We construct a composite variational functional K = K_ent + K_rec + K_bdry on the space of density operators, built from quantum relative entropy, record-consistency penalties, and boundary-compression divergence. We prove that K recovers three pillars of physics in three limits: free-energy minimization (exactly), environment-induced superselection (for commuting record operators), and the Einstein field equations (via the Jacobson route, with the Clausius relation and Bianchi identity derived rather than assumed). Beyond recovering known results, we show that the three sectors of K cycle through dominance as physical conditions change, and that the crossover regimes, where two sectors are simultaneously relevant, yield new, quantitative predictions inaccessible to any single-sector theory. We derive the crossover scales separating the thermodynamic, measurement, and gravitational regimes and identify a concrete laboratory prediction: population-dependent decoherence rates for quantum systems, governed by the competition between K_ent and K_rec. The gradient flow of K with respect to the Bures metric is proven well-posed with exponential convergence. The Lagrange multipliers in the record term are determined by physically measurable decoherence timescales, introducing no constants beyond those already present in quantum mechanics and general relativity.
We construct a composite variational functional K = K_ent + K_rec + K_bdry on the space of density operators, built from quantum relative entropy, record-consistency penalties, and boundary-compression divergence. We prove that K recovers three pillars of physics in three limits: free-energy minimization (exactly), environment-induced superselection (for commuting record operators), and the Einstein field equations (via the Jacobson route, with the Clausius relation and Bianchi identity derived rather than assumed). Beyond recovering known results, we show that the three sectors of K cycle through dominance as physical conditions change, and that the crossover regimes, where two sectors are simultaneously relevant, yield new, quantitative predictions inaccessible to any single-sector theory. We derive the crossover scales separating the thermodynamic, measurement, and gravitational regimes and identify a concrete laboratory prediction: population-dependent decoherence rates for quantum systems, governed by the competition between K_ent and K_rec. The gradient flow of K with respect to the Bures metric is proven well-posed with exponential convergence. The Lagrange multipliers in the record term are determined by physically measurable decoherence timescales, introducing no constants beyond those already present in quantum mechanics and general relativity.
We construct a composite variational functional K on the space of quantum density operators, built from three information-theoretic components: quantum relative entropy (measuring deviation from global consistency), record-consistency variance penalties (enforcing definiteness of macroscopic records), and boundary-compression divergence (penalizing deviations in boundary encoding). We prove that this single functional recovers three distinct pillars of physics in three limiting regimes: Helmholtz free-energy minimization in thermodynamic equilibrium (exactly, via the Jaynes identity), environment-induced superselection in the quantum measurement limit (for commuting record operators, reproducing Zurek's einselection), and the Einstein field equations in the semiclassical gravitational limit (via the Jacobson thermodynamic route, with the Clausius relation derived as a fixed-point condition rather than assumed). The gradient flow of K with respect to the Bures-Riemannian metric on density-operator space is proven to be well-posed for finite-dimensional systems, with exponential convergence following from the lambda-convexity of quantum relative entropy. We establish a uniqueness characterization: K is the unique functional (up to monotone reparameterization) satisfying scale universality, boundary sensitivity, record consistency, CPTP monotonicity, and reduction to known physics. The Lagrange multipliers in the record-consistency term are not free parameters but are determined by physically measurable decoherence timescales, yielding a framework with no adjustable constants beyond those already present in quantum mechanics and general relativity. This work provides a unified variational language for thermodynamics, decoherence, and gravitation, grounded in established mathematics: quantum information geometry, gradient flows in metric spaces, and thermodynamic derivations of gravitational dynamics.
We propose a thermodynamic variational framework in which quantum mechanics, classical dynamics, and gravitation emerge as equilibrium regimes of a single free-energy functional defined on probability distributions rather than on trajectories, wavefunctions, or spacetime metrics. The functional balances Fisher information, potential energy, and Shannon entropy, encoding an exploration–exploitation trade-off uniquely fixed by information-theoretic considerations. Matching the Fisher term to the quantum kinetic energy fixes its coefficient without free parameters. Extremization of the functional yields the continuity equation and the quantum Hamilton–Jacobi equation, and thus reproduces the Schrödinger equation as a thermodynamic equilibrium condition. At mesoscopic scales, competition between Fisher information and entropy introduces a characteristic quantum–classical crossover length that provides a thermodynamic perspective on decoherence. Measurement is interpreted as an irreversible thermodynamic transition, with energetic costs bounded by Landauer's principle. In the macroscopic regime, we show that requiring thermodynamic stability and local boundary response selects area-law entropy scaling as the leading contribution under stated assumptions. Given an area-law entropy, standard local arguments recover Einstein's field equations. The framework yields falsifiable predictions across quantum, mesoscopic, and gravitational regimes.
We propose a self-consistent axiomatic framework that unifies quantum unitary evolution, thermal dissipation, and spacetime geometry based on three fundamental principles: the \emph{complex-time principle}, the \emph{information-geometry equivalence principle}, and the \emph{maximal entropy gradient principle}. Starting from these axioms, we rigorously derive three core equations: a complex-time quantum-thermal master equation, an information-induced gravitational field equation, and a unified gradient-flow dynamics that couples quantum states, thermodynamics, and curvature. From these, we further derive four key derived equations: a generalized Einstein field equation in the low-energy limit, a quantum-thermodynamic uncertainty relation, a relation between entropy production and curvature scalar, and a conservation law for quantum information current. The framework naturally recovers standard quantum mechanics, thermodynamics, and general relativity in appropriate limits, while offering new predictions about the relation between quantum entanglement and spacetime structure. All derivations are carried out in full mathematical detail, and the resulting equations are original, closed-form, and internally consistent.
Any finite-resolution description of a physical system incurs two costs: an information cost, quantifying microscopic complexity via relative entropy, and an ordering cost, quantifying geometric or structural organization. The Entropy–Order Balance (EOB) framework postulates that realized physical laws correspond to Lyapunov-stable equilibria of the total cost functional. Part I showed that the gravitational sector — Einstein's field equations and the cosmological constant — emerges as the stable ordering response. This paper shows that quantum mechanics emerges as the stable information response. Starting from the same five operational axioms applied to resolution windows with smooth Riemannian structure, supplemented by Lyapunov stability and two standard GPT-to-quantum hypotheses (reversible subgroup, local tomography/purification), the paper derives: State-space convexity from variational stability — a result that standard quantum reconstruction programs (Hardy 2001, Chiribella–D'Ariano–Perinotti 2011) assume as a postulate Hilbert space structure and completely positive trace-preserving (CPTP) dynamics Born rule probabilities from decision-theoretic stability over KL divergence Measurement as a barrier-dominance phenomenon: inter-sector transitions in the EOB functional are exponentially suppressed in apparatus size, producing effective state reduction without a collapse postulate A universal decoherence timescale τ_dec = d_S/c depending only on system dimension and the same constant c that determines Λ = cH₀² in Part I Pointer basis selection via Holevo quantity maximization Conditional extensions requiring additional AQFT inputs (Bisognano–Wichmann property, Haag–Kastler net structure) yield an area-law entropy exponent d = 2 and identification of the EOB balance scale with the Planck length. Part I (companion paper) derives Einstein's field equations and the cosmological constant from the same axiomatic framework.
We present a fundamental unified framework based on information geometry, in which quantum mechanics, statistical thermodynamics, and general relativity emerge from a single information-theoretic ontology. Starting from three rigorous axioms that establish quantum information as the primary physical substrate, we construct the universal information action functional using the Bures–Fisher–Rao metric on the manifold of quantum density matrices. Full variational derivation yields a unified field equation that governs the evolution of quantum information and emergent spacetime geometry. In appropriate limiting regimes, we recover the Schrödinger equation, equilibrium statistical mechanics, and the Einstein field equations. This work provides the foundational layer of a complete theory of everything based on the principle: geometry is information.
This work presents an extended, mathematically complete formulation of the Information Geometry Unification Principle (IGUP), a foundational framework in which quantum mechanics, statistical thermodynamics, general relativity, and spacetime geometry emerge from a single information-theoretic ontology. We begin with rigorous axioms defining information as the primary physical substance, then systematically construct the Bures–Fisher–Rao metric on the manifold of quantum density matrices. We derive the universal information action functional and perform full variational derivations to obtain the unified field equations. We show explicitly how standard physical theories including the Schrödinger equation, Heisenberg picture, Liouville mechanics, classical thermodynamics, and Einstein field equations are recovered in appropriate limiting regimes. We further derive geometric expressions for entanglement, entropy, temperature, energy, and force. The theory resolves longstanding conceptual tensions between unitary quantum evolution and gravitational irreversibility, provides a natural resolution to the black hole information paradox, and yields eight concrete, experimentally testable predictions. This extended version includes full tensor calculus, covariant derivatives, curvature expansions, and limit theorems, amounting to a self-contained theory of everything based on Geometry=Information.