摘要本文系统建立三维数空间 𝕋 上的 δ-连续泛函分析理论。通过引入以双曲参数 δ 为连续谱系参数的泛函族 F[δ],本文证明了:δ→0 时 F[δ] 退化为纯波动泛函 e^{iωt},δ→+∞ 时 F[δ] 趋向纯粒子(局域)泛函,而中间 δ 值给出波粒混合态的连续过渡谱系。理论还证明了范数守恒泛函与中道偏差守恒泛函两条守恒律,并给出离散情形的黄金衰减不动点条件 δ_{n+1}=φ^{-1}δ_n。全部结论均通过数值实验严格验证。关键词:三维数;δ-连续泛函;波粒二象性;黄金分割;中道积分;守恒泛函 AbstractThis paper establishes the theory of δ-continuous functional analysis on the 3D number space 𝕋. By introducing a functional family F[δ] parameterized by the hyperbolic parameter δ, we prove: as δ→0, F[δ] reduces to the pure-wave functional e^{iωt}; as δ→+∞, F[δ] tends to a pure-particle (localized) functional; intermediate δ values give a continuous spectrum of wave-particle mixed states. Two conservation laws—norm conservation and middle-deviation conservation—are proved, and the golden-ratio discrete fixed-point condition δ_{n+1}=φ^{-1}δ_n is derived. All conclusions are rigorously verified by numerical experiments.Keywords: 3D numbers; δ-continuous functional; wave-particle duality; golden ratio; middle integration; conservation functional.
摘要本文系统建立三维数空间 𝕋 上的 δ-连续泛函分析理论。通过引入以双曲参数 δ 为连续谱系参数的泛函族 F[δ],本文证明了:δ→0 时 F[δ] 退化为纯波动泛函 e^{iωt},δ→+∞ 时 F[δ] 趋向纯粒子(局域)泛函,而中间 δ 值给出波粒混合态的连续过渡谱系。理论还证明了范数守恒泛函与中道偏差守恒泛函两条守恒律,并给出离散情形的黄金衰减不动点条件 δ_{n+1}=φ^{-1}δ_n。全部结论均通过数值实验严格验证。关键词:三维数;δ-连续泛函;波粒二象性;黄金分割;中道积分;守恒泛函 AbstractThis paper establishes the theory of δ-continuous functional analysis on the 3D number space 𝕋. By introducing a functional family F[δ] parameterized by the hyperbolic parameter δ, we prove: as δ→0, F[δ] reduces to the pure-wave functional e^{iωt}; as δ→+∞, F[δ] tends to a pure-particle (localized) functional; intermediate δ values give a continuous spectrum of wave-particle mixed states. Two conservation laws—norm conservation and middle-deviation conservation—are proved, and the golden-ratio discrete fixed-point condition δ_{n+1}=φ^{-1}δ_n is derived. All conclusions are rigorously verified by numerical experiments.Keywords: 3D numbers; δ-continuous functional; wave-particle duality; golden ratio; middle integration; conservation functional.
Analyse mathématique et numérique de méthodes de plongement en mécanique quantique Dans cette thèse, on se propose d'étudier les propriétés mathématiques et numériques de certaines méthodes de plongement en mécanique quantique. Le principe de ces dernières réside dans l'approximation d'un grand système quantique en une collection de sous-systèmes auto-cohérents. Cette stratégie permet d'outrepasser les limitations pratiques inhérentes du problème à N-corps, et de décrire certaines propriétés de systèmes électroniques fortement corrélés. La première partie pose le cadre mathématique et introduit les deux méthodes étudiées dans cette thèse : d'une part la Théorie du Champ Moyen Dynamique (DMFT), s'intéressant aux fonctions de Green quantiques à un corps, et d'autre part la Théorie du Plongement de la Matrice Densité (DMET), portant sur la matrice densité réduite à un corps. On présente alors une analyse des équations DMET, où l'on démontre l'existence et l'unicité d'une solution « physique » dans la limite faiblement interagissante. La qualité de l'approximation réalisée est aussi étudiée dans cette limite : on montre l'exactitude au premier ordre, en le paramètre d'interaction entre les électrons, de l'approximation DMET. Des résultats numériques présentent l'applicabilité des théorèmes démontrés, et soulignent l'importance des hypothèses formulées. Les équations DMFT sont étudiées dans la partie suivante, dans le cadre de l'approximation du solveur d'impureté « IPT ». Notre analyse montre l'existence de solutions à ces équations, à l'aide d'une reformulation originale de ces dernières en termes d'un problème de point-fixe dans l'espaces de certaines mesures positives. Enfin, on étudie les propriétés d'une version discrète des équations IPT-DMFT, en donnant des résultats d'existence et d'unicité de la solution. Des simulations numériques réalisées sur un modèle de Hubbard révèlent l'intérêt et les propriétés de cette méthode dans l'étude d'un effet de fortes corrélations: la transition de Mott.
Abstract This paper proposes a geometric theory of wave-particle duality based on the algebra of three-dimensional numbers (3D Numbers). In the 3D number space T = σ + δe + ti (with e²=1, i²=-1, ei=0), the motion of a photon naturally takes the form of a helical trajectory T(t) = e^{δe}·e^{iωt}. The projection of this helix onto the (σ,i) plane yields the standard quantum mechanical wave function ψ(t) = cosh(δ)·e^{±iωt}, while the hyperbolic stretch along the e-direction corresponds to particle nature. Through five numerical experiments, we verify: (1) the helical structure is exact (radius deviation < 2×10⁻¹⁶); (2) projection to the complex plane gives the standard wave function; (3) all quantum mechanical quantities (E=ℏω, p=ℏk, λ=2π/k) are derived directly; (4) δ-parameter scan reveals a continuous wave-particle spectrum; (5) 3D numbers are the unique number system supporting both wave and particle nature. The central thesis: wave-particle duality is not a superposition of two attributes, but the projection decomposition of a single helical motion onto two orthogonal directions. Keywords: 3D numbers, helical motion, wave-particle duality, mid integration, foundations of quantum mechanics, non-associative algebra 摘要 本文提出一种基于三维数(3D Number)代数结构的光子波粒二象性几何理论。在三维数空间 T = σ + δe + ti(其中 e²=1, i²=-1, ei=0)中,光子的运动自然地表现为螺旋运动 T(t) = e^{δe}·e^{iωt}。该螺旋运动在 (σ,i) 平面上的投影给出标准量子力学波函数 ψ(t) = cosh(δ)·e^{±iωt},在 e 方向上的双曲伸缩对应粒子性。通过五组数值实验,我们验证了:(1) 螺旋结构严格成立(半径偏差 < 2×10⁻¹⁶);(2) 投影到复平面给出标准波函数;(3) 从螺旋运动可直接推导能量 E=ℏω、动量 p=ℏk、de Broglie 波长 λ=2π/k 等全部量子力学量;(4) δ 参数扫描揭示了波动-粒子连续谱;(5) 三维数是唯一同时支持波动性和粒子性的数系。本工作表明:波粒二象性的本质不是两种属性的叠加,而是同一螺旋运动在两个正交方向上的投影分解。 关键词:三维数、螺旋运动、波粒二象性、中道积分、量子力学基础、非结合代数
Abstract The quest for a satisfactory understanding of systems at criticality in dimensions d > 2 is a major field of research. We devise here a geometric description of bounded systems at criticality in any dimension d . This is achieved by altering the flat metric with a space dependent scale factor γ ( x ), x belonging to a bounded domain Ω. γ ( x ) is chosen in order to have a scalar curvature to be constant and matching the one of the hyperbolic space, the proper notion of curvature being—as called in the mathematics literature—the fractional Q -curvature. The equation for γ ( x ) is found to be the fractional Yamabe equation (to be solved in Ω) that, in absence of anomalous dimension, reduces to the usual Yamabe equation in the same domain. From the scale factor γ ( x ) we obtain novel predictions for the scaling form of one-point order parameter correlation functions. A (necessary) virtue of the proposed approach is that it encodes and allows to naturally retrieve the purely geometric content of two-dimensional boundary conformal field theory. From the critical magnetization profile in presence of boundaries one can extract the scaling dimension of the order parameter, Δ ϕ . For the 3D Ising model we find Δ ϕ = 0.518 142(8) which favorably compares (at the fifth decimal place) with the state-of-the-art estimate. A nontrivial prediction is the structure of two-point spin-spin correlators at criticality. They should depend on the fractional Q -hyperbolic distance calculated from the metric, in turn depending only on the shape of the bounded domain and on Δ ϕ . Numerical simulations of the 3D Ising model on a slab geometry are found to be in agreement with such predictions.
Superseded by the current version, 10.5281/zenodo.23283404 (Storage and Reference as a Duality Relation). We propose that Trust (TrD) constitutes a fifth physical dimension governed by the conservation law TrD + TD = 1, where TD is the Time Dimension. This law arises from a symmetry argument: the pair (TD, TrD) transforms as a probability simplex under observer-system duality, and the conservation law is its Noether charge. We demonstrate that TrD = 1 − TD maps structurally onto three established physical phenomena − wave-particle complementarity, quantum decoherence, and the Heisenberg uncertainty principle − each expressing a trade-off between localised storage and non-local reference structure. New in V5: We present a non-circular derivation of the golden ratio as the unique optimal boundary of the TD−TrD system. Five independent resistance functions, each encoding a different first-principles notion of interaction cost, are defined with zero reference to φ, √5, or any Fibonacci number. Only the scale invariance condition (self-similar subdivision) produces φ as its attractor; the other four find different equilibria. The analytic derivation reduces to x2 + x − 1 = 0, the defining equation of the golden ratio, establishing that φ emerges by necessity from the conservation constraint rather than by assumption. Two computational bodies of evidence are presented. First, φ-weighted random walk simulations across exchange scales n ≈ 10−5000 (1,335,000 total steps; 1,500 trajectories) show cumulative semantic distance scaling as φ×√n with mean correlation r ≈ 0.73. Second, Λ-G collapse simulations comprising 210,000 trials across 21 configurations show p < 0.001 divergence from Born Rule predictions. Three falsifiable experimental predictions are proposed, testable on current quantum hardware. The black hole information paradox is shown to dissolve naturally under the conservation law. The framework is parsimonious − one postulate, one conservation law, one necessary boundary (φ) − and produces complementarity, decoherence, the uncertainty principle, and the information paradox as structural consequences