Finite bit construction: coupled R439 helper triples save 140 further ADDs (no new kappa)

[RossRichardson]

Finite bit-circuit construction: replace 40 of PR926's disjoint helper pairs plus 40 unused helpers by coupled three-helper bases. The complete R439 word falls from 11,980 to 11,840 ADDs, saving 140 after charging every preparation and restoration. No new kappa. The selected PR890 result is unchanged.

The complete D bill is 234 + 306 + 4,540 + 306 + 234 = 5,620 ADDs. All non-D fields, every register's MOVE path, all COPY/ERASE records, 439 helpers, nine copied totals and rank mass 5,129 remain unchanged. The semantic endpoint holds for arbitrary target and helper dirt. F = 72,399 and B = 10,464,656 pass the exact local 104-bit and nine-stage 192-bit inequalities; the latter uses about 99.6193% of its ceiling.

This extends PR915/926's fixed-target compensation beyond disjoint helper pairs. Seed 12 exhausts 192 pairs × 55 unmatched helpers × 168 GL(3,2) matrices = 1,774,080 scores, using one shortest CNOT representative per matrix, then one guarded greedy pass. Of 935 strict-gain nominees, 40 are accepted, 115 fail a guard and 780 conflict. This is complete coverage of the declared domain and representatives, not a global circuit optimum. Concurrent PR941's reported 11,948-ADD disjoint-pair floor excludes these overlapping bases; its matching proof is credited as context, not independently certified here. PR952 concerns phase A and is not composed.

Reproduce from the repository root (Python 3.12+, C++17 and Boost):

python3 -I -B research/r439-helper-triples/verify.py

Expected: PASS_R439_HELPER_TRIPLES_FINITE: 11,980 -> 11,840 ADDs; 40 helper triples; no new kappa. Set BOOST_INCLUDE for a nonstandard Boost installation.

Validation passed on CPython 3.14.6 / macOS arm64 / clang 17 / Boost 1.87:

• Complete certificate and canonical word regeneration; full expected receipt equality in a fresh ordinary verification run.
• Independent direct group enumeration, Gaussian inverse, set-valued census, 1,008 shortest-distance edges, and 3,688,220 literal positive instructions.
• 661 separate arbitrary-input D columns in both directions for four words; unchanged owning E8/frame/endpoint checks.
• Fresh unchanged native legality and 5/9/11-stage checks for PR589, PR915 and the candidate; 883/1,327/1,549 columns per word; 144 omitted-bridge controls.
• Ten local control categories, a correctly charged cancelling pair, and make selected-record-check.

Scope: original syntactic H2 and literal K_equals_D remain false, as in PR915/926. Full hosted emission/audit, complete precision invoices, semantic supplier admission and outer assembly remain open. Inherited compiler, common-label/mutable-source, hosting, precision/tape and analytic assumptions remain. No full make verify, cold supplier reconstruction, Lean multiplication theorem or remote CI pass is claimed.

Exact base: 0076643. Only research/r439-helper-triples/ is added; upstream/ and the selected frontier are unchanged. Source/commit pins, proof, complete receipts, stopping rule, local claim and validation record are included. No sprint coordination issue was pinned. The open queue was refreshed through #962 and an exact helper-triples search found no overlap.

Credit Gareth Davidson's PR915/926, Scott Woolsey Biggart's PR733, DreamingOfClouds' PR589, eumemic's PR555/543, Sagar Thakrar and the retained circuit/native-checker lineage. Notices and licenses are preserved. Prepared by Ross Richardson (@RossRichardson) with substantial OpenAI Codex assistance in research, implementation, verification and writing; no independent human review claimed.

Published head: 6292e37. The full implementation and normal replay are at 82f2b20; the final commit only records the result/link in CLAIM.json. Final source pins, selected-record verification, clean-worktree and whitespace checks pass. GitHub checks were queued at 2026-10-13 15:05 UTC; no hosted CI success is claimed.