USE CASES
One question: how to provide spatial truth to an artificial intelligence. No volume, no intelligent decision. Five ways were tried. One path is integer-exact. Four fields are already written on it — and four case studies are detailed below, computed on real data.
FOUR CASE STUDIES · REAL DATA & PROOFS
Before the four fields, four studies you can examine: a cube inflated into its sphere across seams, a CT registered on a radiograph with a Lean certificate, an audit of Google ForTy global forest rasters across 1.39 trillion pixels, and three CERN benchmarks on decentralised medical AI compliance, lesion leakage, and data-free curvature.
Indexation
A 10 m tile is not 10 m at the pole, and two neighbouring grids do not agree at their seam. One Gray metacurve walks the six faces of the cube and gives every cell one address, from Geneva to Svalbard.
Modelling · Simulation
A pre-operative CT registered on an intra-operative radiograph: six exact numbers, a bounded error, and a Lean verdict that certifies the pose or refuses it. Twelve real cases, replayable.
Earth Observation · Inverse POC
Auditing Google DeepMind / WRI ForTy 2020 across 1.39 trillion valid pixels: 0 ha defect, 57.9 Mha of zone duplicates removed, and EUDR plot certificates proved by Lean 4.
Medical AI · CERN Collaboration
Auditing 2,131 chest radiographs in DeMICAF with zero training, unveiling lesion leakage in HAM10000, and providing data-free non-stationary geometric priors for DP-KFC curvature.
The journey to a spatial fact. Four of the ways stop short of a decision an AI can trust. The fifth was built for the problem.
Four critical scenes ask the same question: an operating room, a drone corridor over a city, a mine gallery, a radar. A probability cloud describes them. An exact solid decides.
Surfaces for the eye — perspective (Uccello, c. 1420), shaded display (Gouraud, 1971), then GPUs and splats. Image, not matter. No construction, no volume.
Set logic applied to volumes (Requicha, 1977). Classifying a point as in, on or out breaks on floating point.
Boundary surfaces for CAD (Braid, 1973). Point-in-solid is costly and unstable. Fit for an engineer, not for a real-time AI.
The cell, from crystallography (Haüy, 1784). Friendly to an AI, and without logic or topology — sound when the Arithmetics of Forms pilots the output.
SGDL
Geometry as primitive recursive functions (Péter, 1951; Rotgé, 1997). Integer-exact, polynomial, ready to run in parallel. Built for the question.
PRIMITIVE RECURSIVE FUNCTIONS
Primitive recursive functions are the geometry (Péter, 1951; Rotgé, 1997). The lambda calculus carries them into a programming language (Church, 1936). It is a symbolic calculus: the λ notation, and formal rules of evaluation and substitution, give a mathematical function one reading. Functional programming stands on both — recursive functions (Gödel, Herbrand, Kleene) and the lambda calculus.
The writing is Scheme, the Lisp dialect built as an interpreter of the lambda calculus (Sussman and Steele, 1975). The thesis writes the recursive arithmetic and the forms in that language. Procedures of the system — the gluing of simplices, the computation of densities — are built with the form lambda. That is SGDL-Scheme, 2000.
Church, 1936 · Gödel, Herbrand, Kleene · Péter, 1951 · Sussman and Steele, 1975 · Rotgé, Université de Montréal, 1997 · SGDL-Scheme, 2000.
This indexation is the work of 2025–2026. The approach is patented: a method and system for indexing an n-dimensional object.
Modelling writes a solid. Simulation runs it. Communication carries it to another machine. Encryption protects the signal with the same geometry. None of these gives the solid an address. Indexation does. It was the missing pillar beside modelling, simulation and communication: space-driven, or data-driven, on the theory of Gray metacurves.
A Gray curve visits every cell and steps to a neighbour, so cells that touch in space touch in the index. Peano, Hilbert and Wunderlich are the historical Gray curves. Cantor, Lebesgue, the projective Cantor curve and the semi-Gray curves also fill a square, without that one-step neighbourhood. The theory takes both lines. It classifies the turns, the motifs those turns produce, then composes the motifs into metacurves — a curve whose step is itself a curve — and assembles them as composites and rotors. Enumeration builds the path in integers. The same construction leaves the square for n dimensions: the curves, the metacurves, the volumes an address can name, the cuts through those volumes, and the curves that follow a surface rather than a flat grid. The cubo-sphere is that last case.
A query follows the space, or the data. The index is the address either way. No data is captured: real data stays where it is, and a synthetic solid is something an AI can check. The exolanguage is the engine that describes, indexes and infers those solids. It is not a note attached to a mesh. Its name is Spatial General Descriptive Language, and it now also infers.
Historical two-dimensional filling curves, lifted to n-dimensional hyperstructures. Gray and non-Gray curves feed the families. The families become two-dimensional metacurves, then n-dimensional ones.
Peano, Hilbert, Wunderlich, and what exploration still finds.
Cantor, Lebesgue, projective Cantor, semi-Gray.
As set out for the Open Geospatial Consortium, 28 August 2026.
Exclusion zones, corridors and safety volumes as named solids, not as metadata on a mesh.
GNSS-free autonomy on exact terrain, from the Moon outward.
Observational AI and attested synthetic data, in one scene an AI can verify.
The autonomous mine: crush zones, galleries and pit shells as written solids, not meshes.
