CLAIM: clean ancillas in the two-register m=3 abelian bit model

[waynenilsen]

CLAIM
Question: in the abelian bit register model of #468 (as extended to dirty auxiliaries in merged #1882), do clean (zero-initialized, zero-restored) auxiliary registers let two invertible rational 3×3 data targets succeed in fewer than six rank-one calls? #1882 and #1822 explicitly exclude clean ancillas.
Base commit / predecessor: main e112d71c; #1882 (research/bit-dirty-auxiliary-registers, engine and controls reused unchanged), #468, #288 §1.3(b) for ancilla pricing.
Method and bounded domain: #468's low-span lemma survives clean ancillas (it uses only the data columns), which leaves relation rank ≤ 1 at five calls. The #1882 engine is rerun with the acceptance predicate relaxed to data columns only; exhaustive for one and two clean ancillas (n=3, n=4), with exact rational realization of sampled clean-only terminals and a symbolic cross-check. Early result: n=3 (one clean ancilla) gives 4,051,444 terminals and 0 candidates; the n=4 run is in progress.
Expected artifact: a scoped package research/bit-clean-ancilla-registers/ and a focused PR; no new κ.
Claim expires: 2026-10-14T21:45Z

Prepared by waynenilsen with substantial assistance from Claude (Anthropic).

[waynenilsen]

DONE: PR #2167 (research/bit-clean-ancilla-registers/)

Result: in #468's abelian bit model, two invertible rational 3×3 targets still need six rank-one calls with one or two clean ancillas (zero on input, restored to zero). Six suffice.

• Written: the low-span lemma reads only the data columns, so it survives clean ancillas. At five calls this leaves relation rank ≤ 1.
• Exhaustive: the Research: dirty auxiliary registers do not save in the two-register m=3 bit model (no new kappa) #1882 engine with a data-column predicate finds 0 candidates.
  • n=3: 4,051,444 clean terminals.
  • n=4: 24.9e9 words and 4,012,291,456 clean terminals. The full replay and a standalone run gave the same digest, c060adb1….
  • Almost all terminals are clean-only: they fail the dirty predicate.
• Controls: byte-for-byte Research: dirty auxiliary registers do not save in the two-register m=3 bit model (no new kappa) #1882 regression, positive controls, rank-two failure control, exact rational realisation of clean-only terminals, symbolic cross-check, tamper rejection.

Open: any number of clean ancillas, because #1882's Lemma 1 reads auxiliary columns. Also W=3 with a clean ancilla at eight calls. No new κ. Claim released.