This work studies the emergence of geometric diffusive operators as macroscopic limits of symmetric discrete dynamics. The analysis begins with families of discrete Markov processes defined on countable structures with scale–dependent interactions. No metric, manifold, or coordinate structure is assumed at the microscopic level. Instead, geometric structure arises from the energetic properties of the limiting Dirichlet form. Under locality of interactions, diffusive time scaling, symmetry of transition kernels and stability of trajectories, the macroscopic limit is shown to belong to the class of symmetric second–order strongly local diffusion operators. The limiting structure induces simultaneously a metric geometry, a differential structure and a diffusion operator on the limit space. From this result a structural dichotomy follows. Microscopic models satisfying the structural hypotheses correspond exactly to macroscopic limits belonging to the class of geometric diffusive operators. Conversely, if the macroscopic limit is not geometric, at least one of the structural hypotheses must fail. This establishes a logical equivalence between the microscopic structural conditions and the geometric nature of the macroscopic dynamics. The work further proves a closure theorem showing that the structural properties appearing in the rigidity theorem arise dynamically from microscopic symmetry, shrinking interaction scale and uniform second–moment bounds. In this regime jump contributions collapse, antisymmetric components of the generator disappear and tightness of trajectories emerges from quadratic energy bounds. Finally, an inverse structural rigidity theorem establishes the necessity of the structural hypotheses. Whenever the macroscopic limit is geometric, the microscopic system must belong, up to asymptotic equivalence, to the class of local symmetric quadratically normalized models. Together these results provide a structural classification of diffusive limits of symmetric discrete dynamics. Geometry therefore appears not as a primitive assumption but as a consequence of structural rigidity in the scaling limit. __________________________________ Description: This document is a divulgative introduction to Universal Informational Dynamics (UID), written in Italian. UID interprets physical reality as an informational dynamics: where information concentrates or disperses, an informational curvature (the Laplacian of entropy, ∇²S) arises, slowing down local time. Negative curvature corresponds to attractive zones (positive mass, gravity), while positive curvature corresponds to dispersive zones (dark energy). In both cases, time dilation depends only on the intensity |∇²S|, leading to a dynamic balance between attraction and dispersion. The text explains how UID uses Connes’ spectral geometry to fix scales without circularity, introduces an informational Dirac operator deformed into a weighted Dirac (DW (γ) ) that serves as a Hilbert–Pólya candidate, and shows how its spectral zeta function produces zeros consistent with Riemann’s ζ (s). An operational protocol without artefacts validates both local (GUE statistics) and global (Riemann–von Mangoldt, affine alignment) properties. The document also presents a null test (Orch-OR Dual): UID runs show a significantly lower final entropy (≈1. 057) than random trajectories (≈1. 561, p-value ≈ 0. 05), confirming greater informational stability not explainable by chance. Language: Italian. This work presents a unified theory of quantum gravity and spacetime curvature based on the principles of informational entropy. In this framework, gravity is not simply a consequence of mass-energy, but emerges from the distribution and coherent gradients of information within quantum systems. By extending General Relativity into the domain of information, the theory introduces the new concept of informational curvature (∇²S) and formulates new field equations governed by entropy gradients. Extensive computational simulations, carried out using a rigorously structured, high-fidelity quantum model, provide strong empirical support for this approach. The resulting picture is that of a dynamic, self-organizing informational cosmology, in which space, matter, and gravity naturally arise as logical consequences of entropic structure and coherence. This preprint extends the UID framework with a spectral formulation of entropic time inspired by Connes’ noncommutative geometry. Previous UID works showed that informational curvature acts as the source of gravitational coherence and that entropic time responds symmetrically to both attractive (positive Laplacian) and repulsive (negative Laplacian) regimes. The present contribution introduces a spectral Dirac operator, where geometry is reconstructed from its spectrum through exact factorization and convergence to the continuum. The geometric scale is fixed spectrally via Connes distance, removing circularity with the time channel. Using the UID–Poisson mapping, the informational potential is directly linked to the classical Poisson equation, allowing calibration of the UID constant with Newton’s constant. This update provides a rigorous UID–Connes formulation, reproducible logs, and quantitative predictions testable with optical clocks and relativistic geodesy. We present a formal extension of the Universal Informational Dynamics (UID) framework, establishing a quantitative relationship between entropic curvature and the flow of proper time, which generalizes gravitational time dilation beyond the domain of General Relativity (GR). In UID, local entropic curvature—defined as the Laplacian of a coherent informational field—slows proper time regardless of the curvature’s sign. Both attractive regimes (positive curvature) and repulsive regimes (negative curvature) show a statistically significant inverse correlation (p < 0. 005) between curvature intensity and temporal flow. The model introduces informational curvature tensors and coherent latency tensors that generate emergent metrics, recovering GR-like behavior in the weak-field limit without imposing relativistic constraints. A synthetic test with 64 neurons shows spontaneous time dilation compatible with GR, with slope sign and monotonicity matching Newtonian potential predictions. We also interpret astrophysical observations of cosmic time dilation—such as the (1+z) scaling in Type Ia supernovae and quasars—as qualitatively consistent with the UID curvature symmetry principle. This suggests that time slowdown may result from the intensity of informational curvature rather than exclusively from the metric expansion of spacetime. The results position UID as a predictive extension of gravitational theory, integrating quantum informational geometry with cosmological phenomenology. Future work will include 3+1D tensor simulations, calibration in physical units, and validation on astrophysical datasets. __________________________________________________________________________________________________________ This work extends the UID Entropic Time Theorem, where the pure informational Dirac operator DI satisfied the exact factorization DI squared equals minus LambdaI times the discrete Laplacian, for gamma equal to zero. Here we introduce the weighted conformal Dirac operator DW (gamma), defined on a cyclic graph with informational weights derived from ultra–low entropy quantum states (4 by 4, almost pure). The weighting preserves self-adjointness but deforms the spectrum, which can be aligned with the non-trivial zeros of the Riemann zeta function. We define the UID spectral zeta, compute its zeros without artifacts, and verify local GUE statistics. Three computational certificates—betaₑff (T), deltaₘax (T), and Dᵢnfinity (T) —demonstrate the affine alignment of UID zeros with classical Riemann zeros. From the data, two emergent structures arise: (i) an abelian U (1) symmetry that removes the spectral over-density and restores the Riemann–von Mangoldt density, the strong functional equation, and the Euler product; (ii) a Bell–Boole coherence criterion combining global counting alignment (Boole) and pairwise phase coherence (Bell), offering a test of UID–Riemann rigidity. Analytically, we extend domain and self-adjointness in the continuum, prove strong convergence from the discrete case, establish a canonical affine gauge, construct the completed function XiUID (s) with symmetry s mapped to 1 minus s, and derive a UID trace formula with pseudo-primes and a Hadamard-type rigidity lemma. Important Note — Gauge Symmetry U (1) The observed discrepancy aligned perfectly with an issue previously encountered in the study of the emergence of the informational soliton, primarily linked to the U (1) symmetry and the measurement of the photonic tail. In that context, triplet and doublet configurations appeared, but the U (1) symmetry was globally broken, displaying behavior analogous to what emerged in the analysis of the Hilbert–Pólya operator. In this scenario, projecting onto that hidden degree of freedom eliminates the spectral over-density and restores the Riemann–von Mangoldt profile, the strong functional equation, and the Euler product. The closure is therefore not imposed externally, but arises as the only coherent resolution permitted by UID dynamics. While a rigorous proof remains open, the U (1) mechanism appears to be the sole pathway through which the UID spectrum can align with the Riemann zeros without introducing artifacts. Contatti mail: aleilquantumnerd@gmail. com
Aleil Quantum Nerd-Alessio (Tue,) studied this question.