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

Version 1.6 of 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 fixed background attribute. The model is based on the finite product complex (KN,M=C_N³ C_M), with three visible relational cycles and one finite hidden relational fiber. The analytic core diagonalizes the scalar Laplacian by finite Fourier modes and shows that the complete relational spectrum exhibits the dimension flow (0→4→3→0), while ideal low-energy readout keeps only the hidden zero mode and yields a common three-dimensional readout window for scalar, transverse one-form, and transverse-traceless tensor probes. This version also adds finite-size numerical stability tests beyond the product benchmark. Weak weighted-base perturbations preserve the diagnostic chain (d_R3), (d_f3), (d_w2), and (α_G1). Strong lognormal disorder shows a staged failure pattern: heat-readout warnings and rare low-mode localization appear before clear walk-dimension drift, while Green inverse-distance scaling remains robust in the tested window. Non-product fibered-network ablations identify the direct hidden-visible mixing channels and show that strong global mixing can coexist with weak low-window leakage and a stable three-dimensional visible readout sector. 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.