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.