No Singularities (Bounded Characters)

τ-Formula

All boundary characters bounded → no divergence

Derivation

The No-Singularity Theorem (V.T31) is proved in Book V, Chapter 46 (The Big Bang as Regime: Same Laws, Extreme Parameters) and Chapter 50 (BH Birth and Topological Events). It establishes that singularities — points of infinite curvature or density — are structurally impossible in the τ-framework.

The argument has two faces. For cosmological singularities (the Big Bang): the τ-framework treats the Big Bang not as a singular event where physics breaks down, but as a regime — the opening regime at the ignition depth n_ign where the boundary holonomy algebra first supports stable oscillating modes. The proper time function t(n) is bounded below by the arc-length structure of the α-orbit (V.T08), so no sequence of refinement levels can “go below” the ignition depth. The singular limit a = 0 of classical Friedmann cosmology is unreachable because the refinement tower is discrete and bounded.

For black hole singularities: the τ-framework replaces the S² horizon topology of orthodox GR with T² (the torus fiber of τ³). The key difference is that T² is non-contractible — it has non-trivial fundamental group π₁(T²) ≅ ℤ². A sphere S² is simply connected and can collapse to a point; a torus cannot. The boundary characters on T² remain bounded by the spectral purity guarantee (III.T14), so curvature invariants never diverge. There is a torus core, not a point singularity.

Both arguments rest on the same structural feature: the discrete, bounded, non-contractible character of the τ³ arena. Singularities require a continuum limit that the framework does not admit.

Source

Cosmological singularity avoidance: Book V, Part 6, Chapter 46. Black hole singularity avoidance: Book V, Part 6, Chapter 50. The bounded-character guarantee: Book III, Part 4, III.T14 (Spectral Purity).