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

[waynenilsen]

Scoped lower bound, no new κ. In the abelian bit model of #468, two arbitrary invertible rational 3×3 data targets need six rank-one calls even with any number of dirty auxiliary scalar registers. These registers have arbitrary input and may end at any diagonal monomial, including exact restoration. Six suffice. #468 and the open #720 state their results for exactly three scalar registers; this changes that hypothesis.

Why dirty auxiliaries: #288 §1.3(b) records nested dirty auxiliaries as deficit-neutral, unlike clean ancillas, so they are the relevant extra registers in this model.

What is new

• Lemma 1, for any register count n. If the frame relation lattice has rank ≤ 1 and the product is a diagonal monomial matrix, every endpoint is a sum over pairwise disjoint frame subsets. For n=3 this is Prove nine calls are necessary for three abelian bit registers #468 §4 / Research theorem: nine-call bound for arbitrary invertible three-register targets #720 §2. The new case is a common off-diagonal pattern from projectors whose supports meet in at least three coordinates. That case replaces the three-register J argument for every n: the projectors must all be one N₀, the relation is all-plus, and each non-commuting outside factor lies outside the N₀ run.
• Lemma 2. Prove nine calls are necessary for three abelian bit registers #468's low-span lemma survives auxiliary registers unchanged.
• Theorem. For every (m,W) and any number of dirty auxiliaries, a word with s < mW calls has frame span m+1 ≤ d ≤ s−2. At (3,2) this is empty for s ≤ 5. No target triangularity is used.
• Independent exhaustive engine (C++). It checks every ordered word for one and two auxiliaries: 17,210,368 words for n=3 and 24,883,200,000 for n=4. It covers every rank-≤1 collision pattern and assumes only Lemma 2. Result: zero candidates. This is a second, computational proof for n ≤ 4 that does not use Lemma 1's pattern analysis.

Controls

• Six calls with one auxiliary: 4,852 candidates are reported. W=1 with three calls: the expected candidate is reported.
• Adding the out-of-model rank-two scalar idempotent E₀₀+E₁₁ produces 13,067 candidates at five calls.
• A separate symbolic Laurent implementation, with no shared pairing tables, matches the engine on all n=2 words, all n=3, s=4 words (full mode) and 35,080 sampled n=3, s=5 words.
• 300 engine terminals are realized with exact rational rank-one frames having exactly the stated relation lattice. Each formal group-algebra product equals the reported diagonal endpoint, with a rank-≤2 data target.
• Exact formal identities: the six-call construction with a restored auxiliary passes; a one-frame mutation fails; a J block B,−B is the identity, and separating it breaks diagonality.
• Tampered engine output is rejected: a dropped terminal line, an altered class representative, or a false candidate flag.

Reproduction

python3 -B research/bit-dirty-auxiliary-registers/verify.py          # ~45 s
python3 -B research/bit-dirty-auxiliary-registers/verify.py --full   # ~1 h, adds n=4

Expected: PASS_DIRTY_AUXILIARY_SIX_CALLS. RESULT.json is the recorded full run (macOS, Python 3.11.9, Apple clang). A standalone engine run and the full replay ran separately; they produced identical n=4 terminal digests.

Scope / not covered

• Clean (zero-initialized) ancillas: excluded. The derivative step fails there.
• Three data registers + one dirty auxiliary at eight calls: Lemmas 1–2 exclude spans ≤3 and ≥7. Spans 4–6 remain. The next step is an n-register version of the two-relation argument (Prove nine calls are necessary for three abelian bit registers #468 §5, Research theorem: nine-call bound for arbitrary invertible three-register targets #720 §§3–4) with 120 projectors, then span four and the span-five catalogue. This is recorded as a handoff in the README.
• Other primitives, non-abelian frames, free address maps between calls. Not a Lean proof and not a multiplication theorem. The selected record, upstream/ and frozen evidence are untouched.

Task: catalogue card bit-new-register-model (scope key 7f5f4426ef796683c485, seed 27c624ea8d72df2a037db8762e0bb8a6), research base 1b59b624d21924f8e3537cdc1ac52d3be6eda2fe, rebased onto current main bce54fb3 (quick replay re-passed). Open PRs and discussions were checked at start and again before submission; no overlap found. No discussion claim was posted.

Credit: Ali Romman (#468: model, projection, Laurent and J methods), eumemic (#288: model and auxiliary pricing), huxint (#203: flag method) and David Govea (open #720: disjoint-subset formulation; cited, not a dependency). Prepared with substantial assistance from Claude (Anthropic). No independent human review is claimed.

🤖 Generated with Claude Code

[Mister-Meeseeks]

Batch review at a9c146e2919c82065e157b3e9bc050f38ac816db.

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

Reviewed scope: Written arbitrary-register Laurent-pattern proof shows two invertible3x3 targets require6 rank-one calls despite any number of dirty auxiliaries. Quick C++/independent symbolic/rational controls pass; the author80-minute four-register census was not rerun. Clean ancillas and other paid/free alphabets remain outside scope.

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.