Noether Conservation Laws and Relational Phase Dynamics from Discrete Zero Closure: A Zero-Closure-Preserving Discrete Self-Map, the N→∞ Continuum Field Equations, Local Gauge Geometry, and the Standard-Model One-Generation Representation with Chiral

In the preceding paper (Concept DOI 10.5281/zenodo.22028072) many symmetry structures were derived from a few closure conditions (complex zero closure ΣXa2=0, finite order UN=I, simplex closure, self-consistency) without presupposing a background spacetime or a symmetry group. Two problems remained: Noether-type local conservation laws, and the connection to a dynamics that determines the next state. This paper constructs, under equal amplitude Xi=Aeiφi, an oriented discrete relational current Jij=A2sin(φj−φi) and a discrete action whose stationarity is the discrete continuity equation. The central claim on dynamics is that state rewriting is not admitted implicitly: a dynamics of this axiom system must be a self-map of the admissible state space. Projecting the unconstrained relational force onto the tangent space of the zero-closure manifold and using a finite retraction yields a self-map FN: ZN→ZN that preserves ΣXi2=0 exactly at every finite iteration. The self-map parameter s is a construction/selection parameter and is not physical time; physical time lies inside the Lorentz readout of the fixed-point configuration. In the high-resolution limit N→∞ the discrete continuity equation becomes ∂μJμ=0 and the action becomes the standard massless phase-field action, with the constrained wave/Laplace equation as the general form. Localising the phase origin forces an edge connection; simplex faces give curvature; the continuum limit yields the covariant derivative, Fμν, and Maxwell/Yang–Mills type actions. Combined with the five complex degrees of freedom and the 3⊕2 decomposition of the preceding paper, S(U(3)×U(2)) fixes the hypercharge ratio by the trace-zero condition 3y3+2y2=0, and V*⊕Λ2V decomposes into the 15 left-handed Weyl components dc, L, uc, Q, ec of one Standard-Model generation; all perturbative gauge anomalies and the SU(2) global anomaly are verified to cancel directly. The conjugate two-Weyl-sector selection is identified with the earlier A/B two-channel selection system, giving the minimal normal form dSχ/ds=λJ+gSχ(1−Sχ2). A numerical verification specification, including an axiom-preservation audit as a necessary condition for any update rule to count as dynamics, is given for the existing A/B Fermi-type experimental system. Japanese and English versions (md/tex/pdf) are included. Part of the “Generative Structure of Dimensions” series.