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This scoped interface theorem gives the exact maximum number of simultaneous dyadic helper channels at a fixed cut. Given saturated dyadic bases of the eligible prefix and suffix modules, their pairing matrix has capacity equal to the number of Smith invariant factors that are powers of two. The algorithm returns an attaining unit block and an obstruction to any larger block. Complete kappa is unchanged.
Proof, exact algorithm, receipt, source pins. Base: canonical main 6cf7a68, priority 7. The consumed helper-handoff proof is pinned to PR #5602, head 6c224aa.
A four-ADD causal prefix and its inverse give pairing [5]: rational rank one, dyadic capacity zero. Conversely diag(3,5) has capacity one despite having no unit entry. These examples identify two distinct failure modes in productive-helper searches. Smith normal form itself is classical; the new result is its exact application and constructive certificates for this cut interface.
Normal and optimized registered checks pass on all 2401 integer 2-by-2 matrices with entries from -3 through 3, plus 75 rectangular dyadic matrices through 5-by-5. A separate determinant implementation recomputes every minor gcd. It checks both unimodular factors, the complete matrix identity and the attaining unit block. Five invalid-input or corrupted-certificate controls reject. Committed-byte and Git-free optimized export checks pass.
Python 3.12 and the standard library suffice. Expected markers:
PASS_DYADIC_HELPER_CHANNEL_CAPACITYandPASS_REGISTERED_RESEARCH_PACKAGE.This is algebraic interface capacity. Physical carriers, common frames, permutations, scalar growth and costs, native maps, ownership and the complete caller/compiler invoice still need validation. No current P104 cut census, selected cold replay, full source-history ancestry, outer funding or independent human/formal review was run. The checkout is shallow; hosted CI is separate.
Credit the PR #5599 helper-split line, the pinned causal proof above, PR #5609 for the field-pairing criterion, PR #5611 for dual-pair completion, and Nick Petalas's PR #5115 for prior Euclidean row preparation. Retain their P104/native/controller/compiler lineage. Coordination excludes active physical construction searches. Ali Romman with substantial OpenAI Codex assistance; Apache-2.0.