Research construction: shorten R439 block 4 from 41 to 31 XORs (no new kappa)

[gautamgpt1]

Finite bit-circuit construction; no new kappa. This replaces PR589's R439 phase-B block 4 (frame 520, B[242:283]) with 31 additions instead of 41. Rebuilding its required literal reverse cleanup saves ten more: complete local additions fall 24,305 -> 24,285. Only research/r439-block4-schedule/ is added.

The forward integer maps of both fixed factors are identical. The reverse positive matrix is entrywise dominated (33 entries decrease), all 661 register frame paths agree, and the local paid histogram and rank mass 5,129 are unchanged. No helper, copy, frame transition or recursive child is added. The reverse prefix falls 1,113,070 -> 1,084,714; the local payload remains 93 bits under the 104-bit cap.

The frozen circuit attains exact 17 + 14 minima within the two fixed factors' destination-once, positive-addition class. An independent verifier exhausts 8,192 states, 49,152 state/destination pairs and 39,168 feasible edges using integer subset sums and backward Bellman recursion. This is not unrestricted XOR optimality or a formal multiplication theorem.

Research note, proof and accounting · Executed validation

python -I -B research/r439-block4-schedule/verify.py

Expected:

PASS_R439_BLOCK4_SCHEDULE gates=41->31 ADDs=24305->24285 columns=661 factor_minima=17+14 controls=13 rank_mass=5129 kappa=not_recomputed

Validation passed on Windows x64 / Python 3.13.3:

• Normal, optimized and isolated minimal-input replays; byte-identical deterministic candidate regeneration. Thirteen built-in controls plus four external source/certificate/coverage mutations are rejected.
• Unchanged native C++ legality and five/nine/eleven-stage checks on both words: 883 / 1,327 / 1,549 columns, with all 6 / 18 / 24 omitted-bridge controls per word rejected. Independent and owning native record emitters agree byte for byte. The complete native receipts are retained, excluding only timing and the local stream pathname.
• Selected-record integrity at the experimental base and staged whitespace/source-preservation checks.

make verify could not start because GNU Make is not installed. No full-suite, selected cold replay, global hosted/invoice/outer assembly, Lean or remote CI pass is claimed. All-size/compiler, common-label/mutable-source, hosting, precision/tape, signed decoder and analytic assumptions remain inherited. No composition with another pending circuit PR is certified.

Base: 7ee02b7c96d85f22b38d28bd7ed49bb8c1aeb261. Reviewed supplier PR589: a5afc8c6169ea61ba7c51dfd3cade02cc7b13b46; nine consumed files have exact SHA-256 pins. Main advanced to f567884aeb8851b6e794af1184119a3bfbe1e631 (#799/#589, kappa 0.001661634653669567); its new review was read and all consumed bytes are unchanged. The experimental base is preserved; upstream/ and the selected frontier are untouched.

OS-random assignment seed 04, bit-supplier track, selects block 4. Three candidate tasks, the local scope claim, and its result update are included. There was no pinned coordination issue. The initial 2,000-trial randomized search found no reverse-dominated candidate (20 identity factorizations); one pivot to exhaustive fixed-factor scheduling produced this witness. This failed search is not an impossibility theorem.

Overlap was refreshed immediately before submission. No R439 block-4 change was found. Sagar Thakrar's #728 / Ryan Haines' #792 change R439 phase-A mini-nodes. Yuzhou Wang's #789, commdata2338's #790, and Jerry Lin's #810 provide related methods on distinct historical R420 blocks and are credited; no pending implementation is imported. #661's fixed-order frame class and #827's cleanup-frame lower bound do not exclude this scalar gate-list change.

Credit eumemic (#543/#555), DreamingOfClouds (#545/#589, #503/#510), Dylan Crow, DaysSky, and the retained construction/checker lineages. Prepared by Gautam Gupta (@gautamgpt1) with substantial OpenAI Codex assistance in analysis, code, verification and writing, under Apache-2.0. No independent human review is claimed.