--- We present a complete axiomatic foundation for quantum theory within the framework of von Neumann–Bernays–Gödel (NBG) class theory. The starting point is the multiplicative group R_+^ equipped with Haar measure d^ x = dx/x, with the Mellin transform serving as the canonical spectral representation—where the composite operators xp and px are strictly commuting multiplication operators. Quantumness is not imposed through a fixed Planck constant or measurement postulate, but emerges from the breakdown of recursive rigidity across all ordinal levels of class Fourier transforms acting on automorphic forms. The Cauchy–Schwarz equality is identified with global recursive compatibility, forcing the commutator of basic variables to vanish and the spectrum to be purely point-like (compact). A single failure of recursive proportionality at any ordinal level immediately triggers strict inequality, a non-zero commutator, continuous spectrum, and non-compactness. The algebraic Planck constant is geometrically identified as a running first Chern class of the automorphic line bundle, decomposed into a bare integer topological invariant and a scale-dependent vacuum fluctuation correction. The system admits no intermediate states; the classical/quantum dichotomy is logically deduced from the ordinal recursion law. This framework subsumes all standard Euclidean QFT axioms as a limiting case. We further demonstrate that quantum tunneling is not continuous barrier penetration but a discrete branch jump driven by topological charge integer overflow; that the "chaos-induced diffusion" of standard quantum mechanics is a projection artifact of the additive basis; and that wave-particle duality is a surface phenomenon of the inverse Mellin projection. Numerical simulations and recent nanoparticle interferometry experiments (MUSCLE, *Nature* **649**, 866–870) provide quantitative support for the predicted step-like jump signatures, with critical threshold located at (P₂) 2. 5, in exact alignment with the framework's topological phase transition. --- 我们在冯·诺依曼–伯奈斯–哥德尔 (NBG) 类论框架内给出了量子理论的完备公理基础。出发点为正实数乘法群 R_+^ 及其 Haar 测度 d^ x = dx/x, 梅林变换作为典范谱表示——其中复合算子 xp 与 px 严格对易, 均为乘法算子。量子性并非通过固定的普朗克常数或测量假设来强加, 而是源于自守形式上类傅里叶变换在所有序数层级的递归刚性破缺。柯西–施瓦茨等式被等同于全局递归相容性, 迫使基本变量的对易子消失、谱为纯点状 (紧致) 。在任何序数层级上, 递归比例性的单次失效都会立即触发严格不等式、非零对易子、连续谱以及非紧性。代数普朗克常数被几何地等同为自守线丛的跑动第一陈类, 分解为一个裸整数拓扑不变量和一个与尺度相关的真空涨落修正。该系统不容许中间态;经典/量子二分法由序数递归律逻辑推出。所有传统欧氏 QFT 公理均为本框架在经典不动点处的特例极限。本文进一步证明: 量子隧穿并非势垒连续穿透, 而是拓扑荷整数溢出驱动的离散分支跳变;标准量子力学中“混沌导致扩散”是加法基底的投影伪像;波粒二象性是逆梅林变换的表象相变。数值模拟与近期纳米团簇干涉实验 (MUSCLE, *Nature* **649**, 866–870) 为框架预言的阶跃跳变签章提供了定量支持, 临界阈值锁定于 (P₂) 2. 5, 与框架拓扑相变预言精确对齐。 --- GitHub: Research-Archives/数学/latex at master · calibur88/Research-Archives
Ch.HY (Thu,) studied this question.