Within the kernel transport framework of T17 - T34, it reduces the scalar return sequence to the minimal complementary-sector data that governs it. Defining the complementary-sector orbit: 𝒪q (s): = spans, qs, q²s, q³s,. . . as the smallest q-invariant subspace containing s, three results are established: (1) The scalar return sequence (βₖ) is completely determined by the scalar block β, the action of q on 𝒪q (s), and the restriction of r to 𝒪q (s). Components of the complementary sector outside 𝒪q (s) are invisible to scalar return. (2) Any proof of strong closure of the scalar observable layer reduces to proving finite-dimensional collapse of 𝒪q (s), equivalently, a finite recurrence for the observable sandwich terms r·qᵐ·s. (3) If dim 𝒪q (s) ≤ 2, the scalar return sequence satisfies a second-order linear recurrence. The theorem does not assert finite-dimensional collapse at this stage. It isolates exactly the subspace on which closure must be proved, serving as the bridge between T34's recursive structure and any later defect-sector dimensional reduction result. Status: Orbit minimality, scalar return reduction, invisibility of orthogonal complement, and finite-recurrence implication are all solid by direct algebraic argument. Second-order closure corollary, conditional on the later proof of the two-dimensional observable defect sector. All results inherit T16/T17/T20 conditionality. Dependencies: T14, T15, T16, T17, T18, T19, T20, T26, T27, T28, T29, T30, T31, T32, T33, T34.
Craig Edwin Holdway (Sat,) studied this question.