The physical world sometimes manifests as force (irreversible energy transfer), sometimes as geometry (local phase constraints without energy transfer), and sometimes as entanglement (nonlocal phase correlations without energy transfer). This paper proves that these three representations are three gauge projections of a single underlying structure---the order parameter space metric---at different values of the entropy production rate. Starting from CTP (Closed Time Path) nonequilibrium field theory, we rigorously derive the universal form of the order parameter effective action in the infrared slow-dynamics limit. The metric g_ii measures the "irreversible distance" between adjacent points in order parameter space, and its singularity structure exhaustively classifies all physical effects into five projection types: dynamical projection, critical slowing-down, instantaneous equilibrium, frozen projection, and cyclic projection. The variational principle yields a unified irreversible equation of motion containing an exclusive dynamical prediction---the metric gradient force. The information bottleneck theorem rigorously proves that in the frozen projection, the quantum mutual information irreversibly written into the environment is identically zero, yielding the thermodynamic criterion for the quantum-classical boundary. The core contribution of this paper is a unified framework with "information-entropy production rate" as its criterion. It provides four exclusive quantitative predictions testable on platforms such as cold atoms and superconducting qubits. This paper is the third in a series. The microscopic foundation is provided in a companion paper (DOI: 10.5281/zenodo.21325754), and the axiomatic framework is established in a second companion paper (DOI: 10.5281/zenodo.21325460).
The physical world sometimes manifests as force (irreversible energy transfer), sometimes as geometry (local phase constraints without energy transfer), and sometimes as entanglement (nonlocal phase correlations without energy transfer). This paper proves that these three representations are three gauge projections of a single underlying structure---the order parameter space metric---at different values of the entropy production rate. Starting from CTP (Closed Time Path) nonequilibrium field theory, we rigorously derive the universal form of the order parameter effective action in the infrared slow-dynamics limit. The metric g_ii measures the "irreversible distance" between adjacent points in order parameter space, and its singularity structure exhaustively classifies all physical effects into five projection types: dynamical projection, critical slowing-down, instantaneous equilibrium, frozen projection, and cyclic projection. The variational principle yields a unified irreversible equation of motion containing an exclusive dynamical prediction---the metric gradient force. The information bottleneck theorem rigorously proves that in the frozen projection, the quantum mutual information irreversibly written into the environment is identically zero, yielding the thermodynamic criterion for the quantum-classical boundary. The core contribution of this paper is a unified framework with "information-entropy production rate" as its criterion. It provides four exclusive quantitative predictions testable on platforms such as cold atoms and superconducting qubits. This paper is the third in a series. The microscopic foundation is provided in a companion paper (DOI: 10.5281/zenodo.21325754), and the axiomatic framework is established in a second companion paper (DOI: 10.5281/zenodo.21325460).
A central challenge at the frontier of theoretical physics is transforming qualitative analogies connecting quantum information, spacetime geometry, non-equilibrium thermodynamics, and quantum measurement into exact, falsifiable dynamical equations with proven domains of validity. In this paper, we establish Unified Information–Geometry–Dynamics Physics (UIGDP), a deductive physical framework anchored in our 71-page ZFC Master Reference Monograph (Lee, 2026, DOI: 10.5281/zenodo.23486681) [1] and synthesized with the Extended Quantum Information Geometry System (EQIGS, DOI: 10.5281/zenodo.23557698) [6], the High-Dimensional Geometric Probability System (HDGPS, DOI: 10.5281/zenodo.23557360) [4], and the Dynamic Topological Critical Transition System (DTCTS, DOI: 10.5281/zenodo.23557507) [5]. Condensing local Type III₁ quantum field algebras onto the Takesaki Type II₁ ground factor (ℳ_ground, τ_ground) governed by the universal quadruple 𝔘 = (𝒯_0x8A2C, 𝐂_UPTC, η_friction, ϵ_SHID), we formulate six core axioms (UIGDP-1 through UIGDP-6) and prove six fundamental theorems: (1) Necessary and Sufficient Curvature–Entanglement Conditions for reconstructing spacetime metric tensors g_μν^spacetime from Petz quantum Fisher information Hessians; (2) The Exact Unified IGD Master Cost Identity and Optimal Finite-Time Control Horizon τ⋆ = √((d_g² + Δ_ctrl²)/(2η_friction)); (3) Emergent Einstein–Cartan–Information Field Equations from modular entanglement equilibrium and Area-to-Volume phase transitions at D_crit = e^(𝐂_UPTC); (4) Dynamical Resolution of Quantum Measurement Projection via the URRHC Idempotent Monad 𝕄_URRHC (T² = T) with a testable dephasing floor; (5) Universal State Classification for Far-from-Equilibrium and Active Matter via Thermodynamic Length and Topological Persistence Dissipation; and (6) Quantum Simulator Advantage Thresholds under noise and control deficits. Complete Python numerical verification is provided in Appendix A.
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.
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.
This paper proposes a fundamental framework based on information geometry that unifies quantum mechanics, thermodynamics, and spacetime geometry into a single foundational theory. Starting from three elementary axioms regarding information as the primary physical ontology, we derive the universal information metric, construct the information action functional, and show that all known physical laws emerge as extremal conditions of this functional. The theory recovers standard quantum mechanics, classical thermodynamics, and general relativity in appropriate limits, while predicting new relations between quantum entanglement, gravitational curvature, and entropy. We conclude that all physical phenomena—matter, energy, spacetime, and interactions—are equivalent to geometric structures on the space of quantum information states.
Abstract We propose physical distinguishability as the fundamental primitive underlying quantum mechanics, general relativity, and thermodynamics. A state is defined operationally: two states are physically distinct if a physical interaction or measurement can reveal a difference between them. From this single principle, the distinguishability divergence D generates a smooth state space endowed with a Riemannian metric , a geometric velocity, an accessibility scale Λ, an accessibility ratio , and a geometric time . The framework provides a unified geometric description of the three major domains of physics. Quantum mechanics emerges through the Fubini–Study geometry of projective Hilbert space, thermodynamics through Fisher–Rao and Ruppeiner information geometry, and general relativity through the geometry of geodesic deviation. The threshold condition A=1acquires a universal physical meaning and appears in all three realizations as the boundary between physically accessible and inaccessible evolution. Variational formulations of both the Einstein and Schrödinger equations are constructed within the same geometric framework. The Lorentzian signature of spacetime emerges from a single sign reversal associated with the geometric-time direction, while the coupling constant of the distinguishability action is connected to the observed vacuum energy density. The framework further predicts a characteristic vacuum distinguishability scale ≈8.8×10^(-5) m, corresponding to an energy scale≈2.25" " meV. The theory is tested experimentally on IBM Quantum superconducting processors. Across one-qubit, two-qubit, 127-qubit, and 156-qubit experiments performed on three independent Heron-generation backends, excitation probabilities collapse onto universal curves when parameterized by endpoint Fubini–Study distinguishability. Protocols with identical laboratory control parameters but different endpoint distinguishabilities produce excitation probabilities differing by up to a factor of 3.8, while protocols with identical distinguishability produce identical outcomes across independent processors. Cross-backend transfer errors remain below 1%, providing direct experimental support for endpoint distinguishability as a universal physical variable. Taken together, the theoretical construction, variational formulations, vacuum-sector predictions, and large-scale quantum-processor experiments support distinguishability geometry as a unified and experimentally testable foundation for physical dynamics.
Building on the evolutionary time field theory, this paper proves that the core mathematical structures of quantum mechanics can be derived from the constrained geometry of the evolutionary time field. Starting from the symplectic structure of the constrained geometry of the evolutionary time density and the order parameter, we establish effective canonical commutation relations and derive the time-energy uncertainty relation. The quantum excited states form a Fock space, whose linear structure yields the superposition principle, and the effective dynamics reduces to the standard Schrödinger equation in the flat-space limit. The measurement postulate corresponds to the coupling of the order parameter to an external dissipative channel—the contrapositive of the information bottleneck theorem applies directly, and the Born rule is reduced to an application of the Principle of Minimum Irreversible Cost. This paper further reveals the topological origin of the imaginary unit i, the geometric nonlocality mechanism of many-body entanglement, and the collapse threshold dissipation effect. The conclusion is that the core structures of quantum mechanics—Hilbert space, operator algebra, superposition principle, uncertainty relation, Schrödinger equation, measurement postulate, and Born rule—can all be derived from the constrained geometry of the evolutionary time field. This paper is the tenth in a series. The evolutionary time field theory is established in a companion paper (DOI: 10.5281/zenodo.21436542), the gauge field origin in a second (DOI: 10.5281/zenodo.21438149), the Hawking radiation quantum effects in a third (DOI: 10.5281/zenodo.21436715), and the axiomatic framework in a fourth (DOI: 10.5281/zenodo.21325460).