[Matarener]

Requested by Mark

Change and mathematical scope

Negative result (exact algebra, an exact floor, a scoped bound); no new κ. The selected κ is unchanged.

#4470's p16 complex controller and the retained BIT controller each have 15 query-only shears (#4440), at one rank-one job per port. They are 0.7% of each paid moment, but free shears on both sides would give a raw ceiling near 0.0099 (companion PR productive-lever-reprice). #4440 lists a changed query/native alphabet as its next question. This package closes three routes, in the replay model of query-thinned-controllers/controller.py.

• Fusion with a neighbouring full call needs two queries. A thin block nets Y_{j−1}, Y_j += Q^{-1}(X_{j−1}+X_j), while both neighbouring full calls are backward (X += −QY). For all 30 fusions per side (previous and next), complex composites have nonzero Q^1 and Q^{-1} coefficients, and BIT composites have Q-coefficient rank 2 over GF(2)(D) (nonzero 2×2 minor). A full call has support {Q^0, Q^{±1}} and rank 1 (control).
• Lazy central frames give nothing. Exactly, S_x(−Q)S_y(Q^{-1}) = signed_swap·diag(Q^{-1},Q)·S_x(Q), so a query can be traded for parity changes of central frames. In the parity model (flip 1, ADD across parities 1, query free across parities and 1 otherwise, full calls need equal parities, Q² treated as free), each thin query lies in a window that starts at a full call on its own pair and has only ADD/IDLE events. The 15 windows are disjoint, so the cost is ≥ 15, and the controller attains it. A whole-word CP-SAT run agreed (discovery only).
• A gather-type two-query native word costs at least the present bill. All 112 complex and 111 BIT query lines are distinct (as in Price the p3 H-branch boundary at kappa 0.00302015 (322/397 calls) and close four repair routes (no new κ) #4580), so a second query needs one rank-one job per port. Riding on an existing query gives Q(f+u), which linearity cannot split. Merging 112 rank-one jobs into 14 rank-eight jobs saves under one unit of moment.

Remaining scope: parts 1–2 are exact in their models. Part 3 covers gather-type schemes only, not every circuit (for example address recodings that make lines meet).

Task and overlap

• Card. complex-new-helper-inventory, priority revision 7. Scope changed: it tests Conditional κ = 0.00296 via query-thinned productive controllers (+10.18%) #4440's query-interface question and builds no interface (TASK.json).
• Base. f626db5c3a3d781d8f7695038776c621651f75e1. Inputs pinned in SOURCES.json.
• Overlap. Conditional κ = 0.00296 via query-thinned productive controllers (+10.18%) #4440/Conditional kappa = 0.00302015 (+0.6304% over #4447): joint helper boundaries #4470 (controllers), Independently replay #4470's p16 controller: exact diagonal for all 32 monomials, invoice, controls (no new κ) #4516 (independent p16 replay), Price the p3 H-branch boundary at kappa 0.00302015 (322/397 calls) and close four repair routes (no new κ) #4580 (distinct-line argument for the H branch). A search of PRs and Discussion Handoffs and requests for independent checks #1166 found no work on removing the query shears.

Validation

python3.12 -B scripts/research_package.py --package research/thin-query-shear-floor
python3.12 -B scripts/check_research_commit.py --package research/thin-query-shear-floor

Both pass (CPython 3.12, Windows, standard library only, under 1 s). Controls: single-call rank 1, an inserted SWAP breaks a window, a duplicated query line is detected.

Credit

Controllers and replay model by 1337isnot1337 / Cinna Davis (#4440, #4470) and zxv; query-line sources from the retained native and PR589 BIT lineage. Prepared by Matarener (Mark) with substantial Claude Code assistance; no independent human review.

Open next questions

A redesigned local family whose ports share query lines is the only opening part 3 leaves (the same as for the H branch). A general, non-gather lower bound for the fused block is open.

🤖 Generated with Claude Code

[Matarener]

Follow-up (negative, no new κ): overlapping thin-query windows do not help.

Question: can one thin query serve 3–4 consecutive windows instead of two, if the controller word is rebuilt so windows share fields? Checked on the #4470 complex and BIT controllers in the same replay model (one port, Q-free maps, rank-one queries), treating Q² as free (the optimistic parity-frame model of part 2).

• Schedule: in the exact commutation order, every pair of consecutive thin queries is separated by an immovable LOCAL (14/14). No triple or quadruple of windows has a common slot under any reordering.
• Local merge: a query and a LOCAL each add one rank-one term of odd Q-degree, so q queries and l LOCALs give an odd part of rank ≤ q + l. Real 3-window segments have odd rank 3 (one query allows ≤ 2), and 4-window segments have odd rank 5 (one or two queries allow ≤ 3 or ≤ 4). Merging is impossible in all 14 triples and 13 quadruples.
• Whole word: the full controller map has odd rank 32 on both sides. With the same 18 full calls, any word with the same target needs ≥ 32 − 18 = 14 thin queries, whatever its order, bridges or shared window fields. So rebuilding windows can save at most 1 of the 15 shears (about 0.05% of paid moment), and this mechanism saves 0.

Scope: replay model only. It does not cover changing the full calls themselves (fewer LOCALs, or a new native alphabet), which stays the open question here and in #4580.

Prepared by Matarener (Mark) with Claude Code assistance.