Papers
Four papers in the core Substrate Geometry sequence plus a methodology paper. Three are live on arXiv, one is in moderation, the methodology paper awaits endorsement for math.NA submission. A chemistry preprint on hERG channel QSAR for iboga alkaloids is drafted in parallel and queued for ChemRxiv.
Paper I · Oloid as Local Optimum
Introduces the Contact Distribution Score (CDS), an area-weighted variance of contact time across a rolling body's surface. CDS is the program's first formal operational invariant expressible as a computable function on a watertight mesh. The oloid scores CDS = 8.2 × 10⁻⁷ at 1,198 faces against the cylinder's 4.75 × 10⁻⁵, a 58× discrimination, and is shown to occupy a local minimum within the developable roller family across a 1,430-genome parametric sweep varying angle, offset, and radius ratio.
The two-tier invariant transfer structure is the load-bearing finding beyond CDS itself. The four linear variance metrics — CDS, Hertz SDS, thermal TDS, wear WDS — cluster within measurement precision at approximately 8 × 10⁻⁷, while the nonlinear Basquin fatigue metric FDS diverges to 2.42 × 10⁻⁶ due to S–N amplification. The structural distinction between linear and nonlinear invariants is methodological as well as descriptive: it tells the framework what to expect from a candidate primitive across operational regimes without requiring each regime to be measured independently.
The v2 revision applies the trajectory-coupling correction from the oracle hardening program. The original v1 SDS/CDS = 0.98 transfer was partly an artifact of shared rolling trajectory between oracles; geometry-only SDS reframes to 4.8 × 10⁻⁸ while preserving the 58× discrimination headline. The correction is documented in the methodology paper and traced in the v2 revision history. Read on arXiv.
Paper II · Sloan's Catalog Extension
The thirteen-member mono-monostatic catalog. The mathematical finding underneath: surface critical points are necessary but not sufficient for mono-monostatic equilibrium. The center-of-mass height function on the unit sphere exhibits four to eleven local minima despite Sloan's analytical phase function having exactly two surface critical points. Sloan's parameterization is extended with a Fourier term and optimized via differential evolution, producing the first openly available catalog of computationally verified instances.
The catalog is positioned as a sharpening of Domokos and Várkonyi 2006, not a refutation. Surface critical points remain the foundational mathematical object; what the catalog adds is the empirical observation that the center-of-mass height landscape on the unit sphere can carry equilibrium structure that the surface itself does not predict. Closing this gap requires the Fourier extension; the catalog is the family of solutions in which the gap is closed.
Paper III · Engineering Applications
Measures the engineering payoff of computationally verified mono-monostatic geometries across four application domains. The headline finding is the IMU calibration housing: 349× precision improvement over conventional cylindrical housings, 0.006° versus 2–3° initial orientation uncertainty under simulated MEMS-IMU deployment.
The Gömböc baseplate eliminates orientation uncertainty mechanically, before any software calibration stage. The mechanism is structurally analogous to ZUPT for velocity in inertial navigation: a physical pre-fixing that removes a degree of freedom from the calibration problem before algorithmic correction begins. Conventional cylindrical housings cannot achieve the same mechanism through ballast alone; the paper demonstrates this by exhaustive bottom-weighted mass-concentration sweep up to 30%, none of which produces ECS = 1.
Three additional application domains are characterized in less detail. Aerial seed pods achieve 100% correct orientation against an 83.5% cylindrical baseline. Marine buoy self-righting is scoped but not measured experimentally. Standalone capsule comparison gives BOA = 1.000 with SRE = 0.028, a 27× gentler self-righting trajectory than the capsule baseline. Read on arXiv.
Paper IV · TPMS Electrodes
A 240-run FEniCS surface-PDE regeneration across eight triply-periodic minimal surface geometries — gyroid, diamond, primitive, IWP, FRD, Neovius, plus conventional reference geometries — at three cooling regimes spanning three orders of magnitude in heat transfer coefficient. The methodology pivots from peak-of-time-averaged-temperature to expected-peak-temperature across arc positions, addressing a 41.6% coefficient of variation on peak temperature in the original framing that the single-arc Monte Carlo evaluation resolves.
Diamond edges out gyroid on mean temperature at all three cooling regimes, inverting the gyroid-default framing common in the TPMS thermal literature. The 1.16× within-category constancy and 1.4× cross-category TPMS-versus-conventional topology offset are consistent with surface-area-mediated thermal advantage, with three-dimensional channel-network heat spreading contributing the remaining factor. The face-normal-distribution-uniformity hypothesis remains open as an alternative explanation for the topological residual; discriminating between mechanisms is preserved as future work.
Methodology Paper · Oracle Hardening
Documents the oracle hardening program that ran in parallel with Paper I. The methodology paper makes five load-bearing claims, each supported by two or more independent empirical data points, and includes six explicit "does NOT claim" statements plus six deferred findings preserved for theoretical follow-up. The most striking single result is the discrete Gaussian curvature failure: |Kf| = 1.07 on the oloid mesh at all tested resolutions, while the analytical Gaussian curvature of the oloid is identically zero. The standard discrete K estimator has a topology-mismatched failure mode on developable surfaces; this is documented here for the first time in this specific form.
See the methodology page for full treatment of the five claims and the operating principles they underpin.