All-Depth Carrier Accessibility for Routed Composition

Boolean support paths can overapproximate nonzero routed operator compositions: consecutive supported blocks may lie on different invariant carriers, or a routed prefix may enter the kernel of the remaining suffix. This paper gives an all-depth carrierwise factorization for finite routed products over a declared invariant direct-sum decomposition. It separates aggregate support paths, carrier-resolved support paths, and active routed products; proves exact vanishing from disjoint endpoint carrier support; characterizes cutwise vanishing by carrier-local image--kernel containment; and gives a survivor recursion for exact activity. An exact four-dimensional Z2 construction separates Boolean support from composition. A complete one-carrier depth-two census records eight strict within-carrier image--kernel obstructions among 24 supported candidates. The accompanying S3 and Rubik censuses are bounded computational companions. Their floating-point classifications are not exact-zero certificates or all-depth theorems. Paper XX begins an independently scoped post-protocol mathematical line in the Representation-Induced Mechanics and Evolution (RIME) program. It does not continue or reopen the SOF protocol authority chain closed by Paper XV. Source repository: https://github.com/dooven-prime/rime-lite Independent researcher; no institutional affiliation.