[petalas]

Summary

Close PR3589's open sharpness question for mixed-negative Q8 endpoints, in the t <= 2 regime.

inherited PR3589 bound:   5n >= 11t,   inherited parity:  n = t mod 2

t = 1   (zi), (zj)        minimum 3    attained
t = 2   (zi,zi),(zi,zj),(zj,zj)   minimum 6    attained

The lower bound is inherited and role-free, so these minima are complete over all role counts and over any finite restored dirty workspace. The upper bound is an explicit word: a paid factor with u = v = e_a multiplies role a by its label g, and i^3 = zi, j^3 = zj, so three calls per target role realize the endpoint in 3t calls — which for t = 1, 2 lands exactly on the bound.

Two consequences, in the direction that narrows the question:

• The uniform 11t/5 constant is attained at t = 1 and t = 2. No strictly stronger necessary function of t can hold at those points. This is PR3589's open question 2, answered affirmatively for t <= 2 only.
• The bound is not attained by this witness family for t >= 3. t = 3 admits bound value 7 against a 9-call witness; balanced t = 4 admits 10 against 12. Those minima stay in {7,9} and {10,12} and are not decided here.

This is a mathematical side result for the nonabelian bit-register model. No new kappa, physical supplier, invoice, native conversion or selected-frontier change is claimed.

Evidence

• Before: PR3589 gives a necessary region and explicitly states the factor 11/5 is not claimed sharp. After: the constant is shown attained at t = 1 and t = 2, so no improvement of n as a function of t is available there.
• All 64 Q8 group products agree between a basis bit-arithmetic product and an independent signed-unit table, so the group law behind the witnesses is not assumed from the routine it tests. i^3 = zi, j^3 = zj, z^2 = 1 are checked directly.
• Eight explicit witnesses are replayed to their exact diagonal endpoints; each witness's call count is compared against the independently computed parity-rounded bound.
• Five complete breadth-first searches exhaust every legal word on 1 and 2 roles up to lengths 2 and 5 — 6 and 56,084 distinct states — and find none. These are a cross-check of the model and the witnesses; the theoretical bound is what supplies completeness at t <= 2.
• Behavioural controls run under -O and must produce real rejections for a corrupted Q8 product law, a corrupted z involution, a bad paid label, a zero or out-of-range projector, even v^T u overlap, role capacity, an empty or unknown endpoint target, a truncated witness, a shrunk census limit, a corrupted parity bound, and a receipt round trip. No check uses assert.
• Registered replay passes at the publication commit:

python3.12 -B scripts/research_package.py --package research/q8-min-length-census
python3.12 -B scripts/check_research_commit.py --package research/q8-min-length-census --check-source-history

giving PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES (097d500ac). The package is standard-library only and replays without Git or network.

Search failure, recorded as such

A randomised hill-climbing search for a ten-call balanced four-target word was run over 4–6 roles (18 independent runs, best cost 6, no cost 0). A positive control at twelve calls reached only cost 8, so the search is known to be weak. Its failure is written into VALIDATION.md only to stop a later reader mistaking it for an exclusion. It excludes nothing.

Limits and unrun obligations

t >= 3 is not decided; complete enumeration at four or more roles exceeded this session's budget. Nothing here re-proves PR3589, and no physical scalar, routing, restoration, hosting, precision, recursive or supplier-admission cost is computed. Lean, Docker, the selected full cold replay, ruff (not installed in the authoring environment), independent human review and formal proof-kernel acceptance are unrun.

Credit

Nick Petalas/PR3589 for the mixed-negative boundary-rank bound and the parity, with PR3570 (mixed cubic identity), PR3578 (two-orientation boundary-rank method), PR3536 (sign-sensitive chronology) and PR3558 (cyclic subspace argument). Only the witnesses, the complete small-case census and the sharpness consequence are new here.