CLAIM: projected helper-SWAP loop with arbitrary dirty helpers

[an664]

CLAIM - arbitrary dirty-helper boundary for the PR1420 projected SWAP loop

Question: With exactly one dirty target and any finite number k>=2 of dirty helper registers, can a unit scalar XOR word have raw endpoint I and projected endpoint swapping two designated helpers while fixing every other register in fewer than seven gates? Projection deletes target-destination gates in their original chronological order. All registers must be restored by the raw word; all extra helpers must also be fixed by its projected word.

Base: main 410026d. Predecessor: Gareth Davidson/@bitplane, PR1420 at d19a944, research/projected-cleanup-fibers/. The existing three-wire minimum is seven, extracted from its complete published paired-state census; both predecessor implementations reproduce the committed receipts. That finite case and its seven-gate witness are credited prior results, not this contribution.

Method/domain: an all-register short-word lower bound or an exact at-most-six-gate counterexample. Exactly one target; every ordered distinct-register XOR is allowed and costs one; arbitrary independent dirty inputs; no clean allocation, copying, erasure, free scalar changes, free swaps or group-algebra address labels. Any finite fallback will declare its dimension and length bound and will not imply an unrestricted theorem.

Expected artifact: one scoped proof/certificate with exact replay and meaningful omitted-hypothesis controls, or a precise finding/handoff. This refines PR1420's explicit general projected-defect minimum question to one non-elementary defect. It changes this session's initial complex-masked-decoder selection to a distinct bit-helper-contract prerequisite. No physical frame/hosting admission, selected supplier change, new kappa or complete multiplication theorem.

Overlap: fresh complete snapshot at 01:19-01:20 UTC contains 822 open PRs and 166 Discussion records; no direct matching claim found. PR1105 supplies the earlier three-wire quotient and raw-feedback obstruction with a different minimum-three target. PR1306 concerns complete D-local shears. PR1414 concerns coordinate-address SWAPs in a formal group algebra with free scalar mixing. They are credited and are not repeated. Existing claims1448 and1470 are different scopes.

Credit Gareth Davidson/@bitplane (PR1420), @woklerx (PR1105), and retained source lineage. Substantial GPT-6 Pro assistance through ChatGPT with Codex orchestration/local verification will be disclosed. Apache-2.0.

Claim expires: 2026-10-14T03:23:47Z.

[an664]

FINDING / DONE: the projected helper-SWAP minimum remains seven for every finite k >= 2

This resolves the scoped question in this claim. It is a small all-helper corollary of Gareth Davidson / @bitplane's PR1420, pinned at d19a944897714f2ecae72720b7dafc3e256f2f8d. The seven-gate witness, three-wire minimum, and write-count observation are credited predecessor results. Earlier raw-feedback lineage: @woklerx, PR1105.

Model. Exactly one target y and k >= 2 helpers, all arbitrary independent dirty inputs. Every distinct-register scalar XOR costs one. The raw word must be I. Projection syntactically deletes y-destination gates, in order, and must swap two designated helpers a,b while fixing every other register. It is not a block of the raw matrix.

All-k lower bound. Suppose a word of at most six gates meets those endpoints. Each written register in a raw identity word must be written at least twice: a single write adds a nonzero donor row (every prefix is invertible), which no later write to that register can remove. Both swapped helpers are written, and y is written because otherwise projection equals the raw word. Thus the word has exactly six gates, each of a,b,y is written twice, and no extra helper is written.

Projection leaves four gates, with destinations a,b. Call a gate internal if it is a+=b or b+=a. Restricting to original a,b coefficients, external reads do nothing. Each internal operation acts as a transposition on the three nonzero vectors of F2^2, as does the desired swap. Therefore the number t of internal gates is odd. One such gate is not the swap, so t=3. Exactly one kept gate reads an external register c, which is fixed throughout the projected word (including c=y). That read injects a nonzero coefficient column for the independent dirty input c into the two helper rows. Subsequent internal operations are invertible and cannot annihilate the column. The endpoint therefore cannot be a pure swap. Contradiction. This proof is independent of the number of extra helpers.

Alternatively, the saturated six-write budget reduces immediately to the predecessor's three-wire minimum: restrict to original a,b,y coefficients and delete reads from the unwritten extra helpers. This yields a word of at most six gates with the same raw and projected endpoints. The self-contained parity argument above avoids relying on the finite census.

Matching upper bound, credited to PR1420. With labels 0=a,1=b,2=y, use the chronological word

[(0,1), (1,0), (1,2), (2,0), (1,2), (2,0), (0,1)]

Writing A=(0,1), B=(1,0), C=(1,2), D=(2,0), the block C,D,C,D implements B and restores y for arbitrary dirty inputs. The raw word A,B,C,D,C,D,A is A,B,B,A=I. Projection is A,B,C,C,A=A,B,A=SWAP. Replace target label 2 by k and leave extra helpers untouched to obtain the same seven-gate witness for every k.

Local corroboration. An independently implemented standard-library checker passed normal and optimized Python 3.12.14 replay, with identical outputs. It checked all 256 four-kept-gate words on three wires with a packed-row implementation and a separate Boolean interpreter on all 8 inputs (2,048 assignments). Sixteen words have the desired helper submatrix; all sixteen retain a nonzero dirty-target column; none is a pure swap. The seven-gate witness passed all 88 inputs across k=2,3,5 embeddings. Controls distinguish syntactic projection from a raw block, reject a plain three-XOR swap's raw endpoint, expose a false swap obtained by zeroing the dirty target, and accept the prior five-gate elementary defect. These finite checks corroborate the proof; they do not establish arbitrary-k coverage.

Boundary. This does not classify other projected defects, multiple-target words, altered operation sets or clean allocation. It supplies no frame/hosting admission, supplier, new kappa, or multiplication theorem. No selected-frontier files changed and no prior work is closed.

Overlap refreshed 2026-10-14 01:40-01:41 UTC: 745 open PR bodies and 174 Discussion records, complete pagination without warnings; no competing scope found. Main then ef5f39403610d231a0c27e8a6b4f08c10209bcd6; selected record unchanged.

Prepared by Andrey Mas with substantial GPT-6 Pro assistance through ChatGPT; Codex coordinated, reviewed the bounded argument and independently replayed the checker. AI-assisted work, not independent human review. Apache-2.0.