[unpaidmoon]

Change and mathematical scope

Scoped proof, no new κ. In the abelian scalar-idempotent model, t invertible m×m rational targets plus any number of clean or dirty monomial helper registers still need at least m t paid calls when the frame span has dimension at most m+1 and the normalized extra frame satisfies |K| ≠ 2. The direct coordinate word attains m t and restores every untouched helper to the group identity.

This answers open question 3 of research/span-lemma-every-dimension: Lemma A's line projection survives identity helper targets. Extra registers do not escape that part of the three-register obstruction. The |K| = 2 flag case is not extended. The selected record stays 781451/250000000.

Task and overlap

Base commit: 626e936ce4d61f2584316914fc71baf96940c512 on canonical CrocSwap/integer-mult-bounds main. Priority revision 8.

Session r74, item 7 of the ordered eligible list. The 19 October handoffs are already delivered. Later handoffs whose canonical artifact is an open PR, including the H25 requests on #6321, #6326, #6328, #6332 and #6335, are excluded. The revision-8 next-decision table then supplies six rows, naming p104-side-correction, coverage-law-change and bit-helper-contract. Catalogue cards not named by those rows follow. Item 7 is bit-new-register-model.

Scope status: refined. The card allows a fixed larger register count. The alphabet stays the rational matrix group; the new hypothesis is r helper registers with arbitrary monomial outputs. The result is a lower bound, not a native word.

Nearest work, and why this is distinct:

• Nick Petalas, research/span-lemma-every-dimension: span at most m+1 forces m W calls when every register is an invertible target, and explicitly leaves extra registers open.
• Ali Romman, PR Prove nine calls are necessary for three abelian bit registers #468 / research/bit-three-register-lower-bound: the m = 3 case with no helper registers.
• research/bit-one-relation-any-registers: arbitrary register count, but only when the frames have at most one rational relation.
• research/three-data-aux-direct: eight coordinate projectors on four roles. Not the general line projection.
• Open PR Add S3 three-register rank lower bound #3893: one F_2[S_3] target. Different alphabet.
• The Research: the 916 open mixed GL3 target pairs need exactly their direct cost (no new kappa) #1974 GL3 census and the Research: sharp mixed UT4 root tariffs with arbitrary dirty workspace #2027 UT4 mixed-root tariff use different alphabets and do not imply this bound.

Validation

CPython 3.12, standard library, Linux. From the repository root:

python3 -B scripts/research_selected_baseline.py
python3 -I -W error -B research/span-lemma-idle-helpers/verify.py
python3 -I -O -W error -B research/span-lemma-idle-helpers/verify.py
python3 -B scripts/research_package.py --package research/span-lemma-idle-helpers

Orientation printed PASS_RESEARCH_BASELINE_BINDINGS_ONLY with selected κ 781451/250000000. Both verifier modes printed PASS_SPAN_LEMMA_IDLE_HELPERS with idempotents_w3=28, idempotents_w4=120, pair_products=14400, eight_call_line_covers=0. The package command printed PASS_REGISTERED_RESEARCH_PACKAGE with outcome completed.

The pair check evaluates every 3×3 minor of (I+aN)(I+bM)-I at a,b ∈ {1,2,3,5}. Degree at most 3 in each scalar makes that grid an identity certificate for Lemma R's k=2 case. The induction in PROOF.md is the general argument. No Lean replay and no selected-frontier checker were run.

Credit

eumemic's PR #288 supplies the model. Ali Romman's PR #468 supplies the low-span lemma and the |K|=2 flag bound that this note does not extend. Nick Petalas's span lemma states the open helper question. The one-relation arbitrary-register theorem and the coordinate auxiliary check are retained as neighbouring scope, not as inputs that were replayed.

Prepared by unpaidmoon with substantial Cursor Grok assistance. Apache-2.0. No independent human or formal-kernel acceptance is claimed.

Open next questions

1. Does the |K|=2 flag bound still give k >= m t for clean identity helpers? First step: recompute the dimension-two flag count at m=3 with t invertible targets and r zero blocks. Blocker: Lemma C fails at |K|=2, and the cited 2W bound assumes invertible blocks.
2. Can one idle register and two independent frame relations beat m t? First step: fix m=3, t=3, r=1 and one explicit span-5 frame list, and decide whether an eight-call word exists. Blocker: the two-relation Laurent kernel is not principal.