The Informational-Quantum Paradigm of the Cosmological Cycle: Axiomatic Scale-Norming of the Pregeometric Network and the Entanglement-Density Nature of Gravitation and Mass

This manuscript presents a complete, mathematically closed cosmological framework under the paradigm of “Timeless Informational Realism. ” We demonstrate that the continuous spacetime continuum, macroscopic matter fields, and gravitational attraction are emergent, secondary interface configurations compiled from the fundamental background matrix of the Primary Quantum-Information Register (R₀) and its topological projection, the Pregeometric Quantum Network. Within this framework, Newton's gravitational constant G is derived as an emergent macro-scaling conversion factor mapping microscopic quantum phase fluctuations to macroscopic geometric deformations (G = lPl² c³ /). To establish a rigorous foundation, the spatial pixel step is formalized as a self-contained spectral object via the Sokolov Algebraic Volume Invariant using Connes-Dixmier spectral calculus over the discrete Dirac operator, neutralizing the critical vulnerabilities of tautological dimensional analysis. The minimum volumetric step metric lPl³ is derived analytically from the ultraviolet boundary limit of Quantum Field Theory using Weyl's asymptotic distribution and the Dixmier trace, establishing it as the hardware-stabilized data buffer of the primary processor. We resolve the physical nature of the velocity of light c by proving it to be the exact, scaled limit of the maximum density of quantum uncertainty states (4 qubits) executing phase-refresh cycles (1. 85 x 10⁴3 Hz) within the minimal quantum pixel. Concurrently, the hidden background entropy of this localized cell is derived strictly from the natural logarithm of a unit sphere's area, ln (4), enabling a first-principles theoretical derivation of Boltzmann's constant kB. We introduce the Full-Property Sokolov Uncertainty Step Operator, which hardcodes the canonical Heisenberg non-commutative commutation relation directly into each discrete spatial node. Furthermore, we resolve the origin of matter fields by deriving the Sokolov Isomorphic Projector and computing the running connection factor via the Functional Renormalization Group under the non-equilibrium dissipative dynamics of Lindblad operators. This identity transition operator mathematically transforms the non-commutative phase divergence of microscopic uncertainty loops into a deterministic, scalar physical mass kg in the macro-world by filtering and averaging the wave spectrum of Quantum Field Theory, matching the exact Schwarzschild boundary constraints of General Relativity. The instant cross-influence and absolute identity of all fundamental particles across cosmological scales are mathematically formalized via Category-theoretic Functors and Symplectic Floer intersections, revealing that the 5th internal logical Bulk index projects a zero temporal coordinate where linear time degenerates into an “eternal Now. ” This interconnectedness is formalized via the Sokolov Spherical Web Trace Identity, proving that the Universe functions as a closed spherical network of interconnected quantum balances where a perturbation at any single node is instantaneously registered and stabilized across the entire topology. The model systematically proves that the quantum of action () cancels out during macro-coarse-graining due to open-system environment-induced decoherence, explaining why quantum phase chaos vanishes under macroscopic observation. Finally, the model resolves the 10¹20 cosmological constant paradox via algorithmic data compression at the cosmological horizon and closes the global evolutionary cycle of the cosmos without physical singularities through mass-zero conformal invariance, ensuring absolute quantum unitarity and information conservation across sequential cosmic aeons.