{"schema_version":"dataset-v1","id":"schiffer-tenfold-domain","version":1,"title":"The certified tenfold domain","description":"The boundary of the computer-assisted counterexample: the image of the unit circle under the conformal map φ(z) = z + Σ_{j≥1} p_j/(p_0(10j + 1)) z^{10j+1}, with the 31 shape coefficients of the certificate's centre. Its radius ranges from 0.971 to 1.045; the unit circle is drawn for comparison.","license":"CC0-1.0 (values computed from the public certificate)","problems":["schiffer-pompeiu"],"statements":[{"problem":"schiffer-pompeiu","statement":"schiffer-analytic-tenfold-witness","role":"This domain carries a nonconstant solution with u = 1 and ∂νu = 0."},{"problem":"schiffer-pompeiu","statement":"schiffer-planar-disk-rigidity","role":"The conjecture said only a disk could; this domain is not a disk."}],"figure":{"family":"region-plot","title":"The certified tenfold domain","caption":"Ten shallow petals: the boundary is real-analytic and not a circle (the certificate proves |p₁| ≥ 12, so φ is not linear). It is drawn from the certificate's centre; the certificate proves a true solution within radius 10⁻⁶ of that centre in its weighted coefficient norm.","x_label":"Re","y_label":"Im","x_range":[-2.2,2.2],"y_range":[-1.15,1.15],"y_scale":"linear","x_ticks":[[-1,"−1"],[0,"0"],[1,"1"]],"y_ticks":[[-1,"−1"],[0,"0"],[1,"1"]],"layers":[{"key":"circle","label":"Scope: the unit circle, for 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31 shape coefficients of the certificate centre (pinned commit 2fb3f2e, data/center_L60_S40_R30.hex, sha256 a30353f2…3d29) are read as binary64; φ is evaluated at 360 equally spaced points of the unit circle. Σ_{j≥1} |p_j/p_0| = 0.6492 (certificate bound 0.65).","software":{"python":"3.12.13"},"precision":"binary64 evaluation of the published centre; a plot, not a proof.","generator":"openmath/scripts/build-datasets.py","generated_on":"2026-10-21"},"data":{"series":[{"key":"boundary","label":"φ(e^{iθ}), θ = 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