Readout Dimensions, Hidden Relational Modes, and Blackbody Spectra on Finite Relational Networks

This technical preprint develops an exactly computable finite relational-network benchmark in which dimension is treated as a probe-dependent spectral readout rather than as a unique prior attribute of a background container. The underlying finite product complex is K_{N,M}=C_N^{\square 3}\square C_M, with three visible relational cycles and one finite hidden relational fiber. The scalar Laplacian is diagonalized by finite Fourier modes. Under scale separation, the complete scalar heat trace displays the dimension flow 0→4→3→0. Ideal low-energy readout is implemented by averaging over the hidden fiber, keeping exactly the hidden zero mode. The visible scalar, transverse one-form, and transverse-traceless tensor heat traces share a common three-dimensional readout window, while complete probes read the full relational spectrum. The paper also formulates blackbody spectra as visible spectral density multiplied by the Bose–Planck occupation factor. Weak leakage of hidden nonzero modes into the visible sector produces threshold corrections at hidden-mode frequencies. Further sections discuss hidden-gap stability, long-range Green readouts, torsion-locked leakage coefficients, observer-relative readouts, Bell–CHSH probability gluing, and decoherent path readouts. This is a technical preprint and has not undergone journal peer review. The finite product complex is used as a flat solvable benchmark, not as a complete microscopic spacetime theory.