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https://github.com/Cosmolalia/akataleptos-geodesic-constants... https://github.com/Cosmolalia/akataleptos-geodesic-constants... We present observational and computational evidence that the large-scale filamentary structure of the universe is topologically equivalent to a Menger sponge at finite iteration depth. Recent work has demonstrated that thirteen fundamental physical constants can be derived from the seven structural parameters of the Menger sponge (S=5, P=2, b=3, d=3, Δ=17, removed=7, kept=20) with zero free parameters and sub-parts-per-billion precision for dimensionless quantities. If these constants genuinely originate from Menger geometry, the physical universe should exhibit Menger-like topology at observable scales. We compile observational data from WMAP, Planck, NEXUS+, IllustrisTNG, EAGLE, and other surveys and simulations to test this prediction across eight independent metrics. We find: (1) the cosmic energy budget (WMAP: 73% dark energy, 27% matter) matches the Menger first-iteration void/structure ratio (74.07/25.93) to within 1%; (2) the cosmic web volume void fraction (NEXUS+: 76%) corresponds to Menger iteration 4.7, consistent with 13.8 billion years of finite-time evolution; (3) cosmic filaments are one-dimensional at their core, carry over 50% of total mass in under 6% of total volume, and exhibit hierarchical self-similarity down to at least 10 parsec scales, matching the Menger construction algorithm at every tested scale; (4) the mathematical tools used to identify cosmic web structures (Morse theory, persistent homology, discrete topology) are the same formalism used to characterize Menger-type fractals. We propose that discrepancies between observed and ideal Menger ratios arise from the universe being at finite iteration depth with spatially varying iteration rates due to gravitational time dilation, yielding testable predictions including a spatial dipole in the fine structure constant correlated with local matter density. |