{ "meta": { "generated_from": "scoping/FLAGSHIP-MANUSCRIPT-2026-11-06.md (sections 3, 5 to 8) and flagship_repro/README.md", "as_of": "2026-11-08", "wording_rule": "computer-assisted interval computation; the words certified and proved appear only on rows whose tier allows them", "release_values": { "note-pending": "in the standalone convex-range note, not yet posted", "flagship-held": "in the flagship manuscript, not yet posted", "public": "in a public preprint", "software-local": "in local software that is not yet released" } }, "tiers": { "hand": { "label": "Proved by hand", "short": "HAND PROOF", "cls": "t-proved", "text": "A proof written out on paper. A computer may have checked the numbers, but the argument does not depend on one." }, "independent": { "label": "Computer-assisted + independent second leg", "short": "2 METHODS", "cls": "t-cert", "text": "An interval-arithmetic computation, repeated by a second method that shares no chart, symmetry reduction or polar cap with the first, over the whole statement." }, "computer": { "label": "Computer-assisted", "short": "COMPUTER-ASSISTED", "cls": "t-samp", "text": "A statement about ball-arithmetic enclosures, backed by a script and log. Where a second method covers part of it, the row says exactly how much." }, "exploratory": { "label": "Exploratory (floating point)", "short": "EXPLORATORY", "cls": "t-conj", "text": "A float scan. It shows where to look. It is not a certificate and is never cited as one." } }, "results": [ { "id": "A", "name": "Convex range of a spherical-harmonic perturbation", "kind": "Theorem A", "tier": "hand", "release": "note-pending", "statement": "Let g be a spherical polynomial of degree at most L with max |g| = 1. The body r = 1 + eps g is strictly convex for every eps < 1 / (L^2 + 1), and the constant cannot be improved: a sectoral harmonic reaches it.", "numbers": [ { "k": "Sharp threshold", "v": "1 / (L^2 + 1)" }, { "k": "Key input", "v": "Bernstein-Riesz inequality along great circles" } ], "method": "Bernstein's inequality along great circles bounds the second derivative of g by L^2 max|g|; inserting that into the convexity matrix M = r^2 sigma + 2 grad r (x) grad r - r Hess r gives positive definiteness.", "second_leg": null, "evidence": [ "convex_range_note/main.tex" ], "not_covered": [ "A literature check for prior statements of this inequality is still open.", "It bounds the convex range from below only; Table B shows the true range is larger for non-sectoral bodies." ] }, { "id": "B", "name": "Convexity thresholds for 27 harmonic bodies", "kind": "Table B", "tier": "computer", "release": "note-pending", "statement": "For each of the 27 bodies Y_l^m with 2 <= l <= 7, an interval bracket of the largest amplitude that keeps the body strictly convex, with relative width at most 0.5 percent.", "numbers": [ { "k": "Bodies", "v": "27" }, { "k": "Non-sectoral, eps0 (l^2+1)", "v": "1.077 to 1.472" }, { "k": "Sectoral, eps0 (l^2+1)", "v": "0.996 to 1.002" }, { "k": "Max bracket width", "v": "0.5 %" } ], "method": "Branch and bound with a centered form from forward-mode automatic differentiation, in ball arithmetic.", "second_leg": "Cartesian cube-face method on three bodies: Y33, Y44, Y31. The other 24 have the primary method only.", "evidence": [ "convex_range_note/repro/", "convex_range_note/repro/table_final.json" ], "not_covered": [ "24 of the 27 brackets have no second method.", "l above 7." ] }, { "id": "C", "name": "Equilibrium counts for the m = 1 harmonic bodies", "kind": "Theorem C", "tier": "computer", "release": "flagship-held", "statement": "For l = 2, 3, 4, 5, 6, 7 and every amplitude from just above 0 to the stated upper end, the body has exactly S = U = l stable and unstable equilibria and H = 2l - 2 saddles, all nondegenerate, with none in the polar caps.", "numbers": [ { "k": "l = 2", "v": "eps above 0 to 0.1994, (2, 2, 2)" }, { "k": "l = 3", "v": "eps above 0 to 0.1062, (3, 3, 4)" }, { "k": "l = 4", "v": "eps above 0 to 0.0668, (4, 4, 6)" }, { "k": "l = 5", "v": "eps above 0 to 0.0452, (5, 5, 8)" }, { "k": "l = 6", "v": "eps above 0 to 0.0327, (6, 6, 10)" }, { "k": "l = 7", "v": "eps above 0 to 0.02465, (7, 7, 12)" } ], "method": "Amplitude intervals are split into tiles. On each tile the centroid and the equilibrium conditions are evaluated with interval amplitude; Krawczyk uniqueness boxes and exclusion boxes cover the sphere. Below about 60 percent of the range the same is done on the rescaled potential (rho - 1) / (2 eps), which stays smooth down to eps = 0.", "second_leg": "Point checks by the cube-face method at one amplitude each for l = 3, 4, 5, 6, 7 (l = 6 and 7 inside the rescaled cover); inconclusive for l = 2. These are not interval covers.", "evidence": [ "flagship_repro/m1/m1_interval.py", "flagship_repro/m1/m1_independent.py", "flagship_repro/m1/m1_interval_Y71_merged.json", "flagship_repro/m1/m1_small_eps.py" ], "not_covered": [ "l = 8 and above.", "m = 0 and m of 2 or more.", "Second method inconclusive for l = 2 (equilibria on chart seams); only point checks, no second-method cover of the rescaled tiles.", "Stable and unstable labels in the second method are floats, checked through S - H + U = 2." ] }, { "id": "D1", "name": "Sloan shape 2: upper bound on the convex range", "kind": "Closed form", "tier": "hand", "release": "flagship-held", "statement": "At the point of least radius the Gaussian curvature of Sloan's second shape is an explicit function of beta and is negative on (beta*, 0.2474), with beta* = 0.0355047718919948. The convex range stated in the original note (beta up to 0.17) is therefore not convex.", "numbers": [ { "k": "beta*", "v": "0.0355047718919948" }, { "k": "Stated in the note", "v": "beta <= 0.17" } ], "method": "Direct evaluation of the curvature at (theta, phi) = (pi/2, pi), where R^4 = 1 - 4 beta.", "second_leg": null, "evidence": [ "scoping/FLAGSHIP-MANUSCRIPT-2026-11-06.md section 5.2", "scoping/SLOAN-CONVEXITY-THRESHOLD-CLOSED-FORM-2026-11-02.md", "flagship_paper/verify_beta_star.py" ], "not_covered": [ "Independently re-derived with symbolic algebra on 2026-11-07: the root matches the closed form to 33 digits. A human reading of the derivation is still wanted." ] }, { "id": "D2", "name": "Sloan shape 2 is strictly convex for beta up to 0.03550477", "kind": "Theorem D", "tier": "computer", "release": "flagship-held", "statement": "K > 0 on the whole surface for every beta in [0, 0.03550477]. The remaining gap to beta* is 1.9e-9.", "numbers": [ { "k": "Primary range", "v": "0 to 0.03550477" }, { "k": "Second method range", "v": "0 to 0.0355047" }, { "k": "Gap to beta*", "v": "1.9e-9" }, { "k": "Boxes (primary)", "v": "26,535,337 + 2,797" } ], "method": "Compact radial-graph chart plus polar caps, centered form from automatic differentiation.", "second_leg": "Cartesian cube-face method over beta from 0 to 0.0355047. A third route (convexity matrix in (t, psi, beta)) covers the whole range up to 0.0355047718 as a replayable record of the box search, re-validated by a separately written checker; the gap to beta* is 9.2e-11.", "evidence": [ "flagship_repro/sloan/certify_fine.py", "flagship_repro/sloan/gap_fine_b03550477.log", "flagship_repro/sloan/sloan_audit.py" ], "not_covered": [ "The last 7e-8 of the range by the cube-face method (the third route covers it).", "The single value beta = beta*." ], "headline": { "k": "Sloan shape 2 stays convex up to beta", "v": "0.03550477", "d": "The note states 0.17. The closed-form limit is 0.0355047718919948." } }, { "id": "D3", "name": "Sloan shape 2 has the origin as centroid and exactly two equilibria", "kind": "Derivation", "tier": "hand", "release": "flagship-held", "statement": "For every 0 < beta < 1/4 the centroid is the origin (the integral of sin^3 t exp(i eta) vanishes) and the radius function has exactly two nondegenerate critical points, a minimum at (pi/2, pi) and a maximum at (pi/2, 0). With strict convexity these are the stable and unstable equilibria.", "numbers": [ { "k": "Equilibria", "v": "2, nondegenerate" }, { "k": "Valid for beta", "v": "0 < beta < 1/4 (centroid, critical points)" } ], "method": "Hand derivation of the centroid integral and the gradient of R on the sphere.", "second_leg": null, "evidence": [ "scoping/FLAGSHIP-MANUSCRIPT-2026-11-06.md section 5.4" ], "not_covered": [ "Re-derived with no algebra errors found. A human reading is still wanted.", "The original note states these facts without derivation." ] }, { "id": "D4", "name": "Sloan shape 1: upper bound on the convex range", "kind": "Closed form", "tier": "hand", "release": "flagship-held", "statement": "At the point of least radius the Gaussian curvature of Sloan's first shape is (125 beta^2 - 35 beta + 1) / (1 - 4 beta)^(5/2), negative between the roots (7 - sqrt 29) / 50 = 0.03229670385730 and 0.24770329. The convex range stated in the original note (beta up to 0.15) is therefore not convex, too large by a factor of about 4.6.", "numbers": [ { "k": "beta*", "v": "0.03229670385730" }, { "k": "Stated in the note", "v": "beta <= 0.15" } ], "method": "Symbolic curvature of the implicit surface at (theta, phi) = (pi/2, -pi/2), where R^4 = 1 - 4 beta; the sign is checked by ball arithmetic on [0.0324, 0.15] and [0.0001, 0.032].", "second_leg": null, "evidence": [ "scoping/FLAGSHIP-MANUSCRIPT-2026-11-06.md section 5.5", "flagship_repro/sloan/shape1_pole_K.py" ], "not_covered": [ "One point of the surface: this refutes the stated range. Convexity below the root and the equilibria of this shape are the subject of D5.", "A human reading of the derivation is still wanted." ] }, { "id": "D5", "name": "Sloan shape 1 is strictly convex and mono-monostatic for beta up to 0.03229670385", "kind": "Theorem", "tier": "computer", "release": "flagship-held", "statement": "The convexity matrix of Sloan's first shape is positive definite on the whole sphere, poles included, for every beta in [0, 0.03229670385]. With the origin as centroid and exactly two nondegenerate critical points of the radius, the body has one stable and one unstable equilibrium. The remaining gap to the closed-form limit is 7.3e-12.", "numbers": [ { "k": "Range covered", "v": "0 to 0.03229670385" }, { "k": "Gap to beta*", "v": "7.3e-12" }, { "k": "Boxes", "v": "11,629,511 nodes, 5,814,756 leaves" }, { "k": "Smallest lower bound of det M", "v": "1.2e-16" } ], "method": "Branch and bound in ball arithmetic on M11 > 0 and det M > 0 over (t, psi, beta), with psi = p - 5t. The matrix is polynomial in sines and cosines, so the poles, where the phase 5t makes the body only C^1, are covered by the same boxes. Centroid and the two equilibria follow by hand.", "second_leg": "The matrix agrees in sign with the curvature of the implicit surface (automatic differentiation) at 6000 of 6000 random points, 667 of them with K < 0. This is a formula check, not a second box computation. A separately written checker replays the box-search record.", "evidence": [ "flagship_repro/sloan/shape1_convexity_cert.py", "flagship_repro/sloan/shape1_M_vs_implicit.py", "flagship_repro/sloan/shape1_cert_runs.log" ], "not_covered": [ "The last 7.3e-12 below the closed-form limit.", "A negative control: the same program proves det M < 0 on boxes for beta in [0.0324, 0.0326].", "A human reading of the derivation of the matrix is still wanted.", "The surface is only C^1 at the poles; local convexity there uses the standard fact that a C^1 surface with positive definite M at every neighbouring point is convex." ], "headline": { "k": "Sloan shape 1 stays convex up to beta", "v": "0.03229670385", "d": "The note states 0.15. The closed-form limit is 0.0322967038573." } }, { "id": "E1", "name": "Phase-1 body: strictly convex", "kind": "Theorem E1", "tier": "independent", "release": "flagship-held", "statement": "The Phase-1 body, r^4 = 1 + 4 beta sin t cos(p - eta - a1 sin eta) with beta = 23149/10^6 and a1 = 234433/10^6, has K > 0 on the whole surface.", "numbers": [ { "k": "Primary boxes", "v": "1,089 + 367 caps" }, { "k": "Second method boxes", "v": "119,142" }, { "k": "Parameters", "v": "exact rationals" } ], "method": "Compact chart plus polar caps with plain interval evaluation.", "second_leg": "Cube-face Cartesian method with a centered form, no chart, symmetry or cap shared with the first. Full coverage of the surface.", "evidence": [ "flagship_repro/phase1/phase1_convexity.py", "flagship_repro/phase1/phase1_audit.py", "flagship_repro/phase1/phase1_audit.log" ], "not_covered": [ "Only positivity of K is established, not a stated margin." ] }, { "id": "E2", "name": "Phase-1 body: exactly one stable and one unstable equilibrium", "kind": "Theorem E2", "tier": "independent", "release": "flagship-held", "statement": "With its centroid enclosed by exact integrals (c_x = -0.0026955695464916...), the Phase-1 body has S = U = 1 and H = 0: exactly two equilibria, both nondegenerate.", "numbers": [ { "k": "Centroid c_x", "v": "-0.00269556954649164614590662..." }, { "k": "Centroid width", "v": "below 1e-40" }, { "k": "Counts (S, U, H)", "v": "(1, 1, 0)" }, { "k": "Jacobian determinants", "v": "0.042 and 0.047" }, { "k": "Exclusion boxes (second method)", "v": "51,346" } ], "method": "Interval counter with Krawczyk uniqueness boxes on the compact chart and the polar caps.", "second_leg": "Cartesian implicit surface on six cube faces, equilibrium condition N x (x - c) = 0, Krawczyk uniqueness in two boxes and an exclusion proof elsewhere.", "evidence": [ "flagship_repro/phase1/phase1_centroid.py", "flagship_repro/phase1/phase1_count.py", "flagship_repro/phase1/phase1_count_audit.py" ], "not_covered": [ "The parameters come from an earlier optimiser run that does not reproduce from the text of the earlier preprint. The object certified here is the one with these exact parameters." ], "headline": { "k": "Phase-1 equilibria (S, U, H)", "v": "(1, 1, 0)", "d": "One stable, one unstable, no saddles. Two independent methods agree." } }, { "id": "F1", "name": "Phase-1 gradient floor", "kind": "Margin", "tier": "computer", "release": "flagship-held", "statement": "Away from the two equilibria the round-metric gradient of the squared centroid distance has a computer-assisted floor m(r0) outside caps of radius r0.", "numbers": [ { "k": "r0 = 0.4", "v": "2.3e-3" }, { "k": "r0 = 0.2", "v": "1.2e-3" }, { "k": "r0 = 0.1", "v": "6.2e-4" }, { "k": "r0 = 0.05", "v": "3.0e-4" }, { "k": "r0 = 0.02", "v": "1.3e-4" }, { "k": "r0 = 0.005", "v": "2.5e-5" } ], "method": "Forward-mode automatic differentiation in ball arithmetic, bulk chart plus both polar caps in a stereographic chart.", "second_leg": null, "evidence": [ "flagship_repro/margins/phase1_robust.py", "flagship_repro/margins/robust_sweep.log" ], "not_covered": [ "Phase-1 only; no other body has margins.", "No second method." ] }, { "id": "F2", "name": "Phase-1 Hessian at the equilibria: flat in one direction, steep in the other", "kind": "Margin", "tier": "computer", "release": "flagship-held", "statement": "At the stable equilibrium the Hessian eigenvalues lie in [0.0066, 1.741]; at the unstable one in [-1.596, -0.0068]. The landscape is flat in one direction and steep in the other, with a condition number near 260.", "numbers": [ { "k": "Stable eigenvalues", "v": "0.0066 to 1.741" }, { "k": "Unstable eigenvalues", "v": "-1.596 to -0.0068" }, { "k": "Condition number", "v": "about 260" }, { "k": "Definite on the box for", "v": "r0 <= 0.005" } ], "method": "Coordinate Hessian enclosed on boxes around each equilibrium.", "second_leg": null, "evidence": [ "flagship_repro/margins/phase1_hessian.py", "flagship_repro/margins/phase1_hessian.json" ], "not_covered": [ "Definiteness is not established on the box at r0 = 0.01.", "No second method." ], "headline": { "k": "Flat versus steep at the equilibria", "v": "about 260", "d": "Condition number of the Hessian: curvature 0.0066 in one direction, 1.741 in the other." }, "hess": { "stable": [ 0.0066, 1.741 ], "unstable": [ -1.596, -0.0068 ] } }, { "id": "F3", "name": "How much shape error Phase-1 tolerates", "kind": "Proposition", "tier": "computer", "release": "flagship-held", "statement": "If a radial perturbation of degree at most k has amplitude a below the stated bound, the perturbed body is still strictly convex and still has exactly one nondegenerate minimum and one nondegenerate maximum of the squared centroid distance. The bounds are sufficient conditions, not sharp ones.", "numbers": [ { "k": "k = 1", "v": "a <= 2.0e-6" }, { "k": "k = 5", "v": "a <= 1.1e-6" }, { "k": "k = 10", "v": "a <= 6.7e-7" }, { "k": "k = 50", "v": "a <= 1.4e-7" }, { "k": "Convexity margin of perturbed body", "v": "lambda_min(M) >= 0.0766" } ], "method": "A perturbation argument on the two boxes, with every constant enclosed in interval arithmetic: gradient bound, Hessian bound, centroid shift, Bernstein-Riesz.", "second_leg": null, "evidence": [ "flagship_repro/margins/prop_constants_cert.py", "flagship_repro/margins/prop_gr_cert.py", "flagship_repro/margins/prop_tolerance_cert.py", "flagship_repro/margins/prop_convexity_cert.py" ], "not_covered": [ "A human reader of the proposition and its constants is still wanted.", "Phase-1 only.", "Not a physical tolerance for any material." ], "headline": { "k": "Tolerated radial error, degree 1", "v": "2.0e-6", "d": "A sufficient condition. About 0.1 micrometre on a 5 cm body." } }, { "id": "F4", "name": "Is Phase-1 close to the best this family can do?", "kind": "Scan", "tier": "exploratory", "release": "flagship-held", "statement": "Over a grid of one extra Fourier harmonic added to Sloan's phase, the best gradient robustness among convex (1,1,0) bodies is about 2.9e-3 against 1.6e-3 for Phase-1, and the best Hessian margin about 9.6e-3 against 6.6e-3. Re-optimising inside this family gains less than a factor of two.", "numbers": [ { "k": "Best gradient robustness (scan)", "v": "2.9e-3" }, { "k": "Phase-1 gradient robustness", "v": "1.6e-3" }, { "k": "Best Hessian margin (scan)", "v": "9.6e-3" }, { "k": "Phase-1 Hessian margin", "v": "6.6e-3" } ], "method": "Grid candidates refined by a continuous root solve. A first version that counted grid extrema reported 21 minima and 19 maxima on flat ridges for a body with exactly one of each; it was discarded.", "second_leg": null, "evidence": [ "flagship_repro/margins/robust_map.py", "flagship_repro/margins/robust_map2.py" ], "not_covered": [ "A coarse float scan of one phase function. It is not a bound on all gombocs." ] }, { "id": "G1", "name": "Phase-1 equilibrium count as a replayable record", "kind": "Software result", "tier": "computer", "release": "software-local", "statement": "For every centroid in a stated box around the Phase-1 centroid, the squared centroid distance of the Phase-1 body has one minimum, one maximum and no saddle, and a second, independent program re-evaluates the whole record.", "numbers": [ { "k": "Result (S, U, H)", "v": "1, 1, 0" }, { "k": "Euler characteristic", "v": "2" }, { "k": "Leaves", "v": "16053 (16051 Empty, 2 One)" }, { "k": "Search and replay", "v": "17 s and 5 s" }, { "k": "Replay in a browser", "v": "28.9 s (VALID, on a busy machine); a copy claiming 2 stable positions is rejected" }, { "k": "Centroid box half-width", "v": "1e-24" } ], "method": "The sphere is covered by boxes on the six faces of a cube. A box is Empty when the gradient of the squared centroid distance is provably nonzero for every centroid in the box, and One when Krawczyk's operator shows a single critical point, typed by the interval Hessian. The search is untrusted; a short separate checker re-evaluates every box in ball arithmetic and recomputes the totals.", "second_leg": null, "evidence": [ "software/eqcert/certificates/phase-1_count_exact_point.json", "software/eqcert/src/eqcert/check_count.py", "software/eqcert/tests/test_count_certificates.py", "scoping/SOFTWARE-AND-TRIALS-DATASHEET-2026-11-08.md" ], "not_covered": [ "Strict convexity is an input; row E1 covers it.", "The centroid box is the paper's printed value widened by 1e-24, not an enclosure computed here.", "The checker is a short program, not a formally verified one.", "Nothing is claimed about a printed object." ], "runs": [ "R-2026-11-08-002", "R-2026-11-08-003" ] }, { "id": "G2", "name": "Single-harmonic m=1 counts from the same checker", "kind": "Software result", "tier": "computer", "release": "software-local", "statement": "At one small amplitude for each degree l = 2 to 5, the body r = 1 + e P_l'(u_z) u_x has (l, l, 2l-2) stable, unstable and saddle positions, the class reported in row C.", "numbers": [ { "k": "l = 2, e = 1/50", "v": "(2, 2, 2), 2688 leaves" }, { "k": "l = 3, e = 1/100", "v": "(3, 3, 4), 4509 leaves" }, { "k": "l = 4, e = 1/200", "v": "(4, 4, 6), 24622 leaves" }, { "k": "l = 5, e = 1/400", "v": "(5, 5, 8), 140313 leaves" } ], "method": "The same cover and checker as row G1, with the radial formula of the body in place of the Sloan form and an exact rotation of the cube so that no critical point lies on a cube edge (the l = 2 body has its extrema exactly on edges).", "second_leg": null, "evidence": [ "software/eqcert/tests/test_harmonic_counts.py", "software/eqcert/src/eqcert/check_count.py", "software/eqcert/src/eqcert/count_cert.py", "scoping/SOFTWARE-AND-TRIALS-DATASHEET-2026-11-08.md" ], "not_covered": [ "One amplitude for each degree; row C covers the whole range.", "Degrees 6 and above were not run here.", "The centroid box is exact quadrature padded by 1e-12, not an enclosure.", "Convexity of these bodies comes from row G3 and Table B, not from this record." ], "runs": [ "R-2026-11-08-005" ] }, { "id": "G3", "name": "Convexity of radial bodies, checked against the published thresholds", "kind": "Software result", "tier": "computer", "release": "software-local", "statement": "A convexity matrix for any radial body, evaluated on the cube-face charts, changes sign at the thresholds of Table B (l = 2, 3, 4, m = 1), and strict convexity of those bodies is shown by interval computation for every amplitude up to a stated fraction of the threshold.", "numbers": [ { "k": "Sign check l = 2", "v": "+0.00042 at 0.99 of the threshold, -0.00113 at 1.01" }, { "k": "Sign check l = 3", "v": "+0.00213 at 0.99, -0.00254 at 1.01" }, { "k": "Sign check l = 4", "v": "+0.00301 at 0.99, -0.00624 at 1.01" }, { "k": "Convex up to, l = 2", "v": "e = 0.106341 (80 percent), 612392 leaves, replay 80 s; e = 0.12628 (95 percent), 2588916 leaves, replay 292 s" }, { "k": "Convex up to, l = 3", "v": "e = 0.041712 (80 percent), 233540 leaves, replay 34 s" }, { "k": "Convex up to, l = 4", "v": "e = 0.020245 (80 percent), 138220 leaves, replay 23 s" } ], "method": "Convexity of a radial graph is positive definiteness of M = r^2 g + 2 dr (x) dr - r Hess r on the tangent plane. The matrix is built from the second-order jets of the unit vector in each cube-face chart, and the amplitude is an interval, so one record covers a whole range of amplitudes. The matrix is cross-checked against automatic differentiation of the implicit surface (the sign of its determinant agrees with the Gauss curvature at every checked point) and against the thresholds of Table B.", "second_leg": null, "evidence": [ "software/eqcert/src/eqcert/check_radial_convex.py", "software/eqcert/tests/test_radial_convexity.py", "software/eqcert/certificates/harmonic-l2-m1_convex_e_0_0.12628.json", "software/eqcert/certificates/harmonic-l3-m1_convex_e_0_0.041712.json", "software/eqcert/certificates/harmonic-l4-m1_convex_e_0_0.020245.json", "scoping/SOFTWARE-AND-TRIALS-DATASHEET-2026-11-08.md" ], "not_covered": [ "Up to 95 percent of the lower bracket for l = 2 and 80 percent for l = 3 and 4; the last few percent need a much larger search.", "The hand derivation of the matrix is not machine checked; it is cross-checked against automatic differentiation and the published thresholds.", "Only the m = 1 harmonic bodies were run." ], "runs": [ "R-2026-11-08-004" ] }, { "id": "G4", "name": "Mesh trials: how much shape error a printed gomboc can take", "kind": "Floating-point trial", "tier": "exploratory", "release": "software-local", "statement": "In 20 random smooth radial perturbations per level of a Phase-1 mesh, most meshes keep one stable and one unstable position at a relative error of 1e-4 and most do not at 1e-3.", "numbers": [ { "k": "Phase-1, error 1e-5", "v": "20 of 20 keep one of each" }, { "k": "Phase-1, error 1e-4", "v": "18 of 20 (6-step test), 20 of 20 (10-step test)" }, { "k": "Phase-1, error 1e-3", "v": "4 of 20 and 5 of 20" }, { "k": "Phase-1, error 3e-3 and 1e-2", "v": "0 of 20; 9 of 20 meshes still convex at 1e-2" }, { "k": "Sloan shape 2 at the edge of its convex range", "v": "20 of 20 at 1e-5, 17 of 20 at 1e-4, 0 of 20 from 1e-3" }, { "k": "Scale", "v": "1e-4 of the radius is about 5 micrometres on a 100 mm body" } ], "method": "A smooth random radial error of sup-norm eps is added to the mesh, the mesh centroid is recomputed, and minima and maxima of the distance from the centroid are counted within a neighbourhood of 6 and of 10 grid steps. Convexity of each perturbed mesh is tested by checking that every vertex lies on the convex hull.", "second_leg": null, "evidence": [ "thread2_empirical_tolerance/mesh_tolerance_trials.py", "thread2_empirical_tolerance/run_p1.log", "thread2_empirical_tolerance/run_s2.log", "scoping/SOFTWARE-AND-TRIALS-DATASHEET-2026-11-08.md" ], "not_covered": [ "One kind of smooth error; layer steps and surface roughness are not modelled.", "A floating-point scan on a coarse grid, not an enclosure.", "Shape 2 was run at the edge of its convex range only, the weakest choice.", "No physical object was printed or measured." ], "runs": [ "R-2026-11-08-017", "R-2026-11-08-018" ] }, { "id": "G5", "name": "A nearest-neighbour scan miscounts a flat body", "kind": "Floating-point trial", "tier": "exploratory", "release": "software-local", "statement": "A nearest-neighbour scan of the Phase-1 landscape reports several minima and maxima that vanish when the neighbourhood is widened, while the body has one of each.", "numbers": [ { "k": "Neighbours within 1 grid step", "v": "9 minima, 7 maxima" }, { "k": "Within 2 steps", "v": "7 and 5" }, { "k": "Within 3 steps", "v": "3 and 3" }, { "k": "Within 6 and 10 steps", "v": "1 and 1" } ], "method": "The surface is sampled on a 91 by 180 grid and grid points that are the strict minimum or maximum of the distance from the centroid among grid points within r steps are counted.", "second_leg": null, "evidence": [ "software/eqcert/notebooks/when_plots_lie.ipynb", "software/eqcert/notebooks/wpl_helpers.py", "scoping/SOFTWARE-AND-TRIALS-DATASHEET-2026-11-08.md" ], "not_covered": [ "A larger neighbourhood does not make a scan right in general: an extra position closer than the neighbourhood would be missed.", "One body and one grid.", "The enclosure counts in rows E2 and G1 are the reference, not this scan." ], "runs": [ "R-2026-11-08-019" ] } ], "papers": [ { "id": "arXiv:2604.12238", "title": "Computational Validation of the Oloid as a Local Optimum in the Developable Roller Family", "status": "posted", "note": "Foundational paper: substrate geometry as classification by operational invariants." }, { "id": "arXiv:2604.17095", "title": "Computational Construction and Engineering Evaluation of Verified Mono-Monostatic Bodies", "status": "posted", "note": "Introduces the Equilibrium Count Score and the Fourier-extended Sloan family that gave the Phase-1 parameters." }, { "id": "arXiv:2604.17120", "title": "Sloan's Analytical Gomboc at Published beta: A Strict-Convexity-Constrained Reanalysis", "title_flag": "title retired and corrected in the staged revision: the published beta attribution is wrong", "status": "posted", "note": "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." }, { "id": "arXiv:2608.11213", "title": "Static Equilibria of Perturbed Spheres: A Single-Harmonic Class Map, a Parity Obstruction, and a Certified Counter for the Mono-Monostatic Regime", "status": "posted", "note": "Title as posted on arXiv (verified against the arXiv page on 2026-11-07). Equilibrium counts for harmonic near-spheres; listed under math.GM.", "title_verbatim": true }, { "id": "Standalone note", "title": "Sharp convex range of a spherical-harmonic perturbation", "status": "ready", "note": "Four pages. Contains Theorem A and Table B. Not yet posted." }, { "id": "Flagship manuscript", "title": "Margins for an explicit smooth gomboc, by interval computation (working title)", "status": "drafting", "note": "Typeset draft, 13 pages. Awaiting human readers, a literature check and the last interval run. Not yet posted." } ], "repro": [ { "item": "Phase-1 centroid", "script": "phase1/phase1_centroid.py", "command": "python -X utf8 phase1/phase1_centroid.py", "time": "seconds" }, { "item": "Phase-1 convexity, primary", "script": "phase1/phase1_convexity.py", "command": "python -X utf8 phase1/phase1_convexity.py", "time": "under 1 s" }, { "item": "Phase-1 convexity, second method", "script": "phase1/phase1_audit.py", "command": "python -X utf8 phase1/phase1_audit.py", "time": "326 s" }, { "item": "Phase-1 count, primary", "script": "phase1/phase1_count.py", "command": "python -X utf8 phase1/phase1_count.py", "time": "139 s" }, { "item": "Phase-1 count, second method", "script": "phase1/phase1_count_audit.py", "command": "python -X utf8 phase1/phase1_count_audit.py", "time": "about 190 s" }, { "item": "Sloan convexity, primary", "script": "sloan/certify_fine.py", "command": "python -X utf8 sloan/certify_fine.py", "time": "about 2,600 s" }, { "item": "m = 1 counts, interval", "script": "m1/m1_interval.py", "command": "python -X utf8 m1/m1_interval.py 6 0.004303 0.0086075 28", "time": "hours per l" }, { "item": "m = 1 counts, second method", "script": "m1/m1_independent.py", "command": "python -X utf8 m1/m1_independent.py 3 1 25", "time": "190 to 245 s" }, { "item": "Gradient floor", "script": "margins/phase1_robust.py", "command": "python -X utf8 margins/phase1_robust.py 0.2 0.0012", "time": "10 to 20 s" }, { "item": "Hessian margin", "script": "margins/phase1_hessian.py", "command": "python -X utf8 margins/phase1_hessian.py", "time": "seconds" }, { "item": "Perturbation tolerance table", "script": "margins/prop_tolerance_cert.py", "command": "python -X utf8 margins/prop_tolerance_cert.py", "time": "seconds" }, { "item": "Phase-1 count record (G1)", "script": "software/eqcert/examples/make_count_certificate.py", "command": "python examples/make_count_certificate.py out.json (from software/eqcert)", "time": "about 10 s to search, 3 s to replay" }, { "item": "Harmonic m=1 count records (G2)", "script": "software/eqcert/tests/test_harmonic_counts.py", "command": "PYTHONPATH=src python -m pytest tests/test_harmonic_counts.py (from software/eqcert)", "time": "about 3 minutes" }, { "item": "Radial convexity records (G3)", "script": "software/eqcert/examples/make_radial_convexity.py", "command": "python examples/make_radial_convexity.py (from software/eqcert)", "time": "about 25 minutes" }, { "item": "Mesh tolerance trials (G4)", "script": "thread2_empirical_tolerance/mesh_tolerance_trials.py", "command": "python mesh_tolerance_trials.py phase-1 20 120 (then sloan-shape-2)", "time": "about 15 minutes each" }, { "item": "Scan miscount (G5)", "script": "software/eqcert/notebooks/when_plots_lie.ipynb", "command": "open the notebook, run section 2", "time": "seconds" } ], "environment": "Python 3.11.9, python-flint 0.9.0, mpmath 1.3.0, sympy 1.14.0, numpy 2.4.6, scipy 1.17.1 (Windows). Timings are from one desktop machine with other jobs running; treat them as orders of magnitude." }