Certify fixed-basis R420 target matching and the eight-ADD gap

[thakrarsagar]

This contribution certifies the optimal disjoint target pairing for PR #694's fixed helper basis and applies it to PR #681's R420 construction.

Results:

• Complete local cost: 18,362 ADDs
• Published Consolidate six circuit PRs: 31.72% fewer bit ADDs and 22–25% fewer complex primitives #681: 18,378 ADDs
• Existing Research: compose R420 bases for 8,368 fewer XORs #733 matching on the same Consolidate six circuit PRs: 31.72% fewer bit ADDs and 22–25% fewer complex primitives #681 body: 18,370 ADDs
• New incremental saving: eight ADDs beyond that existing recipe

A 58-pair witness and 120 integer vertex prices certify the exact saving of 1,442 within this fixed-basis pairing class. All 7,140 edge inequalities and equality of the primal/dual totals are checked without relying on a numerical solver.

Recorded validation:

• Exact reconstruction of the Consolidate six circuit PRs: 31.72% fewer bit ADDs and 22–25% fewer complex primitives #681 predecessor receipt
• Forward/reverse checks on all 660 inputs and restoration controls
• Separate matrix-based verification and certificate mutation tests
• Fresh native 5/9/11-stage and hosted replays comparing published Consolidate six circuit PRs: 31.72% fewer bit ADDs and 22–25% fewer complex primitives #681 with this replacement
• Relocation and corruption checks

The #733 matching on the same #681 body is checked locally; a separate native/hosted replay for that comparator is not claimed.

Reproduce from the repository root:

python3 -I -B research/r420-optimal-composed-pairs/verify.py
python3 -I -B research/r420-optimal-composed-pairs/independent_check.py

Local precision remains 104 bits. Hosted precision remains 196 bits charged at 256; two-level precision remains 432 charged at 512. This establishes neither a new kappa nor global circuit optimality. All inherited assumptions remain; the selected frontier and upstream/ are unchanged.

See research/r420-optimal-composed-pairs/README.md and research/r420-optimal-composed-pairs/VALIDATION.md for pinned sources, predecessor credits, complete checks and limitations.

Prepared for Sagar Thakrar with substantial OpenAI Codex assistance, under Apache-2.0.

[Mister-Meeseeks]

100-PR batch review at 3147cac35c659152d36db1e0e60b9ce4b152c008.

This remains open after scoped triage. Next review step: Local matching certificate plus hosted/native evidence: source-bound native/cohort coverage needs a dedicated continuation with Boost; do not infer hosted admission from the local pairing proof.

This is a specific continuation point, not a rejection or a request to redo already recorded checks. Completed local evidence is retained separately from outstanding whole-head obligations.

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