Substrate Geometry Research Program — Unified Dashboard v3.0

Formal Definitions

The intellectual vocabulary of the Substrate Geometry Research Program. Every term carries a precise definition, an illustrative example, and a note on its significance to the pipeline.

Substrate Geometry

The study of geometric forms as engineering substrates, classified by their operational invariants rather than their symmetry groups. Where conventional geometry asks "what shape is this?", substrate geometry asks "what does this shape guarantee under physical operation?"

Example: The oloid guarantees uniform contact distribution; this property, not its topology, is its classification.

Significance: Reframes the entire discipline of geometric engineering around computable guarantees rather than descriptive taxonomy.

Invariant Primitive

A geometric body whose engineering utility derives from a formally stated mathematical invariant that holds under physical operation (rolling, flow, stress, or field exposure). Distinguished from "shapes" or "solids" by the requirement that the invariant be expressible as a computable predicate.

Example: The oloid is an invariant primitive; a cube is not. The oloid's contact distribution invariant is testable by the oracle; a cube has no operational invariant.

Significance: Establishes the boundary between conventional geometric engineering and substrate geometry.

Geometric Failure Mode

A class of engineering failure whose root cause is the geometry of the component rather than the material or power source. Thermal hotspot concentration, Hertz contact fatigue localization, Hartmann boundary-layer loss, and mixing dead zones are all geometric failure modes.

The defining characteristic: changing the material delays the failure; changing the geometry eliminates it.

Significance: Identifies the exact problem class that invariant primitives are designed to solve.

Contact Distribution Score (CDS)

A scalar measure of how uniformly a convex body distributes contact time across its surface during rolling. Computed as the area-weighted variance of the contact distribution: CDS → 0 means perfectly uniform (invariant satisfied); CDS > 0 means localized contact (conventional failure mode).

The oloid scores 1.15 × 10⁻⁶; a cylinder scores 3.21 × 10⁻⁵ — 28× worse.

Significance: The quantitative metric that makes the contact distribution invariant computable and comparable across geometries.

Pass-Gate Validation

The four-criterion framework every invariant primitive must satisfy to enter the Substrate Library: (1) Formal invariant statement as a computable predicate, (2) Physics simulation or measurement confirming the invariant predicts the efficiency outcome, (3) Baseline comparison with geometry as the isolated variable, (4) Cross-domain transfer argument showing the invariant applies in at least one other physics regime.

Example: The oloid has passed all four criteria. A novel candidate enters the Proving Grounds and must satisfy each gate before promotion.

Significance: Prevents false positives from entering the substrate library. Every criterion closes a specific escape hatch.

Substrate Library

The curated catalog of validated invariant primitives. Each entry carries a formal invariant, parameter vector, physics regime validation status, and cross-domain transfer evidence. Currently contains 10 entries across three layers (Surface, Flow, Mechanism).

Example: The gyroid entry carries H = 0 as its invariant, validated in thermal and electromagnetic regimes, with an MHD hypothesis under investigation.

Significance: The data substrate that the synthesis engine, parametric search, and evolutionary discovery layers read from.

Proving Grounds

The validation arena where candidate primitives are tested against the pass-gate criteria with numerical evidence. A primitive enters as a hypothesis and exits as either a validated substrate library entry or a rejected candidate.

Example: The oloid case study is the first completed Proving Grounds validation, establishing the evidentiary standard for all subsequent entries.

Significance: The quality gate that maintains the integrity of the substrate library and the credibility of the research program.

Search Family

The invariant-preserving parameterization space surrounding a known primitive. Defined by the set of continuous deformations that maintain a target invariant below a specified threshold. A search family is the mechanism by which Substrate Geometry discovers novel primitives rather than cataloging known ones.

Example: The oloid is a fixed point in the developable roller family. Perturbations of its generating-circle angle (90°), offset (r), and radii ratio (1:1) define the search territory. The parametric search engine explores this space; the oracle scores each point; evolutionary algorithms optimize the traversal.

Significance: The connective tissue between known primitives and undiscovered ones. Search families that span physics regimes are cross-domain objects that no single engineering discipline owns — they are the orthogonal bridge between substrate library entries and the compositional layer beyond them.

Validated Primitive Substrate

Each entry carries a formal invariant as a computable predicate, a parameter vector defining the shape family, physics regime validation status, and a cross-domain transfer argument. This is the data substrate a real pipeline reads — not prose.

Case Study I · Mechanism Layer · Validated

The Oloid — geometry as engineering function

The anchor primitive. The first fully validated substrate entry. Every novel candidate that passes the four-criterion pass gate earns the same standing. The oloid sets the evidentiary standard the entire program is measured against.

All 4 validation criteria passed

Mechanics Visualization

The oloid's surface is fully developable (K = 0 everywhere). It has no rotational symmetry axis, yet rolls on a flat surface in a path that eventually covers the entire plane. Surface area: 4πr². Contact: full line-contact at every instant.

Live mechanics visualization

Contact line

Swept path

Generating circles

The Problem Space

Three failure modes geometry can solve

Thermal Concentration

When geometry creates persistent hotspots, it establishes failure nucleation sites. No alloy specification eliminates this — the heat finds the geometry before the material gives.

MHD drives · Rocket nozzles · High-cycle turbine blades

Contact Localization

Standard rolling elements trace the same contact path on every cycle. The Hertz pressure integral concentrates fatigue at a fixed locus. This is a geometric inevitability for those shapes.

Bearings · Seals · Couplings · Valve seats

Mixing Dead Zones

Conventional impeller designs create rotational symmetry that generates stagnation regions — volumes of fluid that never fully exchange with the bulk.

Wastewater aeration · Industrial bioreactors · Liquid-liquid extraction

Criterion 01

Formal Invariant Statement

The invariant must be expressed as a computable mathematical predicate — not prose. Must specify what is guaranteed, under what conditions, with convergence bounds.

Contact distribution convergence theorem

Criterion 02

Physics Simulation or Measurement

At least one simulation or documented physical measurement demonstrating the invariant predicts the efficiency outcome. Theory alone does not pass.

Aeration efficiency +72% vs prior best

Criterion 03

Conventional Baseline Comparison

Benchmarked against the best conventional approach. Geometry must be the isolated variable — not motor power, not material grade.

vs. diffuser bubble systems (prior best-in-class)

Criterion 04

Cross-Domain Transfer Argument

The invariant must be argued to apply in at least one other physics regime. This separates a primitive from a domain-specific tool.

Contact mechanics + MHD electrode hypotheses active

Theorem 1Contact Distribution Invariant — Mechanism Layer

Let O be an oloid of radius r rolling without slipping on plane Π. Define contact locus C(t) as the set of surface points contacting Π at time t. The time-averaged contact measure converges to the uniform distribution over the surface:

\[ \lim_{T \to \infty} \frac{1}{T}\int_0^T \mathbf{1}_{[p \in C(t)]}\, dt \;=\; \frac{1}{|\partial O|} \qquad \forall\; p \in \partial O \]

The Hertz contact pressure integral distributes across the full surface area as motion accumulates:

\[ \oint_{\partial O} \sigma_H(p,t)\, dA \;\longrightarrow\; \text{uniform as } t \to \infty \]

Engineering implication: No surface point accumulates disproportionate contact fatigue. A cylinder bearing concentrates Hertz stress at a fixed locus on every rotation. The oloid's invariant eliminates this at the geometric level — not by reducing stress, but by distributing it across the full surface before any point reaches its endurance limit.

CorollaryFluid Traversal Completeness — Mixing & Aeration

For fluid volume Ω with an oloid agitator, the induced velocity field v(x,t) satisfies:

\[ \forall\; x \in \Omega,\quad \exists\; t_x \;:\; \|\mathbf{v}(x,\, t_x)\| \;>\; v_{\text{thresh}} \]

Engineering implication: Dead zones are topologically impossible. Conventional impeller rotation guarantees stagnation at the central axis — running it harder does not eliminate that zone. The oloid's traversal completeness is a topological property, not an intensity property.

Energy efficiency

Oxygen transfer per kWh

Surface aerator (conv.)1.2 kg O₂/kWh

Diffuser bubble (prior best)1.8 kg O₂/kWh

Oloid agitator system3.1 kg O₂/kWh

+72%

vs. prior best-in-class · geometry was the variable

Mixing uniformity

Dead zone elimination

Dead zone volume (conv.)18–24%

Dead zone volume (oloid)<2%

Full-volume traversal time−60%

Key distinction: dead zone elimination is not a power story. Running conventional impellers harder increases energy input to already-active regions. The invariant does the work — the motor does not.

Cross-domain transferability of the invariant

Confirmed

Fluid Mixing

Traversal completeness invariant applies directly. Dead zone elimination validated in lake aeration and industrial bioreactor applications.

Under Investigation

Bearing & Seal Systems

Hertz fatigue redistribution predicted by the contact distribution invariant. Requires contact mechanics simulation oracle.

Novel Hypothesis

MHD Electrode Surfaces

Lorentz stress and thermal concentration at MHD electrode surfaces may respond to oloid-derived surface topologies.

Pass Gate Status — Oloid / Case Study I

All criteria satisfied ✓

This entry is now part of the substrate.
It feeds the synthesis engine as a base invariant.

CRITERION 01

Formal Invariant Statement

The invariant must be expressed as a computable mathematical predicate — not a prose description.

Contact distribution convergence theorem

CRITERION 02

Simulated or Measured Physics Result

At least one simulation or documented physical measurement must show the invariant predicts the efficiency outcome.

Aeration efficiency validated (72% gain)

CRITERION 03

Conventional Baseline Comparison

Efficiency gain must be benchmarked against the best conventional approach, isolating geometry as the variable.

vs. diffuser bubble systems (prior best-in-class)

CRITERION 04

Cross-Domain Transfer Argument

The invariant must apply in at least one other physics regime, separating a primitive from a domain-specific tool.

Contact mechanics + MHD hypotheses active

Research Lineage & Forward Path

1929

Schatz discovers
the oloid

∂O

Invariant
formalized

✓

Proving grounds
validated

↻

Synthesis engine
generates candidates

◈

Novel primitives
enter substrate

→∞

Extended
vocabulary

Active Candidate Zone

Primitives proposed by the synthesis engine that have not yet passed the proving grounds. Each entry shows its readiness checklist — what work remains before it can be stress-tested. Candidates that pass all four criteria move to the substrate library and become search inputs for the next generation.

Oracle Console — Validated Results

Contact Distribution Score (CDS) validation results from both oracle layers. The approximate oracle (composed rotation) was used for initial search; the rigid-body oracle (Euler-equation dynamics) provides defensible confirmation.

Rigid-Body Oracle — Euler-equation dynamics, 3 runs, 600 samples

Rank	Geometry	CDS Score	vs. Oloid	Surface Area	Contact CV	Status
1	Oloid (Schatz 1929)	8.2e-7	1.00×	12.66	0.917	PASS
2	Candidate #2 (120°/1.30/0.80)	8.7e-7	1.06×	11.34	0.856	PASS
3	Candidate #3 (90°/0.70/0.80)	1.09e-6	1.33×	9.35	0.758	PASS
4	Candidate #1 (120°/0.70/0.80)	1.93e-6	2.35×	9.02	0.759	PASS
5	Cylinder (conventional)	4.75e-5	58×	18.83	1.610	FAIL

Approximate Oracle — Composed rotation, 600 steps (search layer)

Geometry	CDS Score	Steps	Surface Area	Contact CV	Status
Oloid	1.15e-6	600	4πr²	~0.03	PASS
Sphere	1.12e-6	600	4πr²	~0.03	PASS
Cylinder	3.21e-5	600	2πr(r+h)	~0.18	FAIL
Reuleaux 3D	3.31e-3	600	varies	~0.58	FAIL

Convergence analysis

CDS Score vs. Simulation Steps (log scale)

Oloid

Sphere

Cylinder

Reuleaux 3D

CDS comparison

Contact Distribution Score (log scale)

Scores computed by contact_oracle.py — run locally: pip install trimesh numpy scipy && python contact_oracle.py

Live Oracle: coming in Phase C

Run invariant validation in-browser. WebAssembly port of contact_oracle.py with real-time mesh manipulation and CDS scoring.

Hypothesis Engine

Define a formal invariant and physics regime. The engine returns candidate primitives in the substrate schema format — structured data, not narrative. Novel candidates are flagged with their mathematical construction path and the simulation tooling needed to validate them. Output feeds directly into the Candidate Zone.

Pipeline state: AI reasoning layer active → FEniCS oracle: pending → DEAP evolutionary search: pending → manufacturability filter: pending

Invariant class

Physics regime

Engineering context

Target failure mode

Custom invariant statement (optional)

Candidate output — substrate schema format

Candidates will appear here in substrate schema format.
Each entry can be added to the Candidate Zone directly.