Every result, with its tier and its limits

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.

None for this row.

• 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.

Branch and bound with a centered form from forward-mode automatic differentiation, in ball arithmetic.

Cartesian cube-face method on three bodies: Y33, Y44, Y31. The other 24 have the primary method only.

• 24 of the 27 brackets have no second method.
• l above 7.

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.

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.

• 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.

Direct evaluation of the curvature at (theta, phi) = (pi/2, pi), where R^4 = 1 - 4 beta.

None for this row.

• 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.

Compact radial-graph chart plus polar caps, centered form from automatic differentiation.

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.

• The last 7e-8 of the range by the cube-face method (the third route covers it).
• The single value beta = beta*.

Hand derivation of the centroid integral and the gradient of R on the sphere.

None for this row.

• Re-derived with no algebra errors found. A human reading is still wanted.
• The original note states these facts without derivation.

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].

None for this row.

• 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.

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.

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.

• 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.

Compact chart plus polar caps with plain interval evaluation.

Cube-face Cartesian method with a centered form, no chart, symmetry or cap shared with the first. Full coverage of the surface.

• Only positivity of K is established, not a stated margin.

Interval counter with Krawczyk uniqueness boxes on the compact chart and the polar caps.

Cartesian implicit surface on six cube faces, equilibrium condition N x (x - c) = 0, Krawczyk uniqueness in two boxes and an exclusion proof elsewhere.

• 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.

Forward-mode automatic differentiation in ball arithmetic, bulk chart plus both polar caps in a stereographic chart.

None for this row.

• Phase-1 only; no other body has margins.
• No second method.

Coordinate Hessian enclosed on boxes around each equilibrium.

None for this row.

• Definiteness is not established on the box at r0 = 0.01.
• No second method.

A perturbation argument on the two boxes, with every constant enclosed in interval arithmetic: gradient bound, Hessian bound, centroid shift, Bernstein-Riesz.

None for this row.

• A human reader of the proposition and its constants is still wanted.
• Phase-1 only.
• Not a physical tolerance for any material.

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.

None for this row.

• A coarse float scan of one phase function. It is not a bound on all gombocs.

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.

None for this row.

• 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.

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).

None for this row.

• 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.

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.

None for this row.

• 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.

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.

None for this row.

• 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.

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.

None for this row.

• 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.