A structural derivation of spectral structure and quantization is obtained from minimal requirements on discernibility and persistence under variation. Rather than postulating quantum degrees of freedom, operators, or Hilbert space, the analysis identifies which assumptions are required for eigenstructure to arise as a consequence of stability under irreversible loss. Building on prior results establishing accumulated loss, path dependence, and curvature as necessary consequences of persistence, the paper introduces a clear admissibility ledger. Under explicit conditions—ordered variation, local loss density, and a quadratic stability regime—the second variation of accumulated loss defines a curvature operator that classifies admissible deviations by their susceptibility to persistence loss. Eigenmodes emerge as stability classes selected by persistence filtering, not as fundamental postulates. Discrete spectra appear only when admissibility conditions enforce effective compactness. Additional restrictions concerning closed continuations and phase-valued transport lead to a closure condition under repeated traversal, reproducing Bohr-type quantization as a conditional consequence rather than an axiom. Global spectral weights and regularization are introduced solely as technical bookkeeping devices when persistence comparison is extended to families of modes. The framework deliberately separates structural derivation from physical interpretation. Hilbert space structure, probabilistic interpretation, and standard quantum dynamics arise only under further assumptions and are not required for spectral stability itself. Standard quantum mechanics is recovered as a specialization of the general framework under additional linearity, normalization, and interpretive assumptions. The paper isolates the minimal structural content underlying eigenstructure and quantization, making explicit which assumptions are responsible for each step. It stops short of system-specific applications, but the entropy-weighted variational object derived here serves as the foundation for predictive extensions developed in companion works, including corrections to tunneling behavior, interference suppression, and collapse-like phenomena emerging from resolution instability.
A structural derivation of spectral structure and quantization is obtained from minimal requirements on discernibility and persistence under variation. Rather than postulating quantum degrees of freedom, operators, or Hilbert space, the analysis identifies which assumptions are required for eigenstructure to arise as a consequence of stability under irreversible loss. Building on prior results establishing accumulated loss, path dependence, and curvature as necessary consequences of persistence, the paper introduces a clear admissibility ledger. Under explicit conditions—ordered variation, local loss density, and a quadratic stability regime—the second variation of accumulated loss defines a curvature operator that classifies admissible deviations by their susceptibility to persistence loss. Eigenmodes emerge as stability classes selected by persistence filtering, not as fundamental postulates. Discrete spectra appear only when admissibility conditions enforce effective compactness. Additional restrictions concerning closed continuations and phase-valued transport lead to a closure condition under repeated traversal, reproducing Bohr-type quantization as a conditional consequence rather than an axiom. Global spectral weights and regularization are introduced solely as technical bookkeeping devices when persistence comparison is extended to families of modes. The framework deliberately separates structural derivation from physical interpretation. Hilbert space structure, probabilistic interpretation, and standard quantum dynamics arise only under further assumptions and are not required for spectral stability itself. Standard quantum mechanics is recovered as a specialization of the general framework under additional linearity, normalization, and interpretive assumptions. The paper isolates the minimal structural content underlying eigenstructure and quantization, making explicit which assumptions are responsible for each step. It stops short of system-specific applications, but the entropy-weighted variational object derived here serves as the foundation for predictive extensions developed in companion works, including corrections to tunneling behavior, interference suppression, and collapse-like phenomena emerging from resolution instability.
This paper derives a persistence-weighted path integral from first principles of coherent comparison under composed admissible variation. Rather than postulating probabilistic, geometric, or dynamical structure, the analysis asks which bookkeeping devices are unavoidable if comparison is to remain well defined when admissible variations are recomposed, refined, or reordered. Starting from minimal operational requirements—discernibility, admissible variation, and persistence—the paper shows that path dependence, accumulated loss, stationarity, and quadratic stability classification are structurally forced. An accumulated loss functional emerges as the minimal additive bookkeeping required for coherence, and persistence weighting arises as a monotone representation of the induced ordering on admissible paths. Under additional stability requirements, the exponential weighting appears as a canonical representative compatible with arbitrary recomposition. Second-order variation around persistence-stationary paths yields a quadratic stability form, which can be repackaged as a curvature operator acting on admissible perturbations. Spectral structure then appears as a classification of persistence-relevant deviations, without presupposing geometry, dynamics, or operator postulates. Familiar formalisms—path integrals, entropy-weighted amplitudes, and spectral descriptions—arise only as restricted realizations once further admissibility, locality, and representational constraints are imposed. The entropy-weighted Feynman path integral is recovered explicitly as such a downstream specialization, with all additional assumptions labeled. The result is not a constructive physical model but a boundary-forming derivation: it fixes which structures are forced by coherence of comparison and which questions are deferred to later, more constrained analyses. The paper provides a foundational anchor for subsequent applications that treat persistence-weighted amplitudes and curvature operators as fixed inputs rather than postulates.
This paper analyzes quantum tunneling and bound states as empirical stress tests for a persistence-based comparison framework. Rather than postulating wave mechanics or operator dynamics, the analysis takes as fixed a small set of structural ingredients: admissible variation, accumulated loss under recomposed change, and a persistence requirement selecting stable continuations. Within this framework, persistence-weighted path amplitudes and an associated quadratic stability operator arise as derived objects. The paper examines what is forced when these objects are applied to barrier penetration and confinement, and which standard quantum-mechanical results are recovered only as correspondence limits under additional conditions. Tunneling is treated as a local suppression problem, while bound states are analyzed in terms of closure and stability rather than operator postulates. Explicit worked examples demonstrate how WKB exponents and discrete spectra emerge in entropy-flat limits, and how controlled deviations arise when admissibility or curvature conditions are relaxed. No new principles are introduced. The results isolate the structural footing of familiar semiclassical constructions and delineate a clear window of falsifiability for persistence-based comparison beyond standard wave-mechanical representations.
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.
The mathematical structure of quantum theory is usually postulated rather than explained—Hilbert spaces, operators, and probabilistic structure—without explaining why this arena should exist in the first place. In this work we address a more primitive question: starting from the absence of any prior structure, what conditions must be satisfied for coherent persistence to be possible at all? We begin from a state of zero determination (the Void), understood not as physical emptiness but as the absence of admissibility constraints. From this starting point, we show that the possibility of persistence forces a minimal triadic structure consisting of reference, transformation, and return. This structure enables self-verification without external supervision and distinguishes genuine continuity from accidental resemblance. The requirement of self-verifying persistence forces deviation to be representable quantitatively and to compose across histories. This necessity leads uniquely to a linear vector space structure. Stability further imposes a knife-edge viability condition: identity-preserving and identity-violating contributions must equalize at closure. From this condition, an inner product and norm are forced, yielding Hilbert space geometry and operator representations as consequences of viability rather than axioms. Within this framework, familiar features of quantum kinematics—such as uncertainty, unitary evolution, and probabilistic weighting—arise as structural necessities rather than postulated rules. The present work establishes the kinematic foundations of coherence, deriving the mathematical arena of quantum theory from the requirement that something persist at all. Applications to mass relations, symmetry, temporal ordering, and composite systems are developed separately.Version 1.0. This preprint establishes the structural foundations of Coherence Algebra and derives Hilbert space from self-verification and persistence.
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.