{ "what": "the effective Hamiltonian H̄(P) of the one-dimensional cell problem H(x, P + u′) = H̄(P) with H = p²/2 + V(x) (Gomes–Yang, arXiv:1810.03483, §6), enclosed as a band: the flat part decided exactly, the rotating part bracketed by bisection on a rigorously enclosed integral, the degeneracy at P₀ met as a widening band and an undecided point; the paper's Table 1 (and the Gomes–Oberman value it cites) and Table 2 held against the enclosures", "statement": "For V = sin 2πx: 9 of 22 sampled P are FLAT (H̄ = 1 exactly, |P| ≤ P₀ with P₀ ∈ [1.273238819, 1.273240333] ∋ 4/π), 12 ROTATING with H̄ bracketed to at most 2.0e-6 (at P = 3, nearest the degeneracy), 1 UNDECIDED (P = 4/π itself, inside the P₀ enclosure). Table 1: H̄(1.5, 2.5) for the separable cosine Hamiltonian is 1.244637626..1.244637655 + 3.165327608..3.165327639 = [4.409965234, 4.409965294]; the value 4.4099660 the paper quotes from Gomes–Oberman lies ABOVE the enclosure by 7.1e-7; the paper's own k-sequence lies BELOW, BELOW, BELOW, BELOW (Newton) — below, as the entropy-penalized H̄^k must. Table 2: at P = 0.5 on −sin 2πx the exact value is 1 (FLAT, decided) and the printed H̄◇ = 0.96476 at k = 100 sits 0.03524 below it — the entropy penalization's gap, not the mesh's.", "verdict": "DECIDED (band), MEASURED (their numbers)", "PS": [ 0, 0.25, 0.5, 0.75, 1, 1.1, 1.2, 1.25, 1.27, 1.2732395447351628, 1.275, 1.28, 1.29, 1.3, 1.35, 1.4, 1.5, 1.75, 2, 2.25, 2.5, 3 ], "K": { "sweep": 1024, "table": 8192, "P0": 4096 }, "printed": { "table1": { "P": [ 1.5, 2.5 ], "potential": "cos 2πx₁ + cos 2πx₂ (separable)", "cited": 4.409966, "citedFrom": "Gomes–Oberman, SIAM J. Control Optim. 43 (2004), as quoted on p. 20", "k": [ 10, 100, 1000, 10000 ], "HRF": [ 4.40251, 4.40916, 4.40989, 4.40996 ], "NM": [ 4.40935, 4.40994, 4.40996, 4.40996 ] }, "table2": { "P": 0.5, "potential": "−sin 2πx", "k": 100, "N": [ 15, 30, 60, 120 ], "Hdiamond": [ 0.964609, 0.964754, 0.96476, 0.96476 ], "exactStated": 1 }, "P0stated": "4/π (p. 17)" }, "p0": { "sin": { "value": [ 1.273238818791572, 1.2732403330965356 ], "width": 0.0000015143049636456853, "contains4overPi": true }, "cos": { "value": [ 1.27323881879147, 1.2732403330966384 ], "width": 0.000001514305168370811, "contains4overPi": true }, "msin": { "value": [ 1.2732388187913537, 1.2732403330967537 ], "width": 0.000001514305399963334, "contains4overPi": true } }, "band": [ { "P": 0, "regime": "FLAT", "value": [ 1, 1 ], "width": 5e-324, "stalled": false, "tightened": false, "it": null, "qWidth": null, "why": null, "mather": "Dirac at x = 0.25" }, { "P": 0.25, "regime": "FLAT", "value": [ 1, 1 ], "width": 5e-324, "stalled": false, "tightened": false, "it": null, "qWidth": null, "why": null, "mather": "Dirac at x = 0.25" }, { "P": 0.5, "regime": "FLAT", "value": [ 1, 1 ], "width": 5e-324, "stalled": false, "tightened": false, "it": null, "qWidth": null, "why": null, "mather": "Dirac at x = 0.25" }, { "P": 0.75, "regime": "FLAT", "value": [ 1, 1 ], "width": 5e-324, "stalled": false, "tightened": false, "it": null, "qWidth": null, "why": null, "mather": "Dirac at x = 0.25" }, { "P": 1, "regime": "FLAT", "value": [ 1, 1 ], "width": 5e-324, "stalled": false, "tightened": false, "it": null, "qWidth": null, "why": null, "mather": "Dirac at x = 0.25" }, { "P": 1.1, "regime": "FLAT", "value": [ 1, 1 ], "width": 5e-324, "stalled": false, "tightened": false, "it": null, "qWidth": null, "why": null, "mather": "Dirac at x = 0.25" }, { "P": 1.2, "regime": "FLAT", "value": [ 1, 1 ], "width": 5e-324, "stalled": false, "tightened": false, "it": null, "qWidth": null, "why": null, "mather": "Dirac at x = 0.25" }, { "P": 1.25, "regime": "FLAT", "value": [ 1, 1 ], "width": 5e-324, "stalled": false, "tightened": false, "it": null, "qWidth": null, "why": null, "mather": "Dirac at x = 0.25" }, { "P": 1.27, "regime": "FLAT", "value": [ 1, 1 ], "width": 5e-324, "stalled": false, "tightened": false, "it": null, "qWidth": null, "why": null, "mather": "Dirac at x = 0.25" }, { "P": 1.2732395447351628, "regime": "UNDECIDED", "value": null, "width": null, "stalled": false, "tightened": false, "it": null, "qWidth": null, "why": "|P| lies inside the enclosure of P₀ = [1.273227941186796,1.273252146955114]", "mather": null }, { "P": 1.275, "regime": "ROTATING", "value": [ 1.0009692461384376, 1.0009707459259831 ], "width": 0.000001499787545489184, "stalled": true, "tightened": true, "it": 19, "qWidth": 0.000002482334150011312, "why": null, "mather": "absolutely continuous, density 1/(T(c)√(2(c − V)))" }, { "P": 1.28, "regime": "ROTATING", "value": [ 1.0042826135863812, 1.0042841293119766 ], "width": 0.0000015157255954800775, "stalled": true, "tightened": true, "it": 19, "qWidth": 0.000002150434686321035, "why": null, "mather": "absolutely continuous, density 1/(T(c)√(2(c − V)))" }, { "P": 1.29, "regime": "ROTATING", "value": [ 1.0118348070778, 1.0118363892837576 ], "width": 0.0000015822059575842928, "stalled": true, "tightened": true, "it": 18, "qWidth": 0.0000019876533419171953, "why": null, "mather": "absolutely continuous, density 1/(T(c)√(2(c − V)))" }, { "P": 1.3, "regime": "ROTATING", "value": [ 1.020098638788358, 1.0201002679975604 ], "width": 0.0000016292092024361438, "stalled": true, "tightened": true, "it": 19, "qWidth": 0.000001908225157221111, "why": null, "mather": "absolutely continuous, density 1/(T(c)√(2(c − V)))" }, { "P": 1.35, "regime": "ROTATING", "value": [ 1.067535156919521, 1.0675369113813622 ], "width": 0.0000017544618411413637, "stalled": true, "tightened": true, "it": 19, "qWidth": 0.0000017106392715682264, "why": null, "mather": "absolutely continuous, density 1/(T(c)√(2(c − V)))" }, { "P": 1.4, "regime": "ROTATING", "value": [ 1.121762747185396, 1.121764563890093 ], "width": 0.000001816704697121807, "stalled": true, "tightened": true, "it": 17, "qWidth": 0.0000015956343206724457, "why": null, "mather": "absolutely continuous, density 1/(T(c)√(2(c − V)))" }, { "P": 1.5, "regime": "ROTATING", "value": [ 1.2446366988236932, 1.2446385824276311 ], "width": 0.0000018836039379355185, "stalled": true, "tightened": true, "it": 17, "qWidth": 0.0000014361929485673388, "why": null, "mather": "absolutely continuous, density 1/(T(c)√(2(c − V)))" }, { "P": 1.75, "regime": "ROTATING", "value": [ 1.6159051794714918, 1.6159071297669128 ], "width": 0.000001950295420938275, "stalled": true, "tightened": true, "it": 20, "qWidth": 0.0000011852080099572506, "why": null, "mather": "absolutely continuous, density 1/(T(c)√(2(c − V)))" }, { "P": 2, "regime": "ROTATING", "value": [ 2.063794435233037, 2.0637964104911455 ], "width": 0.0000019752581086329712, "stalled": true, "tightened": true, "it": 20, "qWidth": 0.000001021622283214896, "why": null, "mather": "absolutely continuous, density 1/(T(c)√(2(c − V)))" }, { "P": 2.25, "regime": "ROTATING", "value": [ 2.581256122937218, 2.581258110021989 ], "width": 0.00000198708477094911, "stalled": true, "tightened": true, "it": 20, "qWidth": 9.014179060962136e-7, "why": null, "mather": "absolutely continuous, density 1/(T(c)√(2(c − V)))" }, { "P": 2.5, "regime": "ROTATING", "value": [ 3.1653266265918236, 3.1653286199745865 ], "width": 0.0000019933827628548784, "stalled": true, "tightened": true, "it": 19, "qWidth": 8.079522952897379e-7, "why": null, "mather": "absolutely continuous, density 1/(T(c)√(2(c − V)))" }, { "P": 3, "regime": "ROTATING", "value": [ 4.527885155369878, 4.527887154612475 ], "width": 0.000001999242596362194, "stalled": true, "tightened": true, "it": 15, "qWidth": 6.705960631236963e-7, "why": null, "mather": "absolutely continuous, density 1/(T(c)√(2(c − V)))" } ], "table1": { "c15": [ 1.244637626079612, 1.244637655177193 ], "c25": [ 3.1653276077695307, 3.165327638798285 ], "sum": [ 4.409965233849142, 4.409965293975479 ], "width": 6.012633679830515e-8, "cited": 4.409966, "citedPlace": "ABOVE", "citedGap": 7.060245206247373e-7, "HRF": [ { "v": 4.40251, "place": "BELOW" }, { "v": 4.40916, "place": "BELOW" }, { "v": 4.40989, "place": "BELOW" }, { "v": 4.40996, "place": "BELOW" } ], "NM": [ { "v": 4.40935, "place": "BELOW" }, { "v": 4.40994, "place": "BELOW" }, { "v": 4.40996, "place": "BELOW" }, { "v": 4.40996, "place": "BELOW" } ] }, "table2": { "regime": "FLAT", "value": [ 1, 1 ], "Hdiamond": [ { "v": 0.964609, "gap": 0.03539099999999995, "place": "BELOW" }, { "v": 0.964754, "gap": 0.035246, "place": "BELOW" }, { "v": 0.96476, "gap": 0.03524000000000005, "place": "BELOW" }, { "v": 0.96476, "gap": 0.03524000000000005, "place": "BELOW" } ] }, "mather": [ { "P": 1.5, "T": [ 0.7624764664209688, 0.7624781543105226 ], "density": [ [ 0.8312582472550789, 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"rotating": 12, "undecided": 1, "refused": 0 }, "widest": { "P": 3, "width": 0.000001999242596362194 }, "provenance": { "code": "instruments/hbar/exact.js", "sha256": "2070aca1c7d6d1e272ba4b64ebd7db6ffb98ca5e35786ca136f710db84bbbc08", "derive": "instruments/hbar/derive.js", "deriveSha256": "83b8bbe7972e6170c6610470b4a06af74f7ad54fdcddfa3fd53af27b8e3c1d77", "taylor2Sha256": "e8fe058b351adebdca8a79b3ee7e34a72ba27989b4c3103c7ef260fbefc86f2c", "paper": "D. A. Gomes, X. Yang, The Hessian Riemannian flow and Newton's method for effective Hamiltonians and Mather measures, arXiv:1810.03483v2; ESAIM M2AN 54 (2020) 1883–1915 — §6.1 (the one-dimensional formula), §6.2 (Table 1), §6.4 (Table 2)" }, "generatedBy": "node instruments/hbar/run.js" }