Four-Dimensional Scale Space: A Theoretical Framework

A comprehensive theoretical framework in which physical scale — the logarithm of apparent object size — is treated as a genuine fourth spatial coordinate s, extending the familiar three spatial dimensions (x, y, z) to a four-dimensional space (x, y, z, s). The exponential scaling requirement uniquely determines the metric to be anti-de Sitter space AdS4 in Poincaré coordinates with curvature radius L = Rc²/GM. The framework is developed in eight parts covering: classical field theory and the Newtonian limit; canonical quantisation and the scale uncertainty principle; biological applications including cross-scale geodesics, protein dynamics, and morphogenesis; four-dimensional extended objects with scale pressure and boundary conditions; comparison with related frameworks (Lindeberg, Nottale, Kaluza-Klein, AdS/CFT, Vick, Šestáková); limitations; and an appendix documenting the parallel conjecture that a new complex numeric representational system — with integration and differentiation as primitive operations — is needed to fully measure across scale. The founding hypothesis was first articulated by the author in 1977. This document is the first complete mathematical realisation of that hypothesis. Developed in dialogue with Claude.ai (Anthropic), March 2026.