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) = X₃→₍X₍→₃ decomposes as: Pₛc·T (a) ·Pₛc = Pₛc·X₃→₍·Pₛc·X₍→₃·Pₛc + Pₛc·X₃→₍·P⊥·X₍→₃·Pₛc 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 Pₛc + 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 Pₛc ⊕ 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.