A Covariant Variational Architecture for the Carrier Action

This paper formulates a covariant variational architecture for the continuous, lossless LC carrier of the Signature Framework. The stationary Bulk and leading translational Shift sectors are treated as configurations of a single constitutive field rather than as independent additive media. The carrier-frame field action is normalized by the constitutive relations D = C₀E and H = B/L₀. Evaluated on the actively translated Bulk configuration, the constitutive law reproduces the leading Shift response H_Sh^(1) = v × D_B, together with its anisotropic local energy density, constitutive momentum identity, and integrated response P_Sh^(1) = (4 E_Bulk / 3c²) v. The stationary Bulk flux invariant is retained explicitly. Conservation of this invariant over enclosing spherical representatives produces the Gauss-surface-like 1/r² dilution of the Omnidirectional Spherical Process-Load Domain (OSPLD), the corresponding 1/r⁴ energy-density profile, and finite exterior Bulk energy. For the closed-circulation Spin sector, the displacement flux is represented by a genuine spatial two-form and connected to the constitutive field through a first-order Lie-advection architecture. The Spin profile is determined by a leafwise Hodge–Neumann projection enforcing solenoidality, vanishing normal boundary flux, and orthogonality to harmonic Neumann modes. The circulation frequency is derived from a collective coordinate, so the reduced Spin action yields conservation of the canonical circulation momentum p_α = I_Sp^(H) ω rather than treating ω as an independently varied field. The reaction-world-tube action is retained explicitly but is not assigned an assumed microscopic surface equation of state. Consequently, covariant conservation of the total bulk-plus-boundary Hilbert tensor is formulated as a conditional completion statement. The paper therefore provides a mathematically controlled variational architecture for the carrier action while identifying the reaction-surface dynamics required for its full closure. This work constitutes Paper V of the Bulk–Shift–Spin development within the Signature Framework. Related Publications: Paper I — Bulk Sector: Firtl, M. A. (2026). A Constitutive Derivation of Stationary Bulk Fields from a Topological Boundary-Flux Invariant. Zenodo. doi: 10.5281/zenodo.21612107 Paper II — Shift Sector: Firtl, M. A. (2026). A Constitutive Field Derivation of the Shift Sector in a Lossless Continuous Carrier Vacuum. Zenodo. doi: 10.5281/zenodo.21613055 Paper III — Spin Sector: Firtl, M. A. (2026). A Constitutive Field Derivation of the Spin Sector as the Hodge-Projected Circulation of Advected Carrier Flux. Zenodo. doi: 10.5281/zenodo.21648295 Paper IV — ADM Alignment: Firtl, M. A. (2026). Constitutive Alignment of Bulk, Shift, and Spin Process-Load Sectors with ADM Source Projections. Zenodo. doi: 10.5281/zenodo.21670272 Foundation Lemma: Firtl, M. A. (2026). Foundation Lemma: Constitutional Principles and Nomenclature of the Signature Framework. Zenodo. doi: 10.5281/zenodo.20575293