Emergent Spacetime Rigidity and the Regularization of the Big Bang Singularity (v4)

In standard Friedmann cosmology, the energy density and curvature invariants diverge as the scale factor approaches zero. This behaviour reflects the extrapolation of a continuous geometric manifold into a regime where the available informational degrees of freedom are insufficient to support it. In the rigidity framework developed in HLR and QUR, every causal domain possesses a finite informational capacity, and the rigidity density satisfies the constitutive identity: Sext(L) · ρrig(L) ≈ MP⁴ For a Planck-sized region this implies Sext = O(1) and ρrig = O(MP⁴), enforcing a finite ultrametric core in place of the classical singularity. The effective curvature emerging from this domain is capped by: |Reff| ≤ ℓP⁻² so that the Big Bang corresponds not to a divergent geometric state but to a phase of informational saturation. As the causal horizon expands, the extensive entropy grows, the rigidity density relaxes from its saturated value, and the curvature falls below the cap. This transition naturally generates an early quasi–de Sitter plateau: a regime where the constant rigidity component V₀ dominates once the derivative stiff component K(a) ∝ a⁻⁶ has diluted. No inflaton field or potential is introduced; slow-roll arises statistically from the structural decay of K(a), and primordial perturbations originate from spatial variations of the geometric coherence field χgeo(x). The resulting picture provides a fully regular Big Bang with:• finite entropy and finite curvature at the origin,• a smooth transition from an ultrametric Planck core to an effective FRW geometry,• a structural mechanism for an early quasi–de Sitter phase,• natural initial conditions for the inflationary epoch and standard cosmology.This framework preserves continuity with the late-time interpretation of rigidity—responsible for the observed vacuum-energy scale—while replacing the classical singularity with a finite and informationally coherent origin consistent with the fundamental constraints of the rigidity theory.The inflationary dynamics and the detailed generation of primordial perturbations are deferred to a dedicated companion work. Editorial Note —December 2025 This version incorporates the full structural advances achieved in the recent development of the rigidity framework.The principles guiding the regularized Big Bang — informational, quantum, and geometric — all trace back to the same microscopic structure of Planck-scale coherent domains. The results presented here follow from these structural constraints alone, without introducing extra fields, potentials, or model-dependent assumptions.Inflation is not analyzed in this paper: only the conditions for its onset are established, with the full treatment deferred to a separate companion work.