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 βₖ by Pₛc·Tᵏ·Pₛc = βₖ·Pₛc, the scalar return sequence satisfies the exact recursion: β₊+₁ = βₖ·β + rₖ·s The scalar return at every order is determined by this block recursion. Complementary-sector dynamics enter through the excursion term rₖ·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 Pₛc: 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.