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Explicit Quantum Calculations from Entropy Geometry: Worked Examples in the TEQ Framework

David Sigtermans

Published Jun 13, 2025• 1 citation
Publisher

Abstract

We present a sequence of explicit, worked-out examples illustrating how canonical quantum phenomena arise from the Total Entropic Quantity (TEQ) framework, which reconstructs quantum theory as a consequence of entropy-weighted resolution geometry. Focusing on tunneling, quantization, uncertainty, interference, and decay, each example is calculated directly from the entropy-weighted action, without recourse to wavefunctions, operator postulates, or probabilistic axioms. The paper also provides a general algorithm for constructing the entropy metric Gij, clarifying its structural relation to entropy curvature and dynamical stability. This approach demonstrates that quantum effects—typically treated as fundamental mysteries—emerge naturally from the geometric selection and suppression of entropy-stable trajectories. The results confirm that TEQ is not only a structural reformulation but a computationally viable foundation for quantum mechanics. Further details on the axiomatic basis and broader implications of TEQ are provided in foundational work; here, we complement those results by emphasizing explicit calculations and systematic metric construction, bridging principle and prediction within the TEQ framework.

DOI
10.20944/preprints202506.1187.v1

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We present a sequence of explicit, worked-out examples illustrating how canonical quantum phenomena arise from the Total Entropic Quantity (TEQ) framework, which reconstructs quantum theory as a consequence of entropy-weighted resolution geometry. Focusing on tunneling, quantization, uncertainty, interference, and decay, each example is calculated directly from the entropy-weighted action, without recourse to wavefunctions, operator postulates, or probabilistic axioms. The paper also provides a general algorithm for constructing the entropy metric Gij, clarifying its structural relation to entropy curvature and dynamical stability. This approach demonstrates that quantum effects—typically treated as fundamental mysteries—emerge naturally from the geometric selection and suppression of entropy-stable trajectories. The results confirm that TEQ is not only a structural reformulation but a computationally viable foundation for quantum mechanics. Further details on the axiomatic basis and broader implications of TEQ are provided in foundational work; here, we complement those results by emphasizing explicit calculations and systematic metric construction, bridging principle and prediction within the TEQ framework.
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.
We derive quantum tunneling probabilities and bound-state quantization directly from the entropy-weighted path integral and entropy curvature principles of the Total Entropic Quantity (TEQ) framework. Instead of invoking traditional quantum postulates such as wavefunctions, operator algebra, or boundary-condition quantization, we demonstrate that exponential suppression of entropy-unstable trajectories in high-curvature entropy geometries naturally produces canonical tunneling profiles and energy discretization. Standard quantum results—including the WKB tunneling formula and the Bohr–Sommerfeld quantization rule—are recovered explicitly as special limiting cases of a broader variational principle grounded in entropy geometry. These results provide structurally grounded, empirically falsifiable predictions distinct from conventional quantum mechanics, testable in engineered nanoscale and quantum optical systems.
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 refers to a curvature-functional over distinguishability configurations, maintaining only those structural patterns that remain stable or well-posed under finite informational resolution—that is, structures that are not destroyed or dissolved by coarse-graining or limited observability. Axiom 1, the Minimal Principle (MP), selects from these configurations those that remain maximally resolvable under finite entropy resolution. From this entropic foundation, core features of quantum theory—including the Born rule, Schrödinger dynamics, quantization, and gravitational structure—emerge as consequences of variation within an entropy-curved geometry. TEQ unifies quantum coherence, classical emergence, and spacetime curvature under a single informational principle, replacing conventional postulates with entropy-stabilized structure. The framework predicts deviations in high-curvature regimes and reinterprets physical law as the geometry of distinguishability rather than imposed dynamics.
We show that the mathematical structure of Hilbert space—typically postulated in quantum theory—emerges as a consequence of two fundamental principles derived within the TEQ framework: (1) entropy as a generative structural principle; (2) a minimal principle selecting stable distinctions. These principles, and the resulting entropy-weighted action functional, are derived from first principles in prior work. Here, we analyze the entropy-weighted path integral and the geometry of entropy curvature, and prove that the space of entropy-stabilized modes satisfies the axioms of a Hilbert space: linearity, inner product structure, norm completeness, and probabilistic interpretation. In this view, the Born rule itself emerges from entropy-weighted path selection. Importantly, this result shows that the TEQ framework fully recovers the standard Hilbert space formalism—not by postulate, but as a structural consequence of its core principles—thus preserving compatibility with quantum theory while providing it with a deeper generative foundation. Our results strengthen the explanatory power of the TEQ framework and offer a structural derivation of core quantum features from a common thermodynamic-geometric basis.
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.