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 more I'm keeping out of the public summary until the arXiv preprint lands.

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.

Four papers are now in the program: two published on arXiv and two compiled. The research has earned endorsement from Hellmuth Stachel (Professor Emeritus at TU Wien, one of the few living geometers who has published on the oloid), who provided the contact information of the grandson of the original figure who constructed the oloid in 1929. That lineage handoff validated the research direction more concretely than any peer review could.

The methodology paper is pending endorsement for arXiv submission. Papers I v2 and II v2 are in final pre-submission review. The frozen Paper I codebase in core/ is never edited -- it serves as the canonical reference implementation.

Physics is the purest test of the Gardener framework because the medium refuses to cooperate with Architect-style specification. You cannot tell an oloid to be optimal. You can only build the conditions under which optimality is detectable, then watch which shapes survive.

The lesson that transfers back to the other Deep Synthesis projects: every domain has a set of physical invariants you're either honoring or ignoring. The Gardener move is to make the invariants visible and let the medium arbitrate. The Architect move is to make the invariants conform to the spec.

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.