The Spacetime Dynamics on the Substrate: Discrete Dispersion, Quantized Wavelengths, and Finite-Difference Group Velocity — A Wave-Equation Companion to the Spacetime Constants Paper at (N_w, N_c) = (2, 3) with Zero Free Parameters

Description 

The substrate's natural wavelength has a period equal to its own modular temperature.

On the 42-layer Crystal Topos tower at the Mihailescu-Catalan integer pair (N_w, N_c) = (2, 3), periodic boundary conditions in the tradition of Born and von Karman 1912 quantize the allowed wavelengths to the finite set lambda_n = D / n with tower depth D = 42 forced by the Heegner closure 1807 = 13 times 139. By the Nyquist symmetry n <-> D - n, the substrate supports exactly twenty-one distinct physical wavelengths. There is no twenty-second wavelength and no fractional wavelength. The shortest wavelength is lambda = 2 cells (Nyquist limit); the longest is lambda = 42 cells (full tower).

The dispersion relation, evaluated by exact algebra at the 21 allowed integer indices, is omega_n = 4 * sin^2(pi * n / D). This is the Bloch 1928 / Slater-Koster 1954 tight-binding spectrum, with hopping strength J = 1/N_w and action quantum hbar = 1/N_w read from the substrate's coarse-graining isometry. No continuous wavenumber k, no derivative d-omega/d-k, no continuum limit appears anywhere in the construction. The half-angle identity 1 - cos(theta) = 2 * sin^2(theta/2) is used as exact finite algebra, not as a Taylor expansion.

The structural revelation. At the natural wavelength lambda = chi = 6 (one rectangle in the substrate's tiling), index n = 7 on D = 42 yields

  omega_7 = 4 * sin^2(pi/6) = 4 * (1/2)^2 = 1 exact
  T_7 = 2*pi / omega_7 = 2*pi exact

The period is the Bisognano-Wichmann modular temperature beta = 2*pi. The substrate's natural spatial unit (one rectangle = chi cells) and natural temporal unit (one modular cycle = 2*pi ticks) are tied through the dispersion. The match is forced by sin(pi / chi) = sin(pi/6) = 1/2, an exact algebraic identity that holds only at chi = 6 = N_w * N_c with (N_w, N_c) = (2, 3). At any other atom pair the self-consistency fails. Substrate algebra produces substrate dynamics.

Two further indices give exact rational frequencies. At n = 14 the wavelength is lambda = 3 = N_c (column count) and omega = 4 * sin^2(pi/3) = 4 * (sqrt(3)/2)^2 = 3 exact. At n = 21 the wavelength is lambda = 2 = N_w (Nyquist) and omega = 4 * sin^2(pi/2) = 4 exact. The three exact-frequency indices {7, 14, 21} form the arithmetic progression 7n, where 7 = D / chi is the number of rectangles stacked into the tower.

Group velocity. The standard definition v_g = d-omega / d-k is undefined on a discrete substrate. The framework computes the forward finite difference

  v_g(n) = (omega_{n+1} - omega_n) * D / (2 * pi)

between adjacent allowed wavelengths, recovering Newton's seventeenth-century method (Newton 1664 finite-difference manuscripts; Brook Taylor 1715 Methodus Incrementorum Directa et Inversa) that predates Leibniz-Newton infinitesimal calculus. The maximum group velocity across all 21 transitions is bounded above by the Lieb-Robinson velocity c = N_w = 2 sites per tick established in the companion paper Spacetime Constants. At the zone edge the group velocity is zero by aliasing: the Nyquist mode is a standing wave that does not propagate. Phase velocity v_p = omega_n / k_n approaches c at long wavelength and equals 4/pi at Nyquist.

Methodological discipline. Wavelength is quantized to 21 values. Frequency is a finite sequence of 21 numbers. Group velocity is a finite difference. The discrete second difference psi(x+1) - 2 psi(x) + psi(x-1) replaces the Laplacian. No d-omega/d-k. No integral over a continuous wavenumber. No continuum limit. The mathematics throughout is closer to Newton 1664 than to Leibniz 1684.

Four independent classical traditions overdetermine the wavelength quantization: Born-von Karman 1912 (crystals), Bloch 1928 (electrons in periodic potentials), Brillouin 1922 / 1930 (first Brillouin zone), and Bisognano-Wichmann 1976 (modular flow). Cross-domain anchors: Debye 1912 (phonon mode counting on lattices), Maxwell 1865 (antisymmetric Re-Im coupling as the substrate-level analogue of the electric-magnetic cross-coupling), Cheneau et al. 2012 (experimental Lieb-Robinson cone observed in ultracold-atom optical lattices), Wilson 1974 (lattice gauge theory), Nyquist 1928 / Shannon 1949 (sampling theorem and UV cutoff), and Aubry 1990 / MacKay-Aubry 1994 (discrete breathers, the lattice mass-gap mechanism).

Eleven receipts verify the main result with mean gap zero percent: all by exact algebraic identity or by-construction integer arithmetic. Six predictions and falsifiers are locked, including: any 22nd distinct physical wavelength on a substrate-realized system falsifies D = 42; any deviation from omega = 1 at lambda = chi falsifies the dispersion form; any group velocity exceeding c falsifies the Lieb-Robinson bound.

The framework's contribution is not the dispersion relation, which is textbook Bloch tight-binding from 1928. The contribution is reading the dispersion at the substrate's specific 21 allowed wavelengths and finding that one of them, the natural rectangle at lambda = chi = 6, produces exactly the modular temperature 2*pi already used throughout the framework. Continuum physics misses this because it takes D approaching infinity, in which limit the wavelength set becomes continuous and the structural revelation at lambda = chi is lost in a thicket of nearby modes. The framework's discreteness is essential.

External inputs: none beyond what the companion paper Spacetime Constants establishes. Zero free parameters. Twenty-one wavelengths, eleven receipts, one structural revelation.

Built on the Crystal Topos foundation: the discrete core (Montgomery 2026, Discrete Core), the MERA scaling spectrum at chi = 6 (Montgomery 2026, MERA Scaling Spectrum), the Type II-infinity core structure (Montgomery 2026, Type II-infinity Core), and the triangular duality (Montgomery 2026, Triangular Duality). Direct companion to the spacetime constants paper (Montgomery 2026, Spacetime Constants). The natural-wavelength self-consistency at lambda = chi ties the spatial substrate to the Bisognano-Wichmann modular flow established by the recursive 2-pi tower construction (Montgomery 2026, Recursive 2-pi).

The code package contains numerical receipts in Python (exact rational arithmetic at the three special points; floating-point verification across all 21 wavelengths; 22 PASS checks), kernel-checked formal proofs in Lean 4 (vanilla, no mathlib) and Agda 2.6 (pure Nat with refl proofs), and an interactive HTML widget with live n slider, discrete dispersion plot showing 21 dots (not a curve, by design), three exact-wavelength cards, finite-difference group velocity panel, and the structural-revelation banner.

Keywords

lattice dispersion, tight-binding model, Bloch theorem, Born-von Karman boundary conditions, Brillouin zone, Nyquist limit, sampling theorem, discrete substrate, lattice causality, Lieb-Robinson velocity, group velocity, phase velocity, finite differences, no-calculus methodology, Newton finite differences, wavelength quantization, modular flow, Bisognano-Wichmann temperature, discrete breathers, MacKay-Aubry breathers, ultracold-atom lattices, Cheneau experiment, photonic crystals, lattice gauge theory, Wilson loop, phonon dispersion, Debye model, Maxwell equations, Slater-Koster method, Pontryagin duality, spinor cube, antisymmetric coupling, Crystal Topos, substrate adjacency, Heegner closure