First Principles Exploration of Quantum Geometry: From Cantor Space to Calabi-Yau Manifolds

Background: Modern physics has established two pillars—general relativity and quantum field theory—in the macroscopic and microscopic domains respectively. However, these two frameworks are incompatible at a fundamental level, and the core postulates of quantum mechanics (such as the uncertainty principle and superposition) have long been accepted as axioms without inquiry into their physical origin. Objective: This paper aims to derive, from minimal assumptions, how spacetime geometry emerges from the discrete Cantor set through topological gluing to form a continuum, and subsequently generates higher-dimensional geometry via recursive folding, culminating in a natural derivation of Calabi-Yau manifolds. This study seeks to provide a geometric origin for the key features of quantum mechanics and to propose experimentally testable predictions. Methods: Following the logic of physical construction constrained by the principle of least action and symmetry, we start from a primordial symmetric field with Z₂ symmetry: (1) Spontaneous symmetry breaking introduces a primitive binary distinction (labeled 0 and 1); (2) Scale self-similarity drives recursive breaking, generating the Cantor set of all infinite binary sequences; (3) Based on the requirements of uniqueness, continuity, order, and measurability for physical spacetime, we define a topological gluing operation on the Cantor set, yielding the one-dimensional continuum; (4) Recursive folding operators extend the one-dimensional structure to higher dimensions; (5) We argue that under fluctuations, only certain folding modes remain stable, and the geometric conditions satisfied by these stable modes naturally lead to Calabi-Yau manifolds (Ricci-flat, Kähler, and SU(3) holonomy). Results: This study yields the following conclusions: (1) Spontaneous breaking of the primordial symmetric field necessarily produces a discrete binary structure; (2) Scale self-similarity forces the recursive process to proceed infinitely, establishing the Cantor set as the fundamental discrete substrate; (3) The axiomatic requirements of physical spacetime select a unique gluing rule, which inevitably introduces a minimum scale ε and densely distributed topological defects—providing a geometric basis for quantum uncertainty; (4) Recursive folding is truncated at the Planck scale, with the cutoff parameter ε self-consistently mapping to the Planck length; (5) Stable folding modes under fluctuations must satisfy Ricci-flatness, Kähler structure, and SU(3) holonomy—precisely the defining characteristics of Calabi-Yau manifolds; (6) The discrete topological invariants of Calabi-Yau manifolds originate from the folding cutoff, potentially offering a geometric explanation for quantum numbers in particle physics; (7) Based on this geometric framework, this paper proposes a particle spectrum conjecture with explicitly testable predictions, including mass ratios and generational features. Conclusion: This paper establishes a complete geometric path from first principles to Calabi-Yau manifolds. The study suggests that the Cantor set, as the mathematically most complete "space of possibilities," evolves through a selection mechanism (topological gluing) imposed by physical laws into the spacetime structure that supports reality. The imperfections introduced during gluing give rise to a minimum scale and primordial fluctuations—potentially the geometric origin of quantum effects. The selected paths, through recursive folding, generate the observed four-dimensional spacetime and its internal compactified Calabi-Yau manifolds. This framework attempts to unify discrete geometry and continuous spacetime, provides falsifiable predictions, and lays a geometric foundation for subsequent dynamical theories. 背景:现代物理学在宏观与微观领域分别建立了广义相对论与量子场论两大支柱,但二者在基本框架上互不兼容,且量子力学的核心假设(如测不准原理、叠加态)长期作为公理存在,其物理起源尚待阐释。 目的:本文旨在从极简预设出发,推导时空几何结构如何从离散的康托尔空间(Cantor Space)经拓扑粘合涌现为连续统,进而通过递归折叠生成高维几何,最终自然导出卡拉比-丘(Calabi-Yau)流形。本研究试图为量子力学核心特征提供几何起源解释,并提出可被实验检验的初步预言。 方法:遵循最小作用量原理与对称性约束的物理建构逻辑,本文从具有 对称性的原初对称场出发:(1) 通过对称性自发破缺引入原始二元区分(标记为 0 与 1);(2) 利用尺度自相似性驱动递归破缺,生成由无限长二进制序列构成的康托尔空间;(3) 基于物理时空的唯一性、连续性、有序性及可度量性要求,在康托尔空间上定义拓扑粘合操作,涌现出一维连续统;(4) 引入递归折叠算子,将一维结构扩展至多维几何;(5) 论证在涨落作用下,只有某些折叠模式能够保持稳定,这些稳定模式所满足的几何条件自然导出卡拉比-丘流形(里奇平直、凯勒结构及 和乐群)。 结果:本研究得出以下结论:(1) 原初对称场的自发破缺必然可导致离散二元结构的生成;(2) 尺度自相似性强制递归过程无限进行,确立康托尔空间为底层离散基底;(3) 物理时空的公理化要求筛选出唯一的粘合规则,该过程不可避免地引入最小尺度 及稠密分布的拓扑瑕疵点,构成了量子不确定性的几何基础;(4) 递归折叠受普朗克尺度截断,且截断参数 与普朗克长度存在自洽映射;(5)在涨落中保持稳定的折叠模式必须满足里奇平直、凯勒、SU(3)和乐等条件,这正是卡拉比-丘流形的定义特征;(6) 卡拉比-丘流形的离散拓扑不变量源于折叠截断,为粒子物理量子数提供了潜在的几何解释;(7) 基于此几何架构,本文提出了一个包含具体质量比关系与谱系特征的粒子谱猜想,具备明确的可检验性。 结论:本文建立了一个从第一原理推导至卡拉比-丘流形的完整几何路径。研究表明,康托尔空间作为数学上最完备的“可能性集合”,通过物理规律的筛选机制(即拓扑粘合)演化为支持现实存在的时空结构。粘合过程中的非完美性引入了最小尺度与原初涨落,这可能是量子效应的几何本源。经筛选的路径通过递归折叠,生成了观测到的四维时空及其内部紧致化的卡拉比-丘流形。该框架在尝试统一离散几何与连续时空,提供了可证伪预言,为后续动力学理论提供了几何基础。