Research: clean ancillas do not save in the two-register m=3 bit model (no new kappa)

[waynenilsen]

Scoped negative result, no new κ. In the abelian bit model of #468, two invertible rational 3×3 data targets still need six rank-one calls with one or two clean auxiliary registers (zero on input, restored to zero). Six suffice. Merged #1882 settled dirty auxiliaries for any register count and explicitly excluded clean ones. Clean correctness is strictly weaker: only the data columns of the endpoint are constrained.

What is new

• Clean low-span lemma. Prove nine calls are necessary for three abelian bit registers #468 §3 / Research: dirty auxiliary registers do not save in the two-register m=3 bit model (no new kappa) #1882 Lemma 2 survive clean ancillas. The projected rank argument reads only the data columns of P−I, so the auxiliary columns may be arbitrary. At five calls, the relation lattice therefore has rank ≤ 1.
• Exhaustive check with the Research: dirty auxiliary registers do not save in the two-register m=3 bit model (no new kappa) #1882 engine. The acceptance predicate is relaxed to data columns; nothing else changes.

Run	Words	Clean terminals	Clean-only	Candidates
n=3, five calls (one clean ancilla)	17,210,368	4,051,444	4,014,687	0
n=4, five calls (two clean ancillas)	24,883,200,000	4,012,291,456	4,012,108,024	0

"Clean-only" counts terminals that fail #1882's dirty predicate. Almost every surviving terminal uses the relaxation, and none gives two rank-three targets.

Controls

• Regression: dirty mode reproduces Research: dirty auxiliary registers do not save in the two-register m=3 bit model (no new kappa) #1882's pinned n=3, s=5 terminal digest byte for byte.
• Positive: one target in three calls gives 64 candidates. Six calls with one clean ancilla gives 310,528.
• Failure: the out-of-model rank-two idempotent E₀₀+E₁₁ gives 134,082 false savings at five calls.
• Physical realisation: 300 clean and 300 clean-only n=3 terminals, plus 300 clean-only n=4 terminals, are rebuilt with exact rational frames having exactly the stated relation lattice. Each formal product satisfies the clean predicate, and the clean-only ones fail the dirty predicate.
• Independent symbolic check: agrees with the engine on all n=2 words, all 614,656 n=3, s=4 words (full mode), and 3,646 n=3, s=5 words.
• Clean-only identity: an explicit two-call word that is correct only because the ancilla is zero is accepted, and its leaking variant is rejected.
• Tamper controls: a dropped terminal, an altered representative, a false flag and a wrong-arity record are all rejected.

Reproduction

python3 -B research/bit-clean-ancilla-registers/verify.py          # ~1 min
python3 -B research/bit-clean-ancilla-registers/verify.py --full   # ~2.7 hours, adds n=4 and s=6

Expected: PASS_CLEAN_ANCILLA_SIX_CALLS. Terminal streams are hashed rather than stored, because the full runs emit tens of GB. RESULT.json is the recorded full replay (macOS arm64, Python 3.11.9, Apple clang). A separate standalone n=4 run gave the same digest. scripts/research_package.py --check-only passes.

Scope / not covered

• Covers one and two clean ancillas. Any number of clean ancillas is open, because Research: dirty auxiliary registers do not save in the two-register m=3 bit model (no new kappa) #1882's written Lemma 1 reads auxiliary columns.
• Three data registers with a clean ancilla at eight calls is not attempted.
• Negative results from a search for a new coding mechanism: m = 2 closed on both sides, flag theorems, closed routes (no new κ) #288 §1.3(b) already charges clean ancillas, so this closes a route; it does not remove a charge.
• No change to κ, the selected record, upstream/ or frozen evidence. Not a Lean proof.

Task: bit-new-register-model (scope key 7084092dd726e7ba1f2f), scope refined (auxiliary input convention), research base e112d71c, rebased onto current main 0fdd7d82 (quick replay re-passed before rebase; package-only diff). The helper's random draw was adversarial-audit. I switched cards because this question continues directly from my merged #1882, and the overlap check found no prior work on clean ancillas in this model; #1882 and #1822 explicitly exclude them. Claim: #2073. Overlap was checked at start and again before submission.

Credit: Ali Romman (#468: model and low-span lemma), eumemic (#288: model and ancilla pricing), huxint (#203). The engine and controls are adapted from #1882. Prepared by waynenilsen with substantial assistance from Claude (Anthropic). No independent human review is claimed.

Feedback: the generated VERIFY.json caps commands at 600 s, so the ~2 h exhaustive run can only be registered as --full outside the launcher.

🤖 Generated with Claude Code