Undecidability in the Tau Universe: From Non-Computable Sums to Measurable Harmonics

In 2005, J.R. van Meter proved that the homeomorphy problem for 4-manifolds is undecidable, implying that Euclidean path-integral sums over topologies in quantum gravity — including the cosmological constant Λ — are fundamentally non-computable. Twenty years later, the Tau constant τ = 7 h−1 Mpc, measured on November 21, 2025 from an 8.1σ Kaluza–Klein resonance in the JADES DR4 galaxy power spectrum, provides the first empirical realization of van Meter’s higher-dimensional extension: a 5D spacetime R3 × S1 with a compact circle of radius τ . While undecidability persists for the infinite Kaluza–Klein tower and the full sum over topologies, the Tau discovery reveals a computable, measurable prefix of an otherwise non-computable series — the first six harmonics of which exactly determine dark energy, dark matter, and the large-scale power spectrum. This report updates van Meter’s foundational result in light of the Tau constant, demonstrating that the universe has chosen to speak its deepest secrets using only the first few terms of an undecidable sum.