Finite bit construction: phase-B block programs on the R439 helper-star D, 11,783 -> 11,500 ADDs (no new kappa)

[thakrarsagar]

Status: finite bit-circuit construction; no new κ. The selected result, the selected bit supplier and upstream/ are unchanged; the bit side does not bind.

Track 00–19 (bit supplier), OS-random seed 1, base 2921874911956718fcaaaf9396fe8267b3cfa9f0. Only research/r439-bsection-stars-seed01/ is added.

What changed

The incorporated helper-star word (PR967, research/r439-helper-stars-seed19, 11,783 ADDs) rebuilt D only and kept PR589's phase-B blocks. The shortest recorded block programs (#1015, #923) were built on the 24k-ADD PR589 word. #1015 and the batch-952 review both say they were not composed with the D-substitution line. This PR composes them:

complete local word	ADDs	D	F	B	local / nine-stage bits (caps 104/192)
helper stars (main)	11,783	5,563	72,847	10,394,084	102 / 192
+ ten spliced phase-B blocks	11,515	5,563	24,625	5,846,566	95 / 181
+ unchanged seed-19 star pass, rerun	11,500	5,548	25,717	5,911,840	95 / 182
pair control (rerun)	11,715	5,763	24,563	5,892,602	95 / 181

1. Composition (−268 ADDs). Nine Finite bit construction: multi-write shear programs for all ten R439 phase-B blocks (24,305 -> 24,053 ADDs, no new kappa) #1015 multi-write programs plus Finite bit construction: R439 frame520 uses 29 ADDs, four fewer complete ADDs than PR844 (no new kappa) #923's frame-520 program replace all ten blocks (388 → 254 gates), and C is rebuilt by the literal-cleanup rule. Each replacement has the original F₂ block map on all 2^n inputs, an equal integer transfer and a smaller reverse transfer.
2. Freed guard (−15 more). The helper-star word sat at bit-length 192 under the 2^192 nine-stage cap, and its seed-19 pass rejected 24 nominees on that guard. The spliced blocks lower both positive bounds. The unchanged builder, called as a library and rerun on the same PR915 parent with the same blocks, then makes 0 guard rejections and admits 236 edges instead of 221.

For both words, PR589's A, scatter, copied totals, register frame paths, nine COPY/ERASE pairs and paid profile (rank mass 5,129) are unchanged. The pinned owner checker reports the same H1–H5 pattern as the incorporated helper-star word: only its known syntactic-H2 and K_equals_D flags are false.

Reproduce

python3 -I -B research/r439-bsection-stars-seed01/verify.py
python3 -I -O -B research/r439-bsection-stars-seed01/verify.py

Expected (about 20 s each, stdlib only):

PASS_R439_BSECTION_STARS blocks=10 gates=388->254 ADDs=11783->11515->11500 guard_rejections=24->0 columns=661 controls=8 no_new_kappa

The verifier rebuilds all payloads and search receipts byte-for-byte. It replays all 661 formal columns through the literal COPY/ERASE stream in both directions for four words and checks the D dirty boundary against PR589. It runs the pinned xcheck_p.py and recomputes the profile. It bills the pinned export_records.py records, whose bounds equal the literal-stream bounds, and checks both guards. It also confirms that the original pass still gives 11,783 with 24 rejections.

Eight controls are rejected: mutated block source, dropped block gate, block gate at frame 0, stale cleanup (F₂-exact but rejected by the literal-cleanup rule), omitted cleanup gate, omitted star restoration, a 105-bit local bound, and the 236-edge D without the spliced blocks, which fails the nine-stage guard. That last control shows the extra stars depend on the composition. The pushed branch was also replayed from a clean sparse clone.

Scope

• No κ change, and no proposal to adopt this as the supplier. Hosted invoice, binding and outer assembly were not run.
• The native 5/9/11-stage C++/Boost replay was not run (no Boost headers here).
• Block programs are best-found results, not minima; the star pass is one deterministic greedy pass.
• Not composed: Local bit construction: target retuning after helper mixing saves three paid R439 ADDs #1040 (−3) and Local bit construction: save 14 ADDs with one restored post-PR967 helper shear #1047 (−14) on the 11,783 word, and the Finite bit construction: certify R439 matching floor and save 31 further ADDs #941/Finite bit construction: exact R439 four-helper fusion below the pair floor #1045 helper-matching line.

Overlap and credit

Before submission I checked open PRs through #1060. #1015 and #989 compose blocks only on the 24k PR589 word. #1040 and #1047 are single D edits. None combines phase-B blocks with the helper-star D.

Credit: block programs by Aaron Wang (#1015, open; inputs/PR1015-CANDIDATE.json copied unchanged from 7588c797) and Ross Richardson (#923, frame 520). The helper-star builder/D is from PR967 (prepared for @theon199), generalizing Gareth Davidson's #926 on the #915 target basis. The underlying word is DreamingOfClouds' #589 on eumemic's #555, with the retained checker lineage.

Prepared by Sagar Thakrar with substantial assistance from Anthropic Claude (Claude Code).