A Conditional Proof Framework for the Non‑Existence of Odd Perfect Numbers Category‑Theoretic, Logical, and Analytic Unified Formalization

This paper establishes a rigorous three‑dimensional conditional proof system addressing the classic odd‑perfect‑number problem. We construct a four‑dimensionalreal vector‑space embedding for positive integers, a family of idempotent constraintprojection operators, and a complete Chebyshev‑metric threshold‑bounding system. The whole number‑theoretic problem is reduced to three explicit verifiablehypotheses. All classical foundations including multiplicative arithmetic‑functionproperties, Euler’s structural theorem, abundance inequalities, and numerical lowerbounds are given with full derivations. Under the three core hypotheses, we rigorously prove the non‑existence of odd perfect numbers. A complete Lean4 formalization skeleton is provided for computer‑aided verification. This framework containsoriginal, unique constructions and establishes a novel paradigm for Diophantineexistence problems 本文建立一套严谨的三维条件证明体系来处理经典奇完全数问题。构造4维实向量空间对正整数进行嵌入、约束幂等投影算子族、完整切比雪夫度量阈值边界系统。将整个数论问题归约为三条可检验的显式假设。全部经典基础,包括积性算术函数、欧拉结构定理、丰度不等式、数值下界均附带完整推导。在三条核心假设成立前提下严格证明奇完全数不存在。附带Lean4形式化骨架供计算机验证。本框架具备原创独有结构,为丢番图存在性问题提供全新范式