Research: optimal disjoint R420 target pairs save 3,250 XORs (no new kappa)

[thakrarsagar]

This research construction with a restricted optimality certificate saves 3,250 binary XOR additions in eumemic's PR543 R420 bit word: 26,916 → 23,666 overall, with frame-zero D 19,922 → 16,672. It uses 60 disjoint target pairs and includes every preparation and restoration operation. No new kappa is claimed.

A small independent certificate proves 3,250 is the exact optimum within the disjoint target-pair substitution class. Its 120 nonnegative doubled vertex prices satisfy q_i+q_j >= 2*w_ij for all 7,140 unordered pairs and sum to 6,500. Every disjoint pairing therefore saves at most 3,250. The frozen 60-pair circuit attains that bound. The shipped standard-library checker verifies the inequalities and construction without a matching solver.

Concurrent #713, by Scott Woolsey Biggart, is credited. It appeared during the final refresh and independently achieved the original 3,150-addition greedy saving. This final submission adds 100 saved additions and the exact upper-bound certificate. No priority for target-row factoring is claimed. The result concerns this four-operation pair identity only, not arbitrary D circuits.

Every helper remains untouched during D. The complete dirty-input operator, physical frame paths, 420 helpers, nine copies and rank mass 5,112 are preserved. The reverse positive-prefix bound rises from 2,270,064 to 3,703,810; the local payload allowance rises from 100 to 102 bits, below the inherited 104-bit cap. No hosted invoice is reused.

The original H2 syntax and literal K extractor reject this word as expected. The package records both rejections and proves the semantic replacement: D executes at frame zero and has exactly (x,s,y) -> (x,s,y+Ks). Predecessor checkers are unchanged.

python3 -B research/r420-target-pairs/verify.py
# Optional unchanged native cohort checks, with C++17 and Boost headers:
BOOST_INCLUDE=/path/containing/boost python3 -B research/r420-target-pairs/verify.py --native

Expected: PASS_R420_TARGET_PAIRS D=19922->16672 ADDs=26916->23666 saved=3250 columns=660 rank_mass=5112.

Validation on macOS arm64, Python 3.14.8, Apple Clang 21.0.0 and Boost 1.92.0:

• Exact source pins, 660 arbitrary-input columns, complete frame paths, paid costs and independent positive-prefix calculations.
• Standard and isolated optimized Python replay; exact frozen matching/dual/word certificate.
• Fresh unchanged native legality and 5/9/11-stage checks on 900/1,380/1,620 formal columns, with 6/18/24 omitted-bridge controls, for both source and final word. Records fall from 29,874 to 26,624.
• Nine local circuit rejection controls, an underpriced-dual rejection control and a paid neutral-pair positive control. Two native corruptions pass legality but fail every stage's formal-column check.
• Independent endpoint/frame/dual checks, gzip recompression acceptance, altered-payload rejection, selected-record integrity, documentation links and whitespace checks.

Base: 66849e1475c930fd7732b1a0067ffee7e1e95f60. PR543 source: 2065fe4f72a914d907f1046603f5a36f17018535. Assignment seed 13. All 14,280 orientations were scored. Following the late #713 overlap, one maximum-weight matching nomination replaced the greedy pass; an independently derived integer assignment dual closes the whole stated class. The accepted frontier at this base remains #626 with #543, conditional kappa 0.001635527380375392.

Dugongue's #476 supplies the reversible target-factoring precedent; #450 is earlier target mixing. Concurrent #686/#694 factor helper columns and motivate the semantic D contract. This target-row package imports none of their code. The pre-submission overlap refresh covered 534 open PRs and found #713; no pinned coordination issue existed, so the local claim records the result and this PR.

Only research/r420-target-pairs/ is added. Linux execution, hosted-parent reconstruction, whole-owner/finite invoices, acceptance of semantic H2' and changed-word outer assembly remain open. The all-size/common-label/mutable-basis, precision/tape and analytic assumptions are inherited. Repository-wide regeneration and the unchanged selected cold replay were not run. This is finite evidence and a scoped matching bound, not a formal proof of integer multiplication or a practical runtime claim.

Contributor: Sagar Thakrar, with substantial OpenAI Codex assistance. New material follows Apache-2.0 and retains predecessor credit.

[Mister-Meeseeks]

100-PR batch review at afeb452f58ff0cbf109172335e2ba282aac3dcde.

Merged to main, preserving the original contributor commits and package attribution.

Reviewed scope: Disjoint target pairs save 3,250 XORs; exact matching dual, complete F2 endpoint and paid paths. Generalized semantic D interface, not literal H2 admission.

The pinned scoped command passed; the batch ledger contains its command, input hashes, environment and log. This incorporates the research/tooling contribution and does not change the selected κ or discharge the inherited multiplication-theorem assumptions.

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