Outer-Support Induction of the Scalar Return Weight

Within the kernel transport framework of T17 - T31, establishes that the scalar-channel component of the round-trip transport operator T(a) = Xd→nXn→d decomposes as: P_sc·T(a)·P_sc = P_sc·Xd→n·P_sc·Xn→d·P_sc + P_sc·Xd→n·P⊥·Xn→d·P_sc The first term is transport remaining entirely within the scalar channel. The second term is support-mediated transport, trajectories that exit the scalar channel, propagate through the complementary sector P⊥, and return. The decomposition is an algebraic identity requiring only P_sc + P⊥ = I and linearity: no further assumptions. Whenever the mixed term is nonzero, the scalar return weight is not intrinsic to the scalar channel alone. This provides the structural mechanism underlying the coefficient β(a) in T31: β(a) depends on support-mediated transport whenever the mixed term is nonzero. Status: Decomposition identity solid: algebraic, no conditions. Nonvanishing of the mixed term conditional on assumption (ii), transport maps not block-diagonal with respect to P_sc ⊕ P⊥ removes the obstruction but does not guarantee nonvanishing; explicit verification from T17 kernel geometry remains open. β(a) dependence inherits T31 diagonalizability assumption. All results inherit T16/T17/T20 conditionality. Dependencies: T14, T15, T16, T17, T18, T19, T20, T26, T27, T28, T29, T30, T31.