Discrete Cavity Modes and Logical Density of States in the PDL Framework

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 vertices, edges, and triangles, introduces a photon-like logical excitation, and specifies a strictly local update rule indexed by integer triples (nx,ny,nz)n_x,n_y,n_z(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)Nn_x,n_y,n_zN(nx,ny,nz) and a triangle-violation fraction ε(t)\varepsilon(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 = 4n_x+n_y+n_zN=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\nu \propto 1/Nν∝1/N, the combinatorics of integer triples with bounded nx+ny+nzn_x+n_y+n_znx+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\rhonu \propto \nu^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.