The Scale Space–CNRS Programme: 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 u — the physical scale coordinate. A point (x, y, z, u) has not only a spatial location but a scale address. Equal steps in u produce equal ratios of apparent physical size. The unit of u is the nat, with u = 0 corresponding to 1 metre (human scale). The founding hypothesis — that physical scale constitutes a genuine fourth dimension — 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 [1, 2]. The present document translates that hypothesis into a specific mathematical framework developed in dialogue with Claude (Anthropic) and ChatGPT (OpenAI) in 2026. The framework comprises the kinematic geometry of four dimensions: the metric, geodesics, curvature, and a local kinematic 1/R projection sector (Track 1, Part I; see Table 1 for the full Part structure) — a scalar-kernel result, not itself the programme’s selected gravitational mechanism. A separate dynamical architecture supplies the programme’s candidate low-energy dynamical closure (Part II): a leafwise construction in which each scale position carries its own four-coordinate gravitational geometry, with a protected massless two-polarization tensor zero mode in its leading gravitational sector. Quantum theory, biology, scale-extended objects, and comparison with related frameworks (Parts III–VI) build on the kinematic geometry. Track 2 (CNRS, Parts VII–IX) is an independent mathematical programme: a positional number system in complex base z0 =−2 + i in which complex numbers are single digit strings. Its results are stated in technical papers and, where appropriate, checked by independent formalization. The connection between the two tracks (Part X) is presently limited to a single structural parallel, stated without technical dependency; four previously proposed technical contact points relied on an architecture since retired. Stated plainly: the programme’s kinematic geometry, by itself, does not produce a distinct falsifiable prediction. Empirical consequences require additional dynamical, source, and measurement assumptions, whose status and limitations are stated separately in Parts II–V. The technical papers and the programme books (The Forgotten Coordinate: The Missing Geometry of Size, from Atoms to Galaxies and The Forgotten Coordinate: Technical Companion) are indexed at https://www.nul1.com. Earlier numbered Scale Space papers are retained as a historical development record. Current claim level sources and supersession mappings are identified at the programme index.