This paper develops a toy model of a three-dimensional cavity within the Projective Dynamic Logo (PDL) framework, where physical reality is represented by minimal logical closures on finite signed graphs rather than by fields on a continuous spacetime. The author constructs a fixed “PDL cavity” graph with 27 vertices, 63 edges, and 49 triangles, introduces a photon-like logical excitation, and specifies a strictly local update rule indexed by integer triples (nx, ny, nz) (nₓ, nᵧ, nᵦ) (nx, ny, nz). For each index triple, the joint (excitation + graph) dynamics defines a discrete-time trajectory with a well-defined minimal period N (nx, ny, nz) N (nₓ, nᵧ, nᵦ) N (nx, ny, nz) and a triangle-violation fraction ε (t) (t) ε (t) measuring coherence leakage at each step. Numerical exploration over many low-lying index triples shows that the dynamics is strictly periodic with uniformly small leakage, and reveals a robust empirical law N=4 (nx+ny+nz) N = 4 (nₓ+nᵧ+nᵦ) N=4 (nx+ny+nz) for a large family of modes, alongside a highly degenerate subset with minimal period N=4N=4N=4. Interpreting NNN as a logical wavelength and defining a logical frequency ν∝1/N 1/Nν∝1/N, the combinatorics of integer triples with bounded nx+ny+nznₓ+nᵧ+nᵦnx+ny+nz leads to a cumulative mode count that grows as the cube of the bound, yielding an effective density of states ρ (ν) ∝ν2 () ²ρ (ν) ∝ν2 in three dimensions—directly analogous to the standard result for electromagnetic or quantum cavities. Although the model does not implement full quantum field dynamics, it demonstrates that a purely discrete, relational, and local PDL framework can sustain cavity-like modes with organised spectra and the characteristic three-dimensional density-of-states scaling, strengthening the case for PDL as a viable substrate for emergent quantum-like behaviour.
Cédric Laubscher (Tue,) studied this question.