Entropy as First Principle: Deriving Quantum and Gravitational Structure from Thermodynamic Geometry

1. Introduction

Quantum mechanics (QM), despite its unrivaled empirical success, is traditionally introduced through axioms whose deeper origins remain opaque [1,2]. Core elements—the Born rule, wavefunction collapse, and quantization—are postulated rather than derived, leaving foundational questions unresolved [3,4,5].

The Total Entropic Quantity (TEQ) framework proposes a reversal of this structure: quantum behavior is not fundamental, but emerges from entropy. Rather than treating entropy as a secondary thermodynamic quantity, TEQ reinterprets it as the generative constraint governing all physical evolution. Quantum coherence, measurement, and classicality arise from a variational principle in which entropy and action jointly constrain viable trajectories.

At the core of TEQ is a decomposition of entropy into three dynamically coupled components:

• Realized entropy — local macroscopic disorder;
• Latent entangled entropy — nonlocal quantum correlations;
• Latent classical entropy — quasi-stable, decohered structure encoding memory.

This triadic view reframes the quantum-to-classical transition as a continuous redistribution of entropy, eliminating the need for discontinuous collapse or observer-centric postulates. TEQ further introduces the concept of entropy dimensions $D_S$, classifying the effective dimensionality of entropy flow:

• $D_S=0$: nonlocal entangled phase space;
• $D_S\in [2,3)$: latent classical correlations (e.g., pointer states);
• $D_S\ge 4$: realized macroscopic structure and spacetime irreversibility.

We reformulate TEQ around a single meta-constraint that replaces traditional axioms:

Minimal Principle (MP):

Physical trajectories are those that remain maximally distinguishable relative to the entropy dimensionality $D_S$ of their domain. (See Section 8 for a unified summary of the principle and its structural consequences.)

This principle defines physical evolution through geometric constraints. Entropy structure and resolution together determine which configurations persist. From this, we derive:

• Derivation of an entropy-weighted path integral from first principles;
• The emergence of $\beta$ as a Lagrange multiplier governing entropy weighting;
• The Born rule and its entropy-curvature corrections;
• The Schrödinger equation and quantization as stability conditions;
• Classicality as the smooth-limit behavior of entropy geometry.

The path integral evaluates relative distinguishability among trajectories, with entropy gradients acting as geometric filters that select thermodynamically viable paths; the associated entropy curvature, which governs deformation of phase structure, is defined formally in Section 4.

Unlike other entropy-based approaches—such as Entropic Dynamics [6] or Jaynesian inference [7]—TEQ does not assume standard quantum structure. It predicts testable deviations in regimes of strong entropy curvature, including:

• Gravitational and cosmological quantum systems [8,9,10];
• Ultra-isolated and far-from-equilibrium quantum systems;
• Nonlinear decoherence and entropy-sensitive corrections to standard dynamics [4,5].

These include modified Born statistics, altered coherence times, and constraints on quantization tied to entropy geometry. TEQ thus opens a new route to probing the entropic foundations of quantum theory.

We refer to the geometrothermodynamic limit as the regime in which entropy curvature governs the dominant structural behavior, allowing simplified descriptions where local fluctuations are subordinated to large-scale thermodynamic geometry. Many of the results in this manuscript, such as quantization, classical emergence, and gravitation, are derived as structural consequences within this regime.

Remark 3 

(Conceptual Intuition and Broader Significance). At its core, the TEQ framework proposes that physical structure arises from the geometry of entropy itself. Instead of treating quantum laws and gravitational dynamics as fundamental, TEQ starts with a simpler idea: systems evolve along paths that remain maximally distinguishable under entropic constraints. This guiding principle leads to the emergence of quantum amplitudes, spacetime structure, and even fundamental constants, as consequences of a deeper entropic geometry.

This perspective matters beyond physics. It suggests that the very notion of “what exists” depends on what can be resolved or distinguished—an idea with deep resonance in information theory, complexity science, and the philosophy of measurement. In TEQ, quantization is not imposed but arises from structural limits on distinguishability. Entropy is not a measure of ignorance, but a generative principle that shapes what can be known, stabilized, or predicted. This reorientation offers a new foundation for understanding coherence, irreversibility, and the emergence of classical reality.

Structure of the manuscript. Section 2 introduces the entropy-weighted path integral and defines the effective action. Section 3 derives structural relations among ℏ, $\beta$, and $k_B$, identifying a two-dimensional dynamical surface. Section 4 interprets quantization as a geometric stability condition. Section 5 derives the Born rule and formulates corrections in high-curvature regimes. Section 6 presents the Schrödinger equation as a limit of entropy-stabilized evolution. Section 7 proposes an entropic basis for spacetime and gravity.

Structure of the manuscript. Section 2 introduces the entropy-weighted path integral and defines the effective action. Section 3 derives structural relations among ℏ, $\beta$, and $k_B$, identifying a two-dimensional dynamical surface. Section 4 interprets quantization as a geometric stability condition. Section 5 derives the Born rule and formulates corrections in high-curvature regimes. Section 6 presents the Schrödinger equation as a limit of entropy-stabilized evolution. Section 7 proposes an entropic basis for spacetime and gravity.

Navigating TEQ

For orientation, we provide two reference tools: a structural overview diagram and a glossary of key terms. Figure 1 illustrates how the core axioms generate the major results of the theory, clarifying the dependency structure between assumptions, derivations, and physical consequences. The accompanying table summarizes central concepts, helping to reduce interpretive load–especially for interdisciplinary readers.

To further support conceptual clarity, Table 1 summarizes key terms used throughout the manuscript, with a focus on their operational meaning within the TEQ framework.

Dynamical Regimes in TEQ. To clarify how standard physical behavior emerges from the TEQ framework, we introduce a structural taxonomy of dynamical regimes based on the dominance of entropy versus action and the geometry of entropy flow. This classification helps distinguish the quantum limit, classical behavior, and novel entropy-dominant phases predicted by TEQ. Table 2 below summarizes these regimes and their associated features.

2. The Entropy-Weighted Feynman Path Integral from a Minimal Principle

This section derives the Feynman path integral as a consequence of TEQ’s structural axioms. Rather than postulating quantum behavior, we show that entropy-weighted amplitudes emerge from the Minimal Principle (MP), which governs stability under finite resolution. Classical action is generalized into a geometric entropy functional, and the resulting dynamics favor entropy-resilient trajectories. The standard quantum amplitude arises in the limit of vanishing entropy curvature. By embedding thermodynamic stability into the variational structure, this approach recasts quantization as a response to entropy-induced instability in trajectory space.

In TEQ, quantum structure is not assumed—it is derived from a single variational constraint:

Minimal Principle (MP):

Physical trajectories maximize distinguishability of entropy flow under structural constraints.

This principle unifies path selection, thermodynamic stability, and quantum amplitudes. The Feynman path integral emerges naturally as a statistical expression of entropy-weighted distinguishability, generalizing classical action into entropy geometry.

2.1. Standard Amplitude Formulation

In standard quantum mechanics, the transition amplitude is given by:

$$\mathcal{A}\propto \int \mathcal{D}\left[\varphi \right]e^{iS\left[\varphi \right]/\hslash },$$

(1)

where $S\left[\varphi \right]$ is the classical action, and $\mathcal{D}\left[\varphi \right]$ integrates over all histories. All paths contribute equally in magnitude; only phase varies [11]. This implicitly assumes no entropy-based preference among paths.

In TEQ, uniform amplitude is recovered only when entropy curvature vanishes. In general, entropy-instability leads to suppression: paths with higher entropy production contribute less. This is not added by hand—it arises directly from the Minimal Principle. Decoherence hints at this mechanism: paths differing significantly in entropy production interfere less, biasing evolution toward entropy-stable trajectories.

2.2. Entropic Weighting as a Consequence of MP

Lemma 1 

(Entropy Bias from the Minimal Principle). The Minimal Principle selects trajectories based on entropy stability. This weighting parallels the Feynman–Vernon influence functional [12]:

$$\mathcal{A}\propto \int \mathcal{D}\left[\varphi \right]e^{iS\left[\varphi \right]/\hslash }\mathcal{F}\left[\varphi \right],\mathcal{F}\left[\varphi \right]=e^{-\mathcal{I}\left[\varphi \right]},$$

(2)

where $\mathcal{I}\left[\varphi \right]$ encodes entropy-producing interactions, structurally analogous to decoherence functionals in open-system quantum theory [4,13].

TEQ does not postulate this suppression—it derives it (see Appendix A). Entropy weighting follows naturally from constrained optimization of distinguishability under MP, introducing the Lagrange multiplier $\beta$. The entropy functional ${\tilde{S}}_{\mathrm{apparent}}\left[\varphi \right]$, defined below, captures the observer-resolvable entropy flow.

2.3. Entropy-Weighted Amplitudes from MP

In Appendix A.5, we derive:

Theorem 1 

(Path Amplitudes from the Minimal Principle). The effective amplitude for a path ϕ is:

$${\mathcal{A}}_{\mathrm{eff}}\left[\varphi \right]\propto exp\left({\displaystyle \frac{i}{\hslash }}S\left[\varphi \right]-\beta {\tilde{S}}_{\mathit{apparent}}\left[\varphi \right]\right),$$

(3)

where $\beta \in \mathbb{C}$ arises from constrained variation. The classical Feynman integral is recovered in the limit $\beta \to 0$.

The parameter $\beta$ interpolates between dynamical regimes. When $\beta ={\displaystyle \frac{i}{\hslash }}$, standard unitary evolution emerges; when $\beta ={\displaystyle \frac{1}{k_{B}T}}$, the system enters a thermal regime. These are not independent assumptions but limiting cases of a unified structure.

As derived in Appendix B, the effective action becomes:

$$S_{\mathrm{eff}}\left[\varphi \right]=\int dt\left(L(\varphi ,\dot{\varphi })-i\hslash \beta g(\varphi ,\dot{\varphi })\right),$$

(4)

where $g(\varphi ,\dot{\varphi })$ is the entropy flux functional, arising naturally from the local geometry of the entropy metric.

Remark 4 

(Entropy Weighting, Measure Instability, and Path Selection). In standard quantum mechanics, all paths contribute equally to the path integral, differing only by phase. TEQ introduces entropy flow as a geometric constraint that modifies this principle. Trajectories are no longer treated symmetrically: paths that lead to steep entropy gradients are exponentially suppressed, as they imply a loss of stable, resolvable distinctions. The entropy weighting term $e^{-\beta {\tilde{S}}_{\mathit{apparent}}\left[\varphi \right]}$ breaks the uniformity of the measure, introducing exponential sensitivity to entropy curvature.

In regimes of strong entropy variation, the unmodified path integral becomes unstable: infinitesimally close trajectories can differ exponentially in weight due to high entropy curvature, rendering the standard measure ill-defined. TEQ resolves this by selecting only entropy-stationary trajectories, around which the weighting becomes sharply peaked and stable. This eliminates the singular behavior and motivates a deformation of the variational geometry itself. Quantization thus emerges not as an axiom, but as a structural response to entropy-induced instability in trajectory space.

2.4. Interpretation and Structural Role of $\beta$

The constants ℏ and $\beta$ emerge as dual Lagrange multipliers (see Appendix A.5):

• $\beta$ suppresses entropy-unstable paths;
• ℏ preserves phase coherence;
• Together, they govern entropy-weighted quantum dynamics.

Corollary 1 

(Born Rule from the Minimal Principle). The Born rule arises naturally in the large-β limit from suppression of entropically unstable paths. Probabilities reflect stability, not fundamental indeterminism. See Section 5.

Paradigm Shift. Quantum amplitudes emerge as structural consequences of entropy-weighted distinguishability. The Minimal Principle selects viable trajectories. $\beta$ and ℏ are thus structural multipliers—not independent parameters.

2.5. Modified Euler–Lagrange Equation

Theorem 2 

(Entropy-Corrected Dynamics from MP). Variation of the entropy-weighted action (4) yields:

$${\displaystyle \frac{d}{dt}}\left({\displaystyle \frac{\partial L}{\partial \dot{\varphi }}}-i\hslash \beta {\displaystyle \frac{\partial g}{\partial \dot{\varphi }}}\right)-\left({\displaystyle \frac{\partial L}{\partial \varphi }}-i\hslash \beta {\displaystyle \frac{\partial g}{\partial \varphi }}\right)=0.$$

(5)

This extends classical extremality by introducing geometric bias favoring entropy-stable trajectories.

Remark 5 

(Intuitive Meaning of Entropy Curvature). Entropy curvature measures how rapidly entropy production changes as we move between different paths. Intuitively, high entropy curvature corresponds to regions where small changes in trajectory lead to large changes in distinguishability. The Minimal Principle favors paths that remain stable under these variations—effectively selecting trajectories that avoid steep entropy gradients. Thus, entropy curvature functions like a landscape of stability: physical paths settle into entropy-stable valleys rather than entropy-unstable peaks.

In the limit $\beta \to 0$, standard quantum or classical mechanics is recovered.

Corollary 2 

(Entropy-Stabilized Classical Emergence). As $\hslash \to 0$, the classical Euler–Lagrange equation reemerges, but entropy curvature continues filtering viable macroscopic paths. Classicality is thus a stabilized limit of entropy geometry.

Paradigm Shift. The least-action principle is generalized by the Minimal Principle. Physical evolution selects entropy-resilient paths. Classical mechanics is thus not fundamental but emerges naturally as the stable, entropy-filtered limit of quantum dynamics.

3. Geometric Structure of Fundamental Constants

This section analyzes how the apparent constants of nature—Planck’s constant ℏ, the inverse temperature parameter β, and Boltzmann’s constant $k_B$—emerge within TEQ as projections of a unified entropy-action geometry. Rather than being fixed, irreducible inputs, these quantities reflect structural relationships that govern resolvability and coherence under entropy curvature. By making this geometry explicit, we show that these constants are neither fundamental nor independent, but arise as limiting behaviors within a minimal two-dimensional framework. The result is a principled reinterpretation of physical constants as emergent from the same entropic structure that underlies quantum and thermal dynamics.

In the Total Entropic Quantity (TEQ) framework, the constants ℏ, $\beta$, and $k_B$ are not fundamental. They emerge as structural parameters governing entropy-resolved path distinguishability.

Remark 6 

(Why Constants Are Not Fundamental). In traditional physics, constants such as ℏ, β, and $k_B$ are taken as irreducible inputs—fixed quantities defining the behavior of quantum, thermal, and statistical systems. In TEQ, these constants instead emerge from the geometry of entropy and action. They are not intrinsic to the universe, but arise as structural parameters that encode how finely distinctions can be resolved, and how entropy suppresses or stabilizes certain paths. Their apparent constancy reflects the stability of entropy geometry under specific regimes, not fundamental necessity.

This structure follows from two generative assumptions:

• Axiom 0: Entropy defines a geometric field over possible configurations, establishing resolvability;
• Axiom 1 (Minimal Principle): Physical paths remain maximally distinguishable within this entropy geometry.

From this foundation, the constants ℏ, $\beta$, and $k_B$ emerge as interdependent projections within a unified geometric structure.

Minimal Principle and Entropic Selector

Consider first the entropy-weighted amplitude introduced in TEQ:

$$\mathcal{A}\left[\varphi \right]=exp\left({\displaystyle \frac{i}{\hslash }}S\left[\varphi \right]-\beta {\tilde{S}}_{\mathrm{apparent}}\left[\varphi \right]\right).$$

This amplitude structurally distinguishes two fundamental aspects:

• an imaginary component, governing quantum coherence via the action $S\left[\varphi \right]$;
• a real component, encoding thermodynamic irreversibility via entropy ${\tilde{S}}_{\mathrm{apparent}}\left[\varphi \right]$.

This duality strongly suggests a simple geometric representation in terms of a two-dimensional entropy-action plane. Specifically, we represent amplitudes through linear projections onto this plane using a complex selector vector $\overrightarrow{\xi }\in {\mathbb{C}}^2$:

$$\overrightarrow{\xi }=\lambda \left[\begin{array}{c}{e}^{i\theta }\\ -e^{-i\theta }\end{array}\right]=\lambda \left[\begin{array}{c}cos\theta +isin\theta \\ -cos\theta +isin\theta \end{array}\right],$$

where the magnitude $\lambda \in {\mathbb{R}}^{+}$ sets the overall scale, and the selector angle $\theta \in [0,{\displaystyle \frac{\pi }{2}}]$ modulates the balance between quantum coherence (imaginary) and entropy-driven dissipation (real).

Remark 7 

(Rationale for the Linear Geometric Assumption). The choice of a linear two-dimensional geometry to represent the entropy-action structure is not arbitrary, but is motivated by simplicity and direct correspondence to the inherently real and imaginary decomposition of amplitudes. While nonlinear or higher-dimensional generalizations could be considered—particularly in regimes of strong entropy curvature or complex multi-scale interactions—the linear geometric assumption constitutes the simplest, conceptually transparent, and analytically tractable choice that fully captures the dual role of entropy and action. Future extensions might explore nonlinear geometric structures or higher-dimensional selectors, particularly where empirical deviations from standard quantum predictions necessitate more complex geometries.

Matching this geometric selector explicitly to the amplitude structure:

$${\displaystyle \frac{i}{\hslash }}=\lambda (cos\theta +isin\theta ),-\beta =\lambda (-cos\theta +isin\theta ),$$

we immediately recover the fundamental parameters in geometric form:

$${\displaystyle \frac{1}{\hslash }}=\lambda sin\theta ,\beta =\lambda cos\theta .$$

Thus, the dimensionless entropy-resolution parameter emerges as:

$$\alpha :=\beta \hslash =cot\theta .$$

This parameterization gives a direct geometric interpretation of $\theta$ as a fundamental dial tuning the balance between quantum coherence and entropic dissipation:

• Quantum-coherent limit ($\theta \to \pi /2$): entropy curvature vanishes; dynamics become unitary.
• Classical-thermal limit ($\theta \to 0$): entropy dominates, coherence is suppressed, and dynamics become fully dissipative.

The selector vector, therefore, is not introduced as an independent postulate. Rather, it explicitly encodes a structure already implicitly present in the TEQ amplitude. Making this structure explicit clarifies the underlying geometric unity, illustrating clearly how the constants ℏ, $\beta$, and $k_B$ emerge naturally from a minimal geometric assumption.

Figure 2. Entropy resolution parameter $\alpha =cot\left(\theta \right)$ as a function of selector angle $\theta$. The quantum limit corresponds to $\theta \to \pi /2$, the classical/thermal limit to $\theta \to 0$.

Figure 2. Entropy resolution parameter $\alpha =cot\left(\theta \right)$ as a function of selector angle $\theta$. The quantum limit corresponds to $\theta \to \pi /2$, the classical/thermal limit to $\theta \to 0$.

Dimensional Consistency and the Role of $k_B$

To restore thermodynamic units, define:

$$\alpha :=\beta \hslash ={\displaystyle \frac{\hslash }{k_{B}T}}\Rightarrow \beta ={\displaystyle \frac{\hslash }{\alpha k_B}}.$$

Theorem 3 

(Unified Constants from Entropy Geometry). The constants ℏ, β, and $k_B$ arise from a two-dimensional entropy-action geometry:

$${\displaystyle \frac{1}{\hslash }}=\lambda sin\theta ,\beta =\lambda cos\theta ,\alpha =cot\theta .$$

Corollary 3 

(Entropy-Coherence Scaling and Limiting Regimes). The dimensionless parameter $\alpha =cot\theta$ governs the balance between entropy suppression and phase coherence. It defines the dynamical regime of the system, with limiting behaviors:

• Quantum (unitary): $\alpha =i\Rightarrow \beta ={\displaystyle \frac{i}{\hslash }}$;
• Thermal: $\alpha ={\displaystyle \frac{\hslash }{k_{B}T}}\ll 1$;
• Intermediate: $\beta \in \mathbb{C}$, partial coherence regimes.

Feynman and Boltzmann weights emerge as limits of the same entropy-weighted structure, and the constants ℏ and $k_B$ act as scale-setting factors for phase coherence and entropy resolution. Representative values are summarized in Table 4.

Figure 3. Entropy—coherence spectrum in the complex $\beta$ plane. The grayscale intensity encodes relative coherence strength, with higher values corresponding to stronger phase coherence. The imaginary axis captures coherent dynamics ($\beta ={\displaystyle \frac{i}{\hslash }}$), while the real axis corresponds to entropy-dominated suppression regimes ($\beta \in {\mathbb{R}}^{+}$). The plot illustrates how TEQ interpolates between quantum and thermodynamic behavior through variations in entropy geometry.

Figure 3. Entropy—coherence spectrum in the complex $\beta$ plane. The grayscale intensity encodes relative coherence strength, with higher values corresponding to stronger phase coherence. The imaginary axis captures coherent dynamics ($\beta ={\displaystyle \frac{i}{\hslash }}$), while the real axis corresponds to entropy-dominated suppression regimes ($\beta \in {\mathbb{R}}^{+}$). The plot illustrates how TEQ interpolates between quantum and thermodynamic behavior through variations in entropy geometry.

Remark on the Emergence of Unitarity. In TEQ, unitarity is not a fundamental postulate but an emergent behavior in the flat-curvature, coherence-dominated limit of the entropy-weighted action. It arises when entropy flow vanishes and the system approaches maximal resolution, with $\beta ={\displaystyle \frac{i}{\hslash }}$. In this regime, amplitudes evolve purely oscillatory and probability is conserved through unitary dynamics.

As entropy gradients activate and curvature grows, coherence is lost and evolution becomes non-unitary. Dissipation, decoherence, and irreversibility are thus not anomalies to be explained away, but intrinsic consequences of entropy-weighted structure. Conservation is not violated, but generalized: replaced by structural distinguishability across entropy-curved paths.

TEQ implies that unitarity is not generic, but a special case realized only when entropy curvature vanishes. In strongly curved regimes—such as early cosmology, black hole evaporation, or macroscopic decoherence—non-unitary evolution is structurally expected. See Appendix F for standard approaches to handling non-unitarity.

Observer Resolution and Gauge Structure

The parameter $\alpha =\beta \hslash$ encodes the entropy resolution available to the observer—how finely entropy-distinguishable structure can be resolved. Shifts in $\theta$ act as gauge transformations in the entropy geometry [14,15], leaving the underlying dynamics invariant but altering the resolution scale.

Structural Role of Axiom 0. The geometric relationships among ℏ, $\beta$, and $k_B$ are not imposed. They arise from a prestructured entropy geometry (Axiom 0), constrained by the Minimal Principle (Axiom 1). Constants are projections of resolution and coherence within this geometry.

4. Quantization as Entropic Deformation of Symplectic Geometry

This section derives quantization as a geometric consequence of entropy-stabilized dynamics. In TEQ, classical symplectic geometry becomes unstable under entropy curvature, and the variational structure deforms to preserve distinguishability. This deformation leads to modified conjugate momenta, noncanonical Poisson brackets, and ultimately to quantum commutators. Quantization thus arises not from postulates or operator formalism, but from the failure of classical resolution in entropy-curved phase space. By tracing how entropy flow modifies the effective action and phase structure, we show that noncommutativity encodes the minimal distinguishability permitted by entropy geometry.

In TEQ, quantization and noncommutativity are not postulated but arise as necessary geometric consequences of entropy structure. The central idea is that entropy flow governs the resolvability of trajectories. When entropy curvature $\kappa \ne 0$, the underlying geometry of phase space is no longer flat and classical symplectic structure fails to describe stable evolution. Instead, TEQ imposes a variational constraint that selects only those trajectories which remain distinguishable under entropy deformation.

This modifies the effective phase space geometry, leading to a generalized conjugate momentum, deformed Poisson brackets, and eventually noncommutative operator structure. Quantization thus becomes a stability condition rooted in thermodynamic distinguishability, not an externally imposed axiom.

4.1. Entropy-Weighted Dynamics and Effective Momentum

Lemma 2 

(Entropy-Weighted Momentum). Given the entropy-weighted effective action

$$S_{\mathrm{eff}}\left[\varphi \right]=\int dt\left[L(\varphi ,\dot{\varphi })-i\hslash \beta g(\varphi ,\dot{\varphi })\right],$$

the effective momentum is

$${\mathsf{\Pi }}_{\mathit{eff}}={\displaystyle \frac{\partial L}{\partial \dot{\varphi }}}-i\hslash \beta {\displaystyle \frac{\partial g}{\partial \dot{\varphi }}},$$

generalizing classical conjugate momentum by incorporating entropy flow as a stabilizing term.

Remark on Hermitian and Non-Hermitian Structure. In TEQ, the effective action is complex-valued, with entropy flow contributing an imaginary deformation. The resulting Euler–Lagrange equation naturally incorporates dissipative structure, and the effective Hamiltonian may become non-Hermitian in regimes of strong entropy curvature. Hermiticity is not postulated, but emerges in the entropy-coherent limit where entropy flow is symmetric or negligible. Thus, both Hermitian and non-Hermitian dynamics are unified within a single variational principle, reflecting local geometry of entropy stability rather than arbitrary assumptions. Dissipation, decoherence, and resolvability loss correspond directly to non-Hermitian deformation.

Definition 1 

(Entropy Curvature). Entropy curvature is defined as

$$\kappa :={\displaystyle \frac{{\partial }^{2}g}{\partial {\dot{\varphi }}^2}},$$

characterizing local constraints on distinguishability in velocity space.

This quantity measures how rapidly entropy flow $g(\varphi ,\dot{\varphi })$ changes with respect to variations in velocity $\dot{\varphi }$. A nonzero $\kappa$ indicates that small fluctuations in trajectory velocity lead to measurable changes in entropy flow, thus affecting the stability and resolvability of the path. When $\kappa =0$, entropy suppression is uniform across paths; when $\kappa >0$, entropy acts as stabilizing curvature, filtering out rapidly diverging trajectories. This curvature term deforms the canonical phase structure, underpinning TEQ’s generalized quantization conditions.

Remark 8 

(Structure and Interpretation of the Entropy Functional $g(\varphi ,\dot{\varphi })$). The entropy functional $g(\varphi ,\dot{\varphi })$ plays a central role in the TEQ framework. It defines the local entropy flux along a trajectory, determining how resolvability varies under coarse-grained evolution. Unlike thermodynamic entropy, which is usually state-dependent, g is a rate-functional—it encodes how entropy is produced or dissipated during transitions between configurations.

Minimal Requirements:

• $g(\varphi ,\dot{\varphi })\ge 0$: entropy is not locally reversed under coarse-graining;
• g is at least $C^2$ in $\dot{\varphi }$: ensures that curvature $\kappa ={\partial }^{2}g/\partial {\dot{\varphi }}^2$ is well-defined;
• g is observer-dependent: it encodes resolution limits, hence may vary with effective coarse-graining scale.

Illustrative Example: Free Particle with Resolution Cost. Consider a simple entropy functional:

$$g(\varphi ,\dot{\varphi })={\displaystyle \frac{1}{2}}{\dot{\varphi }}^2,$$

which penalizes high-velocity transitions due to their instability under coarse resolution. This choice implies:

$$\kappa ={\displaystyle \frac{{\partial }^{2}g}{\partial {\dot{\varphi }}^2}}=1,$$

and leads directly to entropy-curvature-induced deformation of the momentum and commutator structure (see Section 4).

Generalizations: More complex systems may involve higher-order or non-quadratic forms of g, such as:

$$g(\varphi ,\dot{\varphi })={\displaystyle \frac{1}{2}}A\left(\varphi \right){\dot{\varphi }}^2+B\left(\varphi \right)\dot{\varphi }+C\left(\varphi \right),$$

where $A\left(\varphi \right)$ acts as an entropy metric tensor. This form parallels Lagrangians with configuration-dependent mass and introduces spatially modulated entropy curvature.

Interpretive Note: The functional g does not depend on a prior thermodynamic system. It is emergent from the distinguishability structure and encodes the entropy penalty for transitioning between configurations at finite resolution. In this way, TEQ generalizes the notion of entropy flow into a purely geometric field over phase space trajectories.

Remark 9 

(Intuitive Origin of Quantization from Entropy Geometry). Quantization in TEQ emerges because entropy curvature modifies how distinct states in phase space relate to each other. Classically, positions and momenta commute, reflecting infinite resolution of phase space structure. Entropy curvature introduces a fundamental resolution limit: states cannot be infinitely distinguished due to thermodynamic constraints. As a result, positions and momenta no longer commute exactly, manifesting as quantization. Conceptually, quantization is not an imposed rule, but a geometric necessity arising from the limited resolvability of paths under entropy constraints.

4.2. Deformation of Symplectic Structure

In classical mechanics, the Poisson bracket reflects the flat, globally resolvable geometry of phase space–each trajectory is distinguishable, and the symplectic structure remains uniform. But under entropy weighting, this assumption breaks down. When entropy curvature varies rapidly, the effective measure in the path integral becomes sharply nonuniform: trajectories arbitrarily close in configuration space can differ exponentially in entropic weight. This leads to a singular, ill-defined measure and signals a geometric instability in the classical variational framework. Classical resolution fails not due to stochasticity, but because the space of trajectories no longer supports uniform distinguishability. The path integral becomes dominated by a sparse set of entropy-stationary configurations, and the underlying symplectic structure must deform accordingly. Quantization thus emerges as a structural repair–restoring coherent dynamics in the presence of entropy-induced degeneracy.

To preserve well-defined evolution under these conditions, TEQ deforms the symplectic structure itself. This deformation ensures that path stability, distinguishability, and phase volume all remain consistent with the entropy-weighted action. The result is a curved, thermodynamically-constrained geometry of phase space.

In Appendix D.1, we derive:

Theorem 4 

(Entropy-Deformed Poisson Bracket). Entropy curvature modifies the classical Poisson bracket:

$${\{\varphi ,{\mathsf{\Pi }}_{\mathit{eff}}\}}_{\mathit{TEQ}}=1-{\displaystyle \frac{\beta }{m}}\kappa .$$

This deformation reflects the shift from pure Hamiltonian dynamics to entropy-constrained trajectory space. The classical bracket is recovered as $\kappa \to 0$ or $m\to \infty$.

Theorem 5 

(Quantization from Entropy-Curved Geometry). From the entropy-weighted action (Eq. (4)) and the resulting deformation of symplectic structure (Appendix D), the canonical commutator becomes:

$$[\hat{\varphi },{\hat{\mathsf{\Pi }}}_{\mathit{eff}}]=i\hslash \left(1-{\displaystyle \frac{\beta }{m}}\kappa \right),$$

where $\kappa :={\partial }^{2}g/\partial {\dot{\varphi }}^2$ is entropy curvature. The classical commutator emerges when $\kappa \to 0$.

Corollary 4 

(Thermodynamic Origin of Noncommutativity). Quantization is not postulated but structurally required: noncommutativity arises naturally as a geometric constraint from entropy-curved phase space. It encodes minimal resolvability of trajectories under entropy-driven stability.

Remark 10 

(Structural Role of Mass). The parameter m acts not as inertial mass per se, but as a buffer against entropy curvature. Canonical behavior is preserved if:

$${\displaystyle \frac{\beta }{m}}\kappa \ll 1.$$

This suggests a thermodynamic interpretation of mass as resistance to entropic deformation, analogous to how mass resists spacetime curvature in general relativity (see Section ).

Structural Insight. In TEQ, quantization is not imposed but derived. Entropy curvature makes classical phase space structure unstable to perturbations. The only trajectories that remain well-resolved under entropy-weighted variation are those consistent with a deformed, quantized geometry. Quantization thus arises as the only viable structure compatible with entropy-stabilized evolution.

4.3. Quantization from Entropic Mode Stability

Theorem 6 

(Discrete Modes from Entropy Stationarity). In the entropy-weighted path integral

$${\mathcal{A}}_{\mathit{eff}}\left[\varphi \right]=\int \mathcal{D}\left[\varphi \right]e^{iS\left[\varphi \right]/\hslash }{e}^{-\beta {\tilde{S}}_{\mathit{apparent}}\left[\varphi \right]},$$

dominant contributions come from entropy-stationary paths. First-order stationarity yields:

$$p={\displaystyle \frac{\partial S}{\partial x}}=i\hslash \beta {\displaystyle \frac{\partial {\tilde{S}}_{\mathit{apparent}}}{\partial x}},$$

showing quantized momenta arise as entropy-optimized configurations.

Remark 11 

(Derivation and Intuition). Extremizing the effective action yields:

$$\delta S\left[\varphi \right]=i\hslash \beta \delta {\tilde{S}}_{\mathit{apparent}}\left[\varphi \right],$$

balancing dynamical cost against entropy distinguishability. Interpreting $p=\partial S/\partial x$ leads directly to quantized momenta as conditions ensuring stable distinguishability.

Corollary 5 

(Entropy-Curvature Uncertainty Relation). Entropy curvature modifies the uncertainty bound:

$$\Delta \varphi \Delta {\mathsf{\Pi }}_{\mathit{eff}}\gtrsim {\displaystyle \frac{\hslash }{1-\frac{\beta }{m}\kappa }}.$$

Positive curvature increases the lower bound, reflecting thermodynamic limits on resolution.

Structural Insight. The uncertainty principle is entropy-sensitive. In TEQ, thermodynamics and quantization unify as constraints on stable distinguishability.

Remark on Hilbert Space Emergence. In TEQ, the Hilbert space formalism of quantum theory is not assumed but emergent. In the flat entropy-curvature limit and high-resolution regime ($\beta \gg 1$), the space of distinguishable amplitudes becomes effectively linear, with interference defining a well-behaved inner product. Hilbert space thus emerges naturally as the coherence-maximized, flat-curvature limit of entropy geometry. It encodes distinguishable, resolvable configurations under maximal resolution. Outside this regime—where entropy curvature is strong or observer resolution limited—the Hilbert space description breaks down, and TEQ provides a more general variational structure.

5. Derivation of the Born Rule from the Minimal Principle

This section derives the Born rule not as a postulate but as a consequence of TEQ’s Minimal Principle. In standard quantum theory, probabilities are assigned by fiat through squared amplitudes. In TEQ, these probabilities arise from entropy-weighted variation: only distinguishable, entropy-stable paths contribute significantly to the path integral. By expanding around entropy-stationary trajectories, we recover the Born rule as the entropy-flat limit and obtain explicit corrections from entropy curvature. The result grounds probabilistic outcomes in thermodynamic geometry, linking quantum statistics to structural resolvability.

In standard quantum mechanics, the Born rule is postulated:

$$P\left(B_i\right)={\left|\psi \left(B_i\right)\right|}^2,$$

(6)

assigning probabilities to outcomes $B_i$ via the squared wavefunction amplitude [2,16]. In TEQ, this rule is not assumed but derived: probabilities emerge from the entropy-weighted stability of distinguishable paths under the Minimal Principle.

5.1. Path Probabilities from Entropy-Constrained Variation

Lemma 3 

(Minimal Principle and Path Probability). Let $\mathsf{\Pi }\left(B_i\right)$ denote the unnormalized path weight for outcome $B_i$:

$$\mathsf{\Pi }\left(B_i\right):={\left|{\int }_{\varphi \in B_i}\mathcal{D}\left[\varphi \right]e^{{\displaystyle \frac{i}{\hslash }}S\left[\varphi \right]}{e}^{-\beta {\tilde{S}}_{\mathit{apparent}}\left[\varphi \right]}\right|}^2.$$

(7)

Then the normalized probability is:

$$P\left(B_i\right):={\displaystyle \frac{\mathsf{\Pi }\left(B_i\right)}{{\sum }_j\mathsf{\Pi }\left(B_j\right)}}.$$

(8)

Entropy weighting reflects the requirement that only entropy-stable paths contribute. Thermodynamically unstable trajectories are filtered out by the Minimal Principle.

5.2. Saddle-Point Expansion and Entropy-Stable Variation

This expansion parallels the saddle-point and Gaussian path integral techniques used in semiclassical quantum mechanics and statistical field theory [17,18,19].

Theorem 7 

(Born Rule from Entropy-Stabilized Paths). Expanding ${\tilde{S}}_{\mathit{apparent}}\left[\varphi \right]$ near an entropy-stationary path ${\varphi }_{\mathit{st}}$:

$$\begin{array}{cc}\hfill {\tilde{S}}_{\mathit{apparent}}\left[\varphi \right]& ={\tilde{S}}_{\mathit{st}}+{\displaystyle \frac{1}{2}}\int dtd{t}^{\prime }\delta \varphi \left(t\right)H(t,t^{\prime })\delta \varphi \left(t^{\prime }\right)+\mathcal{O}\left(\delta {\varphi }^3\right),\hfill \end{array}$$

(9)

$$\begin{array}{cc}\hfill H(t,t^{\prime })& :={\left.{\displaystyle \frac{{\delta }^2{\tilde{S}}_{\mathit{apparent}}}{\delta \varphi \left(t\right)\delta \varphi \left(t^{\prime }\right)}}\right|}_{{\varphi }_{\mathit{st}}},\hfill \end{array}$$

(10)

leads to:

$$\mathsf{\Pi }\left(B_i\right)\approx e^{-2\beta {\tilde{S}}_{\mathit{st}}}{\left[det\left({\displaystyle \frac{\beta H}{\pi }}\right)\right]}^{-1/2}\left(1+\mathcal{O}\left({\beta }^{-1}\right)\right).$$

(11)

The Hessian $H(t,t^{\prime })$ encodes local entropy curvature and quantifies distinguishability in path space.

Corollary 6 

(Minimal Principle Correction to Born Rule). The effective amplitude becomes:

$$\psi \left(B_i\right):=e^{{\displaystyle \frac{i}{\hslash }}{S}_{\mathit{st}}}{e}^{-\beta {\tilde{S}}_{\mathit{st}}},$$

(12)

yielding probability:

$$P\left(B_i\right)={\displaystyle \frac{|\psi \left(B_i\right){|}^2·{(detH)}^{-1/2}}{{\sum }_j{\left|\psi \left(B_j\right)\right|}^2·{(det{H}_j)}^{-1/2}}}.$$

(13)

In the limit of flat entropy curvature:

$$P\left(B_i\right)\approx {\displaystyle \frac{|\psi \left(B_i\right){|}^2}{{\sum }_j{\left|\psi \left(B_j\right)\right|}^2}},$$

(14)

recovering the standard Born rule.

Interpretive Note: The curvature H modulates path amplitude normalization. TEQ predicts observable corrections in regimes where entropy gradients are strong, nonuniform, or non-negligible.

5.3. Entropy Curvature and Logarithmic Expansion

From Eq. (13):

Lemma 4 

(Path Probabilities in the Large-$\beta$ Limit).

$$\begin{array}{cc}\hfill lnP\left(B_i\right)& =-2\beta {\tilde{S}}_{\mathit{st}}-{\displaystyle \frac{1}{2}}lndetH+\mathcal{O}\left({\beta }^{-1}\right),\hfill \end{array}$$

(15)

$$\begin{array}{cc}\hfill {\tilde{S}}_{\mathit{st}}& =-{\displaystyle \frac{1}{2\beta }}lnP\left(B_i\right)-{\displaystyle \frac{1}{4\beta }}lndetH+\mathcal{O}\left({\beta }^{-2}\right).\hfill \end{array}$$

(16)

Probabilities encode not only amplitudes but entropy curvature. This links statistical outcomes to the geometry of distinguishability.

5.4. Born Rule as Entropy-Flat Limit

Theorem 8 

(Born Rule from Minimal Principle). When entropy curvature is flat ($H\to \mathrm{const}$):

$$P\left(B_i\right)\to {\left|\psi \left(B_i\right)\right|}^2.$$

(17)

More generally:

$$P\left(B_i\right)\propto {\left|\psi \left(B_i\right)\right|}^2·{(detH)}^{-1/2}.$$

(18)

Corollary 7 

(Thermodynamic Correction Factor). Probabilities acquire a curvature-sensitive normalization:

$$P\left(B_i\right)={\left|\psi \left(B_i\right)\right|}^2·Z{\left(B_i\right)}^{-1},Z\left(B_i\right)=\sqrt{det(\beta H/\pi )}.$$

(19)

Paradigm Shift. The Born rule is no longer a postulate. It emerges as the entropy-flat limit of the Minimal Principle. Probabilities reflect structural distinguishability shaped by entropy curvature.   Empirical Implication. The correction factor $Z{\left(B_i\right)}^{-1}$ provides a concrete falsifiability criterion. Controlled deviations from $|\psi \left(B_i\right){|}^2$ in entropy-curved systems would offer direct evidence for or against TEQ.

5.5. Experimental Implications and Regimes of Deviation

• Entropy-flat regime: $detH\approx \mathrm{const}\Rightarrow P\left(B_i\right)\approx {\left|\psi \left(B_i\right)\right|}^2$. Standard quantum behavior holds.
• Entropy-curved regime: Variations in H introduce measurable corrections, particularly in systems with strong gradients or nonequilibrium dynamics.
• Testable domains: High-precision interferometry [20], gravitational decoherence [21], and nonequilibrium quantum thermodynamics [22] offer experimental windows for detecting curvature-induced deviations.

6. Emergence of the Schrödinger Equation via the Minimal Principle

This section derives the Schrödinger equation from the Minimal Principle, without assuming it as a foundational postulate. In TEQ, quantum dynamics emerge from entropy-weighted variation: only those trajectories that remain both phase-coherent and entropy-stable contribute to physical evolution. By analyzing the effective momentum arising from the entropy-weighted action and applying a gradient approximation, we recover the Schrödinger equation as the unique evolution law consistent with entropy-constrained distinguishability. This reframes unitary quantum dynamics as a limiting behavior of entropy-stabilized variational structure.

In conventional quantum theory, the Schrödinger equation

$$i\hslash {\displaystyle \frac{\partial \psi }{\partial t}}=\hat{H}\psi$$

(20)

is introduced axiomatically (see, e.g., [1,23]). In TEQ, it arises from the Minimal Principle: physical trajectories must remain entropy-distinguishable under constrained variation within the entropy geometry defined by Axiom 0. Only entropy-stable, phase-coherent paths satisfy this condition.

6.1. Entropy-Constrained Action and Effective Momentum

We begin with the entropy-weighted action (4):

$$S_{\mathrm{eff}}\left[\varphi \right]=\int dt\left(L(\varphi ,\dot{\varphi })-i\hslash \beta g(\varphi ,\dot{\varphi })\right),$$

where L is the classical Lagrangian, g encodes apparent entropy flow, and $\beta \in \mathbb{C}$ governs entropic suppression.

Variation of this action yields the modified Euler–Lagrange equation (Theorem 5), extending classical dynamics with an entropic stabilizer term.

Definition 2 

(Entropy-Weighted Momentum). The conjugate momentum becomes:

$$p_{\mathit{eff}}={\displaystyle \frac{\partial L}{\partial \dot{x}}}-i\hslash \beta {\displaystyle \frac{\partial g}{\partial \dot{x}}},$$

(21)

introducing a complex deformation of phase space governed by entropy flow.

Example 1 

(Free Particle with Entropy Flow). For $L={\displaystyle \frac{1}{2}}m{\dot{x}}^2$ and $g={\displaystyle \frac{1}{2}}{\dot{x}}^2$:

$$p_{\mathit{eff}}=(m-i\hslash \beta )\dot{x},\dot{x}={\displaystyle \frac{p_{\mathit{eff}}}{m-i\hslash \beta }}.$$

This alters trajectory structure and embeds entropic filtering directly into kinematics.

Entropy Gradient Approximation

Lemma 5 

(Entropy Gradient Approximation). Because $\beta \gg 1$ in typical physical regimes (see Appendix C), and assuming ${\tilde{S}}_{\mathit{apparent}}$ has bounded curvature in $\dot{x}$, we have

$$〈{\displaystyle \frac{\partial {\tilde{S}}_{\mathit{apparent}}}{\partial \dot{x}}}〉\approx {\displaystyle \frac{\partial {\tilde{S}}_{\mathit{apparent}}}{\partial x}}\mathit{up}\mathit{to}\mathcal{O}\left({\beta }^{-1/2}\right).$$

This justifies replacing velocity gradients with spatial ones, enabling a configuration-space representation of entropy-weighted dynamics. A full derivation appears in Section H.

Minimal Principle and Schrödinger Dynamics

Theorem 9 

(Schrödinger Equation from the Minimal Principle). Given the entropy-weighted momentum

$$p_{\mathit{eff}}={\displaystyle \frac{\partial L}{\partial \dot{x}}}-i\hslash \beta {\displaystyle \frac{\partial g}{\partial \dot{x}}},$$

from Definition 2, and applying the entropy gradient approximation (Lemma 5), we find that in the large-β regime:

$$〈{\displaystyle \frac{\partial g}{\partial \dot{x}}}〉\approx {\displaystyle \frac{\partial g}{\partial x}}.$$

This allows us to express $p_{\mathit{eff}}$ as a differential operator in configuration space. In the entropy-stabilized regime, this operator takes the form:

$$p_{\mathit{eff}}\mapsto -i\hslash \nabla ,$$

reflecting stable phase-resolved entropy flow in configuration space.

Substituting into the effective Hamiltonian

$$H={\displaystyle \frac{p_{\mathit{eff}}^2}{2m}},$$

we obtain:

$$H\psi =-{\displaystyle \frac{{\hslash }^2}{2m}}{\nabla }^2\psi ,$$

and hence the evolution law

$$i\hslash {\displaystyle \frac{\partial \psi }{\partial t}}=H\psi .$$

(22)

Corollary 8 

(Schrödinger Equation as Stability Condition). The Schrödinger equation arises as the unique configuration-space dynamics compatible with entropy-constrained distinguishability. It reflects the selection of stable, phase-coherent trajectories under the Minimal Principle.

Paradigm Shift. The Schrödinger equation is not a foundational axiom. It is the dynamical signature of entropy-stabilized, phase-coherent evolution, selected by the Minimal Principle within the entropy geometry defined by Axiom 0.

6.2. From Classical Action to Entropic Coherence

TEQ extends classical extremality by embedding entropy stability into the variational structure. At quantum scales, entropy curvature alters trajectory viability, selecting only phase-coherent, entropy-stable paths. This generalization is summarized in Table 5.

7. Toward Gravitation from Entropic Geometry

This section develops the emergence of gravity in TEQ as a structural response to entropy geometry. Rather than postulating spacetime curvature, we show that gravitational phenomena arise from entropy gradients that bias which configurations remain distinguishable. Axiom 0 introduces curvature into configuration space via entropy flow, while the Minimal Principle filters for stability. Together, these generate effective geometric structure: energy–momentum flow becomes linked to entropy curvature, and mass emerges as resistance to entropic deformation. Gravitation, in this framework, is not a fundamental interaction but the thermodynamic stabilization of resolvable structure.

See also Jacobson [8], Padmanabhan [9], and Verlinde [10] for related views of gravity as emergent from entropy gradients.

Remark 12 

(Conceptual Summary: Gravity from Entropy Structure). In TEQ, gravity is not a fundamental force but a structural response to irreversible entropy flow. Where gradients become non-reversible, curvature arises as resistance to redistributing distinctions—an effect rooted in the entropy-dimensional analysis of Section 2. Spacetime is not a backdrop but an emergent bias: a geometry that favors stable, distinguishable paths. Gravitational phenomena arise when entropy structure becomes rich enough to sustain localized, persistent configurations.

In TEQ, gravitation is not a property of predefined spacetime. It emerges from the interaction of two structural principles. Axiom 0 defines entropy as a generative constraint on the geometry of possible configurations. It endows configuration space with curvature, resolution limits, and the potential for structure. The Minimal Principle (MP) then selects those configurations that remain maximally distinguishable under entropy flow. Together, they imply that classical curvature arises as the entropy-preserving deformation of trajectory space.

7.1. Entropy Dimensionality and Curvature Activation

Entropy redistribution behaves qualitatively differently across regimes of entropy dimensionality:

Definition 3 

(Entropy Dimensional Regimes).

1. 

Entangled Regime ($D_S=0$): Fully nonlocal entropy. No classical support; no spatial geometry.

2. 

Latent Regime ($D_S\in [2,3)$): Stable quasi-classical correlations (e.g., pointer states); approximate geometry.

3. 

Realized Regime ($D_S\ge 4$): Entropy gradients become irreversible and geometric. Spacetime structure emerges.

Remark 13 

(Conceptual Meaning of Entropy Dimensional Regimes). These dimensional regimes classify how much structure entropy can support. In the entangled regime, distinctions are fully nonlocal—nothing is individually identifiable. In the latent regime, approximate classicality begins to stabilize, allowing emergent structure. Only in the realized regime does entropy flow become irreversible enough to generate robust, geometric features like spacetime and curvature. These transitions mark structural thresholds: from quantum coherence to classical decoherence to gravitational geometry.

Only for $D_S\ge 4$ does entropy curvature take geometric form. Below this threshold, distinguishability remains delocalized or latent.

7.2. Noether Balance and Entropy-Stress

The following result extends Noether’s theorem [24] to complex, entropy-weighted actions. Although Noether’s original formulation assumed real-valued Lagrangians, the symmetry argument still yields conservation laws—now encoding the joint flow of energy and entropy.

Lemma 6 

(Diffeomorphism Invariance under Entropic Constraint). Let the entropy-weighted effective action be

$$S_{\mathit{eff}}=S-i\hslash \beta {\tilde{S}}_{\mathit{apparent}}.$$

Under an infinitesimal coordinate shift $x^{\mu }\mapsto x^{\mu }+\delta x^{\mu }$, we obtain:

$$\delta S_{\mathit{eff}}=\int d^{4}x\left(T^{\mu \nu }-i\hslash \beta {\mathsf{\Sigma }}^{\mu \nu }\right){\nabla }_{\mu }\delta x_{\nu },$$

where:

• $T^{\mu \nu }:=\delta S/\delta g_{\mu \nu }$ is the energy–momentum tensor,
• ${\mathsf{\Sigma }}^{\mu \nu }:=\delta {\tilde{S}}_{\mathit{apparent}}/\delta g_{\mu \nu }$ is the entropy-stress tensor.

Remark. This generalizes the one-dimensional action (4), where entropy flow enters through a scalar functional $g(\varphi ,\dot{\varphi })$. In the covariant setting, entropy geometry gives rise to the tensor ${\mathsf{\Sigma }}^{\mu \nu }$, which structurally biases the dynamics toward distinguishable paths.

Corollary 9 

(Entropy-Balanced Continuity). From invariance under the Minimal Principle:

$${\nabla }_{\mu }{T}^{\mu \nu }=0,{\nabla }_{\mu }{\mathsf{\Sigma }}^{\mu \nu }=0.$$

These continuity conditions express a simple idea: energy and entropy are not arbitrarily exchanged—they must flow in ways consistent with the underlying structure. Rather than being imposed, this balance follows naturally from the entropy-weighted geometry and the variational principle.

7.3. Entropy Curvature as Geometric Source

Definition 4 

(Entropy Curvature Tensor). Define:

$${\kappa }^{\mu \nu }:={\nabla }^{\mu }{\nabla }^{\nu }{\tilde{S}}_{\mathit{apparent}},$$

as the second-order entropy curvature. This tensor captures how entropy flow deforms the surrounding structure—it measures how curved the entropy landscape is at a given point.

Relation to Local Curvature: In the one-dimensional or trajectory-based formulation, entropy curvature appears as

$$\kappa :={\displaystyle \frac{{\partial }^{2}g}{\partial {\dot{\varphi }}^2}},$$

which quantifies local constraints on distinguishability in velocity space. The tensor ${\kappa }^{\mu \nu }$ generalizes this to spacetime, encoding how distinguishability constraints vary in all directions and contribute to the emergent geometry.

Theorem 10 

(Emergent Geometry from Entropic Equilibrium). The Minimal Principle requires a local match between energy flow and entropy gradient. In simplified terms:

$$T^{\mu \nu }{\xi }_{\nu }=T{\nabla }^{\mu }{\tilde{S}}_{\mathit{apparent}},$$

where ${\xi }^{\mu }$ represents a small shift in the local frame—a kind of “preferred direction” used to compare changes. From this balance it follows that:

$${\kappa }^{\mu \nu }\propto T^{\mu \nu }.$$

In other words, the curvature of the entropy structure acts as the source of geometry—replacing the role played by the Einstein tensor in general relativity. Gravity, on this view, is not a background condition but a thermodynamic response to structured entropy flow.

Structural Insight. In TEQ, spacetime is not presupposed. It emerges from the structure defined by Axiom 0 and the constraint imposed by the Minimal Principle. Gravity arises as the thermodynamic geometry of resolvable distinctions—curvature is not a backdrop, but the outcome of constrained entropy flow.

Theorem 11 

(Mass as Entropy-Curvature Resistance). To preserve distinguishability under entropy curvature, a system must remain stable against entropic deformation. This sets a structural constraint:

$${\displaystyle \frac{\beta }{m}}\kappa \ll 1,$$

where:

• κ denotes the entropy curvature—i.e., the second derivative of apparent entropy,
• β is the entropy-sensitivity parameter,
• and m represents the system’s effective mass.

This expresses the requirement that mass must be large enough to suppress destabilizing entropy gradients.

To relate this to spacetime structure, consider the entropy-weighted action’s invariance under infinitesimal coordinate transformations. Applying the principle of stationary action (as in Noether’s theorem) leads to a generalized balance condition between entropy-induced stress and entropy curvature:

$$\delta S_{\mathit{eff}}=\int d^{4}x\left(T^{\mu \nu }-i\hslash \beta {\mathsf{\Sigma }}^{\mu \nu }\right){\nabla }_{\mu }\delta x_{\nu }.$$

If we assume the stress tensors are approximately aligned with the entropy curvature in the high-β regime, then structurally:

$${\mathsf{\Sigma }}^{\mu \nu }\propto {\displaystyle \frac{m}{\beta }}{\nabla }^{\mu }{\nabla }^{\nu }{\tilde{S}}_{\mathit{apparent}}.$$

Solving for mass gives:

$$\boxed{m\sim \beta ·\left({\displaystyle \frac{{\mathsf{\Sigma }}^{\mu \nu }}{{\nabla }^{\mu }{\nabla }^{\nu }{\tilde{S}}_{\mathit{apparent}}}}\right)}.$$

Interpretation: Mass arises not as a fundamental quantity but as a measure of a system’s resistance to deformation by entropy gradients. It quantifies how much entropy-stress is needed to sustain a given curvature in the entropy landscape. In this view, inertial mass is a geometrically emergent property linked to entropic stability.

Remark 14 

(Mass as Curvature Buffer). In TEQ, mass quantifies resistance to entropy curvature. Like inertia in GR, it reflects how a system buffers structural deformation—but here it arises from entropy geometry, not spacetime metrics.

Outlook: Toward a Field-Theoretic Formulation

While the present manuscript focuses on semiclassical and particle-based configurations, the TEQ framework extends naturally to field-theoretic domains. In this generalization, entropy is no longer a functional of trajectory variables alone, but becomes a functional over field configurations: $\tilde{S}\left[\varphi \left(x\right)\right]$, with $\varphi$ representing matter or gauge fields over spacetime. Entropy curvature then defines a local geometric structure over field space, suggesting that the TEQ variational principle can be formulated as an extremization over sections of fiber bundles equipped with an entropy metric.

This opens a path to incorporating gauge fields, fermions, and local symmetries into TEQ, potentially aligning with existing entropy-gradient formalisms in gravity and quantum information [8,9,25,26]. The entropy-stress tensor ${\mathsf{\Sigma }}^{\mu \nu }$, introduced here in the gravitational context, may generalize to an entropy current ${\mathsf{\Sigma }}^{\mu }\left[\varphi \left(x\right)\right]$, whose divergence governs the emergence of effective spacetime structure and renormalization behavior in field theories. Such developments are reserved for future work, but they indicate the broader scope of the TEQ framework beyond its point-particle foundations.

This completes the structural unification. Spacetime, curvature, and inertia are not independent elements, but emergent consequences of entropy geometry. Axiom 0 provides the generative field; the Minimal Principle selects the stable distinctions within it.

8. Core Principles from the Minimal Principle (MP)

This section articulates the Minimal Principle (MP) as the sole generative condition from which all physical structure in TEQ emerges. Rather than positing separate principles for quantization, dynamics, or measurement, TEQ derives them as consequences of a single requirement: stability of distinguishable structure under entropy flow. The MP replaces traditional axioms with a geometric criterion of resolvability. From this, the core architecture of physics—dynamics, probability, quantization, and curvature—unfolds as structurally necessary. The result is a radically minimal foundation, in which the diversity of physical phenomena reflects variations in entropy geometry, not multiplicity of laws.

TEQ does not begin with quantum postulates. It derives physical structure from a single generative constraint:

• Minimal Principle (MP):
  Physically realizable trajectories are those that maximize entropy distinguishability under structural constraints.

From this, all features of TEQ follow. Quantization, decoherence, probabilistic measurement, and curvature emerge as structural consequences of entropy-constrained variation.

MP1. Entropy–Action Coupling from Variational Structure

Applying MP to a constrained ensemble of paths yields amplitudes:

$$\mathcal{A}\left[\varphi \right]=exp\left({\displaystyle \frac{i}{\hslash }}S\left[\varphi \right]-\beta {\tilde{S}}_{\mathrm{apparent}}\left[\varphi \right]\right),$$

where $S\left[\varphi \right]$ is the classical action, ${\tilde{S}}_{\mathrm{apparent}}$ is observer-resolvable entropy, and ${\hslash }^{-1}$, $\beta$ arise as Lagrange multipliers.

MP2. Entropy-Weighted Path Selection

The effective amplitude becomes:

$${\mathcal{A}}_{\mathrm{eff}}=\int \mathcal{D}\left[\varphi \right]\mathcal{A}\left[\varphi \right],$$

favoring entropy-stationary trajectories. Destructive interference suppresses unstable paths; coherence emerges from thermodynamic filtering (see Appendix G for related dynamics in entropy-gradient systems).

MP3. Quantization from Entropy Geometry

Canonical commutation relations are deformed by entropy curvature:

$$[\hat{\varphi },{\hat{\mathsf{\Pi }}}_{\mathrm{eff}}]=i\hslash \left(1-{\displaystyle \frac{\beta }{m}}\kappa \right),$$

with $\kappa$ the local entropy curvature. Quantization is thus a geometric stability condition—not an imposed algebra.

MP4. Entropy Resolution as Structural Gauge

The parameter $\alpha =\beta \hslash$ defines observer resolution. Shifts in $\alpha$ interpolate between coherent and dissipative regimes, acting as a gauge on entropy distinguishability.

MP5. Probability as Entropic Flatness

In the large-$\beta$ limit, path probabilities approach:

$$P\left(B_i\right)={\left|\psi \left(B_i\right)\right|}^2+\mathcal{O}\left({\beta }^{-1}\right),$$

with corrections from entropy curvature. Probability reflects geometric distinguishability—not intrinsic randomness.

• Conclusion:
• All structure in TEQ—quantization, measurement, decoherence, gravity—emerges from a single structural condition: the Minimal Principle. This unifies geometry, thermodynamics, and quantum theory within a coherent variational framework.

9. Conclusions and Outlook

This manuscript has introduced the Total Entropic Quantity (TEQ) framework, grounded in two structural axioms. Axiom 0 posits entropy as a generative principle—a geometric constraint defining which configurations can stably distinguish themselves. In TEQ, entropy refers to a curvature-functional over distinguishability configurations, governing which structures persist under resolution. Axiom 1, the Minimal Principle (MP), selects among these those that remain maximally distinguishable under finite entropy resolution. Together, these axioms define a geometry of resolvability that replaces conventional postulates with structural constraints.

From this foundation, canonical features of quantum theory—including the Born rule, Schrödinger dynamics, quantization, and elements of gravitational structure—emerge as consequences of entropy-weighted variation. The result is a unified framework in which quantum coherence, classical emergence, and spacetime curvature arise from the same entropic structure.

Though this work focuses on TEQ in semiclassical regimes, several natural extensions remain open. In quantum field theory, entropy gradients may generalize to field configurations via entropy currents on fiber bundles. In quantum information, entropy-tensor flows may offer geometric analogues to entanglement, particularly in decohering systems. The role of entropy dimensionality $D_S$ in governing causal structure—and potential connections to holographic or thermodynamic gravity models—warrants further exploration.

These directions are not proposed as a personal research program, but as natural extrapolations of the structural logic presented here. This manuscript is offered as one formulation of a deeper principle, to be evaluated, refined, or reinterpreted by others.

Appendix A.1. Path Distribution and Entropy Functional

Let $q\left(t\right)$ be a configuration path and $\mathcal{P}\left[q\right(t\left)\right]$ a probability distribution over such paths. Define the entropy functional:

$$\mathcal{S}\left[\mathcal{P}\right]:=-\int \mathcal{P}\left[q\right(t\left)\right]log\mathcal{P}\left[q\right(t\left)\right]\mathcal{D}q.$$

(A1)

We seek the distribution $\mathcal{P}$ that maximizes this entropy, subject to:

• Normalization: $\int \mathcal{P}\left[q\right(t\left)\right]\mathcal{D}q=1$,
• Average Action: $\int \mathcal{P}\left[q\left(t\right)\right]S\left[q\left(t\right)\right]\mathcal{D}q=\overline{S}$,
  where $S\left[q\right(t\left)\right]$ is the classical action functional associated with each path.

Appendix A.2. Variational Derivation

We introduce Lagrange multipliers $\lambda \in \mathbb{R}$ and $\beta \in \mathbb{C}$, and define the constrained entropy functional:

$$\begin{array}{cc}\hfill \Phi \left[\mathcal{P}\right]=& \mathcal{S}\left[\mathcal{P}\right]-\beta \int \mathcal{P}\left[q\right(t\left)\right]S\left[q\right(t\left)\right]\mathcal{D}q\hfill \\ \hfill & -\lambda \left(\int \mathcal{P}\left[q\right(t\left)\right]\mathcal{D}q-1\right).\hfill \end{array}$$

(A2)

Taking the functional derivative and setting $\delta \Phi =0$, we obtain the stationary condition:

$$log\mathcal{P}\left[q\right(t\left)\right]+1+\beta S\left[q\right(t\left)\right]+\lambda =0.$$

(A3)

Solving for $\mathcal{P}\left[q\right(t\left)\right]$, we find the Gibbs-like distribution:

$$\boxed{\mathcal{P}\left[q\left(t\right)\right]={\displaystyle \frac{1}{Z}}exp\left(-\beta S\left[q\right(t\left)\right]\right),Z=\int exp\left(-\beta S\left[q\right(t\left)\right]\right)\mathcal{D}q}$$

(A4)

Remark A1 

(Beyond Jaynesian Inference). While the form of the distribution resembles Jaynes’ MaxEnt derivation [7], in TEQ the entropy functional is not epistemic or informational in origin. Instead, it reflects a geometric constraint on resolvability itself, derived from the underlying entropy curvature. Thus, the exponential form arises not from inference under uncertainty, but from structural stability under entropy flow.

Appendix A.3. Path Integral Formulation

Given the distribution $\mathcal{P}\left[q\right(t\left)\right]$, the expectation value of an observable $\mathcal{O}\left[q\right(t\left)\right]$ is:

$$\langle \mathcal{O}\rangle =\int \mathcal{O}[q(t)]\mathcal{P}[q(t)]\mathcal{D}q.$$

(A5)

This defines the entropy-weighted path integral, where $\beta$ regulates the contribution of high-action (low-probability) paths.

Lemma A1 

(Entropy-Weighted Path Distribution). Let $S\left[q\right(t\left)\right]$ be the classical action functional. The distribution $\mathcal{P}\left[q\right(t\left)\right]$ that maximizes path entropy subject to normalization and average action constraints is:

$$\boxed{\mathcal{P}\left[q\left(t\right)\right]={\displaystyle \frac{1}{Z}}exp\left(-\beta S\left[q\right(t\left)\right]\right)},Z=\int exp\left(-\beta S\left[q\right(t\left)\right]\right)\mathcal{D}q.$$

Corollary A1 

(Recovery of the Feynman Path Integral). Setting $\beta ={\displaystyle \frac{i}{\hslash }}$ yields:

$$\langle \mathcal{O}\rangle ={\displaystyle \frac{1}{Z}}\int \mathcal{O}\left[q\left(t\right)\right]e^{iS\left[q\right(t\left)\right]/\hslash }\mathcal{D}q,$$

which recovers the Feynman path integral [17,18,19,30].

Appendix A.4. Entropy–Action Correspondence in TEQ

In the TEQ framework, the parameter $\beta$ is not fixed a priori. Rather:

• $\beta ={\displaystyle \frac{i}{\hslash }}$: yields standard unitary quantum mechanics.
• $\beta \in {\mathbb{R}}^{+}$: encodes dissipative entropy suppression, relevant in decoherence and thermodynamic limits [4,5,12].
• $\beta \in \mathbb{C}$: allows interpolations between unitary and non-unitary regimes, depending on the entropy balance.

TEQ generalizes the Feynman path integral by embedding it in a broader entropy-weighted variational structure. This connects coherent quantum dynamics to entropy-driven path selection, and reveals ℏ as the scale at which action and entropy intersect. The influence functional formalism of Feynman and Vernon [12] plays a key conceptual role here. It shows how tracing out environmental degrees of freedom induces effective damping of interference between alternative paths. In TEQ, this suppression is generalized: entropy curvature plays the structural role of biasing path weights, with decoherence arising as a special case of entropy-driven distinguishability loss.

Appendix A.5. Derivation of β as a Lagrange Multiplier from Entropy-Constrained Path Selection

In the TEQ framework, the parameter $\beta$ emerges naturally as a Lagrange multiplier enforcing entropy constraints in the path selection process. This derivation shows how $\beta$ arises from maximizing path entropy subject to constraints on both action and apparent entropy.

Entropy-Constrained Path Variational Principle

Let $\rho \left[\varphi \right]$ be the path probability density functional over histories $\varphi \left(t\right)$. We impose the following constraints:

$$\begin{array}{cc}\hfill \int \mathcal{D}[\varphi ]\rho [\varphi ]& =1,\hfill \end{array}$$

(A6)

$$\begin{array}{cc}\hfill \int \mathcal{D}[\varphi ]\rho [\varphi ]S[\varphi ]& =\overline{A},\hfill \end{array}$$

(A7)

$$\begin{array}{cc}\hfill \int \mathcal{D}\left[\varphi \right]\rho \left[\varphi \right]{\tilde{S}}_{\mathrm{apparent}}\left[\varphi \right]& =\overline{S},\hfill \end{array}$$

(A8)

where $S\left[\varphi \right]$ is the classical action and ${\tilde{S}}_{\mathrm{apparent}}\left[\varphi \right]$ is the entropy functional associated with each path.

We define the entropy of the path distribution as:

$$\mathcal{S}[\rho ]:=-\int \mathcal{D}[\varphi ]\rho [\varphi ]ln\rho [\varphi ],$$

(A9)

and extremize it with Lagrange multipliers $\beta$ and $\lambda$:

$$\delta \left\{\mathcal{S}\left[\rho \right]+\beta \left(\overline{S}-\int \rho \left[\varphi \right]{\tilde{S}}_{\mathrm{apparent}}\left[\varphi \right]\right)+i\lambda \left(\overline{A}-\int \rho \left[\varphi \right]S\left[\varphi \right]\right)\right\}=0.$$

(A10)

Taking the functional variation yields:

$$ln\rho \left[\varphi \right]+1+\beta {\tilde{S}}_{\mathrm{apparent}}\left[\varphi \right]+i\lambda S\left[\varphi \right]=0,$$

(A11)

so that:

$$\rho \left[\varphi \right]={\displaystyle \frac{1}{Z}}exp\left(-\beta {\tilde{S}}_{\mathrm{apparent}}\left[\varphi \right]-i\lambda S\left[\varphi \right]\right),$$

(A12)

with normalization factor:

$$Z=\int \mathcal{D}\left[\varphi \right]exp\left(-\beta {\tilde{S}}_{\mathrm{apparent}}\left[\varphi \right]-i\lambda S\left[\varphi \right]\right).$$

(A13)

Recovery of the TEQ Amplitude and Born Rule

Identifying $\lambda =1/\hslash$, we obtain:

$$\rho \left[\varphi \right]={\displaystyle \frac{1}{Z}}exp\left({\displaystyle \frac{i}{\hslash }}S\left[\varphi \right]-\beta {\tilde{S}}_{\mathrm{apparent}}\left[\varphi \right]\right).$$

(A14)

Thus, the TEQ path amplitude

$$\mathcal{A}\left[\varphi \right]=exp\left({\displaystyle \frac{i}{\hslash }}S\left[\varphi \right]-\beta {\tilde{S}}_{\mathrm{apparent}}\left[\varphi \right]\right)$$

(A15)

is the exponential of the Lagrangian dual function. The modulus squared of this amplitude,

$${\left|\mathcal{A}\left[\varphi \right]\right|}^2=e^{-2\beta {\tilde{S}}_{\mathrm{apparent}}\left[\varphi \right]},$$

(A16)

recovers the path probability density. This confirms that the Born rule arises naturally from the entropy-constrained variational principle, with $\beta$ enforcing the entropy constraint and ℏ setting the scale of phase sensitivity.

Appendix B.1. Geometric Form from Entropy Curvature

Assume that entropy flow defines a Riemannian structure on configuration space $\mathcal{M}$, captured by a positive-semidefinite tensor field $G_{ij}\left(\varphi \right)$. This defines a quadratic form on the velocity space:

$$g(\varphi ,\dot{\varphi }):={\displaystyle \frac{1}{2}}{G}_{ij}\left(\varphi \right){\dot{\varphi }}^i{\dot{\varphi }}^j.$$

(A18)

This expression is:

• Minimal: Involves no higher-order or nonlocal assumptions;
• Coordinate-invariant: A scalar under general reparametrizations;
• Structurally familiar: Analogous to kinetic terms in classical mechanics, but with the metric now encoding entropy curvature.

Appendix B.2. Interpretation as an Entropic Fisher Metric

This construction connects to information geometry, where the Fisher information metric quantifies statistical distinguishability between infinitesimally nearby distributions:

$$G_{ij}\left(\varphi \right)=\mathbb{E}\left[{\partial }_{i}logp(x;\varphi ){\partial }_{j}logp(x;\varphi )\right],$$

(A19)

for a family of distributions $p(x;\varphi )$ on observable configurations [28,31,32]. In this analogy, $G_{ij}$ captures how entropy curvature governs resolvability limits in velocity space. This perspective echoes ideas in thermodynamic geometry [33,34] and recent reconstructions of quantum theory from information-geometric foundations [29].

Appendix B.3. Curvature and Entropic Stability

The second variation of g defines the entropy curvature:

$$\kappa :={\displaystyle \frac{{\partial }^{2}g}{\partial {\dot{\varphi }}^2}}=G_{ij}\left(\varphi \right),$$

(A20)

which appears directly in the entropy-deformed Poisson bracket and generalized quantization conditions (see Section 4). Thus, $G_{ij}$ functions as an entropic analogue of an inertial tensor: it biases trajectory weights according to local geometric stability.

Appendix B.4. Summary and Outlook

We conclude that the entropy deformation term $g(\varphi ,\dot{\varphi })$ should, under general structural and geometric constraints, take the form:

$$g(\varphi ,\dot{\varphi })={\displaystyle \frac{1}{2}}{G}_{ij}\left(\varphi \right){\dot{\varphi }}^i{\dot{\varphi }}^j,$$

(A21)

with $G_{ij}\left(\varphi \right)$ an emergent entropy-derived metric tensor encoding local distinguishability. This form supports rigorous derivations of entropy-weighted dynamics, deformation of canonical structures, and quantization from entropy geometry.

Appendix D.1. Poisson Bracket from Entropy-Weighted Momentum

We begin with the entropy-weighted action functional:

$$S_{\mathrm{eff}}\left[\varphi \right]=\int dt\left(L(\varphi ,\dot{\varphi })-i\hslash \beta g(\varphi ,\dot{\varphi })\right),$$

(A22)

where $g(\varphi ,\dot{\varphi })$ encodes entropy flux along the path (see Appendix B).

The corresponding effective momentum is:

$${\mathsf{\Pi }}_{\mathrm{eff}}:={\displaystyle \frac{\partial L}{\partial \dot{\varphi }}}-i\hslash \beta {\displaystyle \frac{\partial g}{\partial \dot{\varphi }}}.$$

(A23)

To assess deformation of the symplectic structure, we compute:

$${\{\varphi ,{\mathsf{\Pi }}_{\mathrm{eff}}\}}_{\mathrm{TEQ}}={\displaystyle \frac{\partial {\mathsf{\Pi }}_{\mathrm{eff}}}{\partial \dot{\varphi }}}={\displaystyle \frac{{\partial }^{2}L}{\partial {\dot{\varphi }}^2}}-i\hslash \beta {\displaystyle \frac{{\partial }^{2}g}{\partial {\dot{\varphi }}^2}}.$$

(A24)

Assuming a standard quadratic kinetic term $L={\displaystyle \frac{1}{2}}m{\dot{\varphi }}^2-V\left(\varphi \right)$, and $g={\displaystyle \frac{1}{2}}{\dot{\varphi }}^2$, we find:

$${\{\varphi ,{\mathsf{\Pi }}_{\mathrm{eff}}\}}_{\mathrm{TEQ}}=m-i\hslash \beta \kappa ,\mathrm{with}\kappa :={\displaystyle \frac{{\partial }^{2}g}{\partial {\dot{\varphi }}^2}}=1.$$

(A25)

Here, $\kappa$ characterizes local entropy curvature in velocity space, i.e., how distinguishability varies with infinitesimal changes in velocity.

Normalizing by mass and taking the real part:

$${\{\varphi ,{\mathsf{\Pi }}_{\mathrm{eff}}\}}_{\mathrm{TEQ}}=1-{\displaystyle \frac{\beta }{m}}\kappa .$$

(A26)

This deformation recovers the canonical bracket in the limit $\kappa \to 0$ or $\beta \to 0$, and encodes the suppression of unstable paths under entropy-weighted dynamics.

Appendix D.2. Effective Commutator via Path Variation

To derive the entropy-corrected commutator without assuming canonical quantization, consider the entropy-weighted amplitude:

$$\mathcal{A}\left[\varphi \right]=exp\left({\displaystyle \frac{i}{\hslash }}S\left[\varphi \right]-\beta {\tilde{S}}_{\mathrm{apparent}}\left[\varphi \right]\right),$$

(A27)

and expand the effective action to second order near a stable path:

$${\delta }^2{S}_{\mathrm{eff}}=\int dt\left(\delta \varphi {\displaystyle \frac{{\partial }^{2}L}{\partial {\varphi }^2}}\delta \varphi +\delta \dot{\varphi }{\displaystyle \frac{{\partial }^{2}L}{\partial {\dot{\varphi }}^2}}\delta \dot{\varphi }-i\hslash \beta \delta \dot{\varphi }{\displaystyle \frac{{\partial }^{2}g}{\partial {\dot{\varphi }}^2}}\delta \dot{\varphi }\right).$$

(A28)

This modifies the curvature of path fluctuations in momentum space, inducing an effective noncommutativity of the form:

$$[\hat{\varphi },{\hat{\mathsf{\Pi }}}_{\mathrm{eff}}]\sim i\hslash \left(1-{\displaystyle \frac{\beta }{m}}\kappa \right),$$

(A29)

where the coefficient reflects reduced distinguishability between nearby paths under entropy-weighted evolution. This scaling arises from stability analysis of path fluctuations, analogous to the role of fluctuation-action curvature in effective action formalisms [40,41,42,43].

Thus, both the classical bracket and its quantum deformation emerge from entropy geometry rather than canonical assumptions. TEQ thereby unifies dynamical and statistical constraints in a variationally coherent framework, extending quantization to entropy-curved configuration spaces.

Appendix E.1. Quadratic Expansion Around Entropy-Stable Trajectory

Let the path be expanded as $\varphi \left(t\right)={\varphi }_{\mathrm{st}}\left(t\right)+\delta \varphi \left(t\right)$, where ${\varphi }_{\mathrm{st}}$ minimizes the entropy functional. A second-order expansion yields:

$${\tilde{S}}_{\mathrm{apparent}}\left[\varphi \right]={\tilde{S}}_{\mathrm{apparent}}\left[{\varphi }_{\mathit{st}}\right]+{\displaystyle \frac{1}{2}}\int dtd{t}^{\prime }\delta \varphi \left(t\right)H(t,t^{\prime })\delta \varphi \left(t^{\prime }\right)+\mathcal{O}\left(\delta {\varphi }^3\right),$$

(A30)

where the kernel $H(t,t^{\prime })$ is the functional Hessian:

$$H(t,t^{\prime }):={\left.{\displaystyle \frac{{\delta }^2{\tilde{S}}_{\mathrm{apparent}}}{\delta \varphi \left(t\right)\delta \varphi \left(t^{\prime }\right)}}\right|}_{\varphi ={\varphi }_{\mathrm{st}}}.$$

(A31)

Lemma A3 

(Leading-Order Approximation of the Entropy-Weighted Path Integral). To leading order, the entropy-weighted path integral becomes:

$$\mathcal{Z}\approx e^{-2\beta {\tilde{S}}_{\mathit{apparent}}\left[{\varphi }_{\mathrm{st}}\right]}·\int \mathcal{D}\left[\delta \varphi \right]e^{-\beta \int dtd{t}^{\prime }\delta \varphi \left(t\right)H(t,t^{\prime })\delta \varphi \left(t^{\prime }\right)}.$$

Appendix E.2. Reduction via Discretization

Discretizing time into N intervals, the path integral reduces to a Gaussian over ${\mathbb{R}}^N$:

$$\mathcal{Z}\approx e^{-2\beta {\tilde{S}}_{\mathrm{apparent}}\left[{\varphi }_{\mathrm{st}}\right]}·\int d^N\overrightarrow{\delta \varphi }exp\left(-\beta {\overrightarrow{\delta \varphi }}^{\top }H\overrightarrow{\delta \varphi }\right),$$

(A32)

where $H\in {\mathbb{R}}^{N\times N}$ is the discretized Hessian matrix, with entries

$$H_{ij}:={\left.{\displaystyle \frac{{\partial }^2{\tilde{S}}_{\mathrm{apparent}}}{\partial {\varphi }_i\partial {\varphi }_j}}\right|}_{\varphi ={\varphi }_{\mathrm{st}}}.$$

Theorem A2 

(Gaussian Evaluation of Entropy-Weighted Integral). Let $H\succ 0$ be a positive-definite entropy curvature matrix. Then:

$$\mathcal{Z}\approx e^{-2\beta {\tilde{S}}_{\mathit{apparent}}\left[{\varphi }_{\mathrm{st}}\right]}·{(detH)}^{-1/2},$$

up to a normalization factor absorbed during probability normalization [45,46].

Corollary A4 

(Entropy-Curvature Dominance in Path Probability). The dominant contribution to path probability scales as:

$$P\left(B_i\right)\propto e^{-2\beta {\tilde{S}}_{\mathit{apparent}}\left[{\varphi }_{\mathrm{st}}\right]}·{(detH)}^{-1/2},$$

where ${\tilde{S}}_{\mathit{apparent}}$ is minimized and H quantifies local curvature in entropy geometry.

Appendix E.3. Interpretation and Structural Significance

This result is structurally analogous to the semiclassical stationary phase approximation in conventional path integrals, but here weighted by entropy geometry rather than oscillatory phase. It highlights two factors:

• Dominant Suppression: The exponential term reflects entropy minimization across histories.
• Fluctuation Curvature: The determinant encodes the local “stiffness” or curvature of entropy geometry around the dominant path.

Remark A3. 

The curvature correction modifies quantum probabilities and contributes directly to the generalized Born rule derived in Section 5. This reveals a direct link between entropy stability and the statistical structure of quantum measurement.

Paradigm Shift. In TEQ, the dominant contribution to quantum probabilities arises not from phase coherence alone, but from entropic stability. The Gaussian path integral reveals that both entropy minimization and entropy curvature govern the likelihood of quantum outcomes. This reframes quantum amplitudes as structurally constrained by thermodynamic geometry, embedding the Born rule within a variational entropy-weighted framework.

Appendix G.1. Preliminary Assumptions

Consider a dynamical system evolving in continuous time:

$${\displaystyle \frac{d\mathbf{x}}{dt}}=f\left(\mathbf{x}\right),$$

(A33)

with $\mathbf{x}\in {\mathbb{R}}^n$, and f continuously differentiable [57].

Assumption 1 

(Lyapunov Stability [58]). There exists a scalar function $V\left(\mathbf{x}\right)$ such that ${\displaystyle \frac{dV}{dt}}\le 0$ along trajectories. Then $\mathbf{x}\left(t\right)\to {\mathbf{x}}_{*}$ as $t\to \infty$, for some attractor ${\mathbf{x}}_{*}\in {\mathbb{R}}^n$.

Assumption 2 

(Entropy Gradient Flow). The system evolves along the gradient of an entropy-derived potential:

$$f\left(\mathbf{x}\right)=-\nabla \Phi \left(\mathbf{x}\right),\Phi \left(\mathbf{x}\right):=-TS\left(\mathbf{x}\right),$$

(A34)

where $S\left(\mathbf{x}\right)$ is a coarse-grained entropy function and T an effective temperature [59,60].

Assumption 3 

(Local Convexity Near Equilibrium). The entropy potential satisfies $\nabla \Phi \left({\mathbf{x}}_{*}\right)=0$ and ${\nabla }^2\Phi \left({\mathbf{x}}_{*}\right)\succ 0$, implying local stability near the attractor [7].

Appendix G.2. Gradient Flow Interpretation

Theorem A3 

(Monotonic Entropy Increase). Under the above assumptions, the system evolves monotonically toward higher entropy:

$${\displaystyle \frac{dS}{dt}}=-{\displaystyle \frac{1}{T}}{∥\nabla \Phi \left(\mathbf{x}\right)∥}^2\le 0.$$

Corollary A5 

(Stability of Entropic Attractors). If $\nabla \Phi \left({\mathbf{x}}_{*}\right)=0$ and ${\nabla }^2\Phi \left({\mathbf{x}}_{*}\right)\succ 0$, then the trajectory satisfies $\mathbf{x}\left(t\right)\to {\mathbf{x}}_{*}$, a local entropy maximum.

Remark A4. 

This generalizes the second law of thermodynamics into a local dynamical principle, where entropy increase is a structural outcome of the system’s geometry rather than a probabilistic trend or imposed constraint.

Appendix G.3. Extension to Open Systems

In open systems, entropy exchange with the environment modifies the internal entropy potential.

Definition A1 

(Time-Dependent Entropy Potential). Let

$$\Phi (\mathbf{x},t):=-T\left(t\right)S(\mathbf{x},t),{\mathbf{F}}_{\mathit{entropic}}:=-\nabla \Phi (\mathbf{x},t),$$

define the dynamic entropy landscape.

Theorem A4 

(Non-Equilibrium Steady State (NESS) [61]). A NESS satisfies 

$$\sigma ={\displaystyle \frac{d{S}_{\mathit{system}}}{dt}}+{\displaystyle \frac{d{S}_{\mathit{environment}}}{dt}}=0,\mathrm{with}{\displaystyle \frac{d{S}_{\mathit{system}}}{dt}}\ne 0,$$

where σ is the total entropy production rate.

Corollary A6 

(Moving Entropic Attractors [62]). In time-dependent landscapes, dynamic attractors evolve as local minima of $\Phi (\mathbf{x},t)$, leading to entropy-guided adaptation.

Appendix G.4. Implications for TEQ

• Entropy gradients define vector fields over configuration space, determining structural flows without invoking optimization.
• Attractors correspond to entropy-stationary configurations under coarse-grained evolution.
• The arrow of time emerges as an internal feature of entropy geometry, not an externally imposed boundary condition.

Paradigm Shift. In TEQ, attractors are emergent geometric fixed points in entropy-weighted flow, not imposed equilibrium constraints. This unifies thermodynamic, dynamical, and informational evolution under a single variational principle based on entropy geometry.

Appendix H.1. Entropy-Weighted Expectation

Given the entropy-weighted path integral distribution:

$$\mathcal{P}\left[x\left(t\right)\right]\propto exp\left({\displaystyle \frac{i}{\hslash }}S\left[x\right]-\beta \int g(x,\dot{x})dt\right),$$

the expectation of an observable $\mathcal{O}\left[x\right]$ is:

$$\langle \mathcal{O}\rangle ={\displaystyle \frac{\int \mathcal{D}\left[x\right]\mathcal{O}\left[x\right]e^{iS\left[x\right]/\hslash -\beta \int gdt}}{\int \mathcal{D}\left[x\right]e^{iS\left[x\right]/\hslash -\beta \int gdt}}}.$$

(A38)

For $\mathcal{O}=\partial g/\partial \dot{x}$, we compute:

$$〈{\displaystyle \frac{\partial g}{\partial \dot{x}}}〉={\displaystyle \frac{\int \mathcal{D}\left[x\right]\frac{\partial g}{\partial \dot{x}}{e}^{iS/\hslash -\beta \int gdt}}{\int \mathcal{D}\left[x\right]e^{iS/\hslash -\beta \int gdt}}}.$$

(A39)

Using the chain rule:

$${\displaystyle \frac{dg}{dt}}={\displaystyle \frac{\partial g}{\partial x}}\dot{x}+{\displaystyle \frac{\partial g}{\partial \dot{x}}}\ddot{x},$$

(A40)

and taking the expectation:

$$〈{\displaystyle \frac{dg}{dt}}〉=〈{\displaystyle \frac{\partial g}{\partial x}}\dot{x}〉+〈{\displaystyle \frac{\partial g}{\partial \dot{x}}}\ddot{x}〉.$$

(A41)

Appendix H.2. Fluctuation Suppression

Lemma A4 

(Entropy-Curvature Suppression). Let $g(x,\dot{x})$ be locally convex in $\dot{x}$, with:

$$g(x,\dot{x})\approx g(x,{\dot{x}}_{\mathit{st}})+{\displaystyle \frac{1}{2}}\kappa \left(x\right){\left(\delta \dot{x}\right)}^2,\mathrm{where}\kappa \left(x\right):={\displaystyle \frac{{\partial }^{2}g}{\partial {\dot{x}}^2}}>0.$$

Then velocity fluctuations satisfy:

$$〈|\ddot{x}{|}^2〉\le {\displaystyle \frac{C}{\beta \kappa \left(x\right)}},$$

(A42)

for some finite constant C depending on system scale and local smoothness.

(Sketch). 

This follows from standard Gaussian path integral suppression: large $\beta \kappa$ exponentially dampens off-path fluctuations in $\dot{x}$, bounding the variance of second derivatives [18,19]. □

Appendix H.3. Main Result

Theorem A5 

(Entropy Gradient Approximation). Let $\beta \gg 1$, and assume $g(x,\dot{x})$ is $C^2$ in $\dot{x}$ with local entropy curvature $\kappa \left(x\right):={\partial }^{2}g/\partial {\dot{x}}^2$. Then:

$$\left|〈{\displaystyle \frac{\partial g}{\partial \dot{x}}}〉-{\displaystyle \frac{\partial g}{\partial x}}\right|\le {\displaystyle \frac{C}{\sqrt{\beta \kappa \left(x\right)}}},$$

(A43)

where $C\in {\mathbb{R}}^{+}$ depends on the system’s resolution scale and fluctuation amplitude.

Corollary A7 

(Stability of the Hamiltonian Limit). In the limit $\beta \to \infty$, the approximation

$$〈{\displaystyle \frac{\partial g}{\partial \dot{x}}}〉\approx {\displaystyle \frac{\partial g}{\partial x}}$$

becomes asymptotically exact. Entropy-weighted Hamiltonian dynamics remains well-posed and converges to classical form.

Appendix H.4. Interpretation

This result justifies the substitution used in deriving entropy-deformed Hamiltonian dynamics and Schrödinger evolution in TEQ. It confirms that under strong entropy weighting, configuration gradients dominate over velocity gradients, effectively compressing the entropy geometry into configuration space.

Paradigm Shift. In TEQ, velocity-space entropy gradients collapse to configuration gradients in the high-$\beta$ regime. This enables entropy-weighted classical and quantum mechanics to emerge as coarse-grained limits of a unified entropy geometric variational principle.