Wave–Particle Duality and Horizon Thermodynamics from Entropy Geometry: Unifying Quantum Optics and Gravitational Structure in TEQ

1. Axioms of TEQ: Self-Contained Restatement

The TEQ framework is constructed from two explicit axioms:

• Entropy Geometry: Physical structure is governed by an entropy metric $h_{ij}$ on configuration space, quantifying the ability to distinguish states. All observable distinctions are measured relative to this metric.
• Minimal Principle of Stable Distinction: Realized physical trajectories or modes are those which maximize the stability of distinctions under entropy flow; technically, these are critical points (and stable eigenmodes) of the entropy-weighted action functional.

No additional postulates—about dualities, microstructure, or boundary conditions—are assumed. These axioms are sufficient to derive all main results presented here.

2. TEQ Structural Principles

We begin by summarizing the core structural principles of the TEQ framework that underpin the present analysis. These provide the mathematical basis for everything that follows.

2.1. Entropy Geometry and Resolution Line Element

In the TEQ framework, physical structure is defined by the capacity to sustain stable distinctions. This is encoded in the entropy geometry:

$$d{R}^2=h_{ij}d{x}^{i}d{x}^j$$

where $h_{ij}$ is the entropy metric governing how sharply different configurations can be told apart. Intuitively, this metric defines a local resolution structure: it specifies which directions in configuration space admit stable distinctions and which do not. When the entropy metric collapses along a certain direction, no physical structure can be distinguished there.

The construction of $h_{ij}$ is detailed in [4], Sec. 2 and Appendix B, and in [5], Appendix B. Observable patterns correspond to entropy-stabilized structures along trajectories selected by the entropy-weighted effective action:

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

(1)

Here, L is the classical Lagrangian, while the second term introduces entropy weighting, governing how likely a pattern is to persist under the entropy-weighted suppression functional determines which patterns are stabilized and persist under entropy flow.

2.2. Entropy Curvature and Stabilized Modes

The stability of patterns under entropy flow is determined by the entropy curvature operator H, which is derived from the second variation of $S_{\mathrm{eff}}$. The explicit form of H and its eigenmode structure are presented in [5], Theorem 3.2, and [4], Appendix D. The eigenmodes of H represent patterns that can survive the entropic “blurring”: only these modes remain physically meaningful.

In transverse geometry, H dictates the mode profile and Fourier structure of observable phenomena. Thus, TEQ shows that what we actually observe—be it a photon’s interference pattern or the entropy scaling of a black hole horizon—are not arbitrary, but are precisely those structures stabilized by the underlying entropy geometry.

3. Universal Exponential Suppression: Main Theorem and Proof

Theorem 1

(Universal Exponential Suppression of Transverse Modes). Let a system be described by an entropy metric $h_{ij}$ on configuration space, such that $h_{ij}$ degenerates along one direction ($h_{\Vert }\to 0$) and is finite and positive-definite in the transverse directions. Then, the physically realized transverse modes $\psi \left(k_{\perp }\right)$, as selected by the entropy-weighted path integral, are exponentially suppressed in Fourier space:

$$\psi \left(k_{\perp }\right)\propto exp(-\alpha |k_{\perp }\left|\right),$$

with α proportional to the effective entropy curvature.

Sketch of Proof.

• In the degenerate limit, all physically meaningful distinctions are confined to the transverse subspace.
• The entropy-weighted path integral assigns weights $exp(-\hslash \beta \int g(\psi ,\dot{\psi })dt)$, where for stationary (eigen)modes, $g(\psi ,\dot{\psi })\sim {\omega }_{k_{\perp }}$.
• For linear dispersion, ${\omega }_{k_{\perp }}\propto \left|k_{\perp }\right|$, so each transverse Fourier mode receives weight $exp(-\alpha |k_{\perp }\left|\right)$.
• The result is robust to nontrivial transverse entropy curvature; $\alpha$ encodes its magnitude.
• This logic applies to any configuration space with such metric structure, regardless of microphysics.

□

A full technical elaboration appears in Appendix B.

4. Table of Explicit Assumptions and Their Impact

Remarks on Regularity and Robustness.

The derivation assumes:

• The entropy metric $h_{ij}$ is positive-definite and smooth in the transverse directions, and
• The degeneracy (collapse) in the longitudinal direction is well-defined (i.e., $h_{\Vert }\to 0$ continuously).

If the metric is irregular or exhibits discontinuities, the proof applies piecewise in each regular domain; the exponential suppression may be locally modified but the universal feature—sharp cutoff above a certain $|k_{\perp }|$—remains, so long as transverse entropy curvature is finite. Singularities in curvature or metric may produce stronger suppression but cannot generate slower-than-exponential decay. Thus, the prediction is robust to moderate irregularities and nontrivial curvature, but could be violated by non-metric or non-differentiable entropy structures, which would signal a breakdown of TEQ’s applicability.

5. Transverse Entropy Geometry of Photons

We now apply the TEQ structural principles to photon propagation, focusing on how observable wave patterns arise from the underlying entropy geometry.

5.1. Collapse of Longitudinal Resolution

When a photon propagates, the entropy metric degenerates along its direction of motion: longitudinal resolution collapses, and no structure can be distinguished along that axis. All information about the photon’s state that survives is encoded in the transverse entropy geometry, $h_{\perp }$.

Intuitively, this means that a photon does not carry a well-defined structure “along its path,” but exists as a pattern at the threshold of distinguishability in the directions perpendicular to its motion. Observable phenomena such as diffraction, interference, and polarization are determined entirely by this finite, transverse entropy structure. In effect, the photon is best understood as a stabilized transverse pattern at the very boundary of what can be resolved. (For the mathematical derivation of this geometric collapse, see [4], Sec. 6 and [5].)

5.2. Fourier Structure from Entropy Curvature

The transverse modes $\psi \left(r_{\perp }\right)$ are determined by the equation

$$H_{\perp }\psi =\lambda \psi ,$$

where $H_{\perp }$ is the transverse entropy curvature operator, obtained from the second variation of the entropy-weighted action [4,5]. This reflects the general TEQ principle: observable patterns correspond to entropy-stationary modes, or critical points of the entropy-weighted action, whose stability is set by the curvature operator.

The eigenmodes of $H_{\perp }$ are precisely those configurations that remain physically distinguishable under entropy flow. Their eigenvalues quantify the entropy cost of small deviations, making them robust against entropic “blurring.” This reasoning applies both to full configuration space and to any reduced subspace, such as the transverse plane relevant for photons.

In polar coordinates, the transverse Laplacian takes the standard form

$${\nabla }_{\perp }^2\psi ={\displaystyle \frac{1}{r_{\perp }}}{\displaystyle \frac{d}{d{r}_{\perp }}}\left(r_{\perp }{\displaystyle \frac{d\psi }{d{r}_{\perp }}}\right),$$

which is valid for any problem exhibiting radial symmetry in the transverse plane. This structure underlies a wide class of physical systems, from optics to quantum fields near horizons.

The entropy-weighted path integral assigns to each mode an amplitude suppression:

$$w\left(k_{\perp }\right)\propto exp(-\beta {\omega }_{k_{\perp }}).$$

As shown in [4], Theorem 2.2 and Appendix E, for entropy-stabilized transverse modes the entropy flux satisfies $g(\psi ,\dot{\psi })\sim {\omega }_k$, leading to $S_{\mathrm{apparent}}\sim {\omega }_k$. This yields an exponential suppression in Fourier space:

$$\psi \left(k_{\perp }\right)\propto exp\left(-\alpha |k_{\perp }|\right).$$

This exponential cutoff has a direct physical interpretation: it means that fine details—such as very sharp features or tightly focused spots—become exponentially less likely to be realized. As a result, the familiar diffraction limit and uncertainty in photon wave behavior emerge not as arbitrary constraints, but as direct consequences of the structure of entropy geometry.

6. Transverse Entropy Geometry of Black Hole Horizons

We now analyze the case of black hole horizons, where a similar entropy-geometric structure determines the transverse mode profile.

Black hole horizons are renowned for the fact that their entropy scales with the transverse area [1,2,3]:

$$S={\displaystyle \frac{A}{4G\hslash }}.$$

Within the TEQ framework, this scaling is not an isolated law, but follows from the transverse entropy geometry, $h_{\perp }$, governing which distinctions can survive at the horizon. As shown in [4], Sec. 7, and rigorously in [6], all entropy associated with the black hole is encoded in the structure of the horizon’s transverse geometry.

6.1. Transverse Mode Profiles

Quantum fluctuations near the horizon exhibit transverse modes governed by an effective transverse entropy curvature—in direct analogy with the photon case. The entropy-weighted path integral produces the same kind of exponential suppression:

$${\psi }_{\mathrm{horizon}}\left(k_{\perp }\right)\propto exp\left(-\beta {\omega }_{k_{\perp }}\right)\propto exp\left(-\beta c|k_{\perp }|\right).$$

This parallels the structure found for photons: the entropy flux for stabilized horizon modes also satisfies $g(\psi ,\dot{\psi })\sim {\omega }_k$, yielding the same universal exponential cutoff.

Physically, this means that just as photons display a sharp limit to how fine their transverse patterns can be, black hole horizons set an absolute cap on the “surface detail” that can be maintained at the edge of spacetime. The entropy geometry acts as a universal filter, blurring out finer and finer distinctions and leaving only robust, transverse structure.

7. Unification

The parallel structures of photon wave behavior and horizon entropy scaling arise from the same entropy-geometric mechanism, as summarized below.

Aspect	Photon	Black Hole Horizon
Longitudinal resolution	Collapsed	Collapsed inside horizon
Transverse entropy curvature	Finite $h_{\perp }$	Finite $h_{\perp }$
Fourier profile	$exp(-\alpha |k_{\perp }\left|\right)$	$exp(-\beta |k_{\perp }\left|\right)$
Entropy scaling	Diffraction limit	Area law

In both cases, the exponential Fourier cutoff is not coincidental but arises directly from the entropy-weighted path integral and the same transverse entropy curvature operator $H_{\perp }$, as derived in [4,5].

This structural commonality is more than a mathematical analogy: it shows that the limits we encounter in both quantum optics and gravitational physics are manifestations of a single geometric principle. The universe’s “resolution boundary”—set by entropy geometry—governs both the spread of light and the entropy content of black hole horizons.

By tracing both quantum and gravitational features to finite entropy curvature and the collapse of resolution, the TEQ framework proposes a genuine unification of these phenomena at the level of geometric structure.

8. Empirical Predictions, Falsifiability, and Guidance

The TEQ framework makes several concrete predictions that can be empirically tested and, in principle, distinguished from standard quantum or gravitational theories. This section highlights key scenarios, contrasts predictions, outlines feasible experiments, and explicitly addresses falsifiability.

Discriminating Scenarios

A primary structural prediction of TEQ is that the suppression of transverse modes at high resolution follows an exponential profile,

$$\psi \left(k_{\perp }\right)\propto exp(-\alpha |k_{\perp }\left|\right),$$

whereas standard quantum mechanics (in many settings) predicts a Gaussian or power-law suppression. This distinction becomes testable in:

• Ultra-high-resolution photon diffraction: In single-photon or matter-wave interferometry at resolutions approaching the theoretical diffraction limit, TEQ predicts a sharper (exponential) cutoff in the intensity profile of diffracted modes than standard wave theory.
• Artificial or analogue horizons: Laboratory realizations of horizons (e.g., in fluid dynamics or optical systems) can probe the predicted exponential suppression in the spectrum of horizon fluctuations. A sharper transition is predicted as the entropy metric degenerates.

Proposed Experiments and Observational Tests

Concrete experiments could include:

• Photon interferometry: High-precision interferometers (e.g., using entangled photons or ultracold atoms) could directly measure the suppression profile of high transverse modes, distinguishing exponential from Gaussian or power-law decay.
• Analogue gravity platforms: Recent experiments using Bose-Einstein condensates or fluid analogues of black hole horizons are capable of probing fluctuation spectra. Measuring the transverse structure of horizon modes near the collapse of longitudinal resolution could test TEQ’s unique prediction.

Falsifiability Criteria

A unique and falsifiable prediction of TEQ is the exponential suppression of high transverse modes in diffraction and horizon-like systems. Standard quantum wave theory predicts a Gaussian or power-law decay.

Empirical Falsification:

• In any experiment (e.g., single-photon or electron diffraction at ultra-high resolution, or fluctuation spectra in laboratory analogue horizons), if the intensity or probability distribution of high $k_{\perp }$ modes is found to decay slower than exponentially (i.e., follows a power-law or Gaussian form without an exponential tail), this would directly contradict the TEQ prediction and refute the universality of entropy-geometric suppression.
• Quantitatively, fitting the observed profile to $I\left(k_{\perp }\right)\propto exp(-\gamma |k_{\perp }{|}^n)$ and finding $n<1$ (power law) or $n=2$ (Gaussian) would falsify TEQ, which demands $n=1$.
• Specific proposed tests include:

  –

  Ultra-high-resolution photon diffraction (measuring tail profiles).

  –

  Transverse fluctuation spectra in analogue black hole or Rindler horizons.

Experimental Guidance

We encourage experimentalists to:

• Precisely measure the tail of the $k_{\perp }$ distribution in high-resolution diffraction and horizon analog systems.
• Distinguish between exponential ($n=1$), Gaussian ($n=2$), and power-law ($n<1$) suppression.
• Report both the best-fit exponent and confidence interval, to test TEQ against standard theory.

Summary Table: TEQ vs. Standard Predictions

Scenario	Standard Prediction	TEQ Prediction	Observable Difference
Photon diffraction	Gaussian/power-law tail	Exponential cutoff	Sharper suppression of high k modes
Horizon fluctuations	Smooth crossover	Sharp exponential cutoff	Steeper spectral drop-off
Entropy dimensionality	Integer scaling	Non-integer, possibly fractal	Anomalous scaling exponents

9. Philosophical and Physical Consequences

This section explores the deeper philosophical, ontological, and epistemological implications of the structural results above, highlighting how TEQ recasts foundational concepts of quantum theory and gravity.

Ontological and Epistemological Implications

The TEQ framework shifts both the ontology (what exists) and the epistemology (what can be known) of quantum and gravitational systems. Ontologically, what persists is not a `particle” or a “field,” but those patterns that remain distinguishable under finite entropy curvature. For example, a photon is not a tiny object flying through space, but a transverse entropy-stabilized pattern at the edge of resolvability. Similarly, black hole horizons do not conceal inaccessible regions, but mark boundaries where distinguishability itself collapses.

From an epistemological standpoint, TEQ reframes what it means to observe, measure, or know. Information is not simply what is recorded in a measurement, but the result of which distinctions can survive the universal suppression of distinguishability imposed by entropy flow. The Born rule, quantum interference, and even the emergence of causal structure all arise—not as unexplained rules, but as consequences of the geometry of resolution. In this perspective, knowledge is limited by what entropy geometry allows to be discerned: there is no “hidden reality” beyond what can be stably distinguished.

Wave–particle duality, then, is not a paradox or a mystery. It is a structural consequence: “wave” and “particle” are simply different regimes of distinguishability permitted by finite entropy curvature.

Entropy Dimensionality and the Structure of Distinction

A rigorous definition of entropy dimensionality ($D_S$) within TEQ remains an open challenge, but its conceptual role provides a new way to connect the photon and the black hole horizon.

For photons, the entropy metric collapses longitudinally ($h_{\Vert }\to 0$), so all structure is confined to the transverse directions. The effective entropy dimensionality is reduced, potentially even fractal in character, reflecting the scale-free nature of interference and polarization patterns.

In black holes, the horizon acts as a transverse entropy boundary: entropy dimension collapses radially, and the area law expresses the fact that all surviving distinction is transverse. In both cases, collapse of entropy dimensionality gives rise to the same exponential cutoff in Fourier space.

This suggests that entropy dimensionality, even if non-integer or locally defined, offers a unifying language for discernibility in quantum gravity. The photon and the horizon are both best understood as entropy-stabilized patterns at the boundary of what can be resolved.

While not yet formalized, $D_S$ may be connected to existing notions such as the information dimension in fractal geometry, the effective number of distinguishable microstates in coarse-grained statistical mechanics, or the scaling behavior of entropy-weighted measures in information geometry. A promising direction is to define $D_S$ via the local scaling of the entropy measure under coarse-graining:

$$\mu \left(ϵ\right)\sim {ϵ}^{D_S}$$

where $\mu \left(ϵ\right)$ counts the number of resolvable distinctions at resolution scale $ϵ$. Such a definition would provide a geometric, observer-independent account of dimensionality based on entropy structure, consistent with the TEQ principle that all physical observables arise from stable distinctions under entropy flow.

Philosophical Summary Statement

What is usually called “wave–particle duality” is not a mysterious dualism, but the signature of entropy geometry at the resolution boundary. A photon is a transverse entropy-stabilized pattern whose finite entropy curvature dictates its observable wave-like properties. The unity of wave and horizon phenomena is not just a formal resemblance, but a natural consequence of the same entropy-geometric constraint.

10. Unexplained Observations and Potential Applications

While this paper focuses on deriving structural results within the TEQ framework, it is worth noting that several empirical phenomena, currently lacking fully satisfactory explanations, may find a natural structural account in the results presented here.

We emphasize that the following connections are suggested by the formal results but are not yet rigorously derived in full within the TEQ framework. Future work is required to formalize these connections explicitly.

• Universality of the diffraction limit: The derived exponential suppression of transverse modes (Section 5 and Section 6, Appendix B) offers a structural explanation for the universal character of diffraction and resolution limits observed across different quantum systems. The TEQ framework suggests that such behavior is a geometric consequence of entropy structure, rather than an artifact of specific wave equations.
• Horizon entropy area law: The TEQ derivation of the area law scaling for horizons (Section 6) provides a non-microstructural explanation for this universal feature of gravitational systems. This may shed light on why the area law holds so generally, even beyond regimes where a detailed microscopic model is known.
• Unruh radiation and accelerated horizons: The universality theorem proved in Appendix B suggests that any entropy-degenerate system should exhibit similar exponential suppression of transverse modes. This raises the possibility that observed features of Unruh radiation and analog gravity systems may be structurally explained within the TEQ framework. A full derivation of this connection is a subject for future work.
• Entropy-dimensionality effects: The conceptual role of entropy dimensionality $D_S$ introduced in Section 9 suggests new avenues for explaining scaling behavior in systems where effective dimensionality changes near critical surfaces (e.g., horizons, confinement boundaries, condensed matter analogues). Formalizing this connection within TEQ remains an open challenge.

We present these connections not as proven results, but as structurally motivated avenues for further investigation. They point to the potential explanatory power of the TEQ framework beyond the specific derivations presented here.

11. Future Work

While this paper establishes a rigorous structural unification of photon wave behavior and horizon entropy scaling under the TEQ framework, several promising directions remain open for further investigation and formalization.

Entropy Dimensionality

Future work will aim to formalize entropy dimensionality $D_S$ within the TEQ framework, potentially revealing fractal and non-integer structures of discernibility that further unify quantum, gravitational, and informational phenomena. The role of entropy dimensionality in determining the scaling behavior of observable degrees of freedom remains a key question for both theoretical development and empirical testing.

New Phenomena and Predictions

Building on the universality result proven in Appendix B, the TEQ framework suggests several avenues where new physical phenomena may be predicted or existing unexplained observations structurally accounted for:

• Universal exponential cutoff prediction: TEQ predicts that in any system where the entropy metric becomes degenerate along one direction, the surviving transverse structure will exhibit exponential suppression in Fourier space. This could be tested in:

  –

  analog gravity systems (e.g., fluid horizons, optical analogues),

  –

  condensed matter systems with entropy flow constraints,

  –

  high-precision photon or matter-wave diffraction experiments.

• Entropy-dimensionality effects: TEQ suggests that when entropy dimensionality collapses (non-integer $D_S$), physical systems will display novel scaling of their observable degrees of freedom. This could lead to new predictions in:

  –

  near-horizon physics,

  –

  fractal optical systems,

  –

  cosmological horizon studies.

• Entropy curvature as a control parameter: TEQ further suggests that manipulating the effective entropy curvature—through environment, boundary conditions, or flow control—could alter observable wave behavior. This opens possible new directions in:

  –

  quantum control,

  –

  optical design,

  –

  thermodynamic engineering of quantum systems.

These directions represent structural consequences of the TEQ framework that, while not yet fully developed, point toward broad potential explanatory and predictive power. Formalizing these connections and testing their predictions will be a priority in future work.

12. Conclusions

We have shown that wave–particle duality in quantum optics and horizon entropy scaling in gravitational systems both arise from a single structural principle in the TEQ framework: finite transverse entropy curvature governing which distinctions can be stably maintained.

The explicit exponential suppression observed in both photon diffraction and horizon mode profiles follows directly from the entropy-weighted variational principle. This extends the TEQ-based unification of quantum and gravitational structure, offering a structural explanation for phenomena that were previously linked only by analogy.

By recognizing that the same entropy geometry governs both photon wave behavior and black hole horizon entropy, this work provides a unified structural explanation for both domains. The limits set by entropy curvature are not arbitrary but reflect a universal principle about what the universe allows to be resolved, whether at the smallest scales of quantum optics or the largest boundaries of spacetime.

Several empirical phenomena, currently lacking complete explanations, appear structurally compatible with this result and are discussed as potential applications. Finally, the derivation presented here suggests that this exponential suppression of high-frequency transverse modes is a general structural consequence of entropy geometry whenever longitudinal resolution collapses. To make this universality explicit, Appendix B provides the general proof that in any system where the entropy metric degenerates along one direction, the surviving observable structure exhibits the same exponential Fourier cutoff derived here.