Uniform Finite-Field Route Profiles

Boolean support of one-step sector blocks does not determine whether an ordered labelled routed product is nonzero. For a marked projective carrier over a finite field, the missing information is encoded by collisions among the preimages of infinity under word prefixes. This paper studies a declared projective action over finite fields. The carrier is divided into a singleton sector containing the point at infinity and a complementary sector containing the finite field elements. The labelled transformations are S, R, and the inverse of R. Equality and exclusion relations among prefix poles determine whether each supported route survives. The same information also defines finite survivor automata for individual fields and generic route profiles at each fixed depth. At depth two, every finite field with at least three elements has 45 supported labelled candidates and exactly 14 zero routes. At depth three, there are 216 candidates, and the number of zero routes is determined exactly by the field characteristic and cardinality. At every positive depth, the prefix-pole classification gives an exact Fibonacci-based formula for the number of supported candidates, rational transfer series for each fixed field, and a stabilization theorem. Specifically, the complete labelled zero-route set at a fixed depth stabilizes when the field has more elements than that depth and its characteristic is not among the finitely many exceptional primes associated with that depth. The computational package uses exact finite-field arithmetic, integral matrix lifts, source-addressed replay receipts, and a pinned Lean/Mathlib source closure. The compiled Lean subset covers the complete depth-two theorem, the depth-three shape and count package, and the arbitrary-depth Boolean candidate count. The arbitrary-depth prefix-pole theorem, survivor-automaton construction, rationality, and large-field stabilization remain manuscript theorems supported by separate exact replay. All results are relative to the declared action, labelled alphabet, marked partition, composition order, and route semantics. No field-independent automaton, all-depth scalar zero-count formula, presentation invariance, Hecke or modular-form interpretation, renormalization-group universality, or spectral zero-mode claim is made. Paper XXI of the Representation-Induced Mechanics and Evolution (RIME) program. It continues the independently scoped post-protocol carrier and route mathematics initiated by Paper XX. Source repository: https://github.com/dooven-prime/rime-lite Independent researcher; no institutional affiliation.