Research: compose R420 bases for 8,368 fewer XORs

[swbiggart]

This finite bit-circuit construction combines Greer Guthrie's #694 helper basis with a fresh target matching on its transformed matrix. It saves 8,368 native XORs overall: 26,916 → 18,548, including 1,434 beyond #694. All preparation, helper reads and restoration are counted. No new κ is claimed.

The construction fixes #694's 201 helper pairs, then scores all 14,280 target orientations on KQ. One seed-2 greedy disjoint pass selects 57 target pairs. D costs 201 + 57 + 11,038 + 57 + 201 = 11,554 ADDs. The complete MOVE/COPY/ERASE subsequence, 420 helpers, nine copies and paid rank mass 5,112 are unchanged.

The combined middle matrix must be PKQ. This contribution checks the cross term (P-I)K(Q-I) and rejects its omission. Standalone savings cannot simply be added. The reverse positive prefix rises to 4,152,374; the complete local payload bound is 103 bits, below the 104-bit cap. Global precision invoices are not inherited.

The original literal H2 and K extractor still reject the new syntax. An explicit conjugation proof and complete arbitrary-input prefix check establish the semantic replacement, without patching predecessor checkers.

python3 -B research/r420-two-sided-basis/verify.py
# With C++17 and Boost:
python3 -B research/r420-two-sided-basis/native.py

Expected: PASS_R420_TWO_SIDED_BASIS native_ADDs=26916->18548 saved=8368 beyond_PR694=1434 payload_bits=103 kappa=unchanged.

Validation:

• Focused Linux CI passed on Python 3.11 and 3.13 at dfe84e09298abc5c5e0074afda59a821e10abfb4, including fresh C++ multistage verification on 3.13 and clean-tree checks.
• Exact certificates and four obligation tests pass on Python 3.9, 3.11 and 3.13. A direct four-term formula checks all 50,400 middle-matrix coefficients; an independent incidence census checks every target-pair score.
• Original, reconstructed Research: save 6,934 R420 XORs with a restored helper basis #694 and combined words pass all 660 dirty-input columns in both directions, exact geometry and official export comparison. Reconstructed Research: save 6,934 R420 XORs with a restored helper basis #694 matches its author's native record hash and prefix certificate exactly.
• Fresh unchanged C++ legality and 5/9/11-stage checks pass for original and combined words: 900/1,380/1,620 columns, 96 omitted-bridge controls and six omitted-D controls total.
• Ten local rejection controls, relocated minimal/recompressed-source replay, corrupt predecessor/certificate rejection, optimized-mode refusal, selected-record integrity and staged source-blob/whitespace checks pass.

Base: 66849e1475c930fd7732b1a0067ffee7e1e95f60. Direct pins: #694 9732ac127dd4bef98a846aaa904c20b5a84e72c1, #713 1a19af5c5ddb6de9797a6d085e5df0b90ed6d58b, and selected PR543 2065fe4f72a914d907f1046603f5a36f17018535. Source inputs/notices are preserved byte for byte. #686, concurrent corroboration #716, and historical #476/#450 target factoring are credited. This reuses pinned #713 verification utilities and is not a wholly independent implementation.

Only the new package and focused workflow are added. No pinned coordination issue existed; the completed seed-2 claim is local. The final overlap refresh through #730 found no competing two-sided composition. No optimality is claimed.

Full hosted-parent emission, whole-owner allocation, revised global invoices, outer assembly and semantic-interface acceptance remain integration obligations. All inherited compiler/common-label/mutable-basis/physical/hosting/precision/tape/analytic assumptions remain. No repository-wide suite pass, full theorem or measured runtime improvement is claimed. upstream/ and the selected frontier are unchanged.

Prepared by Scott Woolsey Biggart (@swbiggart) with substantial OpenAI Codex assistance, under Apache-2.0. The predecessor mechanisms remain credited to their authors; this contribution is their checked two-sided composition. No independent human review is claimed.

[Mister-Meeseeks]

100-PR batch review at dfe84e09298abc5c5e0074afda59a821e10abfb4.

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

Reviewed scope: Joint helper/target basis rewrite: full cross-term compensation and all dirty columns checked; 8,368 native XOR saving, semantic D extension still required.

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.