Substrate Geometry
Computer-assisted interval computation / private preview
AConvex range of a spherical-harmonic perturbationHAND PROOFBConvexity thresholds for 27 harmonic bodiesCOMPUTER-ASSISTEDCEquilibrium counts for the m = 1 harmonic bodiesCOMPUTER-ASSISTEDD1Sloan shape 2: upper bound on the convex rangeHAND PROOFD2Sloan shape 2 is strictly convex for beta up to 0.03550477COMPUTER-ASSISTEDD3Sloan shape 2 has the origin as centroid and exactly two equilibriaHAND PROOFD4Sloan shape 1: upper bound on the convex rangeHAND PROOFD5Sloan shape 1 is strictly convex and mono-monostatic for beta up to 0.03229670385COMPUTER-ASSISTEDE1Phase-1 body: strictly convex2 METHODSE2Phase-1 body: exactly one stable and one unstable equilibrium2 METHODSF1Phase-1 gradient floorCOMPUTER-ASSISTEDF2Phase-1 Hessian at the equilibria: flat in one direction, steep in the otherCOMPUTER-ASSISTEDF3How much shape error Phase-1 toleratesCOMPUTER-ASSISTEDF4Is Phase-1 close to the best this family can do?EXPLORATORYG1Phase-1 equilibrium count as a replayable recordCOMPUTER-ASSISTEDG2Single-harmonic m=1 counts from the same checkerCOMPUTER-ASSISTEDG3Convexity of radial bodies, checked against the published thresholdsCOMPUTER-ASSISTEDG4Mesh trials: how much shape error a printed gomboc can takeEXPLORATORYG5A nearest-neighbour scan miscounts a flat bodyEXPLORATORYAConvex range of a spherical-harmonic perturbationHAND PROOFBConvexity thresholds for 27 harmonic bodiesCOMPUTER-ASSISTEDCEquilibrium counts for the m = 1 harmonic bodiesCOMPUTER-ASSISTEDD1Sloan shape 2: upper bound on the convex rangeHAND PROOFD2Sloan shape 2 is strictly convex for beta up to 0.03550477COMPUTER-ASSISTEDD3Sloan shape 2 has the origin as centroid and exactly two equilibriaHAND PROOFD4Sloan shape 1: upper bound on the convex rangeHAND PROOFD5Sloan shape 1 is strictly convex and mono-monostatic for beta up to 0.03229670385COMPUTER-ASSISTEDE1Phase-1 body: strictly convex2 METHODSE2Phase-1 body: exactly one stable and one unstable equilibrium2 METHODSF1Phase-1 gradient floorCOMPUTER-ASSISTEDF2Phase-1 Hessian at the equilibria: flat in one direction, steep in the otherCOMPUTER-ASSISTEDF3How much shape error Phase-1 toleratesCOMPUTER-ASSISTEDF4Is Phase-1 close to the best this family can do?EXPLORATORYG1Phase-1 equilibrium count as a replayable recordCOMPUTER-ASSISTEDG2Single-harmonic m=1 counts from the same checkerCOMPUTER-ASSISTEDG3Convexity of radial bodies, checked against the published thresholdsCOMPUTER-ASSISTEDG4Mesh trials: how much shape error a printed gomboc can takeEXPLORATORYG5A nearest-neighbour scan miscounts a flat bodyEXPLORATORY
PAPERS

Preprints and drafts

Posted means on arXiv now. Ready and drafting items are not public yet and are listed so the site never gets ahead of the record.

arXiv:2604.12238
posted

Computational Validation of the Oloid as a Local Optimum in the Developable Roller Family

Foundational paper: substrate geometry as classification by operational invariants.

arXiv:2604.17095
posted

Computational Construction and Engineering Evaluation of Verified Mono-Monostatic Bodies

Introduces the Equilibrium Count Score and the Fourier-extended Sloan family that gave the Phase-1 parameters.

arXiv:2604.17120
posted

Sloan's Analytical Gomboc at Published beta: A Strict-Convexity-Constrained Reanalysis [title retired and corrected in the staged revision: the published beta attribution is wrong]

Title as posted on arXiv. The 'published beta' attribution in that title is RETIRED: the note it refers to states beta up to 0.17 as sufficient for convexity, and the true boundary is near 0.036. A revised version with a corrected title is staged. The paper is a strict-convexity reanalysis of Sloan's second shape.

arXiv:2608.11213
posted

Static Equilibria of Perturbed Spheres: A Single-Harmonic Class Map, a Parity Obstruction, and a Certified Counter for the Mono-Monostatic Regime

Title as posted on arXiv (verified against the arXiv page on 2026-11-07). Equilibrium counts for harmonic near-spheres; listed under math.GM.

Standalone note
ready

Sharp convex range of a spherical-harmonic perturbation

Four pages. Contains Theorem A and Table B. Not yet posted.

Flagship manuscript
drafting

Margins for an explicit smooth gomboc, by interval computation (working title)

Typeset draft, 13 pages. Awaiting human readers, a literature check and the last interval run. Not yet posted.