[unpaidmoon]

Change and mathematical scope

Scoped obstruction in the existing full formal F2[Q8] dirty-register model. No new κ. The selected conditional value stays 0.001672660197076918.

If an exact word uses only t_i calls labelled i, it needs at least 2 t_i calls labelled j. Trace parity raises that floor when 2 t_i and t_j have opposite parity. Dirty coordinates can change the unitriangular factor K_i, but not its invertibility or this active-image count. The label swap gives the symmetric statement.

The inherited inequality 5n >= 11t is therefore never attained on the rays t_i = 4 t_j and t_j = 4 t_i. For 4k copies of [zi] and k copies of [zj], length 11k has a unique cubic split (4k, 7k), and that split is a square i-block with too few j-calls. Every exact word on this ray has length at least 11k+2. The direct three-call word still costs 15k.

At seven [zi] targets and one [zj] target, the same lemma excludes the relaxation witness (n_i, n_j) = (7, 13). The other 20-call splits (9, 11), (11, 9) and (13, 7) remain open. No 20-call word is constructed or ruled out. The direct word has length 24, so the exact minimum stays in {20, 22, 24}.

This is not a new alphabet, a physical caller, a supplier admission, or a multiplication theorem.

Task and overlap

• Scientific base: 12fa8c5c844be21bfefb38a19503fbd867a11244 (canonical main, priority revision 7).
• Package head: da82b198bd06dfc248bdcdf4ab477da50688fb07.
• Package: research/q8-square-iblock-obstruction.
• Card: bit-new-register-model, scope changed. The card asks whether a new alphabet escapes the three-register obstruction. This package does not change the alphabet; it removes unrealizable equality cases inside the model already on main. Reason in TASK.json.
• Handoff: Handoffs and requests for independent checks #1166 (comment) (Nick Petalas, PR Prove a shape-sensitive bound for mixed negative Q8 targets #4234).
• Nearest overlap: open PR Prove a shape-sensitive bound for mixed negative Q8 targets #4234 proves the shape-sensitive relaxation and names (7,13) as a minimizing count witness. It does not decide either open question. This package does not replay its enumeration and does not depend on the unmerged tree. Premises are the pinned proofs research/q8-mixed-boundary-rank, research/q8-homogeneous-mixed-floor and research/q8-count-region-rank-floor.

Item 9 of the chronological #1166 queue continued from unpaidmoon PR #4161, after dropping delivered rows and keeping comment 19035513 because the census lever that #4161 stopped on was closed by #4090. The #4100 audit map is left to open PR #4311.

1. (7,18) / (7,20) functional census (comment 19021170, question 1).
2. Physical entry/exit contract for the bounded grid and C0/C1 scratch (comment 19030434).
3. Independent review of the research: consume P104 residuals in actual cleanup and price source return #3943 cleanup factorization (comment 19032700, task 3).
4. Changed all-role encoding for the research: derive P104 two-axis endpoint intertwiners and cost gate #3954 equations (comment 19034025).
5. Changed chronological P104 caller (comment 19034741).
6. Source-normalization fusion caller (comment 19035513).
7. Donor-sector contract (comment 19078950, question 1).
8. Whether PR915 covers target-read closure (comment 19078950, question 2).
9. This obstruction (comment 19080750).

Validation

Python 3.12, standard library only.

python3.12 -B scripts/research_package.py --package research/q8-square-iblock-obstruction

Passed locally as PASS_REGISTERED_RESEARCH_PACKAGE research/q8-square-iblock-obstruction outcome=partial. The verifier prints PASS_Q8_SQUARE_IBLOCK_OBSTRUCTION.

Not run: an endpoint search of (9,11), (11,9), (13,7) or of the 13-call splits at shape (4,1); Lean; Docker; the selected full cold replay. The image argument is PROOF.md. The executable checks are corroboration.

Credit

Nick Petalas, PR3570 / PR3578 / PR3589 / PR3736 and open PR #4234, for the cubic ranks, the homogeneous quadratic method, the count region, and the two open questions. PR3536 and PR3558 retain the chronological and cyclic methods. The lineage in those proofs stays with PR3517, Tuafo/PR2900, ywz65/PR2587, PR3487, Aaron Wang/PR2318, Cinna Davis, steph4n-gh and the credited Q8 register sources.

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

Open next questions

1. Does shape (7,1) have a 20-call word on (9,11), (11,9) or (13,7)? First step: the cubic-tight split (13,7), where rank(X_j) = rank(Y_j) = 4. Blocker: no endpoint search is included.
2. Is the (4,1) minimum 13 or 15? The square lemma still allows (4,9), (6,7), (8,5) and (10,3) at 13 calls. Blocker: no exhaustive search at that length.
3. A physical caller and a complete invoice are still required before any supplier or kappa claim. None is identified here.