[DreamingOfClouds]

No new κ. For products of rank-one query units, rank(2K - Γ) = N - r when the product is an involution of rank r (characteristic other than two; reflections allowed), which gives N >= r + 1 for square-zero units. In an algebra with a bimodule projection E onto the ordinary controls and E(Q) = 0, a product equal to O1 diag(g) O2 with r entries in ker E satisfies rank ρ(I + FG) = (N - r)d for the strictly lower F = E(W), G = VU. If w consecutive units have zero mutual pairing (diagonal included), then N >= r + w, in any characteristic and at any position in the word.

The projection exists for the 24 uniform single-qubit Clifford cosets on f columns (independent exactly for f >= 5; the eight monomial cosets are the ordinary controls: wrappers, S^{⊗f}, uniform Pauli, q²-dependent gates), so ordinary controls need not commute with each other or with a q-parity automorphism.

Exact checks:

• Conditional κ = 0.00330720 (+3.000e-6, +0.09079% over #6716): two paid idle fragments unlock independent outputs #6726 header at body variables 1 (606 events, 34 fields): 33 square-zero units (18 LOCAL, 15 QUERY), r = 32, rank(I+FG) = 1 = N - r in both sectors q² = ±1, with E(M) equal to the independent replay. Three copies with zero pairing across copies give N - r = 3: the window bound is attained.
• Theorem 1 on the three-unit word on two fields (r = 2), the Conditional κ = 0.00330720 (+3.000e-6, +0.09079% over #6716): two paid idle fragments unlock independent outputs #6726 eight-field root (nine queries, r = 8), and conjugated, padded, direct-sum and reflection words.
• 30 random words in Q[S₄] with B = Q[D₄] (exact E(M) = I + UFV, Sylvester identity); 6000 window instances.
• Controls: non-involution, controls outside B (12 of 12 break the identity), f = 4 dependence (rank 23), non-consecutive and non-orthogonal windows, header with one query removed.

Base: 7c49fe3. Inputs are unmodified copies pinned in retained/SOURCE-PINS.json to #6726 at 435bbe9.

Reproduce (CPython 3.12, standard library):

python3 -B scripts/research_package.py --package research/p104-query-block-bound

Expected: PASS_QUERY_BLOCK_BOUND (normal and -O) and PASS_REGISTERED_RESEARCH_PACKAGE.

Scope: words of units I + uQv with u, v over the ordinary algebra; the statement about the caller identifies a native call of width P with P mutually orthogonal square-zero units sharing one central Q, which is stated and not derived. Not covered: controls that do not commute with the body variables, non-square-zero native central actions, a recursive query supply, positive characteristic for the Clifford instance, targets outside 1 and ker E, column-split units. No pricing, caller or framework assumption is touched; the five assumptions of the cited words remain.

Overlap: #6726 proves the scalar header saturation (one query cannot be removed, q-independent controls); #6735 covers q²-dependent controls sectorwise (M >= r + 1); #6764 treats S-independent controls and coupled width with a common S and a three-query header. This package gives the general rank identity, the window bound at width w, and the projection setting that admits non-commuting ordinary controls. #6717, #6773 (stage-rank budgets) and #6766 (four-query fragment ceilings) address different questions.

Credit zxv (#6726 caller, root and scalar saturation), thu-nmrc (#6717, #6716), CubeJerry (#6764), ToikY (#6735), Ali Romman (#6398, as cited in #6717). Apache-2.0.

Prepared by DreamingOfClouds with Claude Opus 5.5 (Anthropic), via Claude Code.

🤖 Generated with Claude Code