Research: restored decoder stars save 128 R420 XORs (no new kappa)

[thakrarsagar]

This finite bit-circuit construction replaces disjoint pair sharing with eleven restored helper stars in the nine FULL-frame decoder totals of eumemic's selected PR543 R420 word. The decoder costs 321 -> 257 binary XORs, including all preparation/restoration; literal reverse cleanup doubles the saving to 128 native ADDs: 26,916 -> 26,788. No new kappa is claimed.

The complete phase-A operator on arbitrary dirty inputs is unchanged. D compensation, every register frame path, 420 helpers, nine copies, rank mass 5,112 and the original H1-H5 interface are preserved. Positive growth is recomputed: native nine-stage payload rises from 100 to 103 bits, still strictly below 104.

PR708 at 2ec522a466ec448f148d4355dc43b51c0ef1e0b5 is the immediate predecessor. This saves 14 additional native ADDs over its frozen pair recipe. The three-register witness [515,489,347] saves four ADDs with a star; any disjoint-pair recipe restricted to those registers saves at most two. This distinguishes the search class without claiming global optimality.

Base: 66849e1475c930fd7732b1a0067ffee7e1e95f60. Reviewed PR543 source: 2065fe4f72a914d907f1046603f5a36f17018535. Only research/r420-decoder-bundles/ is added. Source pins, exact recipe/certificate, proof, independent interpreter, native runner, receipts and attribution are included.

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

Expected: PASS_R420_DECODER_BUNDLES decoder=321->257 native_ADDs=26916->26788 saved=128 columns=660 controls=8 no_new_kappa.

Validation passed locally on macOS with Python 3.14.8, GCC 16.2.0 and Boost 1.85.0:

• Original E8/frame/interface checker and all 660 arbitrary-input columns; separate set-valued replay of the entire A operator, complete word and inverse, frame paths and paid invoice.
• Fresh unchanged C++ legality and nine-stage replay for both words: 1,380 formal columns, 18/18 omitted-bridge controls each, nine COPY/ERASE lifecycles. Records fall 29,874 -> 29,746.
• Eight circuit/interface/source controls, canonical native-input rejection, explicit optimized-Python refusal and altered search-metadata rejection.
• Exact certificate comparison, relocated minimal Git archive replay, selected-record binding and staged whitespace checks. Separate AI review's metadata-binding and wording findings were fixed.

Assignment seed 19, drawn once from OS randomness. All 6,320 oriented edges on 80 operands were scored; one seeded greedy star packing selected 44 edges. Stop rule: no positive edge remains on unused operands. No restarts or optimality claim. No pinned coordination issue existed, so the claim is local.

Final overlap refresh through #751 found Chinmay Khaladkar's distinct source-coset construction #748, head 0ccc0b425d2b9667108d3a0918a5cfa0aa7a5861, with larger 244-ADD savings. It changes the helper-dirt map and recomputes D; this contribution preserves the full A operator and D and adds fresh C++ multistage evidence. PR733 and PR742 also have larger savings in D. No overall record or composition with concurrent work is claimed.

The full hosted-parent/global invoice, changed-word outer assembly, repository-wide suite, selected cold replay, Lean and remote CI were not run. Inherited all-size/compiler/common-label/mutable-basis/hosting/residual/precision/tape/analytic assumptions remain. This is not a full multiplication proof or measured runtime speedup. The selected frontier and upstream/ are unchanged.

Prepared for Sagar Thakrar (@thakrarsagar) with substantial OpenAI Codex assistance, including separate workers for mathematical review, independent interpretation and native replay, under Apache-2.0. Credit remains with eumemic, Dylan Crow, DreamingOfClouds, Zhihao Chen and the predecessor lineage. PR708's native wrapper is adapted with explicit attribution. No independent human review is claimed.

[Mister-Meeseeks]

100-PR batch review at 2b30b63789f6d9948233fc67d9d79bca6b185c5b.

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

Reviewed scope: Restored helper stars save 128 native XORs; complete finite endpoint and original paid paths retained, prefix allowance checked.

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.