Geometry of Coherence: the QUR Framework and the Informational Origin of Curvature

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. Starting from the invariant of isotropy π, the paper derives the informational π-law and its volumetric extension π3/2, showing that these constants mark the statistical closure of information in space. The resulting sub-isoperimetric geometry, governed by the relation: A/V2/3 ≥ 4π-1/2 , defines a domain finite in area yet infinite in coherence. 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 global 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 appears as the first visible language of information — finite in form, infinite in meaning.