Countable-Mode Completion of the Directed Quantum Field: Geometric Dirac Dispersion, Gateway-Smeared Interaction, and Branch-Dual Fock Dynamics in Six-Throat Directed Spacetime

Paper XXV of the Six-Throat Directed Geometry program completes the transition from the finite-mode directed quantum-field architecture of Paper XXIV to a countable antiperiodic spectral and Fock-space completion. The finite spectral cutoff is retained only as a computational witness and is not promoted to a physical ultraviolet law. The antiperiodic one-particle space is completed on the half-integer momentum lattice and is isomorphic to l2(Z), with no zero Fourier mode. Symmetric and antisymmetric Fock spaces are then completed over all particle-number sectors. Finite spectral cutoffs form an isometric inductive system, while smeared creation and annihilation operators converge on the dense finite-particle domain. In the completed theory, the point-field kernel converges only distributionally and its diagonal diverges with increasing spectral resolution. The quantum field must therefore be treated as an operator-valued distribution defined through Hilbert-space smearing rather than as a bounded point operator. The free spectrum is fixed directly by the inherited directed Dirac structure. The resulting geometric dispersion determines the completed free Fock Hamiltonian and unitary propagator while preserving antiperiodicity and the two directed branches. The finite-mode point interaction is replaced by a square-summable smeared number-changing source. Under the minimal geometric selection rule adopted in the paper, the smearing profile is determined by the inherited gateway gradient together with the antiperiodic half-winding and a fixed Zero-Region phase convention. The two directed branches retain exact operator duality. Number parity maps the positive-branch Hamiltonian to the negative-branch Hamiltonian, and every m-th order interaction amplitude acquires the corresponding directed factor while particle-number probabilities remain branch matched. The paper also establishes a strict separation between the inherited direction-dependent mass potential and the number-changing Fock source. The former is number preserving and number-parity even, whereas the latter changes particle number and is parity odd. Consequently, the inherited mass-gradient response scale cannot be identified with the overall field coupling strength. That coupling remains an explicit open parameter unless a subsequent layer derives a conversion or background law. The accompanying reproducibility package verifies 165 declared conditions across nine structural blocks, with 165/165 checks passing. The scope of Paper XXV is the countable-mode spectral and Fock completion of the directed quantum field, together with its geometric Dirac dispersion, distributional field limit, smeared interaction structure, and exact two-branch dynamics.