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Wave Behavior as Emergent Resolution Stability: Deriving Quantum Structure from Entropy Geometry

David Sigtermans

Published Apr 28, 2025• 0 citations
Publisher

Abstract

We show that wave-like behavior, interference, and quantization emerge necessarily from two axioms: (1) entropy geometry as the generator of physically distinguishable structure, and (2) a minimal principle selecting trajectories stable under entropy flow. Within the Total Entropic Quantity (TEQ) framework, entropy curvature defines the geometry of resolution, and the stability condition selects log-periodic modes as the only entropy-resilient solutions. From this structure, we derive the entropy-weighted path integral, the Born rule as an asymptotic stability condition, the emergence of discrete spectral modes, and the Schrödinger equation as the limiting case of entropy-flat evolution. Quantum wave behavior is thus not postulated, but structurally selected by entropy geometry. The TEQ framework reinterprets quantum theory as a special case of entropy-stabilized dynamics, where physical law arises not from imposed kinematics, but from the geometry of what can stably be resolved under finite informational precision. In this view, precision is not a matter of external control, but a structural limit: resolution is bounded by entropy curvature, which determines what distinctions can persist.

DOI
10.20944/preprints202504.2263.v1

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We show that wave-like behavior, interference, and quantization emerge necessarily from two axioms: (1) entropy geometry as the generator of physically distinguishable structure, and (2) a minimal principle selecting trajectories stable under entropy flow. Within the Total Entropic Quantity (TEQ) framework, entropy curvature defines the geometry of resolution, and the stability condition selects log-periodic modes as the only entropy-resilient solutions. From this structure, we derive the entropy-weighted path integral, the Born rule as an asymptotic stability condition, the emergence of discrete spectral modes, and the Schrödinger equation as the limiting case of entropy-flat evolution. Quantum wave behavior is thus not postulated, but structurally selected by entropy geometry. The TEQ framework reinterprets quantum theory as a special case of entropy-stabilized dynamics, where physical law arises not from imposed kinematics, but from the geometry of what can stably be resolved under finite informational precision. In this view, precision is not a matter of external control, but a structural limit: resolution is bounded by entropy curvature, which determines what distinctions can persist.
We show that wave-like behavior, interference, and quantization emerge necessarily from two axioms: (1) entropy geometry as the generator of physically distinguishable structure, and (2) a minimal principle selecting trajectories stable under entropy flow. Within the Total Entropic Quantity (TEQ) framework, entropy curvature defines the geometry of resolution, and the stability condition selects log-periodic modes as the only entropy-resilient solutions. From this structure, we derive the entropy-weighted path integral, the Born rule as an asymptotic stability condition, the emergence of discrete spectral modes, and the Schrödinger equation as the limiting case of entropy-flat evolution. Quantum wave behavior is thus not postulated, but structurally selected by entropy geometry. The TEQ framework reinterprets quantum theory as a special case of entropy-stabilized dynamics, where physical law arises not from imposed kinematics, but from the geometry of what can stably be resolved under finite informational precision. In this view, precision is not a matter of external control, but a structural limit: resolution is bounded by entropy curvature, which determines what distinctions can persist.
We show that wave-like behavior, interference, and quantization emerge necessarily from two axioms: (1) entropy geometry as the generator of physically distinguishable structure, and (2) a minimal principle selecting trajectories stable under entropy flow. Within the Total Entropic Quantity (TEQ) framework, entropy curvature defines the geometry of resolution, and the stability condition selects log-periodic modes as the only entropy-resilient solutions. From this structure, we derive the entropy-weighted path integral, the Born rule as an asymptotic stability condition, the emergence of discrete spectral modes, and the Schrödinger equation as the limiting case of entropy-flat evolution. Quantum wave behavior is thus not postulated, but structurally selected by entropy geometry. The TEQ framework reinterprets quantum theory as a special case of entropy-stabilized dynamics, where physical law arises not from imposed kinematics, but from the geometry of what can stably be resolved under finite informational precision. In this view, precision is not a matter of external control, but a structural limit: resolution is bounded by entropy curvature, which determines what distinctions can persist.
This paper introduces the Total Entropic Quantity (TEQ) framework, a structural reformulation of quantum theory grounded in two foundational axioms. Axiom 0 posits entropy as a generative constraint: a geometric principle that determines which configurations can stably distinguish themselves, independent of space, time, or dynamics. Here, entropy is defined as a curvature-functional over distinguishability configurations, preserving only those patterns that remain stable under finite informational resolution---that is, structures not dissolved by coarse-graining or limited observability. Axiom 1, the Minimal Principle (MP), selects from these structures those that are maximally stable under entropy-weighted variation. From this entropic foundation, core elements of quantum theory---including the Born rule, quantization, and Schrödinger dynamics---emerge as special cases of entropy-stabilized geometry. The framework derives an entropy-weighted path integral and introduces a corrected Schrödinger equation that governs evolution in regimes of finite entropy curvature. In the high-resolution limit (β→∞ \beta \to \infty β→∞), TEQ reduces to standard unitary quantum mechanics; in more general regimes, entropy flow deforms canonical dynamics, linking decoherence, dissipation, and gravitational curvature. TEQ reinterprets physical law as emergent structure within the geometry of distinguishability, rather than as imposed dynamics on a fixed spacetime background.
This paper introduces the Total Entropic Quantity (TEQ) framework, a structural reformulation of quantum theory grounded in two foundational axioms. Axiom 0 posits entropy as a generative constraint: a geometric principle that determines which configurations can stably distinguish themselves, independent of space, time, or dynamics. Here, entropy is defined as a curvature-functional over distinguishability configurations, preserving only those patterns that remain stable under finite informational resolution---that is, structures not dissolved by coarse-graining or limited observability. Axiom 1, the Minimal Principle (MP), selects from these structures those that are maximally stable under entropy-weighted variation. From this entropic foundation, core elements of quantum theory---including the Born rule, quantization, and Schrödinger dynamics---emerge as special cases of entropy-stabilized geometry. The framework derives an entropy-weighted path integral and introduces a corrected Schrödinger equation that governs evolution in regimes of finite entropy curvature. In the high-resolution limit (β→∞ \beta \to \infty β→∞), TEQ reduces to standard unitary quantum mechanics; in more general regimes, entropy flow deforms canonical dynamics, linking decoherence, dissipation, and gravitational curvature. TEQ reinterprets physical law as emergent structure within the geometry of distinguishability, rather than as imposed dynamics on a fixed spacetime background.
This paper derives the discrete structure of observable quantities—eigenvalues and quantized states—as a natural consequence of the Total Entropic Quantity (TEQ) framework. Starting from two foundational axioms—(1) entropy as geometric structure, and (2) a minimal principle selecting stable distinctions—we show that quantization emerges not as a postulate but as a result of entropy curvature. Eigenstates appear as local minima in entropy-resolvent state space, and eigenvalues define stability classes of distinguishable structure. We further show that the entropy-weighted path integral admits a natural zeta-regularization, and that the spectrum of entropy-stable modes lies entirely on the critical line Re(s) = 1/2. As a consequence, the Riemann Hypothesis (RH) is reinterpreted as a structural condition of entropy stability. Combining this with a contradiction argument—showing that RH and the Goldbach Conjecture (GC) cannot both be false—we conclude: if GC holds and the TEQ framework is valid, then RH must also hold. These results suggest a unified perspective—one in which both quantization and arithmetic regularity arise from the same thermodynamic principle of resolution. Whether one adopts this framework in full or simply considers its coherence, the geometry of entropy flow offers a compelling lens through which physical and symbolic structure may be reinterpreted.
This paper derives the discrete structure of observable quantities—eigenvalues and quantized states—as a natural consequence of the Total Entropic Quantity (TEQ) framework. Starting from two foundational axioms—(1) entropy as geometric structure, and (2) a minimal principle selecting stable distinctions—we show that quantization emerges not as a postulate but as a result of entropy curvature. Eigenstates appear as local minima in entropy-resolvent state space, and eigenvalues define stability classes of distinguishable structure. We further show that the entropy-weighted path integral admits a natural zeta-regularization, and that the spectrum of entropy-stable modes lies entirely on the critical line Re(s) = 1/2. As a consequence, the Riemann Hypothesis (RH) is reinterpreted as a structural condition of entropy stability. Combining this with a contradiction argument—showing that RH and the Goldbach Conjecture (GC) cannot both be false—we conclude: if GC holds and the TEQ framework is valid, then RH must also hold. These results suggest a unified perspective—one in which both quantization and arithmetic regularity arise from the same thermodynamic principle of resolution. Whether one adopts this framework in full or simply considers its coherence, the geometry of entropy flow offers a compelling lens through which physical and symbolic structure may be reinterpreted.