Primordial Black Holes in the Quantum Ultrametric Rigidity Framework

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 coherence thresholds among Planck-scale information patches. Spacetime at the Planck scale is described as an ultrametric network of discrete cells, each carrying an entropy SI and a conjugate rigidity χI obeying the constitutive law: SI · ρrig,I ≃ MP⁴ When coherent clusters of such cells exceed a critical size Ncoll(t), gravitational collapse occurs, 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 of rigidity through an ultrametric tree. Spatially, PBHs form a super-Poisson clustered population, preserving memory of their informational ancestry. The model 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 that of coherence contrast, QUR unifies early-universe collapse, dark-energy behaviour, and the emergence of continuity within a single informational law, expressed by the invariant relation above. 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 introduced in Tosato (2025) and builds upon the Hamiltonian–Lagrangian formulation (HLR), exploring the statistical and physical emergence of primordial black holes as coherence-driven structures.