Scoped negative result: R420 adjacent A swaps force both frames

[thakrarsagar]

R420's adjacent commuting A-gate exchanges cannot reduce its local paid recursive profile: both frames are forced, even when arbitrary rational subspaces are allowed. This is a scoped negative result / research note, with no new κ.

The package exhausts all 2,063 adjacent positions of eumemic's PR543 word. For the 1,735 commuting pairs, all 3,470 gate-occurrences have equal exact lower-span and upper-intersection dimensions. Every legal exchange therefore keeps the original frames and the profile {1:1320, 2:1440, 4:60, 5:120, 8:9}, rank mass 5,112. The cleanup swap needed for literal-reverse H4 is included in the argument and controls.

Only research/r420-adjacent-a-seed00/ is added. It contains the proof, certificate, source pins, two independently written checkers, local sprint claim and validation receipt.

Reproduce

python3 -B research/r420-adjacent-a-seed00/verify.py
python3 -B research/r420-adjacent-a-seed00/independent_check.py

The first command ends with:

PASS_R420_ADJACENT_A_RIGIDITY positions=2063 commuting=1735 forced=3470 gaps=0 rank_mass=5112 controls=8

The second independently checks all 672 closed frames, exact intersections and full symbolic A replays of every accepted swap. It also confirms that all 328 excluded dependency swaps change the A operator.

Passed on macOS arm64 with Python 3.11.15 and 3.14.8, using the standard library. Both checkers also pass under optimized Python. The retained predecessor's exact labels, 660 dirty-input columns and H1–H5 audit passes; eight controls include non-root intersections, noncommuting gates, wrong equal-rank frames, and unmirrored cleanup. The selected-record binding check passes. The unchanged full assembly and repository-wide regeneration suite were not rerun; no remote CI or Lean result is claimed.

Scope and provenance

• Exact base: 66849e1475c930fd7732b1a0067ffee7e1e95f60. PR543 source: 2065fe4f72a914d907f1046603f5a36f17018535. Word and checker Git blobs match that source; SHA-256 pins are included.
• OS-random assignment seed 00, bit-supplier track. No pinned coordination issue existed, so the claim is local. Overlap metadata was refreshed across 528 open PRs through Tooling: cache exact R420 geometry for 6.47x faster mutation replay #709; main was still the exact base.
• This covers one adjacent exchange in the original word, with unchanged surrounding gates, A boundaries, decoder and registers. It does not cover longer reorderings, factored gates, changed circuits, owner maps or call windows.
• PR661 treats fixed-order B frames; this treats A and an order change. PR589's elementary transports use P111. Concurrent Research: shorten R420 phase-A quad 13 from 19 to 12 XORs (no new kappa) #691/Research: save 6,934 R420 XORs with a restored helper basis #694/Research: save 114 R420 XORs by sharing restored decoder operands (no new kappa) #708 change circuits or helper basis and are outside this gate-preserving class. No code from those PRs is imported.
• The selected Conditional κ = 0.001635527380375392: coordinated owner repair on PR623 #626/Conditional κ = 0.001587467473762886: an E8 bit word with 420 helpers, c = 0.001653826988 (+4.18% over #510) #543 conditional κ remains 0.001635527380375392. Uniform compiler, physical realization, hosting, precision/tape and analytic obligations remain inherited.

Credit is retained for eumemic's PR543 construction and audit, Dylan Crow's PR370 format, DreamingOfClouds' nine-stage chain and PR589, Dugongue's PR541/548, Rohan Arun's PR542 and the predecessor lineage. Contributed by thakrarsagar with substantial OpenAI Codex assistance, including a separate worker for the independent implementation and review. Apache-2.0.

[Mister-Meeseeks]

100-PR batch review at ebd567a22d26fadc1c8292b5c57056fb52331266.

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

Reviewed scope: All 2,063 adjacent A positions screened; fixed-order frame bounds force the tested alternatives. No exclusion of arbitrary reorderings.

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.