A Constitutive Field Derivation of the Spin Sector as the Hodge-Projected Circulation of Advected Carrier Flux

This paper derives the leading Spin-sector response of the Signature Framework within a continuous, lossless LC carrier vacuum. Conditioned on a topologically nontrivial tubular defect with an S¹ core, an integral winding class n ∈ ℤ, and a localized circulation envelope a(x), the closed-circulation sector is constructed through local Lie-advection of the stationary Bulk displacement field D_B. Because the raw conjugate response q_n is not generally solenoidal, the physical response is obtained through a complete Hodge-Neumann projection. A scalar Neumann correction eliminates local divergence and normal boundary flux, while an explicit harmonic projection removes independently addable homogeneous Neumann modes. The resulting transverse response profile Q_n is unique within the finite-energy subspace orthogonal to these modes and yields the first-order Spin field H_Sp^(1) = ω Q_n. Structural orthogonality eliminates energy terms linear in the circulation frequency, establishing the quadratic conjugate Spin energy E_Sp^(H) = (1/2) I_Sp^(H) ω² and defining the constitutive circulation-inertia functional I_Sp^(H). A collective-coordinate reduction then gives the canonical circulation momentum p_α = I_Sp^(H) ω. This work constitutes Paper III of the Bulk-Shift-Spin development within the Signature Framework.