The manuscript constructs a free discrete-time electromagnetic theory whose potential lives on the bonds of the diamond lattice (A sites on the even-parity FCC sublattice Lambda of Z^3, B sites shifted by delta_0). Independent gauge functions on the two sublattices act by the bipartite shift (2.4). Magnetic flux is the circulation (2.6) around the elementary puckered hexagons; its Fourier symbol is the antisymmetric 4x4 matrix R of (2.7) that annihilates the gauge plane spanned by 1 and zeta. The discrete action (3.1) yields the two-step update (3.2) whose spatial operator is K = (1/4) R dagger R. Proposition 1, proved by the Cauchy-Binet expansion of the Gram determinant of the 4x3 matrix X = [A 1 zeta] (Appendix A), establishes the central identity R dagger R = Gamma I - M with M = Gamma P_g whenever Gamma != 0. Consequently (1/4) R dagger R = q_E P_ph, where q_E = Gamma/4 is precisely the temporal symbol of the twelve-neighbor cubic-close-packed stencil S_E of (3.12). On the two physical polarizations the dynamics therefore reduce to the scalar recurrence (3.11) at unit Courant number; gauge modes are frozen by the exact Gauss laws (3.5). Faraday's law (3.3) is an identity of the definitions, Ampere's law (3.4) contains no residual Laplacian-mismatch term, and the magnetic cage sums vanish identically by antisymmetry of R. These statements hold under the stated free-field, unconstrained-potential premises and do not require Lorentz invariance or continuum recovery beyond the long-wavelength expansion of q_E.
Proposition 2 confirms spectral stability 0 <= q_E <= 4, exact axial dispersion omega_E(t e_j) = |t|, and the location of the endpoint set q_E = 4 (the lines P_c = P_s = 0). The zero set of q_E on the closed cube B is the eight corners with all |k_j| = pi (equivalent under 2 pi identification), a minor wording clarification already noted as editorial preference; the mathematical content is unaffected. Quantization proceeds from the discrete Legendre transform (4.1) and the symplectic one-step map Phi of (4.2). The canonical bracket (4.3) is taken on all four bond components before Gauss constraints are imposed; it is strictly local. The polynomials beta_m defined by the three-term recurrence (4.5) generate the powers of Phi, yielding the potential commutator (4.7). Time differences and applications of the hexagon curl then produce the three field commutators of Theorem 4. Because K couples only nearest neighbors in the set N of (2.2) and each R factor attaches a hexagon label to bonds at distance at most one in the infinity-norm, the electric support is contained in sums of at most |m| vectors of N while the magnetic and mixed supports are contained in sums of at most |m| + 1 such vectors. Hence all three kernels vanish for ||x - y||_infty > |m| + 1. Direct enumeration for m = 1 ... 4 (Appendix C and Figure 4) shows that the electric bound is attained. The same identity that cancels the projector determinant also makes the Fourier symbols finite trigonometric polynomials, so the finite-range claim is exact vanishing outside a bounded set rather than a Lieb-Robinson tail.
The stationary Gaussian state of each physical oscillator is the unique centered pure state invariant under the Verlet map (4.4); its electric variance is hbar tan(omega_E/2). Proposition 5 proves that the integral of this density over B is finite by partitioning the angle cube and reducing the singularity on the codimension-two endpoint lines to a convergent logarithmic integral (Appendix B). Numerical quadrature yields the constant 3.3565 that appears in the total bond variance (4.15). The state is therefore well-defined as a quasi-free state on the gauge-invariant algebra away from a measure-zero set; finite-volume constructions that avoid the endpoint lines remain open, as the manuscript already states in Section 4.3. Section 5 contrasts the diamond construction with a transverse three-component potential on the edges of Z^3. For any cubic-symmetric compact stencil the electric commutator symbol is q beta_m(q) P_T. Proposition 6 shows, by restriction to the lines k = (t, pi, 0) and k = (t, pi, pi), that the only member of the family for which q (P_T)_11 is a trigonometric polynomial is nearest-neighbor leapfrog (lambda <= 1/sqrt(3)). Consequently every compact cubic stencil that reaches unit Courant number has unbounded electric support on cube edges. The same scalar rule S_E is therefore nonlocal on cube edges and local on diamond bonds; placement decides. A non-compact stencil with hops of length two already realises finite-range commutators at lambda = 1 on cube edges, but loses exact axial dispersion; Table 1 records the trade-offs cleanly. On body-centered bonds the interpolated-wideband stencil becomes local at the price of a third polarization, recovering the known octahedral scheme.
Section 6 supplies two independent results of broader interest for discrete wave equations. Proposition 7 (Bernstein's inequality applied to the cosine sum of a spectrally stable stencil) shows that the directional reach c(u) is at least the directional Courant number; Corollary 8 places the ball of radius lambda inside the convex hull of the hops. Proposition 9 then proves that any spectrally stable, real, inversion-symmetric stencil of unit Courant number on Z^3 must possess a hop of Euclidean length at least sqrt(2), and that the unique such stencil whose hops are no longer is precisely S_E. The proof uses only the quadratic expansion, the stability bounds at the zone corners (pi, pi, 0) and (pi, pi, pi), and the resulting linear algebra on the edge weights; no cubic symmetry is assumed a priori. Proposition 10 establishes the group-velocity bound |nabla_k omega_E| <= 1 by direct differentiation of the symbol of S_E, with equality only on the axes. Thus the exact support of the commutators necessarily extends beyond the Euclidean light sphere (as required by any stable unit-Courant rule), yet no wave packet outruns the long-wavelength speed.
The manuscript is free-field only, sits at the stability boundary, and possesses neither Lorentz symmetry nor continuum Lorentz covariance beyond the long-wavelength limit. These limitations are stated explicitly in Section 7 and do not invalidate the proved identities. Open extensions (finite-volume states, sources, non-Abelian and compact versions, general classification of placements that cancel projector determinants) are correctly separated from the theorems. The deposited scripts and SHA-256 digests (Appendix C) supply a concrete reproducibility path for the figures and the vacuum quadrature; the analytic arguments themselves do not depend on them, though working availability of the deposit was not confirmed in this review. Reference [19] (Boas) is cited for Bernstein's inequality; the bibliographic entry is slightly incomplete but the classical inequality is standard and the proof of Proposition 7 is self-contained. Validated objections confirm no major mathematical defect. The central algebraic identity, the locality theorem, the vacuum integrability argument, the cube-edge obstruction, the least-reach uniqueness result, and the group-velocity bound all hold under the hypotheses stated in the paper. The work therefore supplies a concrete, fully local discrete-time Maxwell theory that simultaneously realises exact axial light-speed and finite-range field commutators, a combination previously obstructed for compact cubic-symmetric stencils on cube edges.
preprint published 2026-11-10 21:41 UTC by preprint-admission
Preprint record published. Integrity screen clean; desk screen PASS. The review summary and available scorecard are attached as public metadata.