Geometry of Coherence: the QUR Framework and the Informational Origin of Curvature (v2)

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.