Cosmological Flatness Ω_k = 0 Exact from τ³ Compactness
Spatial flatness of the universe, Ω_k = 0 exactly, is derived as a structural consequence of the compactness of τ³ — not from inflation or fine-tuning.
Overview
V.T106 proves that the universe is exactly spatially flat (Ω_k = 0) as a direct consequence of the compactness of τ³. In the τ-framework, the spatial hypersurface Σ_now ≅ T² is a compact manifold; compact manifolds have zero curvature in the sense that their profinite structure admits no non-trivial spatial curvature from boundary conditions. This gives Ω_k = 0 exactly — not as a fine-tuned initial condition, not as an inflation output, but as a theorem.
Detail
| The flatness problem is one of the classic puzzles of standard Big Bang cosmology: the universe appears to be spatially flat to very high precision (Planck: | Ω_k | < 0.005 at 95% CL), but in a standard matter-dominated universe, spatial flatness is an unstable equilibrium — any small deviation from Ω_k = 0 at early times grows rapidly. The orthodox resolution is inflation: a period of rapid exponential expansion drives the universe exponentially close to flatness regardless of initial conditions. |
V.T106 resolves the flatness problem without inflation. The τ³ fibration provides a compact space: the now-surface Σ_now ≅ T² is a compact torus (V.D22). The Gauss-Bonnet and Stokes theorems on compact τ³ require the total curvature to vanish. In the τ-Einstein equation, this forces Ω_k = 0 identically — spatial curvature is a global quantity, and on compact τ³ the global curvature integrates to zero.
The result is notable for three reasons: (1) it predicts Ω_k = 0 exactly, not merely ≪ 1; (2) it does so without requiring an inflaton field (C17: No Inflaton No-Go forbids a sixth sector); (3) it is consistent with the No-Singularity Theorem (V.T103) which derives from the same compactness. The flatness prediction is thus tied to the non-singularity and horizon dissolution as a package.
The cross-domain significance is that the same compactness argument that gives Ω_k = 0 also resolves the horizon problem (V.P91: dissolved by τ³ compactness) and yields the No-BH-Evaporation result.
Result Statement
V.T106: Ω_k = 0 exactly, derived from compactness of τ³. Σ_now ≅ T² compact implies total curvature vanishes by Gauss-Bonnet. No inflation required.