USE CASES

Spatial truth for an AI.

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 two case studies run in this page, on real data.

TWO INTERACTIVE CASE STUDIES · REAL DATA

Try the exact computation.

Before the four fields, two studies you can operate: a cube inflated into its sphere with one index walking the six faces, and a CT registered on a radiograph with a Lean certificate that says yes or no. Nothing is a mock-up; each control drives the same integer computation as the case study.

The cube inflating into its sphere, its six faces walked by one Gray metacurve from gold to white, the current cell marked in cyan.

Interactive · Indexation

A 10 m tile on a round Earth.

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.

Move a point from Geneva to the pole
Pull the cube into the sphere
Walk the index across the six faces

A wireframe pelvis CT inside its integer cube, a C-arm, exact gold rays through the bone landing on the detector as a cyan silhouette, a hexagonal certificate seal.

Interactive · Modelling · Simulation

The scanner on the radiograph, to the proven millimetre.

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.

Blend measurement and model
Move the error by hand
Watch the certificate decide

Only one of the five ways is exact

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.

Four scenes in one drawing: a surgical robot, a drone corridor over wireframe towers, an underground mine, a radar dish. A blurred cloud of points against sharp gold outlines.

01 Polygonal Outside.

Surfaces for the eye — perspective (Uccello, c. 1420), shaded display (Gouraud, 1971), then GPUs and splats. Image, not matter. No construction, no volume.

02 CSG Problematic.

Set logic applied to volumes (Requicha, 1977). Classifying a point as in, on or out breaks on floating point.

03 B-Rep Problematic.

Boundary surfaces for CAD (Braid, 1973). Point-in-solid is costly and unstable. Fit for an engineer, not for a real-time AI.

04 Voxels Conditional.

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

05 Arithmetics of Forms The SGDL path

Geometry as primitive recursive functions (Péter, 1951; Rotgé, 1997). Integer-exact, polynomial, ready to run in parallel. Built for the question.

Indexation — Gray metacurves

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.

The taxonomy

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.

Gray

Peano, Hilbert, Wunderlich, and what exploration still finds.

Not Gray

Cantor, Lebesgue, projective Cantor, semi-Gray.

A Generators U, W, Uw, Wu — how a curve enters a cell and leaves it.
B Motifs: lacets, meanders, spirals, stairs, a super-Hilbert, kernels, centres, reflections.
C Two-dimensional Gray metacurves — the motifs composed.
D Composites chain distinct tiles into one path. Rotors nest one curve inside another.
E Enumeration: depth-first search, and backbite. The path is built in integers.
F Enumerative Gray curves in n dimensions — the same one-step neighbourhood, off the square.
G Metacurves in n dimensions — the composition, off the square.
H Hyperblocks and hypershells — the volumes an address can name.
I Hyperplanes, hyperslabs, hypercolumns — the cuts through those volumes.
J Curvilinear Gray curves — the index follows a curve drawn on a surface. The cubo-sphere.
K Exocurves of Gray in n dimensions.
L Xenocurves of Gray in n dimensions. With the exocurves, they close the figure.

As set out for the Open Geospatial Consortium, 28 August 2026.

Four fields

A city drawn as exact solids: flight corridors, safety bubbles around drones, a vertiport volume, and the tunnel under the street.

Low-altitude economy

Exclusion zones, corridors and safety volumes as named solids, not as metadata on a mesh.

Exact 3D and 4D corridors. Dynamic flight corridors and exclusion volumes are strict solids, inferred by the Arithmetics of Forms.
Morphological safety bubbles. A volume around each aircraft that depends on its velocity — a conflict is seen before it happens.
Fleet-scale anti-collision. Real-time logical intersection for a drone fleet over a city.
One model across milieux. The same solid runs from the transit tunnel to the street to the low-altitude corridor.

A lunar surface in contour lines: a crater indexed by a Gray metacurve, a rover path, and a coverage plan drawn as one curve.

Planetary world model

GNSS-free autonomy on exact terrain, from the Moon outward.

Curvilinear indexing. Craters, slopes and cavities described exactly — no triangulation, no GNSS.
A universal projective solid. High-performance planetary simulation, anchored to public LRO/LOLA elevation models.
Synthetic virtual LiDAR. A real capture and a simulated beam land on the same exact chords, with solar tracking.
A frugal loop. Contact-dynamic trajectories and data-driven coverage, on CPU or FPGA, without a GPU.

One scene across four milieux: a terrestrial digital twin, buried infrastructure, lunar habitat modules, and a satellite in orbit.

GEO-TRUST-SPACE

Observational AI and attested synthetic data, in one scene an AI can verify.

Observational AI. Real sensor streams — SAR, thermal, ground-penetrating radar, LiDAR — detect what is there.
A structural filter. The logical engine rejects any generated structure that is physically impossible.
Synthetic data with a provenance. Exact 3D and 4D solids, each with a manifest, ready for training.
Four milieux, one exolanguage. Terrestrial, underground, planetary and orbital infrastructure in a single exact scene.

An open pit as one written solid, the bench band, an autonomous haul fleet, and galleries from the surface to underground.

Mining 4.0

The autonomous mine: crush zones, galleries and pit shells as written solids, not meshes.

Ternary safety verdicts. Interior, frontier or exterior, on integer coordinates. Entering a zone is the verdict frontier.
Exact planning ledgers. The ultimate pit is one written solid. The frontier band is the blast-design line. Counts close on 200,000 panels.
A virtual sensor suite. Bit-exact renders, synthetic LiDAR on the same chords as a real capture, infrared by the nature of the solid.
Frugal at the edge. Integer arithmetic, without a GPU: the language model explains, and never measures.