Bound matched query swaps under commuting phase children

[jmbarrancoml]

I prove a lower bound of 555 single-array phase events for four coherent matched P112 query swaps, allowing shared linear processing, off-frame scalar mixing, COPY, CLEAR and private scratch. Under the declared completed-log profile, I can fit only 497 additional rank-one calls at the comparison saving. This rules out the proposed primitive in the commuting-phase alphabet; I leave the selected exponent unchanged.

The proof expands a fixed linear program around its all-queries-identity baseline. A strictly causal factorization and Sylvester's inequality give k >= (2/3)(d_image+d_row-R_average). I construct real phase characters in a common tensor-Pauli-trivial sector, then prove image and row dimensions112 and defect rank at most16 for the actual matched source matrix. For four banks the dimensions add, even with shared scratch.

I credit eumemic #288 and Ali Romman #468 for the compact-WY, character and image/row-span methods. I add the averaged nonzero-defect form, its extension to singular free programs, the Pauli-sector construction and this source specialization. I also retain Ali Romman's source/phase credit, Rohan Arun's #1931 continuation and Douglas Colkitt's interval-arithmetic credit.

Validation: the registered Python 3.12 replay and direct optimized Python 3.14.6 replay pass. I check 96 direct/causal program maps with 48 singular COPY/CLEAR steps, 3,584 exact phase evaluations, 6,216 pair characters, 57 target block identities, exact source/rank premises, seven assumption controls and the certified conditional budget. The written proofs cover the universal program and tensor-length statements; finite samples do not replace them.

Scope: exact fixed role-linear programs, uniform tensor Paulis and commuting phase children on one array. Noncommuting payload adapters, nonlinear or adaptive operations and new multirole child interfaces remain outside. I do not identify the complete mixed-chart inverse from #1998 with these four matched endpoints, and I do not claim an optimal 555-event construction or a new supplier.

I used substantial OpenAI Codex assistance in mathematics, implementation and prose. I have not obtained independent human review.

Coordination: #1167 (comment)

Reproduce:

python3.12 scripts/research_package.py --package research/paired-log-phase-query-bound