Recursive Structure of Scalar Return Under Iterated Round-Trip Transport

Within the kernel transport framework of T17 - T33, it establishes the exact recursive structure of the scalar return sequence under iterated round-trip transport. Writing the round-trip operator in block form: T = [[β, r], [s, q]] where β = a'a + b'c, r = a'b + b'd, s = c'a + d'c, q = c'b + d'd, and defining β_k by P_sc·T^k·P_sc = β_k·P_sc, the scalar return sequence satisfies the exact recursion: βₖ₊₁ = β_k·β + r_k·s The scalar return at every order is determined by this block recursion. Complementary-sector dynamics enter through the excursion term r_k·s, transport that exits the scalar channel, propagates through q, and returns. The theorem does not assert closure of the scalar observable layer independent of the complementary sector. A forward remark records that if a later result establishes the observable defect sector is two-dimensional, the scalar return sequence satisfies a closed second-order linear recurrence, and strong closure follows immediately. Status: Block recursion and exact formula solid by block induction and rank-one property of P_sc: no conditions beyond block structure. Mirror recursion corollary, conditional on T33 mirror ansatz. Strong closure conditional on later defect-sector dimensional collapse result. All results inherit T16/T17/T20 conditionality. Dependencies: T14, T15, T16, T17, T18, T19, T20, T26, T27, T28, T29, T30, T31, T32, T33.