SS-MULTI: Physical Scale as a Location Coordinate: A Multi-Scale Framework in Scale Space

Physical Scale as a Location Coordinate: A Multi-Scale Framework in Scale Space Multiscale modelling commonly represents physical systems through descriptions at different levels connected by coarse-graining, coupling and closure. This paper develops a conditional mathematical framework in which physical logarithmic scale, (u=\ln(\ell/\ell_0)), is treated as a location coordinate alongside ordinary spatial coordinates. It investigates whether interventions and responses located at different physical scales can reveal structural constraints on the behaviour of an identified system. The framework distinguishes mathematical fields, measurement protocols and empirical evidence. A five-rung construction separates scale location, response, relation, constraint and physical scale extent. It also distinguishes an object's own scale extent from the consistency of nested parts and wholes. The mathematical developments include a classification of scale-covariant second-order operators under specified assumptions, constructions for composing scale profiles and terminal responses across nested systems, and worked reaction–diffusion models with explicit coupling in the scale direction. A Gaussian-carrier example yields a conditional Turing-instability threshold shift and a scale-direction response kernel. A related two-exponent transport formulation separates laboratory-scale diffusion from scale-position transport, subject to its stated coordinate, mode and linearization assumptions. The paper also examines identifiability, conventional growing-domain and hidden-state alternatives, and the role of independent calibration in distinguishing mathematical representations from physically testable claims. Numerical verification supports the stated conditional calculations, while experimental confirmation of physical scale extent and a distinctive Scale Space mechanism remains outstanding. The contribution is a structured, extensible research framework for formulating and testing cross-scale relationships. It provides mathematical constructions and experimental criteria without claiming that the proposed physical scale coordinate has yet been empirically established.