CET Ω: A Modular–Informational Completion of Quantum Theory and Gravity

This paper presents a fully revised and internally consistent formulation of the CET Ω (Causal–Entropic–Tomita Omega) framework, which proposes a modular–informational extension of quantum theory grounded in the Tomita–Takesaki structure of algebraic quantum field theory (AQFT). Using the modular operator \Delta and its Hamiltonian K = -\log\Delta as fundamental informational objects, CET Ω interprets modular flow not as a passive symmetry of a state, but as the underlying physical dynamics that governs stabilization, irreversibility, thermality, and emergent gravity. A central result of the paper is the resolution of the classical measurement problem. Although the modular flow \sigma^\psi_t is unitary on the algebra A, its induced action on the predual state space A^\* is generally mixing in type III von Neumann algebras. This leads naturally to sector stabilization without introducing non-unitary collapse. The author proves a stabilization theorem showing that the long-time limit of the induced predual dynamics converges to a KMS equilibrium state. A major conceptual correction in this formulation is the clarification that the Born rule is not replaced. Instead, the Born rule emerges exactly whenever the system–apparatus pair is in a KMS equilibrium (Hilbert-balanced) state, since modular weights scale as \lambda_i \propto |a_i|^2. Out-of-equilibrium modular configurations lead to controlled, experimentally testable deviations from standard quantum probabilities. The paper also identifies and resolves the longstanding obstacles that prevented modular Hamiltonians from being used as fundamental dynamics: their state-dependence, their nonlocality, the incompatibility of local modular flows, the reversibility of the Tomita–Takesaki automorphism, and the lack of a link to gravity. CET Ω overcomes these by treating informational time \psi as distinct from geometric time, making the modular flow act fundamentally on A^\*, and interpreting gravitational entropy as modular energy. The result is an internally consistent informational dynamics extending quantum mechanics without contradicting experimental tests. The second half of the paper develops the gravitational implications of CET Ω. Using the entanglement first law \delta S = \delta\langle K\rangle, the author derives an emergent Einstein equation supplemented by corrections from modular imbalance. The modular Hamiltonian generates a bulk vector field that drives geometric evolution, unifying spacetime dynamics and informational flow. Finally, the paper outlines a detailed experimental program. CET Ω predicts: stabilization times that scale with (k_B T_{\rm eff})^{-1}, tiny but measurable deviations from Born probabilities in non-KMS regimes, intrinsic decoherence proportional to modular fluctuations, entanglement-spectrum discrepancies at the 10^{-3} level, and thermal offsets in accelerated detectors. These predictions are consistent with all current experiments but falsifiable with next-generation platforms such as trapped ions, superconducting qubits, Rydberg arrays, and cold-atom entanglement tomography. In summary, the paper establishes CET Ω as a comprehensive, mathematically rigorous, and experimentally testable modular–informational extension of quantum theory and gravity, in which measurement, irreversibility, thermality, and spacetime geometry emerge from the same underlying modular structure.