The 37-Dimensional Foam Subspace Why the GHZ Experiment's Hilbert Space Dimension Emerges from 120-Cell Spectral Geometry

A recent experiment (Liu et al., Science Advances 2025) demonstrated theGreenberger-Horne-Zeilinger paradox using a photon in a 37-dimensional Hilbert space, with 3measurement contexts implemented via a time-domain fiber-optic interferometer. The number 37 wasobtained by numerical optimization as the minimum dimensionality for a maximal GHZ paradox with 3contexts. We show that this number arises naturally from the spectral geometry of the 120-celldodecahedral foam, the central geometric object in the Selection-Cost Framework (SCF). Specifically,37 = 36 + 1, where 36 is the dimension of the largest irreducible representation of H4 (thehyperoctahedral symmetry group of the 120-cell) and 1 is the trivial representation. The 36-dimensionaleigenspace occurs at eigenvalue l = -2 of the cell adjacency graph, corresponding to Laplacianeigenvalue 14 — the critical propagation frequency of the foam. Furthermore, the 1200 triangles in theadjacency graph correspond exactly to the 1200 geometric edges of the 120-cell where 3 dodecahedralcells meet, providing the natural 3-context structure required by the GHZ paradox. Every element of theexperimental apparatus maps to foam geometry: fiber loops to cell edges, beam splitters to walljunctions, and time-bin modes to spectral propagation channels.