Boundary of Dimensionless Transformations and Completeness of Product Metric Spaces for Decouplable Engineering Heterogeneous Dimensional Variables

Numerical modeling, metric construction, and iterative computation of finite-dimensional compact static engineering heterogeneous dimensional variables have long suffered from two types of rigorously correctable engineering application defects. First, the engineering field lacks quantitative formal definitions of the topological properties and applicable boundaries of two dimensionless transformations: physical benchmark normalization and extremum scaling normalization. This leads to the mixed use of linear and nonlinear mappings, resulting in insufficient topological compatibility of heterogeneous dimensional variable numerical modeling and poor stability of metric results. Second, there is no analytical conclusion on the metric distortion mechanism of engineering heterogeneous dimensional coupled variables. The determination of coupling strength has long relied on empirical thresholds, lacking parameter-free adaptive judgment criteria based on classical functional perturbation theory. There is no rigorous closed-loop derivation of the scaling law of coupling errors within a compact domain.

Based on the three axioms of metric spaces, finite-dimensional Banach space theory, Buckingham π dimensional analysis theorem, continuous mapping theory on compact sets, and bounded linear operator perturbation theory, under the pre-constraints of finite dimensions, compact closed set domain, no time-varying perturbations, and no strong nonlinear perturbations, this paper strictly distinguishes the topological preservation properties and failure boundaries of two dimensionless mappings for engineering heterogeneous dimensional variables, and formally proves the mathematical rationality of equal-weight and weighted product metric superposition of heterogeneous dimensional variables in the π-induced dimensionless subspace. Aiming at the core gaps in the field, the author completes nine closed-loop rigorous mathematical derivations and two original theoretical system constructions: First, based on the properties of linear isometric isomorphism and the criteria for continuous mappings on compact sets, quantitatively derive the compliant domains and topological failure boundaries of physical benchmark linear normalization and extremum scaling nonlinear normalization, and clarify the algebraic attribute differences between the two types of mappings. Second, step-by-step complete the rigorous deduction of the three-axiom compatibility and finite-dimensional space completeness of arbitrary p∈[1,+∞] order equal-weight and weighted Minkowski product metrics, and establish the compliance of heterogeneous dimensional metric superposition in the dimensionless subspace. Third, relying on the Banach space inverse operator theorem and bounded operator perturbation theorem, reconstruct the metric distortion mechanism of coupled variables in the compact domain, derive the strict scaling analytical formula of full p-order coupling errors without artificial parameters, and establish a coupling topology classification system based on metric topological invariants. Fourth, based on the topological continuity ε-δ criterion, strictly prove the necessary and sufficient conditions for mixed normalization mappings to maintain topological continuity, and supplement the quantitative constraint details for the topological continuity of mixed normalization mappings. Fifth, extend the coupling error scaling theory to weakly nonlinear coupling scenarios satisfying local Lipschitz continuity, and derive the strict upper bound of weakly nonlinear coupling errors. Sixth, extend the coupling error scaling theory to general finite-dimensional Banach spaces, and establish the metric topological invariant theory based on operator norms. Seventh, prove that the coupling topology classification system based on metric topological invariants can be used for linear system robust stability analysis. Eighth, reveal the quantitative relationship between metric topological invariants and matrix condition numbers, and expand them into a universal analysis tool for general linear transformation metric perturbations. Ninth, reveal the essential equivalence between metric topological invariants and the relative distortion of bi-Lipschitz mappings, expand the theoretical system to the field of metric perturbation analysis of nonlinear bi-Lipschitz mappings, and realize the unified characterization of linear and nonlinear mapping metric perturbations.

The full text contains no self-created non-standard academic terms, no empirical fitting constants, no artificial fixed thresholds, and no manual parameter tuning. All derivations rely on classical proven mathematical conclusions, supplementing the original skipped, invalid, and non-closed-loop mathematical proofs in the field. The supporting simulation code fully fits the pre-mathematical constraints, covering five metric working conditions of p=1, p=1.5, p=2, p=3, and p→∞, distinguishing six variable scenarios of decouplable, very weak coupling, weak coupling, medium coupling, strong coupling, and irreversible coupling, comparing three mainstream engineering methods of correlation coefficient method, covariance eigenvalue method, and mutual information method, and introducing two sets of actual engineering data from mechanical vibration and power systems for verification, achieving a strict one-to-one correspondence between theoretical constraints, simulation working conditions, and code parameters. The experimental results show that the parameter-free adaptive coupling strength determination criterion constructed by the author has an accuracy rate of 100% within the constrained domain, the compliant metric numerical deviation is strictly controlled within the theoretical error upper bound, and the coupling error evolution law and spatial topological stability law are completely consistent with the theoretical derivation, which is significantly superior to existing engineering methods in terms of error estimation accuracy and topological consistency. The author completes the standardized reconstruction of the underlying mathematical logic of engineering heterogeneous dimensional modeling under finite-dimensional compact static constraints, systematically establishes the standardized theoretical system of engineering heterogeneous dimensional modeling under full p-order Minkowski metrics in the constrained domain for the first time, fills the systematic theoretical gaps in compliant construction of heterogeneous dimensional metrics, quantification of transformation boundaries, and analysis of coupling errors, and standardizes the underlying application criteria of engineering heterogeneous dimensional modeling in the constrained domain. At the same time, the essential equivalence between metric topological invariants and the relative distortion of bi-Lipschitz mappings revealed by the author provides a unified mathematical tool for metric perturbation analysis in multiple fields such as geometric analysis, computer vision, and machine learning.