Traditional implicit function extremum theory is confined to determining definite numerical maxima and minima for two-dimensional planar curves, with the sole objective of deriving fixed unique values of deterministic variables. This classical approach is fundamentally inapplicable to high-dimensional theoretical systems with no explicit analytical formulation and intrinsically uncertain fundamental parameters, exhibiting narrow applicability, fragmented mathematical logic, and lack of foundational theoretical depth. This paper conducts a rigorous coupling between classical higher mathematical implicit function extremum theory and the independently constructed Primitive Field Unified Steady-State (PFUS) System, which is grounded on the First Cause Axiom, Perturbation Axiom, Light-Dark Sevenfold Symmetric Logical Structure, and Pythagorean Circular Cone Topological Model. Aiming at the inherent uncertainty and non-fixed value property of β₁, the core primitive field parameter, this work reconstructs a complete high-dimensional implicit constraint extremum solution model, and redefines the mathematical connotation, solution criteria and judgment rules of extremum analysis within the PFUS framework. By comprehensive comparison with traditional implicit function extremum methods, this paper clarifies the essential differences between the two paradigms, and elaborates the inherent advantages, theoretical significance and application boundaries of PFUS implicit constraint extremum analysis. Results prove that abandoning the classical fixed-value solving paradigm, and adopting the implicit function extremum framework, can accurately identify the steady-state points, coupling critical points and hierarchical transition critical states of the β₁ primitive field, fully conform to the intrinsic uncertainty of β₁, perfect the rigorous mathematical deduction system of PFUS theory, and fill the theoretical gap in extremum analysis for high-dimensional uncertain implicit constraint systems.
Zhenmin Wang (Fri,) studied this question.