Structure Over Axioms: From Inherent Mathematical Constraints to Three-Dimensional Bases and Omnidimensional Reconstruction

The preceding physical research project, the **Space Interlayer Conjecture / Phase Manifestation Mechanics** (abbreviated as **SIC / PMM**), exposed several blind spots in modern mathematical systems concerning discreteness, numerical definitions, and mathematical representation. Based on these issues, this paper proposes and develops the central principle of **Structure Over Axioms**. This paper argues that axioms should not be regarded as the absolute starting point of mathematical deduction. Rather, they should be understood as expressions of underlying mathematical structures under specific conditions and dimensional constraints. The definability of a mathematical object cannot be established independently of its generative premises, conditions of existence, and structural boundaries. To examine this principle concretely, this paper first analyzes cases such as **TAN90**, investigating whether a numerical value remains definable when its original prerequisite structure disappears. It then addresses issues involving the discrete and the continuous, infinity and boundaries, structural extension, and the criteria by which an extension may be determined to be consistent and well-defined. Within the framework of **Structure Over Axioms**, this paper revisits the three major crises in the foundations of mathematics and investigates whether their underlying mechanisms can be described through a common foundational structure. Subsequently, the paper uses the specialized equations of **PMM** as a concrete construction case. By modifying key extension conditions, it establishes a new mathematical structure and further discovers that this structure can form a **three-dimensional basis** possessing properties of a generative or mother structure. On this basis, corresponding operators, operational rules, and structural representations are defined. After the gradual formation of this mathematical structure, the paper further identifies possible extensions into several fields that were not initially presupposed as research objectives. These include the structural reformulation and dimensional reduction of the seven Millennium Prize Problems, the geometric necessity of probability, and metacognition in artificial intelligence and new AGI architectures. These topics were not the original goals of the study; rather, they emerged as subsequent possibilities through the processes of structural reconstruction and extension. **Keywords**: *Structure Over Axioms*, *Inherent Mathematical Constraints*, *Three-Dimensional Base*, *Foundational Mathematics & Axiomatic Logic*, *Mathematical Crises & Structural Foundations*, *Higher-Dimensional Geometry & Topology*, *Phase Manifestation Mechanics*, *Structural Approach to Millennium Prize Problems*, *Probabilistic High-Dimensional Geometry*, Structural Perspective on AGI Architectures -------------------------------------------------- V1.0.0 • Permanent Archive (IPFS CID):https://teal-giant-starfish-82.mypinata.cloud/ipfs/bafybeidv7hz7isszy6us644vnjxetfltxdkwkqxmiucknkwbr5jg4k6vpe• Integrity Note (SHA-256): C2E6DB27FF449C1A251C84F7D2DB61D04039A552D0ECDB5F76AF00AE7C4465F6 • Governance: Released under AXIOMATIC RECIPROCITY CLAUSE (ARC). Full terms included in the documentation. --------------------------------------------------