Geometric Closure of Black–Body Thermodynamics: Wien Displacement and Stefan–Boltzmann Scaling from Finite–Quality Radiation

Geometric Closure of Black–Body Thermodynamics: Wien Displacement and Stefan–Boltzmann Scaling from Finite–Quality Radiation William Hernandez∗ December 2025 Abstract The universality of black–body thermodynamics has long been regarded as one of the most robust empirical pillars of physics, encapsulated by the Wien displacement and Stefan–Boltzmann scaling laws. In the first two papers of this series, equilibrium radiation was reformulated as a curvature–bounded geometric phenomenon, and the Planck spectrum was shown to emerge as a macroscopic envelope of discrete cavity modes subject to finite–quality boundary response. The present work constitutes the third and final installment of the G–series within the Unified Lattice Framework and addresses the remaining foundational question: the origin and domain of validity of black–body thermodynamics itself. Using no additional assumptions beyond those established in G1 and G2, we examine integrated radiation observables derived from discrete equilibrium mode populations with finite spectral resolution and explicitly separated boundary exchange. We show that Wien displacement and Stefan–Boltzmann scaling arise naturally from temperature–dependent geometric phase–space activation in the regime of dense mode overlap, while predictable deviations occur in finite cavities, at low temperatures, or under restricted spectral resolution. Finite boundary quality enters only through a global leakage timescale that converts stored equilibrium energy into observable flux, and does not impose thermodynamic scaling. Classical black–body laws are thus recovered as asymptotic, coarse–grained limits of curvature–bounded radiation geometry rather than as fundamental microscopic postulates. Together with the preceding papers, these results establish black–body radiation as a fully geometric equilibrium phenomenon within the Unified Lattice Framework. ∗ULF Technologies, LLC Email: [email protected] 1 Contents Global Introduction 3 I Discrete Geometry and Equilibrium Mode Populations 5 Part I Introduction 5 1 Discrete Cavity Geometry and Equilibrium Mode Structure 5 II Macroscopic Aggregation and Spectral Overlap 8 Part II Introduction 8 2 Macroscopic Aggregation of Discrete Equilibrium Radiation 8 III Discrete Stefan–Boltzmann Law from Thermal Phase– Space Activation 11 3 Thermal Phase–Space Activation and Discrete Power Scaling 12 3.1 Thermally Accessible Discrete Modes .................... 12 3.2 Radiated Power from Discrete Mode Populations ............. 12 3.3 Interpretation ................................. 13 Global Conclusion 14 2 Global Introduction Black–body radiation occupies a singular position in the foundations of physics. While the ultraviolet catastrophe exposed the failure of classical continuum assumptions, the empirical success of Planck’s law and its thermodynamic consequences introduced a deeper question: why equilibrium radiation obeys universal scaling laws with such remarkable accuracy across widely different physical systems. The Unified Lattice Framework (ULF) approaches this question from a geometric perspective. In ULF, physical interactions arise from bounded exchange on a discrete substrate endowed with finite curvature rather than from idealized continuum fields [1]. Curvature bounds regulate energy transfer and observability across scales, providing a common geometric mechanism underlying matter stability, gauge dynamics, gravitation, and the emergence of macroscopic continuum limits [2,3,4]. Within this framework, universality is not imposed axiomatically but emerges as a consequence of dense geometric overlap and bounded exchange. This geometric program has been developed across a sequence of rigorous constructions. Finite–curvature bounds have been shown to resolve the matter–stability problem [5], establish existence and mass gap for Yang–Mills theory [6], provide a non–singular formulation of quantum gravity [7], and yield controlled derivations of dark sector phenomena [8]. The same bounded–exchange principle underlies the emergence of macroscopic magnetization and phase behavior from microscopic interaction geometry [9,10], as well as curvature–bounded cosmogenic dynamics [11,12]. Across these domains, classical continuum laws appear as effective descriptions valid only within specific geometric regimes. In the first paper of the present series (G1), equilibrium radiation was reformulated as a curvature–bounded geometric phenomenon [13]. Radiation was modeled as a population of discrete cavity modes whose coupling to the environment is governed by finite boundary quality factors. This construction resolves the ultraviolet catastrophe without invoking ultraviolet cutoffs, continuum densities of states, or a priori quantization assumptions, demonstrating that the classical divergence arises from an unphysical infinite–exchange idealization rather than from equilibrium physics itself. In the second paper (G2), the familiar Planck spectrum was shown to arise as a macroscopic envelope formed by overlapping, finite–width spectral lines associated with these geometry–determined cavity modes [14]. Finite boundary quality fixes the linewidth of each mode, allowing discrete equilibrium populations to merge into a smooth spectral envelope once individual modes are no longer resolved. In this role, the cavity quality factor acts as a geometric exchange and resolution parameter, rather than as a thermodynamic input. The present work completes this program by addressing thermodynamic closure. Rather than postulating Wien displacement or Stefan–Boltzmann scaling as fundamental laws, we examine whether these relations arise from the same discrete equilibrium structure once radiation is integrated over frequency. The central question is therefore not whether classical thermodynamics can be reproduced, but under what geometric conditions it becomes valid. We demonstrate that black–body thermodynamic laws emerge naturally from temperature– dependent phase–space activation of discrete cavity modes in the regime of dense spectral overlap and finite resolution, while controlled deviations are predicted outside this regime. Finite boundary quality enters only through a global exchange timescale that 3 converts stored equilibrium energy into observable flux, and does not impose thermodynamic scaling. Thermodynamic universality is thus revealed as an emergent property of curvature–bounded radiation geometry rather than as an independent postulate. This result places equilibrium radiation on the same conceptual footing as other continuum phenomena within the Unified Lattice Framework, completing a fully geometric account of black–body physics. 4 Part I Discrete Geometry and Equilibrium Mode Populations Part I Introduction The purpose of this part is to establish the microscopic equilibrium structure of radiation used throughout the remainder of the analysis. No thermodynamic laws, continuum limits, or macroscopic scaling relations are assumed or derived here. Instead, we fix the discrete cavity geometry, equilibrium mode populations, and finite boundary response that together define the fundamental equilibrium radiation object within the Unified Lattice Framework. Radiation is modeled as a collection of discrete cavity eigenmodes determined entirely by geometric boundary conditions. Each mode is assigned an equilibrium population according to Bose–Einstein statistics at a fixed temperature, without invoking a continuum density of states or a priori spectral assumptions. The coupling of each mode to its environment is regulated by a finite boundary quality factor, which sets the spectral resolution and defines the observable linewidth associated with that mode. This finite–quality response is treated as a physical property of the boundary, not as a phenomenological smoothing parameter. At this stage, no aggregation across frequency or temperature is performed, and no macroscopic observables are constructed. The resulting frequency–resolved distributions represent the microscopic equilibrium radiation data from which all subsequent macroscopic behavior will be derived. In particular, no assumptions are made regarding the existence of a smooth spectral envelope, Wien displacement, or Stefan–Boltzmann scaling. The role of this part is therefore purely preparatory: to define the discrete, curvature– bounded equilibrium structure that serves as input for macroscopic aggregation and thermodynamic closure in Parts II and III. By isolating these microscopic inputs, we ensure that any emergent thermodynamic behavior arises solely from geometric overlap and bounded observability rather than from implicit continuum idealizations or imposed scaling laws. 1 Discrete Cavity Geometry and Equilibrium Mode Structure We consider a one–dimensional radiation cavity of fixed length Lwith perfectly reflecting geometric boundaries. The electromagnetic field within the cavity admits a countable set of normal modes whose spatial structure is fixed by the boundary conditions and whose frequencies are given by ωn=ckn=nπc L, n = 1,2,..., (1) where cdenotes the speed of light. No continuum limit is assumed; the discrete mode index nlabels physically distinct eigenstates of the cavity geometry. 5 Each mode is populated according to equilibrium Bose–Einstein statistics at temperature T. The mean occupation number of mode nis therefore ¯n(ωn, T) = 1 exp(ℏωn/kBT)−1,(2) and the corresponding equilibrium energy associated with that mode is En(T) = ℏωn¯n(ωn, T),(3) where ℏis the reduced Planck constant and kBis Boltzmann’s constant. The zero–point contribution is omitted, as it does not participate in thermal exchange. Observable radiation is regulated by the finite response of the cavity boundary. Rather than treating each mode as an idealized delta function in frequency space, we associate to each eigenfrequency ωna finite linewidth γn=ωn Q,(4) where Qis a dimensionless boundary quality factor characterizing the rate at which energy is exchanged with the environment. This linewidth reflects finite temporal coherence and finite spectral resolution, and is treated as a physical property of the boundary rather than as a phenomenological smoothing prescription. The frequency–resolved contribution of a single mode to the equilibrium radiation distribution is modeled by a normalized Lorentzian, Ln(ω) = 1 π γn/2 (ω−ωn)2+ (γn/2)2,(5) so that the microscopic equilibrium radiation contribution at frequency ωis given by the discrete sum u(ω, T) = Nmax X n=1 En(T)Ln(ω).(6) At this stage, u(ω, T) is interpreted purely as a microscopic equilibrium object. No macroscopic spectral envelope, peak structure, or thermodynamic scaling is assumed or extracted. Figure 1illustrates the resulting frequency–resolved equilibrium contributions for representative temperatures. Each curve represents the direct construction of u(ω, T) from discrete cavity geometry, equilibrium mode populations, and finite boundary response. The distributions exhibit strong frequency–dependent structure determined by the underlying geometry and mode occupations, without invoking continuum densities of states or thermodynamic laws. These discrete equilibrium distributions constitute the sole microscopic input for the remainder of the analysis. In the following parts, macroscopic observables will be obtained by controlled aggregation of u(ω, T) across frequency and temperature, allowing thermodynamic behavior to emerge without introducing additional assumptions. 6 Figure 1: Discrete equilibrium radiation contributions constructed from cavity geometry and finite boundary response at representative temperatures. Each curve is obtained directly from equilibrium mode populations and finite–Qlinewidths, without continuum assumptions or macroscopic aggregation. These distributions serve as microscopic inputs for subsequent aggregation and thermodynamic analysis. Outline of the paper. Part I establishes the discrete cavity geometry and equilibrium mode populations that serve as microscopic inputs. Part II examines macroscopic aggregation of radiation observables and the onset of spectral overlap. Part III demonstrates the emergence of Wien displacement and Stefan–Boltzmann scaling, identifying the geometric conditions under which classical thermodynamic laws become valid and predicting controlled deviations in finite systems. 7 Part II Macroscopic Aggregation and Spectral Overlap Part II Introduction The purpose of this part is to examine how macroscopic radiation observables arise from the discrete equilibrium structure established in Part I through aggregation alone. No additional physical assumptions are introduced. Geometry, equilibrium statistics, and finite boundary response are held fixed, and the analysis proceeds by summing microscopic contributions without invoking thermodynamic laws, continuum limits, or scaling hypotheses. Starting from the frequency–resolved equilibrium distributions constructed in Part I, we define macroscopic observables by integrating over frequency. This operation corresponds physically to measurements that do not resolve individual cavity modes but instead register total stored or emitted energy. Importantly, aggregation is treated here as a purely mathematical operation applied to previously defined microscopic data, not as a transition to a new physical regime. At this stage, no claim is made regarding the form of the temperature dependence of the aggregated quantities. In particular, no assumptions are imposed concerning Wien displacement, Stefan–Boltzmann scaling, or universality. The resulting behavior reflects finite geometry, finite mode density, and finite spectral resolution, and therefore represents a pre–asymptotic regime in which thermodynamic laws are not yet guaranteed to hold. The role of this part is thus intermediate and diagnostic. By examining how smooth macroscopic behavior emerges from discrete equilibrium inputs through aggregation alone, we isolate the geometric mechanisms responsible for spectral overlap and temperature dependence. This prepares the ground for Part III, where the conditions under which classical thermodynamic laws emerge—and the regimes in which they fail—are examined explicitly. 2 Macroscopic Aggregation of Discrete Equilibrium Radiation Starting from the frequency–resolved equilibrium radiation contributions u(ω, T) constructed in Part I, we now define macroscopic radiation observables by direct aggregation over frequency. This operation corresponds to experimental measurements that do not resolve individual cavity modes but instead register the total energy stored or exchanged by the radiation field. The aggregated equilibrium radiation energy is defined as U(T) = Z∞ 0 u(ω, T) dω, (7) 8 where u(ω, T) is given by the discrete sum over broadened cavity modes, u(ω, T) = Nmax X n=1 En(T)Ln(ω),(8) with En(T) = ℏωn¯n(ωn, T) and Ln(ω) the finite–QLorentzian response associated with mode n. No continuum density of states is assumed, and no analytic approximation is introduced in the aggregation; the integral is evaluated directly from the discrete equilibrium data. Figure 2shows the resulting aggregated equilibrium energy as a function of temperature for a fixed cavity geometry and boundary quality factor. The dependence of U(T) on temperature is smooth and monotonic, reflecting the increasing population of higher–frequency modes and the growing degree of spectral overlap as temperature rises. Importantly, no thermodynamic scaling law is imposed or fitted at this stage, and the plotted behavior represents the direct outcome of geometric aggregation alone. The absence of any singular behavior or discontinuity indicates that macroscopic radiation observables arise naturally from summation of discrete equilibrium inputs, even in a finite cavity with finite spectral resolution. At the same time, the form of the temperature dependence is not yet constrained to follow classical thermodynamic laws. The system remains in a finite–geometry, pre–asymptotic regime in which the density of overlapping modes is increasing but not yet sufficient to guarantee universal scaling. This aggregated equilibrium behavior serves as the bridge between microscopic geometry and macroscopic thermodynamics. In the following part, we examine how the same aggregated observable approaches Wien displacement and Stefan–Boltzmann scaling in the regime of dense mode overlap, and identify the geometric conditions under which classical black–body thermodynamics emerges as an asymptotic limit. 9 radiation, completing—rather than initiating—the geometric description of black–body thermodynamics. References [1] William Hernandez. Unified Lattice Framework: The Finite–Curvature Synthesis of Physics. Zenodo preprint, 2025. [2] William Hernandez. Unified Lattice Framework I: Geometric Resolutions of the Yang–Mills, Matter Stability, and Navier–Stokes Problems. International Journal of Quantum Foundations, 12(1):217–260, 2025. [3] William Hernandez. Unified Lattice Framework II: Curvature–Bound Resolutions of the Gravitational, Dark–Sector, and Singularity Problems. International Journal of Quantum Foundations, 12(1):261–335, 2025. [4] William Hernandez. Unified Lattice Framework III: Quantized Resolutions of the Quantum–Gravity, Cosmogenic, and Continuity Problems. International Journal of Quantum Foundations, 12(1):336–363, 2025. [5] William Hernandez. Finite–Curvature Resolution of the Matter–Stability Problem: A Geometric Proof of Stability for Interacting Quantum Systems. Zenodo preprint, 2025. [6] William Hernandez. Yang–Mills Existence and Mass Gap from a Finite–Curvature Quantum Geometry. Zenodo preprint, 2025. [7] William Hernandez. A Curvature–Bounded, Non–Singular Construction of Quantum Gravity. Zenodo preprint, 2025. [8] William Hernandez. Unified Lattice Framework: A Finite–Curvature Derivation of Dark Energy, Dark Matter, and Black–Hole Seeds. Zenodo preprint, 2025. [9] William Hernandez. The Magnetism Existence, Stability, and Emergence Problem: A Curvature–Bounded Derivation of Macroscopic Magnetization. Zenodo preprint, 2025. [10] William Hernandez. Unified Lattice Framework Magnetism I-III: From Microscopic Coulomb Geometry to Curvature–Bounded Magnetic Operators and Macroscopic Phase Behavior. Zenodo preprint, 2025. [11] William Hernandez. The Impartation Theorem for Curvature–Bounded Quantum Geometry. Zenodo preprint, 2025. [12] William Hernandez. UNIFIED LATTICE FRAMEWORK: SYNTHESIS II — MASS GAP, COSMOLOGICAL CONSTANT, AND NON–SINGULAR COSMOGENESIS. Preprint, 2025. Zenodo preprint. [13] William Hernandez. Curvature–Bound Equilibrium Radiation: A Geometric Resolution of the Black–Body Problem. Preprint, 2025. Zenodo preprint. [14] William Hernandez. Geometric Emergence of the Planck Spectrum: Discrete Cavity Modes and Finite–Quality Boundaries. Preprint, 2025. Zenodo preprint. 16