Geometry of Coherence: the QUR Framework and the Informational Origin of Curvature (v3)

Geometry of Coherence presents the geometric manifestation of the informational principle introduced in Quantum Ultrametric Rigidity (QUR) and rooted in the canonical structure of Spacetime Rigidity (HLR). Here, geometry is not a pre-existing stage but the visible form that coherence assumes when information saturates its degrees of freedom. The paper derives the sub-isoperimetric relation: A / V2/3 ≥ (36π)1/3 (1 + εrig(L)), which quantifies how coherence reduces the effective geometric degrees of freedom and induces mild negative curvature. The correction term εrig(L) expresses the informational deviation from the Euclidean isoperimetric bound, describing domains that are finite in informational boundary yet unbounded in coherence extension. From this framework emerge the coherent pseudosphere, the mild hyperbolicity of the cosmic background, and the duality between rigidity and coherence, expressed by the universal law: S · ρrig ≈ MP⁴ . At cosmological scales, the geometry equilibrates expansion and rigidity, recovering the near-flat curvature of the observed Universe as an informational equilibrium state. Finally, the work bridges holographic and intrinsic perspectives: in QUR, information is not written on the boundary of space but encoded in the curvature of coherence itself. Thus, geometry emerges as the visible form of coherence — finite in capacity, unbounded in expression. Editorial Note — November 2025 Edition Edition v2 and v3 of Geometry of Coherence consolidate and simplify the geometric component of the informational framework developed in Hamiltonian–Lagrangian Rigidity (HLR), Quantum Ultrametric Rigidity (QUR), and Thermodynamics of Coherence (ToC). What was introduced in HLR as a foundational relation and formalized statistically in QUR and thermodynamically in ToC, is here expressed in its spatial form: coherence shapes geometry, sub-isoperimetry encodes the reduction of degrees of freedom, and mild hyperbolicity emerges as the natural equilibrium of large-scale space. With this volume, the framework is recognized not as a trilogy but as a four-part structure, where GoC represents the geometric pillar linking ultrametric discreteness with thermodynamic equilibrium. This edition clarifies the role of coherence as the generator of curvature, positions the coherent pseudosphere as the equilibrium geometry, and prepares the ground for ongoing developments in cosmology and black-hole structure. This v3 edition introduces a minor but conceptually important clarification in the treatment of rigidity density. All heuristic uses of ρrig ∝ 3χ MP² / L² have been removed. Rigidity is now consistently determined by the constitutive balance S · ρrig ≈ MP⁴ and the large-scale scaling ρrig(L) ≈ MP²/ L², while χ is used solely as a coherence parameter shaping the geometry through the field χ(x) and its gradients. The geometric and phenomenological results of the previous version remain unchanged; only the interpretation of χ as a direct density factor has been refined. This edition also streamlines the appendix structure: the material previously listed as Appendix A has been either integrated into the main text or removed where redundant. The technical backbone is now collected in Appendices B–E.