Research: save 114 R420 XORs by sharing restored decoder operands (no new kappa)

[thakrarsagar]

This finite bit-circuit construction reduces PR543 R420's nine phase-A decoder totals from 321 to 264 binary XORs, including all temporary preparation and restoration. Literal reverse cleanup doubles the saving to 114 native ADDs: 26,916 to 26,802. Status: research construction / finite circuit improvement; no new kappa.

The recipe shares 27 disjoint non-retained operand pairs at FULL. It preserves the complete phase-A operator on arbitrary dirty inputs, every register's frame path, 420 helpers, nine copies, rank mass 5,112, and the original H1-H5 interface. The scalar positive bounds increase, so they are recomputed: forward 144,651 to 193,441; reverse 2,270,064 to 2,702,178. Native nine-stage payload rises from 100 to 102 bits, within the 104-bit budget.

Base: 66849e1475c930fd7732b1a0067ffee7e1e95f60. PR543 dependency: 2065fe4f72a914d907f1046603f5a36f17018535. Only research/r420-decoder-sharing/ is added. Source pins, a frozen recipe, exact certificate, independent interpreter, optional native checker driver, written proof and validation receipt are included.

python3 -B research/r420-decoder-sharing/verify.py

Expected: PASS_R420_DECODER_SHARING decoder=321->264 native_ADDs=26916->26802 saved=114 columns=660 controls=7 no_new_kappa.

Validation passed locally:

• Unchanged PR543 label/frame/interface check and all 660 symbolic dirty-input columns, corroborated by a separate set-valued interpreter.
• Unchanged C++ legality and nine-stage replay for both words: 1,380 physical columns and 18/18 omitted-bridge controls each. Actual exported records decrease 29,874 to 29,760.
• Seven circuit/interface negative controls, explicit optimized-Python refusal, and canonical native-input binding control.
• Complete exact-certificate comparison, relocated minimal Git archive replay, selected-record binding, source pins and staged whitespace check.

Assignment seed 15, bit-supplier track. All 6,320 oriented pairs on 80 operands were scored; one seeded greedy disjoint matching was selected. No optimality is claimed. No pinned coordination issue existed, so the claim is local. Final overlap refresh through #704 found no duplicate decoder-total circuit. Concurrent #686/#694 are credited for restored-pair sharing in frame-zero compensation; this applies the mechanism to the separate FULL-frame totals while preserving H2. #670/#674/#688 change phase-B blocks, and #691 changes a phase-A quad. None is imported or composed here.

The full hosted-parent construction, global prefix invoice and changed-word outer assembly remain integration obligations. Repository-wide make verify, Lean and remote CI were not run or claimed. All inherited compiler/common-label/mutable-basis/hosting/precision/tape/residual/analytic assumptions remain. No full multiplication theorem or practical runtime speedup is asserted.

Prepared for Sagar Thakrar (@thakrarsagar) with substantial OpenAI Codex assistance, including separate workers for independent checking and native replay, under Apache-2.0. Credit remains with eumemic's PR543/493, Dylan Crow, DreamingOfClouds, Zhihao Chen and the complete predecessor lineage identified in the package README. No independent human review is claimed.