Primordial Black Holes in the Quantum Ultrametric Rigidity Framework (v2)

Primordial black holes (PBHs) provide a unique observational bridge between quantum gravity and cosmology. In the Quantum Ultrametric Rigidity (QUR) framework, PBHs do not arise from classical overdensities but from threshold events within a discrete hierarchy of Planck-scale informational patches. Spacetime at the Planck scale is described as an ultrametric network of cells, each carrying an informational entropy S and an associated rigidity density obeying the constitutive relation: S · ρrig ≃ MP⁴ When coherent clusters of such cells exceed a critical size Ncoll(t), stochastic activation triggers gravitational collapse, producing PBHs whose mass follows the causal scaling: M ≃ c³ t / G The resulting mass function f(M) is multimodal and log-periodic, reflecting the hierarchical activation steps of the ultrametric tree. Spatially, PBHs form a super-Poisson clustered population that preserves memory of their discrete ancestry. The framework reproduces key observational features—mass banding, non-Poisson variance, and regulated accretion—without introducing new fields or free parameters. By replacing the notion of density contrast with a discrete activation criterion based on informational barriers, QUR unifies early-universe collapse, dark-energy behaviour, and the emergence of macroscopic continuity through a single invariant law. Primordial black holes thus appear not as exotic relics, but as structured manifestations of spacetime rigidity, linking Planckian discreteness to cosmic-scale order. This work extends the Quantum Ultrametric Rigidity (QUR) framework, grounded in the Hamiltonian–Lagrangian Rigidity (HLR) formulation, and applies its informational principles to primordial black-hole formation, developing the statistical and physical emergence of PBHs as activation-driven structures. Editorial Note —November 2025 This second version of Primordial Black Holes in the QUR Framework consolidates the theoretical and observational structure introduced in the first release. The presentation has been streamlined, the hierarchical activation model refined, and the notation aligned with the current conventions of the QUR programme. All chapters have been revised for coherence with the companion works on rigidity, ultrametric geometry, and informational dynamics. A new Appendix U has been added, providing the quantum–hierarchical foundation of the activation process at the level of Planckian patches. This appendix clarifies how ultrametric spectral modes generate structured fluctuations of the Landau barriers, completing the link between the microscopic Hamiltonian and the macroscopic Poisson formulation used in the main text. The results, structure, and scientific scope of the work remain unchanged; this edition improves clarity, internal consistency, and integration across the rigidity programme.