Quantum Ultrametric Rigidity and the Emergence of π in Loop Quantum Gravity (v2)

Quantum Ultrametric Rigidity (QUR) provides a statistical and informational foundation for the discrete geometry used in Loop Quantum Gravity (LQG). Although derived from different principles, the two frameworks quantize the same geometric degrees of freedom and share the same operatorial backbone: the Casimir-type eigenvalue structure of SU(2). The correspondence becomes exact in the regime of suppressed rigidity (χ → 0), where ultrametric coherence collapses and the geometry becomes isotropic. In this limit, the QUR Hamiltonian reduces to the homogeneous quadratic form underlying the LQG area operator. QUR therefore supplies the dynamical origin of the structures that LQG postulates kinematically. A further result concerns geometric normalization. In the isotropic limit, the diagnostic ratio Ξ = A/(4R²) converges to π, identifying π as the invariant that fixes area normalization in the continuum. This geometric completion allows one to relate QUR’s structural coefficient α to the Immirzi parameter γ via a global scaling, without altering the LQG operator algebra. Thus, QUR and LQG form two complementary layers of a single informational–geometric framework: LQG captures the kinematics of the isotropic limit, while QUR explains its statistical origin and identifies the physical regime in which the correspondence holds. In this way, QUR offers a coherent dynamical foundation for the quantized geometry of LQG, clarifying both its structure and its normalization. Editorial Note — November 2025 This Version v2 presents a consolidated and fully updated formulation of the QUR–LQG correspondence. The structure and normalization of the area operator have been revised to align with the refined geometric foundations introduced in: Tosato, G. (2025). Geometry of Coherence: the QUR Framework and the Informational Origin of Curvature (v3). Zenodo. DOI: 10.5281/zenodo.17670007. The present version integrates the Γ/Mosco framework, the ultrametric–to–isotropic transition, and the updated normalization scheme (α, π, γ), providing a coherent and mathematically consistent presentation. Readers are encouraged to use this Version v2 as the reference formulation of the QUR→LQG correspondence. Editorial Clarification — December 2025 The internal Editorial Note in the PDF refers to Geometry of Coherence as “v2.” The correct reference is Tosato, G. (2025). Geometry of Coherence: the QUR Framework and the Informational Origin of Curvature (v3). Zenodo. DOI: 10.5281/zenodo.17670007. This correction concerns only the bibliographic reference; the scientific content of this paper remains unchanged.