Four-Dimensional Scale Space: A Theoretical Framework

This document presents a theoretical framework for four-dimensional scale space, in which the familiar three spatial dimensions (x,y,z) are extended by a fourth dimension s— the scale coordinate. A point (x,y,z,s) has not only a spatial location but a scale address. Equal steps in s produce equal ratios of apparent physical size. The unit of s is the nat (from ‘natural log’), with s= 0 corresponding to 1 metre (human scale).

The founding hypothesis — that physical scale constitutes a genuine fourth spatial coordinate — was first conceived by Donald Palmer in 1977 and developed over nearly five decades, with formal academic statements in two peer-reviewed publications in autumn 2025. The present document translates that hypothesis into a specific mathematical framework developed in dialogue with Claude (Anthropic) in 2026.

The framework is a two-track programme. Track 1 (Scale Space) is developed in Parts I–VI: the metric, geodesics, field equation, action, conservation laws, and a rigorous derivation of the Newtonian weak-field limit (Part I); quantum theory (Part II); biology (Part III); scale-extended objects (Part IV); related frameworks and distinctions (Part V); and the 5D parent theory and dynamical closure (Part VI). The geometry is an AdS-like (Riemannian, positive-definite) 4D space; mass causes motion in s; gravity is the apparent effect of coherent s-motion on scale-stationary observers. Track 2 (CNRS) is developed in Part VII: a positional number system in complex base z0 =−2 + i with digit alphabet {0,1,2,3,4} in which Gaussian-integer complex values are represented as finite single digit strings in CNRS-A, with branch state and calculus structure supplied by the higher CNRS layers; arithmetic is performed by finite automata; and the CNRS-H digit-shift realises ∂/∂ρ exactly. Part VIII traces the five specific contact points between the two tracks.

The companion papers (Scale Space Papers 1–20, 23–24; CNRS papers) and the programme books (Across Scale: A Geometry of Size and Representation, forthcoming; Across Scale: Technical Companion, forthcoming) are indexed at https://www.nul1.com and deposited on Zenodo.