Exclude flagged span-six words with one dirty auxiliary

[unpaidmoon]

Change and mathematical scope

Partial lower bound in the abelian rank-one model. Selected κ stays 0.001672660197076918. The selected record is unchanged.

Three invertible rational 3×3 targets and one dirty auxiliary cannot be reached by eight nonzero calls when either of the following holds:

• The calls are simultaneously permutation-triangular. This holds at every frame span and for every shorter word. Nine coordinate calls still realize the targets, so eight is sharp for this class.
• The frame span is six and the occurring noncoordinate projectors share a complete flag of F₂⁴.

At span six, the cube-rank-at-most-one case reduces to the cited Lemma 1. The remaining cube-rank-two case has off-diagonal pattern dimension at most 2. The finite census leaves only two open span-six families: 72 flagless triples, one GL(4,2) orbit, and 12 cyclic lines.

Task and overlap

Base bb4bccbf42844d2801487450727c2f63e10d1d0f. Package research/auxiliary-span-six-flag. Catalogue card bit-new-register-model, scope refined: the register hypothesis changes from three scalar registers to three data registers plus one dirty auxiliary, the case left open by research/bit-dirty-auxiliary-registers §6.

Nearest work:

• That §6 handoff excludes spans at most 3 and at least 7 and asks for the four-register two-relation classification. 120⁸ stays infeasible; this note supplies the pattern-span census and the flagged span-six bound.
• Open PR Price nine coordinate calls beside one idle dirty auxiliary #3537 is the coordinate-projector pigeonhole and says the 120-projector classification is not done.
• research/two-relation-pattern-census-seed52 (PR Research: exact two-relation four-register pattern census and scoped obstruction #1733) records the pattern counts 105 / 93 / 12 / 196 / 176 / 20 for a different target question (four rank-four targets, fifteen calls). Those counts were recomputed here. The complete-flag column is new: 4892 flagged spans and 72 flagless triples.
• Discussion Handoffs and requests for independent checks #1166 comment 19056267 classifies span four and explicitly leaves spans five and six open. This package takes span six.

Not a selected-frontier update. Clean ancillas, nonabelian frames and free address maps stay out of scope.

Validation

Python 3.12.3, standard library only. Registered replay:

python3 -B research/auxiliary-span-six-flag/verify.py
python3 scripts/research_package.py --package research/auxiliary-span-six-flag --check-only
python3.12 scripts/research_package.py --package research/auxiliary-span-six-flag

Observed: PASS_AUXILIARY_SPAN_SIX_FLAG and PASS_REGISTERED_RESEARCH_PACKAGE research/auxiliary-span-six-flag outcome=partial.

The verifier checks two enumerations of the 120 idempotents, the counts above, 10,080 flag-basis pivots, generation of GL(4,2) (order 20,160), the normal-form orbit of size 10,080, the eight-call coordinate failure, the nine-call sum of three matrix units to I₃, the separation between the 12 cyclic projectors and the coordinate flags, and a one-bit tamper.

The rank bound in README §§2–5 is a written argument. Lemma 1’s internal cases are cited, not re-executed. The predecessor C++ engine, spans four and five, and the 72 triples were not searched.

Credit

Ali Romman, PR #468. eumemic, PR #288. huxint, PR #203. David Govea, PR #720, cited and not a dependency. waynenilsen, research/bit-dirty-auxiliary-registers (PR #1882), for Lemmas 1–2 and the handoff. PR #1733 for the earlier pattern counts. PR #3537 for the coordinate special case. Discussion #1166 comment 19056267 for the parallel span-four classification.

Prepared by unpaidmoon with substantial assistance from Grok 4.7 (xAI). No independent human review is claimed. New work is Apache-2.0.

Open next questions

1. Can the normal-form triple X = e₀(e₀+e₁)ᵀ, Y = (e₀+e₁)e₁ᵀ, Z = (e₁+e₂)(e₀+e₂)ᵀ, interleaved with the four coordinate projectors, realize three invertible blocks in eight calls at span six? One calculation would close all 72 triples. First step: the degree-two bookkeeping of PR Prove nine calls are necessary for three abelian bit registers #468 §5.1 on this seven-matrix alphabet. Blocker: transporting a diagonal target does not keep it diagonal.

2. Can one of the 12 cyclic projectors meet the same bound at span six when the cube rank is two? First step: collapse consecutive copies of the same projector and classify the five-matrix words. Blocker: in pattern dimension one the relation support can meet the coordinate calls.

3. Does span five admit an eight-call word outside the permutation-triangular class? Span four is the parallel note above. First step: the span-five catalogue of PR Prove nine calls are necessary for three abelian bit registers #468 §7 at these 120 projectors. Blocker: relation rank three no longer caps the pattern space at dimension two.

Workflow feedback (optional)

The span-six handoff in the dirty-auxiliary README named the exact finite object (120 projectors, pattern spans, common flag). That made the census bounded. A parallel span-four note landed on #1166 while this package was being checked; citing it in the same PR kept the two spans apart.