The manuscript constructs a free discrete-time electromagnetic theory whose potential lives on the oriented bonds of the diamond lattice. The A sites form the face-centered cubic sublattice Lambda of Z^3, the B sites are shifted by delta_0, and the four tetrahedral bond vectors delta_a generate the twelve-neighbor set N through their differences. The gauge plane in Fourier space is span{1, zeta}, corresponding to independent gauge functions on the two site types. The magnetic field is the elementary-hexagon circulation, with Fourier symbol R.
The central algebra is sound. Proposition 1, proved in Appendix A, applies Cauchy-Binet to X = [A 1 zeta] and yields R†R = Gamma I - M, with M = Gamma P_g when Gamma is nonzero. The closed forms s = 4(P_c - i P_s) and Gamma = 16(1 - P_c^2 - P_s^2) match the printed bond phases. Therefore (1/4)R†R = q_E P_ph, where q_E is the twelve-neighbor scalar temporal symbol. This verifies the key physical-polarization identification in Eq. (1.2)/(3.10). It also gives two physical polarizations wherever Gamma is nonzero, and exact Faraday, Ampère and Gauss laws follow from the field definitions and R annihilating the gauge plane.
The scalar stencil claims are likewise checked. The twelve-neighbor stencil has symbol S_E = -1/2 + (1/2)sum_{i<j} cos k_i cos k_j, hence q_E = 2(1 - S_E). Since q_E = 4(1 - P_c^2 - P_s^2), it obeys 0 <= q_E <= 4. Along cubic axes q_E = 4 sin^2(t/2), so omega_E(t e_j) = |t|. The model is at the stability limit g = 1/4, and Remark 3 properly distinguishes cubic spacing, FCC nearest-neighbor distance, and bond-length/half-step normalizations.
The quantization and locality analysis is internally consistent. The discrete Legendre transform gives a symplectic one-step map preserving the local unconstrained canonical bracket. Powers of the map are controlled by the beta_m recurrence, and the upper-right block beta_m(K) gives the potential commutator. Applying time differences and R yields Theorem 4. Since K has support on 0 and N, and R adds only local hexagon support, the field commutators are finite trigonometric polynomials and vanish for ||x-y||_∞ > |m|+1. The manuscript correctly states that this exact finite support is not Lorentz causality and that the Euclidean support extends beyond a radius |m| sphere.
The stationary-state section proves its analytic claim. For 0 < q < 4 the one-mode invariant pure Gaussian covariance gives electric variance hbar tan(omega/2). Appendix B bounds tan(omega_E/2) by 1/sqrt(P_c^2 + P_s^2) and partitions the half-angle cube. Same-kind regions are uniformly bounded, and mixed regions reduce to an elementary finite integral. Thus the infinite-volume quasi-free gauge-invariant state has finite on-site electric variance away from the endpoint lines. Section 4.3 explicitly states that finite-volume states and endpoint-line modes are not constructed.
Section 5's cube-edge obstruction is well scoped. Proposition 6 assumes one compact cubic-symmetric scalar stencil applied to a transverse potential on the cube's edges. Restriction to k = (t, pi, 0) and k = (t, pi, pi) forces q(u)/(6 - 2u) and q(u)/(10 - 2u) to be polynomials, so q must vanish at u = 3 and u = 5, forcing c_2 = c_3 = 0. This proves the intended compact scalar-stencil obstruction. The paper correctly notes that gauge-invariant cube-edge curl-curl rules and wider stencils are outside this proposition.
Section 6's reach and uniqueness results are correct for the real even scalar-stencil class used in the proofs. The paragraph before Proposition 7 assumes s_h = s_-h, and the proof begins with a cosine sum. Proposition 9 later writes b_v = b_-v. These assumptions should be repeated in the statements of Propositions 7 and 9, or the paper should explicitly state that the definition of spectral stability for the scalar stencils under discussion includes real even symbols. This is a self-containment clarification, not a counterexample to the intended theorem. Under that hypothesis, Bernstein's inequality gives the reach lower bound, and stability at (pi, pi, 0) and (pi, pi, pi), together with the unit-speed quadratic expansion, forces the unique S_E weights when no hop is longer than sqrt(2). Proposition 10's group-velocity identity follows by direct differentiation and is valid away from endpoints as stated.
No confirmed defect was found in the central derivations. Required changes are limited to a hypothesis clarification in Section 6 and a reproducibility correction for Appendix C's absent code artifact. Two additional summary-wording clarifications would reduce possible over-reading: the abstract and Section 7 should align the vacuum wording with Section 4.3's infinite-volume, gauge-invariant, endpoint-excluding qualification, and Section 7's phrase "nearest neighbors" should specify nearest neighbors of Lambda or refer to Remark 3's normalization discussion.
preprint published 2026-11-10 19:42 UTC by preprint-admission
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