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. Movement along s corresponds to exponential scaling as observed from a scale-stationary reference frame: equal steps in s produce equal ratios of apparent physical size. The unit of s is the nat (derived 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 early public articulations in FQXi essay contest submissions and the formal academic statement in two peer-reviewed publications in autumn 2025. The present document translates that hypothesis into a specific mathematical framework developed in dialogue with Claude.ai in March 2026. The framework is developed in eight parts. Part I establishes the classical field theory: the metric, geodesics, field equation, action, conservation laws, and a rigorous derivation of the Newtonian weak-field limit confirming that standard gravity is recovered with no free parameters. The geometry is anti-de Sitter space AdS4, and mass causes motion in s so that gravity is the apparent effect of coherent s-motion on scale-stationary observers. The curvature radius L = Rc2/GM is a property of each gravitating body, taking the universal value L = 2 nats at every black hole horizon. Part II develops the quantum theory, including a scale uncertainty principle, a geometric quantum-to-classical transition requiring no decoherence postulate, and a new quantum number ns. Part III applies the framework to biological systems. Part IV extends to objects with genuine scale extent, introducing scale pressure, new wave modes, and three classes of scale boundary condition. Part V situates the framework relative to five prior bodies of work — Palmer’s own published research, Lindeberg scale-space theory, Nottale’s scale relativity, Kaluza-Klein theory, and AdS/CFT holography. Parts VI, VII, and VIII address limitations, summary, and references respectively. The companion papers providing full derivations are Papers 1–19 (all drafted, May 2026; Papers 1 and 6–19 deposited on Zenodo; Papers 2–5 Zenodo deposits pending confirmation), all deposited on Zenodo. The core papers are:• Papers 1–5: Classical field theory, gravitational predictions, quantum theory, biology I and II.• Papers 6–9: Scale-Kerr perturbation, gravitational waves, interior matching, 5D parent theory.• Papers 10–13: Corrected metric, Einstein tensor, geometric status of c, complex zs conjecture (programme).• Papers 14–15: Dynamical origin of Newton’s law; scale-integration reduction. • Papers 16–17: Discriminating tests of the complex zs conjecture; 5D field equations with complex scale coordinate (Paper 13 Steps 3–4 substantially addressed).• Paper 18: Multi-scale Turing pattern formation via CNRS-H (Implementation 1).• Paper 19: The representational requirement — CNRS as Paper 13 Step 5 (substantially closed).