Substrate Geometry A research program that classifies geometric primitives by what they guarantee under physical operation, rather than by the symmetry groups they inhabit.
Mechanical engineering inherits its foundational primitives — cylinders, spheres, tori — from an era in which a shape's worth was measured by how cleanly it could be written down by hand. The substrates of contemporary practice were selected for analytical tractability under pre-computational constraints, then conserved by inertia long after those constraints lifted. Substrate Geometry investigates what is found when the question of which primitive belongs in which role is asked again, this time with the discipline of computational invariants rather than the convenience of closed-form integration.
The program produced its first results in April 2026 with the computational validation of the oloid as a local optimum within the developable roller family. A thirteen-member catalog of mono-monostatic bodies followed, extending Sloan's analytical Gömböc parameterization. An engineering-applications paper measured the practical payoff: 349× improvement in IMU calibration precision through purely mechanical orientation pre-fixing. A methodology paper documents the oracle-hardening discipline that underwrites every claim in the program.
The papers
58× contact-distribution discrimination between oloid and cylinder, with two-tier invariant transfer across the linear oracle family.
Surface critical points are necessary but not sufficient for mono-monostatic equilibrium. Fourier extension of Sloan's phase function produces the catalog.
349× IMU calibration precision improvement via mechanical orientation pre-fixing. Cylinders cannot achieve ECS=1 through ballast alone.
Diamond edges out gyroid on mean temperature at all cooling regimes. Neovius shows topological bottleneck failure mode.
Discrete Gaussian curvature estimator fails to converge on developable surfaces. The framework's coherence rests on this characterization.
What is found, under the surface
The thesis underneath the papers is older than the program. Every applied field of engineering carries a foundational vocabulary it did not choose freely. The vocabulary was inherited from a period when mathematics ran on paper and a primitive's worth was measured by whether its surface integral admitted a closed form. The cylinder was kept not because the cylinder is right but because the cylinder was solvable. This is true at the level of pedagogy, at the level of CAD default libraries, at the level of textbook conventions in tribology and contact mechanics. It is not a conspiracy. It is sediment.
A computational invariant framework lets the question be asked properly. Given a candidate body and an operational regime, what is the variance of contact time across its surface under rolling motion? What is the distribution of Hertz contact pressure? What is the expected peak temperature under frictional load with arc-position variation? What is the equilibrium count after rest under gravity? These are not philosophical questions. They are computable, falsifiable, and produce numbers that discriminate.
The framework's claim is not that any one shape is universally better. The claim is that the question of which shape belongs in which role can finally be asked.
The program's published findings are first instances of this asking. The oloid for rolling contact. The Gömböc catalog for orientation guarantee. Diamond rather than gyroid for thermal load distribution. None of these displace the existing literature in their domain; each sharpens it by introducing an operational invariant that previous classifications could not express. The methodology paper documents how these invariants are hardened against the silent failure modes of discrete differential geometry on the kinds of surfaces the framework most cares about.
See the papers for full detail, the methodology for the discipline that underwrites them, the lineage for historical context, or the atlas for a visual map of the program's territory.