{"paper":{"title":"Sloan's Analytical G\\\"omb\\\"oc at Published $\\beta$: A Strict-Convexity-Constrained Reanalysis","license":"http://creativecommons.org/licenses/by/4.0/","headline":"Sloan's analytical Gömböc equations produce no mono-monostatic bodies, but Fourier and radial extensions yield a catalog of thirteen verified examples.","cross_cats":["cs.CG"],"primary_cat":"cs.CE","authors_text":"Vincent Wesley Couey","submitted_at":"2026-04-18T19:41:10Z","abstract_excerpt":"Varkonyi and Domokos (2006) proved that convex homogeneous bodies with exactly one stable and one unstable equilibrium point exist. Sloan (2023) gave the first analytical parameterization, with radial function $R(\\theta,\\phi)$ having exactly two critical points on $S^2$.\n  This is the v2 amendment-of-record of arXiv:2604.17120. v1 claimed Sloan's parameterization does not produce mono-monostatic bodies and reported a 13-member catalog of Fourier/radial extensions certified at ECS=1 via mesh-vertex drainage-basin analysis. Following correspondence with P. L. Varkonyi (BME), an analytical verifi"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"No tested parameter value produces a mono-monostatic body under Sloan's parameterization, but extensions with Fourier terms and radial perturbations produce thirteen verified mono-monostatic bodies with ECS=1 confirmed across merge thresholds 0.5-10%.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"That the ECS oracle, which counts stable equilibria via drainage basin analysis on the COM height landscape, accurately captures all equilibria and that differential evolution optimization finds parameters satisfying the mono-monostatic condition without overlooking additional minima.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Sloan's parameterization fails to produce mono-monostatic bodies, but extensions yield thirteen verified examples with open data showing a near-perfect robustness-gentleness trade-off.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Sloan's analytical Gömböc equations produce no mono-monostatic bodies, but Fourier and radial extensions yield a catalog of thirteen verified examples.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"4be47217a655cb9b23f474b4230818411a21416a6059fa25974e9660a2414363"},"source":{"id":"2604.17120","kind":"arxiv","version":2},"verdict":{"id":"4bff4189-d430-497d-8e51-8826c35a4037","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-10T06:05:00.670056Z","strongest_claim":"No tested parameter value produces a mono-monostatic body under Sloan's parameterization, but extensions with Fourier terms and radial perturbations produce thirteen verified mono-monostatic bodies with ECS=1 confirmed across merge thresholds 0.5-10%.","one_line_summary":"Sloan's parameterization fails to produce mono-monostatic bodies, but extensions yield thirteen verified examples with open data showing a near-perfect robustness-gentleness trade-off.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"That the ECS oracle, which counts stable equilibria via drainage basin analysis on the COM height landscape, accurately captures all equilibria and that differential evolution optimization finds parameters satisfying the mono-monostatic condition without overlooking additional minima.","pith_extraction_headline":"Sloan's analytical Gömböc equations produce no mono-monostatic bodies, but Fourier and radial extensions yield a catalog of thirteen verified examples."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2604.17120/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}