{"schema_version":"dataset-v1","id":"chernoff-log-curvature","version":1,"title":"How curved is log f?","description":"The standard Chernoff density f of Z = argmax(W(t) − t²), computed from the cited Airy factorisation f(x) = C_f g(x) g(−x) with C_f = 2^(−1/3)/Ai(0)², and its negative log-curvature κ(x) = −(log f)″(x).","license":"CC0-1.0 (computed values)","problems":["chernoff-density"],"statements":[{"problem":"chernoff-density","statement":"chernoff-strong-log-concavity","role":"inf κ > 0: the theorem."},{"problem":"chernoff-density","statement":"chernoff-curvature-tail","role":"The proved bound κ(x) > 4|x|."},{"problem":"chernoff-density","statement":"chernoff-origin-minimum-curvature","role":"Open: is κ smallest at 0?"}],"figure":{"family":"line-plot","title":"How curved is log f?","caption":"κ(x) = −(log f)″(x) stays above the proved lower bound 4|x| and is about 3.405 at the origin, so it is bounded away from zero: the strong log-concavity theorem.","x_label":"x","y_label":"κ(x)","series":["kappa","bound"],"boundary":"Numerical values on [−2, 2] illustrate the theorems; they prove neither. Whether κ is smallest at 0 everywhere is the open origin-minimum question.","statement":"chernoff-strong-log-concavity"},"provenance":{"method":"g and its first two derivatives by numerical inversion of the bilateral Laplace transform M(s) = Ai(0)/Ai(2^(−1/3)s) along the imaginary axis (mpmath quad, 20 digits) at x = −3, −2.95, …, 3. Checks: Simpson mass 1.000000000, symmetry, and κ(x) > 4|x| at every grid point; variance 0.2635596.","software":{"python":"3.12.15","mpmath":"1.3.0"},"precision":"20-digit quadrature, stored as floating point; a numerical illustration.","generator":"openmath/scripts/build-datasets.py","generated_on":"2026-10-21"},"data":{"series":[{"key":"kappa","label":"κ(x)","x":[-2,-1.95,-1.9,-1.85,-1.8,-1.75,-1.7,-1.65,-1.6,-1.55,-1.5,-1.45,-1.4,-1.35,-1.3,-1.25,-1.2,-1.15,-1.1,-1.05,-1,-0.95,-0.9,-0.85,-0.8,-0.75,-0.7,-0.65,-0.6,-0.55,-0.5,-0.45,-0.4,-0.35,-0.3,-0.25,-0.2,-0.15,-0.1,-0.05,0,0.05,0.1,0.15,0.2,0.25,0.3,0.35,0.4,0.45,0.5,0.55,0.6,0.65,0.7,0.75,0.8,0.85,0.9,0.95,1,1.05,1.1,1.15,1.2,1.25,1.3,1.35,1.4,1.45,1.5,1.55,1.6,1.65,1.7,1.75,1.8,1.85,1.9,1.95,2],"y":[8.265109346118434,8.079716148478314,7.8954353770088925,7.7123609951137775,7.530594474389598,7.350245231010216,7.171431054936108,6.994278524798989,6.818923400219776,6.645510982204707,6.474196431161079,6.305145031018904,6.138532386983218,5.974544543626561,5.81337800942028,5.655239673459852,5.500346600128189,5.348925687826154,5.20121317874137,5.05745400797646,4.91790098225495,4.782813780888691,4.65245777472432,4.5271026623619575,4.407020927002531,4.292486121746439,4.183770995921823,4.081145479924305,3.984874550937857,3.895216006597898,3.812418177964131,3.7367176169041265,3.668336795972113,3.6074818609447,3.554340477219147,3.5090798112008503,3.4718446865572608,3.442755952793767,3.4219091000571957,3.4093731494831587,3.405189842903433,3.4093731494831587,3.4219091000571957,3.442755952793767,3.4718446865572608,3.5090798112008503,3.554340477219147,3.6074818609447,3.668336795972113,3.7367176169041265,3.812418177964131,3.895216006597898,3.984874550937857,4.081145479924305,4.183770995921823,4.292486121746439,4.407020927002531,4.5271026623619575,4.65245777472432,4.782813780888691,4.91790098225495,5.05745400797646,5.20121317874137,5.348925687826154,5.500346600128189,5.655239673459852,5.81337800942028,5.974544543626561,6.138532386983218,6.305145031018904,6.474196431161079,6.645510982204707,6.818923400219776,6.994278524798989,7.171431054936108,7.350245231010216,7.530594474389598,7.7123609951137775,7.8954353770088925,8.079716148478314,8.265109346118434]},{"key":"bound","label":"4|x| (proved bound)","x":[-2,-1.95,-1.9,-1.85,-1.8,-1.75,-1.7,-1.65,-1.6,-1.55,-1.5,-1.45,-1.4,-1.35,-1.3,-1.25,-1.2,-1.15,-1.1,-1.05,-1,-0.95,-0.9,-0.85,-0.8,-0.75,-0.7,-0.65,-0.6,-0.55,-0.5,-0.45,-0.4,-0.35,-0.3,-0.25,-0.2,-0.15,-0.1,-0.05,0,0.05,0.1,0.15,0.2,0.25,0.3,0.35,0.4,0.45,0.5,0.55,0.6,0.65,0.7,0.75,0.8,0.85,0.9,0.95,1,1.05,1.1,1.15,1.2,1.25,1.3,1.35,1.4,1.45,1.5,1.55,1.6,1.65,1.7,1.75,1.8,1.85,1.9,1.95,2],"y":[8,7.8,7.6,7.4,7.2,7,6.8,6.6,6.4,6.2,6,5.8,5.6,5.4,5.2,5,4.8,4.6,4.4,4.2,4,3.8,3.6,3.4,3.2,3,2.8,2.6,2.4,2.2,2,1.8,1.6,1.4,1.2,1,0.8,0.6,0.4,0.2,0,0.2,0.4,0.6,0.8,1,1.2,1.4,1.6,1.8,2,2.2,2.4,2.6,2.8,3,3.2,3.4,3.6,3.8,4,4.2,4.4,4.6,4.8,5,5.2,5.4,5.6,5.8,6,6.2,6.4,6.6,6.8,7,7.2,7.4,7.6,7.8,8]}],"marks":[]},"sha256":"b2b5d1c7aa53e1f6a1c9f434a7e2d48149eea57eb053ccc58a0b72600e16d064"}