The Directed Quantum Field: Fock-Space Emergence, Particle-Number Transfer, and Higher-Order Interaction Networks in Six-Throat Directed Spacetime

Paper XXIV of the Six-Throat Directed Geometry program extends the fixed-particle-number multipartite quantum theory of the preceding paper to a finite-mode quantum-field architecture. The construction begins from the previously established directed many-particle Hilbert sectors and completes them through a direct sum over particle number, producing a field space in which particle number is spectrally resolved while the directed branch and permutation-statistics labels remain distinct. Normalized insertion and removal between adjacent particle-number sectors generate the symmetric and antisymmetric ladder structures. These operators are then lifted to quantum-field operators on the antiperiodic global modes inherited from the closed Six-Throat spacetime. The resulting finite-mode field algebra is controlled by a reproducing kernel, while the zero-particle sector remains a nonzero Hilbert state. Retaining the directed structure produces two orthogonal zero-particle states that are exchanged under branch reversal. Free field evolution is represented by a unitary geometric propagator preserving antiperiodicity, particle number, and the equal-time field algebra. The first interacting layer is constructed from a self-adjoint local number-changing vertex. It couples adjacent particle-number sectors, produces an exact continuity law in number space, and preserves total Hilbert probability while permitting particle number to vary. Multiple local vertices combine coherently into higher-order interaction networks. At each interaction order, the ordered amplitudes retain an exact directed branch dependence. Starting from the zero-particle sector, branch reversal changes the phase structure of odd particle-number sectors while leaving the corresponding particle-number probabilities equal between the two directed branches. The accompanying reproducibility package verifies 187 declared conditions across eight derivational blocks. The scope established here is the finite-mode directed quantum-field architecture: Fock-space composition, ladder algebra, global field modes, the directed zero-particle doublet, free propagation, particle-number-changing interactions, coherent multi-vertex interference, and higher-order directed interaction networks. Continuum completion, general multi-field species coupling, full-spacetime interaction-density completion, and renormalized continuum observables are outside the scope of the present paper.