile is exactly unchanged and the bit side does not bind).

Credit and positioning

Predecessors (not superseded): PR 589 (DreamingOfClouds) / PR 555 (eumemic) (the word), PR 541/PR 548 (Dugongue's transvection move families) and PR 589's tuner, which optimizes the moment to a fixed point and records that its Steinberg step has "nothing to apply that lowers the moment"; equal-moment rewrites are outside its greedy objective, and this package's delta is the ADD-count objective with exact commutation windows. The phase-B block programs (PR873/PR 973 (Cinna Davis)/PR 986 (woklerx)/PR 1001 (ToikY)/PR 1002 (woklerx)/PR 989 (woklerx)/PR 980 (Gareth Davidson), r439-bsection-survey, r439-frame524-seed13, concurrent #1015/PR 1045 (Sagar Thakrar)/PR 1062 (Sagar Thakrar)) are orthogonal regions measured against the same base word; results are not additive across PRs. PR 933 and PR 1048 close moment/reframing classes; this package's rewrites keep the paid profile exactly identical, so they neither raise bit c nor contradict those claims. Concurrent PR 1122 (same track and a coincidentally identical assignment seed, different agent) composes the frozen PR 980/PR 1062 changes on the semantic-D word lineage (11,500 -> 10,652 ADDs there); its object, word and mechanism are different, its results are not additive to this one, and the 3->2 rewrites here are candidates for that lineage as future work. See research/r439-z3to2-seed13/NOTICE.md and CLAIM.md (local sprint claim; no pinned coordination issue exists).

[Mister-Meeseeks]

100-PR batch review at 2e9dc9d3356a6b7258ed869ddcee46352907f746.

Merged to main, preserving the original contributor commits and package attribution.

Reviewed scope: 405 exact commuting-window z-program rewrites preserve all dirty inputs, frame paths and paid profile; gains are specific to the frozen PR589 word.

The pinned scoped command passed; the batch ledger contains its command, input hashes, environment and log. This incorporates the research/tooling contribution and does not change the selected κ or discharge the inherited multiplication-theorem assumptions.

Batch review, attribution and evidence. Thank you for the contribution.