The Discrete Dirac Monopole and the Acoustic Quantum Code: A Spectral Origin for the Twelve GM Frequencies in the Ontological Fundamental Network

This Letter supplements the accompanying paper on the monopole/toroidal generation ofthe Acoustic Quantum Code, which derives the twelve GM frequencies from a continuum Hopf–torus construction. A key open question in that paper is the discrete origin of the topologicalstructure: why does the octahedron have the cycle signature it does, and why do the tritone andfifth ratios follow from it? We show that the Ontological Fundamental Network (OFN) providesa two-layer discrete origin. The dodecahedral layer supplies the spectral gap and the goldenratio, linked to the octahedral gap by an exact identity. The Hamming layer supplies the cyclelength signature (4, 4, 6) and the Betti number b1 = 3, structurally matching the octahedralcycle basis. The isolated singlet acts as the discrete Dirac monopole: it fixes the fractional Aharonov–Bohm holonomy on the cycles, and its amplitudecontrols a critical transition between three dissipation regimes. The critical amplitude is a dynamical threshold, found numerically; an analytical derivation from the cycle topology remainsan open problem. The canonical OFN graph does not reproduce the exact octahedral intersection matrix G∗, but its diagonal (the cycle lengths) matches exactly, and the tritone and fifth ratios follow from the same cycle lengths. We also show that no induced subgraph of Q6 can reproduce G∗exactly. The two layers aretherefore complementary projections of the same octahedral structure: the dodecahedral sidegives the geometry, the Hamming side gives the cycle signature and the dynamics.