Finite bit construction: R439 frame 506 uses 21 ADDs (no new kappa)

[ToikY]

Finite bit-supplier construction: reschedule PR589 R439's frame506 B[102:128) as two exact positive-integer factors with 9+12=21 rather than 26 ADDs. Rebuild required literal full-frame cleanup; the complete local bill is 24,305→24,295 ADDs. No new κ is claimed.

The new evidence is a witness outside Aaron Wang's merged #891 whole-block destination-once class (minimum 24 at 506). Splitting at original gate 114 permits intermediate rows and a second destination write across factors. Discovery exhausts all 25 original cuts103–127; the minimum 21 occurs at114–116. A separate disjoint-donor-subset/backward Bellman implementation confirms the 9/12 minima without importing discovery. This is a restricted-class minimum, not unrestricted circuit optimality.

Validation passed:

• Exact integer endpoints and signed inverses on all ten dirty basis columns; all 1024 local Boolean inputs; six independent arithmetic mutations.
• Pinned owning checker on all 661 dirty columns, H1–H5, complete frame paths, nine COPY/ERASE pairs and unchanged paid profile/rank mass 5129.
• Independently emitted full records equal the pinned exporter byte for byte; six full-word/invoice mutations reject. Forward positive bound 49347; reverse 1113070→1107058; payload 93 bits below 104.
• Unchanged C++17/Boost native legality and 5/9/11-stage replay on both words, 883/1327/1549 formal columns,96 rejected omitted-bridge controls; full local/stage bills and signed prefixes.
• Four packaging tests (optimized/environment replay, minimal relocation, corrupt-source certificate preservation, factor/word binding), regenerated search/certificate checks, selected-record integrity and staged whitespace.

Reproduce with standard-library Python 3.11+:

python3 -I -B research/r439-frame506-two-factor/search.py --check && python3 -I -B research/r439-frame506-two-factor/verify.py

Expected: PASS_R439_FRAME506 block=26->21 ADDs=24305->24295 columns=661 inputs=1024 controls=6 no_new_kappa. Native replay command and exact source/toolchain pins are in the package README; exact receipts, coverage, negative controls and proof are committed.

Research base 407d52f, OS assignment seed 18, bit-supplier track. Selected PR942/T37 remains conditional κ 0.001661960547713947, with unchanged PR589 supplier head a5afc8c. Refreshed main 8bddbee leaves all 18 consumed/context source pins unchanged. Only research/r439-frame506-two-factor/ is added; upstream and selected frontier remain unchanged. No pinned coordination issue was found, so the narrow claim is local.

Overlap: #914 (Ross Richardson) supplies the adapted integer solver/verification pattern on frame524; #891's506 component is improved by three B ADDs while its523 result is not composed. Concurrent #980 changes A and leaves B unchanged. #989 composes 507/508/519/524 and excludes 506. Their savings are not summed here. This is not the smallest known full R439 circuit or a mechanism-priority claim.

Credit eumemic #555/#543, DreamingOfClouds #589/#545 and lineage, Dylan Crow, DaysSky, Zhihao Chen, Aaron Wang #891, Ross Richardson #908/#914, Ali Romman/Alluri native harness ancestry and the selected complex predecessors. Prepared for Eva Ayala (@ToikY) with substantial OpenAI Codex assistance, including one scoped AI worker for independent arithmetic/review. Apache-2.0; no independent human review is implied.

The all-size/compiler, common-label/mutable-source realization, whole-call hosting, precision/tape and analytic obligations remain inherited. No full global hosted invoice, selected cold replay, outer assembly, full-theorem formalization, measured speedup or remote CI pass is claimed. The paid recursive profile is unchanged and complex still binds. Please review as a finite construction, not a new conditional bound. Do not merge automatically.