Light on a Ticking Lattice: Local Discrete-Time Maxwell Evolution and Its Unavoidable Corrections

Washburn's manuscript gives a mathematically coherent analysis of local linear tick rules on a cubic lattice. The central claims are conditional and, on the supplied text, proved under their stated hypotheses: finite translation-invariant stencils, two-step scalar wave updates of the form u(n+2)=2S u(n+1)-u(n), and, for the Maxwell part, a vector potential updated componentwise by one shift-commuting scalar rule S. Validated objections are editorial scope and one unresolved reproducibility gap; none overturn a theorem as stated.

The support and CFL-type part is sound. In Section 2, equation (1) gives the exact one-dimensional lattice d'Alembert solution and equation (2) gives cos omega = cos k, so the exact unit-speed propagation claim follows on the zone convention 0 <= omega <= pi. For the three-dimensional face leapfrog, equation (5) gives sin^2(omega/2)=r^2 sum_j sin^2(k_j/2), and the maximum at R=(pi,pi,pi) yields the stability condition 3r^2 <= 1. The Reach theorem and Face-rules-are-slow corollary use only support addition, not positivity of weights, so they correctly apply to scalar, matrix, and operator weights. The proof technique is the taxicab support bound, and it matches the domain-of-dependence intuition from Courant, Friedrichs and Lewy (1928), Yee (1966), and Taflove and Brodwin (1975), while making a different, non-asymptotic statement about a fundamental grid.

The corner rule analysis in Section 4 is also correct. Equations defining P(k)=prod_j cos(k_j/2), S=2 av_0 av_1 av_2 - 1, and cos(omega/2)=P(k) check out. The stencil weights follow by direct expansion: -3/4 at the centre, 1/8 on face neighbours, 1/16 on edge neighbours, and 1/32 on corners. The corner-rule theorem proves stability, exact axis dispersion, no doubling except the intended omega=pi zone faces, and |grad_k omega| <= 1. The group-velocity proof is a Fourier-symbol calculation plus differentiation; its regularity caveat is the one-sided interpretation at zone faces where omega reaches pi. This is a group-speed theorem, not a causal-reach theorem, and Section 8 correctly separates those notions.

The weight necessity results are among the strongest parts of the paper. For the symmetric 26-neighbour family, stability reduces to inequalities on sigma(k). The edge-and-corner-weight theorem correctly derives a_2+a_3 >= 1/2 from M=(pi,pi,0) and a_3 >= 0 by combining that bound with the R=(pi,pi,pi) constraint. The any-range theorem extends the statement using sign identities at the eight points {0,pi}^3. Its hypotheses are essential: real symmetric scalar weights, stability -1 <= S-hat <= 1, and exact endpoint conditions S-hat(0)=1 and S-hat(pi e_j)=-1. The manuscript's counterexample with weights -5/16 at the voxel, 1/8 on faces, and 3/32 on +/-2e_j is important, since it has long-wave unit speed and no corner-type hops but fails exact axis dispersion at k=pi. W_ij and W_c are signed totals, not probabilistic weights (validated objection O18); the abstract wording is literally correct as a signed statement and benefits from an explicit signed-total remark.

The Maxwell construction in Section 6 is valid and must be summarized with its construction visible (validated objections O1, O5, O8, O13). The update A(n+2)=2S A(n+1)-A(n) assumes one scalar shift-commuting rule applied alike to each component of a vector potential, with backward-transverse initial data div^- A(0)=div^- A(1)=0. Under these premises, the lattice Maxwell theorem proves exact preservation of div^- A, exact Gauss law, div B=0, and Faraday's law by commutation of shifts and the lattice curl-divergence identities. The Ampere correction D=2(1-S)+Delta is derived correctly, and the corner-rule Maxwell-waves theorem computes the symbol d(k)=sigma_corner-sigma_face. This term vanishes on axes and is fourth order at small k. The statement does not cover arbitrary direct updates on E and B or component-mixing electromagnetic rules unless they are separately represented in this potential form. Abstract and overview sentences that say every tick treating all cells and components alike keep three Maxwell equations should name this construction.

The mode quantization in Section 7 is mathematically consistent. Each transverse mode reduces to a one-degree-of-freedom symplectic tick map with q=4 sin^2(omega/2). The mode-vacuum theorem correctly solves Phi Sigma Phi^T=Sigma, obtaining (V,C,W)=V(1,q/2,q), determinant V^2 q(4-q)/4, a unique pure invariant Gaussian for 0<omega<pi, and no finite-variance invariant quantum covariance at q=0 or q=4. The stronger no-invariant-density-operator argument uses compactness of trace-class density operators and absence of eigenvectors for the relevant free-particle shear, citing Reed and Simon and Folland. The physical identification should be qualified (validated objections O2, O4, O9, O14): this is the invariant Gaussian associated with the chosen discrete action and canonical variables, not a derivation of a unique electromagnetic ground state from a conserved Hamiltonian. The manuscript itself notes that q equals magnetic energy only when q=sigma_face, while for the corner rule q=sigma_corner off the axes. Phrases such as the right quantum fluctuations should report the tick-invariant Gaussian of the chosen action.

Section 8's list of unavoidable corrections is well scoped in the scope table. The frequency ceiling follows from tick sampling modulo 2pi. The direction-dependent speed proof is valid for finite real s to -s symmetric two-step stencils, because the face-diagonal symbol is a finite trigonometric polynomial with integer frequencies and cannot equal cos(sqrt(2)t) on an interval. The Ampere-law obstruction is also correct: D=0 would force S=1+Delta/2, whose symbol is -5 at R and is unstable. The precursor language needs care (validated objections O6, O10, O15). The proof establishes nonzero causal support outside the Euclidean light ball for real rules reaching beyond face neighbours, and for the corner rule the far-corner coefficient is 16^{-n}. It does not prove that every such precursor is weak for arbitrary weights or arbitrary initial data. General weak-precursor language should be replaced by the scoped statement. Endpoint stability wording should also say that q=0 and q=4 have a bounded eigen-solution plus a linearly growing generalized solution, so generic endpoint data grow linearly (validated objections O7, O12, O17).

The illustrative computation in Section 5 has a clear design: a 77^3 periodic grid, no wraparound, a smoothed kick initial condition, the face leapfrog at r=1/sqrt(3) as a baseline, and the corner rule as the comparison. However, the supplied review materials did not include figures/make_figures.py or the referenced .dat and image files, so the quoted percentages outside the light sphere and the plotted numerical curves could not be regenerated (validated objections O3, O11, O16, O19; claim C31 unsupported). This is unresolved reproducibility evidence, not a mathematical contradiction. The analytic reach envelopes remain independently confirmed. The reference audit reported no identity mismatches or retraction flags among checked DOIs; the O'Reilly-Beck preprint and Folland book were unavailable to that automated check, which should be handled by stable locator information rather than treated as an integrity problem.

Overall, the manuscript merits acceptance after minor revision focused on scope qualifications (Maxwell construction visible; vacuum as tick-invariant Gaussian of the chosen action; scoped precursor language; generic/shear endpoint growth; signed-total remark on W_ij and W_c) and inclusion or softening of the figure-generation artifacts. No theorem requires mathematical correction.

If the central claims hold, any framework that takes a cubic lattice and discrete ticks as fundamental and requires stable light at one voxel per tick along the axes must include diagonal and corner-type hops; face-only stencils are ruled out by reach alone. The any-range signed-weight theorems turn a known design choice in large-stencil finite-difference schemes into a general necessity result, and the closed-form corner rule supplies an exactly analyzable model linking dispersion, constraint preservation, and mode covariances.

The contribution is mathematical and foundational rather than empirical: it constrains discrete-spacetime models and unifies several CFL and FDTD facts under explicit hypotheses. The physical reading of the tick-invariant Gaussian as the electromagnetic vacuum remains definition-dependent and should be stated as such; the Maxwell exactness claim is conditional on the potential-based componentwise construction. The work does not claim uniqueness of physical light on all possible lattices or new laboratory predictions.

1. Reach theorem and face-rule 1/sqrt(3) corollary hold for scalar, matrix, or operator weights from support alone.
2. Corner rule has closed dispersion cos(omega/2)=P, exact axis speed, real frequencies on the zone, and |group velocity| <= 1 with equality only on axes.
3. Any-range signed W_ij and W_c bounds at the eight points {0,pi}^3 force diagonal and corner-type weight for stable axis-exact two-step rules.
4. Counterexample with hops +/-2 e_j correctly shows corner weight is forced only when axial speed is exact at all wavelengths.
5. Potential-based Maxwell theorem: componentwise shift-commuting scalar S with initial div^- A = 0 yields exact Gauss, div B = 0, and Faraday, with Ampere defect D = 2(1-S)+Delta.
6. Mode-vacuum theorem: unique pure invariant Gaussian for 0 < omega < pi; no finite-variance or density-operator invariant state at omega = 0 or pi (shear).
7. Unavoidable-corrections table pairs each defect with the exact rule class proved.
8. Prior art (CFL, Yee, large-stencil FDTD, BCC QCA) is placed accurately without false equivalence.

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