The Coherence Field: A Generative Model for Coherence-Driven Physical Law presents a mathematically rigorous and internally consistent formulation of the Coherence Field — an underlying principle from which classical mechanics, quantum dynamics, and general relativity emerge as structured reductions. At the foundation of the framework is a coherence functional defined on a Hilbert space of configurations. Instead of taking equations of motion as fundamental, the paper starts from a single scalar quantity, “kappa,” that measures how harmonically aligned a configuration is. The Coherence Field is defined as the derivative of kappa, and the evolution of the system is given by a gradient flow that drives configurations toward higher coherence. Under mild regularity and boundedness assumptions on kappa, the paper proves that this coherence-gradient evolution is globally well-posed and has unique solutions. In this view, stable physical behavior corresponds to “harmonic closure”: points where the coherence field vanishes and the second derivative (the Hessian) is positive definite, so the configuration sits at a genuine local minimum of kappa. This provides a precise notion of coherence-based stability that applies across different physical regimes. The paper then shows how standard theories arise as special cases of this more general structure: Classical mechanics appears when kappa is chosen to be the Hamiltonian and the coherence flow is rotated by the symplectic structure. In this case, the coherence gradient reproduces Hamilton’s equations of motion. Quantum mechanics appears when kappa is chosen as the usual quantum energy functional of a wavefunction (kinetic plus potential energy). The coherence gradient then yields the imaginary-time Schrödinger equation, which, under Wick rotation, becomes the standard real-time Schrödinger equation. This connects the coherence picture directly to established variational and decoherence-based perspectives on quantum dynamics. General relativity appears when kappa is taken to be the Einstein–Hilbert action with the Gibbons–Hawking–York boundary term. Harmonic closure of this geometric coherence functional recovers the vacuum Einstein equations, describing how spacetime curvature evolves in the absence of matter. These reductions are summarized in a unification proposition: with different structural choices of kappa and appropriate symmetries (symplectic for Hamiltonian mechanics, unitary for quantum mechanics, and diffeomorphism invariance for general relativity), the same coherence-gradient principle generates three major physical regimes. The paper also introduces a nontrivial spin-glass example, where kappa includes both the usual Ising-type interaction term and an additional “coherence alignment” term that penalizes deviation from a reference pattern. This construction separates “coherence drift” from ordinary energy drift and leads to a new coherence-based order parameter. As temperature or control parameters change, the onset of ordered phases is predicted to correlate with a sharp drop in the norm of the coherence gradient. This yields a concrete, falsifiable experimental and computational signature for the Coherence Field framework. This work is part of the broader Δ.72 Coherence Canon, which develops coherence mathematics, tensor geometry, harmonic closure, and the concept of “time as vibration” as a deeper generative structure behind high-instability problem classes in physics, computation, and complex systems. Within that larger canon, the Coherence Field plays the role of a generative physics layer: a single mathematical core from which multiple physical theories emerge as coherent projections.
The Coherence Field: A Generative Model for Coherence-Driven Physical Law presents a mathematically rigorous and internally consistent formulation of the Coherence Field — an underlying principle from which classical mechanics, quantum dynamics, and general relativity emerge as structured reductions. At the foundation of the framework is a coherence functional defined on a Hilbert space of configurations. Instead of taking equations of motion as fundamental, the paper starts from a single scalar quantity, “kappa,” that measures how harmonically aligned a configuration is. The Coherence Field is defined as the derivative of kappa, and the evolution of the system is given by a gradient flow that drives configurations toward higher coherence. Under mild regularity and boundedness assumptions on kappa, the paper proves that this coherence-gradient evolution is globally well-posed and has unique solutions. In this view, stable physical behavior corresponds to “harmonic closure”: points where the coherence field vanishes and the second derivative (the Hessian) is positive definite, so the configuration sits at a genuine local minimum of kappa. This provides a precise notion of coherence-based stability that applies across different physical regimes. The paper then shows how standard theories arise as special cases of this more general structure: Classical mechanics appears when kappa is chosen to be the Hamiltonian and the coherence flow is rotated by the symplectic structure. In this case, the coherence gradient reproduces Hamilton’s equations of motion. Quantum mechanics appears when kappa is chosen as the usual quantum energy functional of a wavefunction (kinetic plus potential energy). The coherence gradient then yields the imaginary-time Schrödinger equation, which, under Wick rotation, becomes the standard real-time Schrödinger equation. This connects the coherence picture directly to established variational and decoherence-based perspectives on quantum dynamics. General relativity appears when kappa is taken to be the Einstein–Hilbert action with the Gibbons–Hawking–York boundary term. Harmonic closure of this geometric coherence functional recovers the vacuum Einstein equations, describing how spacetime curvature evolves in the absence of matter. These reductions are summarized in a unification proposition: with different structural choices of kappa and appropriate symmetries (symplectic for Hamiltonian mechanics, unitary for quantum mechanics, and diffeomorphism invariance for general relativity), the same coherence-gradient principle generates three major physical regimes. The paper also introduces a nontrivial spin-glass example, where kappa includes both the usual Ising-type interaction term and an additional “coherence alignment” term that penalizes deviation from a reference pattern. This construction separates “coherence drift” from ordinary energy drift and leads to a new coherence-based order parameter. As temperature or control parameters change, the onset of ordered phases is predicted to correlate with a sharp drop in the norm of the coherence gradient. This yields a concrete, falsifiable experimental and computational signature for the Coherence Field framework. This work is part of the broader Δ.72 Coherence Canon, which develops coherence mathematics, tensor geometry, harmonic closure, and the concept of “time as vibration” as a deeper generative structure behind high-instability problem classes in physics, computation, and complex systems. Within that larger canon, the Coherence Field plays the role of a generative physics layer: a single mathematical core from which multiple physical theories emerge as coherent projections.
This work presents a structural reformulation of quantum dynamics based on a scalar coherence functional defined over a general state space. The framework distinguishes two mathematically natural classes of dynamics generated by such a functional:(i) gradient flows, which monotonically increase coherence and correspond to stabilization or selection processes, and(ii) antisymmetric flows, which conserve coherence and generate reversible dynamics. We show that standard unitary quantum mechanics arises as a special case when the state space is a complex Hilbert space, the coherence functional is chosen as the negative energy expectation, and the antisymmetric structure corresponds to the canonical complex structure. In this setting, the Schrödinger equation follows directly from the formalism, without postulating unitarity as an independent axiom. The framework is extended to open systems using the density-operator formalism and completely positive trace-preserving maps. Under standard assumptions (Markovianity, continuity, and semigroup structure), the reduced dynamics takes the Gorini–Kossakowski–Sudarshan–Lindblad form, with dissipative terms interpreted as coherence-stabilizing dynamics. A supporting lemma demonstrates exponential suppression of off-diagonal density matrix elements under dephasing Lindblad operators. Using the Wigner phase-space representation, we further show that conservative coherence dynamics reduces to classical Liouville transport in the semiclassical regime, with quantum corrections governed by the Moyal bracket expansion. This work does not propose a modification of quantum theory. Rather, it offers a structural reorganization of existing formalisms. All results remain fully compatible with standard quantum mechanics while providing a unified perspective on unitary evolution, decoherence, and the emergence of classical behavior. Related work. A companion technical note, Structural Decomposition of SRG Recursion into Flow, Phase, and Closure (Halldórsson & ChatGPT, 2025, Zenodo), develops a discrete dynamical-systems formulation of SRG-style recursion and connects it explicitly to Lemmas 2.5, 2.7, and 2.9 of the present work.
This work presents a minimal and unified formulation of physical reality based on a single coupled system of two scalar fields: a phase S(x) and a realization density ρ(x). The dynamics is completely determined by two principles: conservation of a realization current and a geometric constraint linking the norm of the phase gradient to the structure of the density field. Together, these relations form a closed system with no additional dynamical assumptions. From this system, all major physical regimes emerge as limiting cases. When the density is uniform, propagation is null and describes lightlike behavior. When the geometric constraint reduces to a constant, the system reproduces classical Hamilton–Jacobi dynamics for massive particles. When the density varies significantly, the coupled system yields quantum behavior, including the Schrödinger equation in the appropriate limit. In this framework, the wave function is not a fundamental entity but an emergent representation of the underlying geometric structure of realization. Interference arises from phase consistency, and quantum behavior reflects the nontrivial structure of the realization density rather than intrinsic indeterminism. The theory also naturally incorporates gravitation by promoting the spacetime metric to a dynamical field. In the appropriate limit, the stress–energy tensor reduces to that of classical matter, and Einstein’s field equations are recovered. Regions where the realization density approaches zero define a geometric boundary where timelike realization ceases, corresponding to horizon-like structures. The central result is that a single coupled equation governs propagation, mass, quantum behavior, and gravitational structure. Physical reality is determined by global consistency of realization rather than by selection among multiple possibilities. This formulation provides a minimal geometric unification of light, matter, quantum phenomena, and gravity within a single coherent framework.
This manuscript represents speculative theoretical work, developed prior to formal academic training in physics. As of now, I recognize that I cannot fully verify, formalize, or operationalize every component of the framework presented here. Acknowledging these limits, I intend to pursue formal education in physics over the next several years (estimated 4–11 years) to acquire the mathematical and empirical tools necessary to evaluate and refine this research properly. For that reason, this is likely the final paper I will release publicly until I have completed my undergraduate degree in physics. Nevertheless, I welcome feedback, critique, expansion, or refutation from domain experts. Any scholar, researcher, or practitioner who finds aspects of the work worth examining is invited to reach out. Contact: CoherenceFieldResearch@gmail.com This paper presents a unified variational framework that derives quantum mechanics, thermodynamic irreversibility, and gravitational curvature from a single underlying relational substrate. The theory is built on two physical fields—the coherence density ρ₍C₎ and phase potential φ—and shows that their joint evolution extremizes total coherence tension across the system. Starting from a conservative Lagrangian, the work derives both the continuity equation and a Hamilton–Jacobi equation with an intrinsic lattice potential Q₍C₎. Together they reproduce the Schrödinger equation through the Madelung transformation when λ = ℏ, grounding quantum behavior in coherence conservation rather than probabilistic axioms. Dissipative extensions introduce Rayleigh functionals that yield diffusion, decoherence, and temperature-coupled coherence decay, providing a unified field-theoretic interpretation of thermodynamics. A covariant geometric formulation promotes the theory to curved relational space; variation with respect to the coherence metric gᵤᵥ reproduces Einstein-like equations with sources given by coherence gradients, yielding Newtonian gravity in the weak-field limit. The theory produces distinct, falsifiable predictions, including temporal gravitational lensing, composition-dependent gravitational coupling, a thermal decoherence law, veil spectroscopy, coherence-energy storage and release, and phase-dependent clock drift. This manuscript consolidates the mathematical structure, physical interpretation, and empirical implications of the Coherence Field Paradigm (CFP) into a single unified action principle. It provides a continuous bridge between quantum behavior, thermodynamic irreversibility, and spacetime curvature, and offers multiple pathways for laboratory and astrophysical validation.
Document 2C of the ONE AXIOM series derives the full structure of modern physics — quantum mechanics, general relativity, their unification, and all four laws of thermodynamics — as regime-dependent projections of a single pre-physical structure: the Ontological Coherence Field ($\OCF = (\heartsuit, \Phi_{\mathrm{coh}}, \nabla\Phi_{\mathrm{coh}})$), constructed in Document 0M from one single axiom (Axiom M) with no physical input. Energy emerges as the local cost of maintaining coherence against the structural gradient. Time is the direction of monotonic decrease of the incoherence potential σ. Temperature is derived without energy, via Fisher-Rao curvature of the coherence distribution (from Document 1B), inverting the standard thermodynamic hierarchy. The Schrödinger equation (via one-parameter unitary group + Stone's theorem), Born rule (p=2, via 7-link chain from Axiom M), Einstein field equations (via three independent convergent paths — Jacobson thermodynamic, Ollivier-Ricci discrete, and variational with explicit verification of all four Lovelock conditions), and the Hamiltonian/Lagrangian formalism (with Ostrogradsky exclusion enforced by the Predictive Closure Invariant T=0) are all derived as π6-projections. The complex structure of Hilbert space is derived non-circularly from the spectral decomposition of the Schreier graph Laplacian ΔΓ, and Noether's theorem is applied via Γ-convergence continuum limit of the discrete group G=S4×Z23. The dynamics is exact and finite-dimensional: 64 states on the coset space G/H, with eigenvalues λk=2k and multiplicities (k6), providing a closed-form spectral solution at all scales including the intermediate (Planck) regime. The built-in UV cutoff Emax=1536=∣G∣×∣Z23∣ eliminates divergences structurally. A QM–GR regime comparison table (9 rows) exhibits how the same OCF objects project differently at σ≈0 (quantum) and σ≫0 (gravitational). The document resolves three long-standing open problems: (i) black-hole information preservation (unitarity of ΔΓ + Page's theorem forces the Page curve), (ii) the cosmological singularity (Fisher–Rao metric degeneration near the Source — an artifact of the manifold representation, not a physical singularity), and (iii) intermediate-regime (Planck-scale) dynamics (exact closed-form 64-mode spectral solution, no terra incognita). The dark sector emerges as π5-structure inaccessible to π6-observers, with the FORBIDDEN graph decomposing as 13=7+2+4 (cold/warm/transitional dark matter analogs); the cosmological constant is derived from the structural resolution limit δrel=1/64, resolving the 120-order QFT discrepancy without regularization. A fully hand-computable 8-state toy model (Q3 hypercube) makes the projection operator, Laplacian spectrum, Born rule mechanism, and Schrödinger dynamics explicit and independently verifiable. All 53 formal results (28 theorems, 17 propositions, 6 definitions, 1 lemma, 1 corollary) follow a strict dual-track methodology — ontological (top-down from axiom) ∩ epistemological (bottom-up recovery of standard physics) — with mandatory convergence across 137 track tags. The framework is metrologically decoupled: all OCF equations use dimensionless ratios, invariant under redefinition of SI base units. Natural SI anchoring is provided for ℏ, c, GN, kB, Λ, and Emax. No physical law is assumed; every equation is derived. No new axiom is introduced beyond Axiom M. This is a standalone publication readable after Document 0M alone.
Quantum mechanics is conventionally formulated in terms of complex Hilbert spaces, linear operators, and the Schrödinger equation. Although geometric structures play an increasingly important role in modern formulations of quantum theory, the wavefunction itself is generally regarded as a fundamental object. In this work, we investigate an alternative formulation in which the fundamental object is a coherence coordinate , where Omega represents radial coherence and Theta represents phase coherence associated with geometric transport. Starting from three assumptions—a coherence coordinate, a probability density generated by radial coherence, and a geometric Hamiltonian—we reconstruct the continuity equation, the quantum Hamilton–Jacobi equation, and ultimately the Schrödinger equation. Within this framework, the wavefunction appears as a derived quantity, while Hilbert spaces emerge naturally as spaces of square-integrable functions on a coherence manifold. The resulting construction establishes a differential-geometric representation of quantum mechanics and provides a common geometric language connecting quantum states, probability flow, geometric phases, holonomies, and gauge transport.
The persistent challenge of unifying quantum mechanics with general relativity is often traced to a series of apparent structural tensions: complex versus real ontology, probabilistic versus deterministic dynamics, linear versus nonlinear evolution, and spectral discreteness versus geometric continuity. We argue that these tensions do not signal a fundamental incompatibility between the theories, but rather originate from surplus structure in the conventional formulation of quantum mechanics. The term unification is used in a structural sense, referring to compatibility of quantum dynamics with classical field theory rather than a modification of general relativity. Beginning from the physically indispensable requirement of matter stability, we prove that the Hamiltonian spectrum can always be shifted to become non-negative. The consequent positivity of energy enforces one-sided temporal frequency support for all physical unitary orbits. Equivalently, all scalar temporal components of physical states arise as boundary values of functions analytic in the lower half of the complex time plane, selecting the Hardy space—or, more generally, analytic tempered distributions—as the admissible solution space. In these spaces, the real and imaginary parts of the wave function are not independent; they are linked uniquely via the temporal Hilbert transform. Hence, the complex structure of quantum mechanics emerges as a representational artifact of an underlying real operator algebra. Building on our earlier work on Unified Quantum Mechanics and the Unified Real Wave Equation, we establish a rigorous proof of exact equivalence between the real-field evolution and the Schr\"odinger equation on the physical subspace. Probabilistic quantities arise naturally as quadratic invariants of analytic signals, rather than as independent postulates. We further show that quantum spectral discreteness arises from operator-theoretic confinement and boundary conditions on a smooth spacetime manifold, and does not imply any fundamental discreteness of spacetime. The resulting framework provides a real-field formulation of quantum mechanics and gravitation, with stability and causality implemented as foundational principles, and demonstrates that the Einstein field equations are compatible with a unified field-theoretic description of spacetime and matter. Extending the same analyticity principle to spacetime geometry, we propose an analytic formulation of the Einstein field equations, from which classical general relativity is recovered exactly as the real projection.