Substrate Geometry

Most geometric primitives used in engineering (cylinder, sphere, torus) were selected because they're easy to parameterize, easy to machine, and have closed-form contact solutions. None of those reasons is about physical performance under load. They're about computational and manufacturing convenience.

The oloid -- a 1929 shape constructed from two perpendicular circles of equal radius with the shared tangent line -- doesn't look like the kind of thing you'd pick for a bearing. But when you feed it through the oracle stack using the single-arc methodology, it outperforms cylinders by 25x on thermal advantage alone, with corresponding improvements in contact distribution, peak Hertz stress, and wear progression.

The interesting claim isn't just "the oloid is good." It's that we've been picking engineering primitives for the wrong reasons and that a systematic shape-classification framework based on physical invariants would produce very different defaults.

Seven physics-based oracles evaluate each candidate geometry independently. Contact distribution (how the load spreads). Hertz stress (peak contact pressure). Thermal (how heat flows at the interface under repeated contact). Wear (volume loss per cycle under the Archard model). Fatigue (cycles to crack initiation). Two further oracles complete the stack; all seven are specified in the preprints.

Each oracle is a physics simulator in its own right, independently validated. The parametric search runs 1,430 variants through all seven and produces a two-tier invariant vector: four linear metrics that cluster tightly at 8×10⁻⁷, and three nonlinear fatigue metrics that diverge sharply. That divergence is the proof that the winning geometry is doing something that isn't reducible to the linear metrics - which is the interesting finding.

Five papers are now in the program. Three are live on arXiv: the developable-roller study (where the oloid surfaces as a local optimum), the engineering evaluation of the mono-monostatic catalog, and the Gomboc catalog paper. Paper I was endorsed on arXiv by Georg Nawratil (TU Wien). He came into it because Hellmuth Stachel, Professor Emeritus at TU Wien and one of the few living geometers to have published on the oloid, read the work, engaged with it, and referred it to Nawratil when his own arXiv category did not qualify him to endorse. Stachel also pointed me toward the grandson of Paul Schatz, who first constructed the oloid in 1929. Having two senior TU Wien geometers take the work seriously validated the direction more concretely than any single review could.

The remaining two papers are on a journal-first track. The methodology paper, which operationalizes the pre-commit + cascade + audit-trail discipline as a reusable cross-domain method, is under review at the SMAI Journal of Computational Mathematics (diamond open access). The TPMS thermal paper, carrying the 25.6x mean-temperature result, is under review at Elsevier Results in Engineering, with a durable SSRN preprint public. The frozen Paper I codebase in core/ is never edited -- it serves as the canonical reference implementation.

The bigger goal here is not one clever shape. It is a way of choosing the geometric building blocks that go into real engineering substrates: bearings, rollers, heat-exchanger channels, anything where a surface meets a load or a flow. If the primitives we default to were chosen for convenience rather than performance, then a systematic, physics-first classification should turn up better ones.

The oloid is the first proof of that. The 25.6x thermal result on a TPMS channel is the second, in a completely different regime. The program keeps testing the same question against new substrates: pick the shape by what it guarantees under real physical operation, and see how much performance was being left on the table by picking it for convenience instead.

The Substrate Geometry repository is public at github.com/gyapaganda-a11y/substrate-geometry under MIT license. CONTRIBUTING.md lists what kinds of extensions are welcome and where coauthorship on the next paper is on the table.

Specifically, three thresholds are named for coauthor invitation: a member #14 of the mono-monostatic catalog, the analog for a related body family (constant-width Reuleaux solids, non-developable rollers, etc.), or a substantive new invariant validated against an existing primitive at the precision documented in the methodology paper.

The repository also ships CITATION.cff for software citation and four issue templates (replication question, citation or acknowledgment, collaboration inquiry, bug report). Replication of any catalog member is welcomed and acknowledged.

All papers in the program are listed on the arXiv author page: arxiv.org/a/couey_v_1.html. Threads on individual papers and replication observations are at x.com/VincentCouey.