Paper T4 Math: Entanglement and Decoherence via Triadic Spin — Formal Layer

We develop the formal layer of Paper T4 from the occupied contact organization of Papers122BB–122BD. The surface energy yields both a local continuity law and a conjugate-coupledfirst-variation equation. The latter identifies the response terms lost when a state-dependenttransport matrix is treated as fixed. A proposed triadic description must satisfy a closurecondition before its three coordinates can be interpreted as a self-contained response. We thenderive a finite-time criterion for which patterns a specified phase-sensitive receiver can distinguish.In a declared uniform, fixed-coefficient comparison on the twelve-guide cycle, one local contactleaves five independent complex combinations dark; two adjacent contacts remove that darksubspace, whereas opposite contacts do not. An exact Gramian gives the response strength anda perturbation bound. These are conditional readout results, not detector outcome probabilities.We also separate relative sign, conserved surface amount, spherical mode selection, and filteredprobability density. An explicit hemisphere calculation shows which additional conditioneddrift produces a squared profile. The complete standard-library program verifies the derivationswithout synthetic observations or dependencies on earlier programs.