{ "what": "the monotone flow of the quadratic stationary MFG (labs/mfg) on the even Galerkin space: at each certified equilibrium, the Lasry–Lions quadratic form evaluated from the interval Jacobian (must return c/2 at δm = cos 2πx) and, where a non-real eigenvalue pair exists, a Krawczyk enclosure of it — which rules out a gradient-flow structure under ANY Riemannian metric near that equilibrium", "statement": "CERTIFICATE 1 (monotone): ⟨DA v, v⟩ = c ∫δm² + ∫ m (δu')² exactly, so the linearised Lasry–Lions operator is positive semidefinite iff c ≥ 0; the Jacobian assembled here returns c/2 at the witness direction on every instance. CERTIFICATE 2 (not a gradient flow): at C1, M1, M2, H1, H2 an eigenvalue μ of the flow Jacobian over the equilibrium's Galerkin box is enclosed with Im μ bounded away from 0, so no metric G and energy E make the flow −G⁻¹∇E near that equilibrium. At C1 the enclosed μ contains the closed-form mode-1 eigenvalue. R1 and H0 are NOT DECIDED: every eigenvalue reached by inverse iteration from the mode shifts is real to 1e-14, and a real spectrum is consistent with a gradient structure; nothing is claimed there. The theorem is about the finite-dimensional Galerkin ODE at the order N recorded; the same candidate is enclosed as a PDE solution by validate.js and both radii are recorded.", "verdict": "VERIFIED", "instances": [ { "id": "C1", "words": "the constant solution, c = 1: the control — every mode has a closed form", "sigma": 0.5, "c": 1, "A": 0, "N": 16, "branch": "aligned", "galerkinRadius": 1.0125233984581429e-13, "pde": { "r": 5.828670879282028e-16, "Z1": 0.01872411095198979, "kappa": 0.01872411095199096, "minM": 0.9999999999999974 }, "form": { "value": [ 0.49999999999999795, 0.5000000000000041 ], "expected": 0.5, "contains": true }, "monotone": true, "modes": [ { "k": 1, "re": -20.239208802178716, "im": 4.414658401527657, "complex": true }, { "k": 2, "re": -79.45683520871486, "im": 8.871687280822929, "complex": true }, { "k": 3, "re": -178.15287921960845, "im": 13.31926721781674, "complex": true }, { "k": 4, "re": -316.32734083485946, "im": 17.764496638938617, "complex": true } ], "tried": [ { "shift": [ -20.239208802178716, 4.414658401527657 ], "mu": [ -20.239208802178716, 4.414658401527666 ] } ], "pairs": [ { "shift": [ -20.239208802178716, 4.414658401527657 ], "muFloat": [ -20.239208802178716, 4.414658401527666 ], "ok": true, "mu": [ [ -20.239208802186273, -20.23920880217116 ], [ 4.41465840150313, 4.414658401552203 ] ], "muWidth": [ 1.5113243989617334e-11, 4.9073634045271326e-11 ], "nonReal": true, "rounds": 1 } ], "certified": [ { "mu": [ [ -20.239208802186273, -20.23920880217116 ], [ 4.41465840150313, 4.414658401552203 ] ], "muWidth": [ 1.5113243989617334e-11, 4.9073634045271326e-11 ] } ], "control": [ { "k": 1, "complex": true, "re": [ -20.239208802178737, -20.239208802178695 ], "im": [ 4.414658401527477, 4.414658401527843 ] }, { "k": 2, "complex": true, "re": [ -79.45683520871495, -79.45683520871478 ], "im": [ 8.871687280821535, 8.871687280824379 ] }, { "k": 3, "complex": true, "re": [ -178.15287921960865, -178.15287921960822 ], "im": [ 13.319267217811399, 13.31926721782222 ] }, { "k": 4, "complex": true, "re": [ -316.3273408348598, -316.3273408348591 ], "im": [ 17.764496638927536, 17.764496638950522 ] } ], "verdict": "NOT A GRADIENT FLOW", "certificate": { "claim": "the monotone flow of the Galerkin MFG at (σ, c, A, N) = (0.5, 1, 0, 16) is NOT a gradient flow under any Riemannian metric near its certified equilibrium: an eigenvalue μ of the flow Jacobian is enclosed with Re μ ∈ [-20.2392088, -20.2392088], Im μ ∈ [4.414658402, 4.414658402] (Im μ ≠ 0); the linearised Lasry–Lions form is positive semidefinite (c ≥ 0): monotone", "verdict": "PROVED", "evidence": { "galerkinRadius": 1.0125233984581429e-13, "pdeRadius": 5.828670879282028e-16, "muRe": [ -20.239208802186273, -20.23920880217116 ], "muIm": [ 4.41465840150313, 4.414658401552203 ], "formValue": [ 0.49999999999999795, 0.5000000000000041 ] }, "assumes": [ "the flow is the L²-paired monotone flow (ṁ, u̇) = −A(m, u) restricted to the even Galerkin space of order N with mass and mean held; the theorem is about that finite-dimensional ODE", "a gradient flow ż = −G⁻¹∇E has a Jacobian similar to a symmetric matrix at an equilibrium, hence real spectrum (linear algebra, not re-proved)" ], "falsifier": [ "the Lasry–Lions form evaluated from the assembled Jacobian at δm = cos 2πx must return c/2 — a sign error in the assembly breaks it", "a symmetric matrix handed to the eigenpair certifier with a complex shift must NOT return a non-real pair", "the eigenpair certifier started 0.5 away from the eigenvalue must return no contraction", "at A = 0 the certified pair must contain the closed-form mode eigenvalue, and the closed form with σ replaced by 1 − σ must be excluded", "the Galerkin equilibrium certifier started from a candidate perturbed by 1e-2 must return no contraction" ], "provenance": { "code": "instruments/monoflow/monoflow.js", "sha256": "4f54364e4932e6ee5c8ed184e44e3e88c84f160cbb35ebe0430fd2bb238d3b4c" }, "why": "" }, "x": [ 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ] }, { "id": "M1", "words": "monotone (c = 1), potential A = 1: the non-constant equilibrium of the observatory", "sigma": 0.5, "c": 1, "A": 1, "N": 16, "branch": "aligned", "galerkinRadius": 1.0114131754335177e-13, "pde": { "r": 1.5543122344752073e-15, "Z1": 0.025990970125087946, "kappa": 0.025990970125091058, "minM": 0.9095460336705414 }, "form": { "value": [ 0.49999999999999795, 0.5000000000000041 ], "expected": 0.5, "contains": true }, "monotone": true, "modes": [ { "k": 1, "re": -20.239208802178716, "im": 4.414658401527657, "complex": true }, { "k": 2, "re": -79.45683520871486, "im": 8.871687280822929, "complex": true }, { "k": 3, "re": -178.15287921960845, "im": 13.31926721781674, "complex": true }, { "k": 4, "re": -316.32734083485946, "im": 17.764496638938617, "complex": true } ], "tried": [ { "shift": [ -20.239208802178716, 4.414658401527657 ], "mu": [ -20.26187729986502, 4.598624618105329 ] } ], "pairs": [ { "shift": [ -20.239208802178716, 4.414658401527657 ], "muFloat": [ -20.26187729986502, 4.598624618105329 ], "ok": true, "mu": [ [ -20.261877299874193, -20.261877299855847 ], [ 4.598624618079274, 4.5986246181313835 ] ], "muWidth": [ 1.834621343732579e-11, 5.2109427883806354e-11 ], "nonReal": true, "rounds": 1 } ], "certified": [ { "mu": [ [ -20.261877299874193, -20.261877299855847 ], [ 4.598624618079274, 4.5986246181313835 ] ], "muWidth": [ 1.834621343732579e-11, 5.2109427883806354e-11 ] } ], "control": null, "verdict": "NOT A GRADIENT FLOW", "certificate": { "claim": "the monotone flow of the Galerkin MFG at (σ, c, A, N) = (0.5, 1, 1, 16) is NOT a gradient flow under any Riemannian metric near its certified equilibrium: an eigenvalue μ of the flow Jacobian is enclosed with Re μ ∈ [-20.2618773, -20.2618773], Im μ ∈ [4.598624618, 4.598624618] (Im μ ≠ 0); the linearised Lasry–Lions form is positive semidefinite (c ≥ 0): monotone", "verdict": "PROVED", "evidence": { "galerkinRadius": 1.0114131754335177e-13, "pdeRadius": 1.5543122344752073e-15, "muRe": [ -20.261877299874193, -20.261877299855847 ], "muIm": [ 4.598624618079274, 4.5986246181313835 ], "formValue": [ 0.49999999999999795, 0.5000000000000041 ] }, "assumes": [ "the flow is the L²-paired monotone flow (ṁ, u̇) = −A(m, u) restricted to the even Galerkin space of order N with mass and mean held; the theorem is about that finite-dimensional ODE", "a gradient flow ż = −G⁻¹∇E has a Jacobian similar to a symmetric matrix at an equilibrium, hence real spectrum (linear algebra, not re-proved)" ], "falsifier": [ "the Lasry–Lions form evaluated from the assembled Jacobian at δm = cos 2πx must return c/2 — a sign error in the assembly breaks it", "a symmetric matrix handed to the eigenpair certifier with a complex shift must NOT return a non-real pair", "the eigenpair certifier started 0.5 away from the eigenvalue must return no contraction", "at A = 0 the certified pair must contain the closed-form mode eigenvalue, and the closed form with σ replaced by 1 − σ must be excluded", "the Galerkin equilibrium certifier started from a candidate perturbed by 1e-2 must return no contraction" ], "provenance": { "code": "instruments/monoflow/monoflow.js", "sha256": "4f54364e4932e6ee5c8ed184e44e3e88c84f160cbb35ebe0430fd2bb238d3b4c" }, "why": "" }, "x": [ 0.9791292615987796, 0.022990899238048382, 0.00014187473200872594, 0.0000014157093787407455, 1.7304505569240597e-8, 2.2366254480033418e-10, 3.01983959223471e-12, 4.1950824333052843e-14, 5.948760440786021e-16, 8.569652047078617e-18, 1.2499573843445132e-19, 1.841585769980094e-21, 2.735853905780639e-23, 4.0927979703112716e-25, 6.159211754985993e-27, 9.316470800836074e-29, 1.4155731193343311e-30, -0.04592024821279444, 0.0007720559262235824, -0.000005975904517784489, 2.209159014595558e-8, -1.2097828563388766e-11, -2.6747248473863766e-13, 1.311444015289324e-15, -2.1783590326144506e-18, -6.588194560654567e-21, 5.203937720269787e-23, -1.3630889849551022e-25, -2.971756229098435e-29, 1.6049837669362648e-30, -6.1234033020006614e-33, 4.092056211510851e-35, -2.237332486662802e-33 ] }, { "id": "M2", "words": "monotone (c = 1), potential A = 0.5", "sigma": 0.5, "c": 1, "A": 0.5, "N": 16, "branch": "aligned", "galerkinRadius": 1.0125233984581429e-13, "pde": { "r": 1.1657341758564057e-15, "Z1": 0.02233703794330956, "kappa": 0.022337037943311894, "minM": 0.9543335965755295 }, "form": { "value": [ 0.49999999999999795, 0.5000000000000041 ], "expected": 0.5, "contains": true }, "monotone": true, "modes": [ { "k": 1, "re": -20.239208802178716, "im": 4.414658401527657, "complex": true }, { "k": 2, "re": -79.45683520871486, "im": 8.871687280822929, "complex": true }, { "k": 3, "re": -178.15287921960845, "im": 13.31926721781674, "complex": true }, { "k": 4, "re": -316.32734083485946, "im": 17.764496638938617, "complex": true } ], "tried": [ { "shift": [ -20.239208802178716, 4.414658401527657 ], "mu": [ -20.244893660749796, 4.461224268773617 ] } ], "pairs": [ { "shift": [ -20.239208802178716, 4.414658401527657 ], "muFloat": [ -20.244893660749796, 4.461224268773617 ], "ok": true, "mu": [ [ -20.244893660758134, -20.244893660741457 ], [ 4.461224268748127, 4.461224268799107 ] ], "muWidth": [ 1.6676438008289555e-11, 5.0979664933947795e-11 ], "nonReal": true, "rounds": 1 } ], "certified": [ { "mu": [ [ -20.244893660758134, -20.244893660741457 ], [ 4.461224268748127, 4.461224268799107 ] ], "muWidth": [ 1.6676438008289555e-11, 5.0979664933947795e-11 ] } ], "control": null, "verdict": "NOT A GRADIENT FLOW", "certificate": { "claim": "the monotone flow of the Galerkin MFG at (σ, c, A, N) = (0.5, 1, 0.5, 16) is NOT a gradient flow under any Riemannian metric near its certified equilibrium: an eigenvalue μ of the flow Jacobian is enclosed with Re μ ∈ [-20.24489366, -20.24489366], Im μ ∈ [4.461224269, 4.461224269] (Im μ ≠ 0); the linearised Lasry–Lions form is positive semidefinite (c ≥ 0): monotone", "verdict": "PROVED", "evidence": { "galerkinRadius": 1.0125233984581429e-13, "pdeRadius": 1.1657341758564057e-15, "muRe": [ -20.244893660758134, -20.244893660741457 ], "muIm": [ 4.461224268748127, 4.461224268799107 ], "formValue": [ 0.49999999999999795, 0.5000000000000041 ] }, "assumes": [ "the flow is the L²-paired monotone flow (ṁ, u̇) = −A(m, u) restricted to the even Galerkin space of order N with mass and mean held; the theorem is about that finite-dimensional ODE", "a gradient flow ż = −G⁻¹∇E has a Jacobian similar to a symmetric matrix at an equilibrium, hence real spectrum (linear algebra, not re-proved)" ], "falsifier": [ "the Lasry–Lions form evaluated from the assembled Jacobian at δm = cos 2πx must return c/2 — a sign error in the assembly breaks it", "a symmetric matrix handed to the eigenpair certifier with a complex shift must NOT return a non-real pair", "the eigenpair certifier started 0.5 away from the eigenvalue must return no contraction", "at A = 0 the certified pair must contain the closed-form mode eigenvalue, and the closed form with σ replaced by 1 − σ must be excluded", "the Galerkin equilibrium certifier started from a candidate perturbed by 1e-2 must return no contraction" ], "provenance": { "code": "instruments/monoflow/monoflow.js", "sha256": "4f54364e4932e6ee5c8ed184e44e3e88c84f160cbb35ebe0430fd2bb238d3b4c" }, "why": "" }, "x": [ 0.9947798420815419, 0.01149883050706588, 0.000035501625461999804, 1.7721071971733185e-7, 1.0835619269481846e-9, 7.005892159315021e-12, 4.7318168883592146e-14, 3.2882034515926645e-16, 2.3324841659957326e-18, 1.6808518615783823e-20, 1.2264094413755126e-22, 9.038705835947015e-25, 6.717090812473406e-27, 5.0266960280397597e-29, 3.7840896656737566e-31, 2.8632654874550062e-33, 2.1762122577987392e-35, -0.022989949824158733, 0.00019335790162218494, -7.483634576373724e-7, 1.382507437684164e-9, -3.740416857148767e-13, -4.199625777012732e-15, 1.0278151328600471e-17, -8.503176837299668e-21, -1.3006474828202823e-23, 5.1093983295186913e-26, -6.70417061415204e-29, -1.263562883131273e-32, 1.3120644749737999e-34, -1.2346603565681277e-36, -1.0245009957399612e-38, -8.737148666522042e-39 ] }, { "id": "R1", "words": "monotone (c = 2), σ = 0.3: every eigenvalue reached is real — NOT DECIDED, and kept as such", "sigma": 0.3, "c": 2, "A": 1.5, "N": 20, "branch": "aligned", "galerkinRadius": 1.9717560917342783e-13, "pde": { "r": 5.551115123125731e-15, "Z1": 0.06442462148407821, "kappa": 0.06442462148408933, "minM": 0.7442792620996302 }, "form": { "value": [ 0.999999999999995, 1.00000000000001 ], "expected": 1, "contains": true }, "monotone": true, "modes": [ { "k": 1, "mu1": -6.217196013219715, "mu2": -35.26122159113772, "complex": false }, { "k": 2, "mu1": -18.045909692007314, "mu2": -141.86776072542241, "complex": false }, { "k": 3, "mu1": -37.782572347040286, "mu2": -319.5231860921766, "complex": false }, { "k": 4, "mu1": -65.41658685171316, "mu2": -568.2380948180057, "complex": false } ], "tried": [ { "shift": [ -6.217196013219715, 2.8651588039659144 ], "mu": [ -6.399255655308502, -2.229557260236146e-16 ] }, { "shift": [ -35.26122159113772, 11.578366477341316 ], "mu": [ -38.35354597649002, -4.510283767506189e-16 ] }, { "shift": [ -18.045909692007314, 6.413772907602194 ], "mu": [ -18.77686554271144, 1.7818318544379552e-15 ] }, { "shift": [ -141.86776072542241, 43.560328217626726 ], "mu": [ -134.02643006506895, 1.2617964216114243e-14 ] }, { "shift": [ -319.5231860921766, 96.85695582765298 ], "mu": [ -302.3659144867712, -1.2124209660857682e-14 ] }, { "shift": [ -65.41658685171316, 20.624976055513947 ], "mu": [ -67.14284849465035, -1.1548922930190598e-14 ] }, { "shift": [ -568.2380948180057, 171.4714284454017 ], "mu": [ -538.1117693552073, 2.3659698608372623e-15 ] } ], "pairs": [], "certified": [], "control": null, "verdict": "NOT DECIDED", "certificate": { "claim": "the monotone flow at (σ, c, A, N) = (0.3, 2, 1.5, 20) is not a gradient flow under any metric — NOT DECIDED: no non-real eigenpair was certified (every float eigenvalue reached was real)", "verdict": "NOT_CHECKED", "evidence": {}, "assumes": [], "falsifier": [ "the Lasry–Lions form evaluated from the assembled Jacobian at δm = cos 2πx must return c/2 — a sign error in the assembly breaks it", "a symmetric matrix handed to the eigenpair certifier with a complex shift must NOT return a non-real pair", "the eigenpair certifier started 0.5 away from the eigenvalue must return no contraction", "at A = 0 the certified pair must contain the closed-form mode eigenvalue, and the closed form with σ replaced by 1 − σ must be excluded", "the Galerkin equilibrium certifier started from a candidate perturbed by 1e-2 must return no contraction" ], "provenance": { "code": "instruments/monoflow/monoflow.js", "sha256": "4f54364e4932e6ee5c8ed184e44e3e88c84f160cbb35ebe0430fd2bb238d3b4c" }, "why": "" }, "x": [ 1.9350037520747698, 0.04053262917858158, 0.0009319132310879299, 0.000026455770304935567, 0.0000010185825292892066, 4.193583653519713e-8, 1.7796352967331536e-9, 7.791245155002852e-11, 3.4841606314803433e-12, 1.5823860693742255e-13, 7.27653282121379e-15, 3.379955081641923e-16, 1.5830701562092054e-17, 7.4664708270187525e-19, 3.542486369436399e-20, 1.6893610476632678e-21, 8.092286485780534e-23, 3.891524855819249e-24, 1.8779054594012953e-25, 9.090137555022451e-27, 4.413845659882257e-28, -0.1334812503823558, 0.005923212279326708, -0.00007796340137757851, -0.0000011088577257218853, 3.715681858642049e-8, -1.1571564922589533e-12, -1.0433045981856086e-11, 8.734488730191733e-14, 1.9442622398662938e-15, -3.734221070065874e-17, -1.7090976373185551e-19, 1.0223445486320596e-20, -4.0756690409063613e-23, -2.03978315520408e-24, 2.5228028977804422e-26, 2.6488643462684225e-28, -7.722234378725374e-30, -8.131903602435955e-33, 4.418809924752343e-31, -1.050267288090289e-29 ] }, { "id": "H1", "words": "herding (c = −12), the symmetry-broken branch of mfg-cap: not monotone, and not a gradient either", "sigma": 0.5, "c": -12, "A": 0, "N": 16, "branch": "herding", "galerkinRadius": 1.961097950697877e-12, "pde": { "r": 1.6246839143199633e-11, "Z1": 0.32017452392132306, "kappa": 0.3201745242872602, "minM": 0.1611959083246188 }, "form": { "value": [ -6.000000000000033, -5.999999999999967 ], "expected": -6, "contains": true }, "monotone": false, "modes": [ { "k": 1, "mu1": 2.779583696855365, "mu2": -30.258001301212797, "complex": false }, { "k": 2, "mu1": -41.59631827253255, "mu2": -104.31735214489717, "complex": false }, { "k": 3, "mu1": -125.09286901089712, "mu2": -218.21288942831978, "complex": false }, { "k": 4, "mu1": -247.97325374258796, "mu2": -371.6814279271309, "complex": false } ], "tried": [ { "shift": [ 2.779583696855365, 1.8338751090566094 ], "mu": [ -9.180456158954527, 1.1455018825853639e-15 ] }, { "shift": [ -30.258001301212797, 10.077400390363838 ], "mu": [ -17.13398246825456, 17.96503279178917 ] } ], "pairs": [ { "shift": [ -30.258001301212797, 10.077400390363838 ], "muFloat": [ -17.13398246825456, 17.96503279178917 ], "ok": true, "mu": [ [ -17.133982468307245, -17.13398246820187 ], [ 17.96503279172871, 17.96503279184963 ] ], "muWidth": [ 1.0537348771322287e-10, 1.2092016277165388e-10 ], "nonReal": true, "rounds": 1 } ], "certified": [ { "mu": [ [ -17.133982468307245, -17.13398246820187 ], [ 17.96503279172871, 17.96503279184963 ] ], "muWidth": [ 1.0537348771322287e-10, 1.2092016277165388e-10 ] } ], "control": null, "verdict": "NOT A GRADIENT FLOW", "certificate": { "claim": "the monotone flow of the Galerkin MFG at (σ, c, A, N) = (0.5, -12, 0, 16) herding branch is NOT a gradient flow under any Riemannian metric near its certified equilibrium: an eigenvalue μ of the flow Jacobian is enclosed with Re μ ∈ [-17.13398247, -17.13398247], Im μ ∈ [17.96503279, 17.96503279] (Im μ ≠ 0); the linearised Lasry–Lions form is indefinite (c < 0, witness value c/2 = -6): not monotone", "verdict": "PROVED", "evidence": { "galerkinRadius": 1.961097950697877e-12, "pdeRadius": 1.6246839143199633e-11, "muRe": [ -17.133982468307245, -17.13398246820187 ], "muIm": [ 17.96503279172871, 17.96503279184963 ], "formValue": [ -6.000000000000033, -5.999999999999967 ] }, "assumes": [ "the flow is the L²-paired monotone flow (ṁ, u̇) = −A(m, u) restricted to the even Galerkin space of order N with mass and mean held; the theorem is about that finite-dimensional ODE", "a gradient flow ż = −G⁻¹∇E has a Jacobian similar to a symmetric matrix at an equilibrium, hence real spectrum (linear algebra, not re-proved)" ], "falsifier": [ "the Lasry–Lions form evaluated from the assembled Jacobian at δm = cos 2πx must return c/2 — a sign error in the assembly breaks it", "a symmetric matrix handed to the eigenpair certifier with a complex shift must NOT return a non-real pair", "the eigenpair certifier started 0.5 away from the eigenvalue must return no contraction", "at A = 0 the certified pair must contain the closed-form mode eigenvalue, and the closed form with σ replaced by 1 − σ must be excluded", "the Galerkin equilibrium certifier started from a candidate perturbed by 1e-2 must return no contraction" ], "provenance": { "code": "instruments/monoflow/monoflow.js", "sha256": "4f54364e4932e6ee5c8ed184e44e3e88c84f160cbb35ebe0430fd2bb238d3b4c" }, "why": "" }, "x": [ -16.624210663731215, -0.3421225797205386, -1.342335345158074e-17, -0.0030677541001681414, -2.9311776616451087e-19, -0.000050921088542587646, -3.609260846937051e-21, -0.0000010062472548219594, -1.419696222130123e-22, -2.165186504006882e-8, -4.9307122611676605e-24, -4.900949277575871e-10, -1.1666614852813635e-25, -1.1472670992280297e-11, -3.585007630828653e-27, -2.750543269957484e-13, -1.0496057721468225e-28, 0.5627690864203121, 0.18216967387350502, 0.04541627905271914, 0.010071841459052112, 0.0020940458316174347, 0.0004179602285864669, 0.0000811053090673284, 0.000015417315741379272, 0.00000288488712363196, 5.331554487449355e-7, 9.75470349616152e-8, 1.769987394896352e-8, 3.1893320626474253e-9, 5.712877485175427e-10, 1.0180040235959682e-10, 1.8190769251129936e-11 ] }, { "id": "H2", "words": "herding (c = −12) with a potential, A = 0.5", "sigma": 0.5, "c": -12, "A": 0.5, "N": 16, "branch": "herding", "galerkinRadius": 1.861621967691463e-12, "pde": { "r": 4.377915128205962e-12, "Z1": 0.3092711097789599, "kappa": 0.3092711099067313, "minM": 0.20514939501317553 }, "form": { "value": [ -6.000000000000033, -5.999999999999967 ], "expected": -6, "contains": true }, "monotone": false, "modes": [ { "k": 1, "mu1": 2.779583696855365, "mu2": -30.258001301212797, "complex": false }, { "k": 2, "mu1": -41.59631827253255, "mu2": -104.31735214489717, "complex": false }, { "k": 3, "mu1": -125.09286901089712, "mu2": -218.21288942831978, "complex": false }, { "k": 4, "mu1": -247.97325374258796, "mu2": -371.6814279271309, "complex": false } ], "tried": [ { "shift": [ 2.779583696855365, 1.8338751090566094 ], "mu": [ -5.891225011988504, 3.155704827882084e-15 ] }, { "shift": [ -30.258001301212797, 10.077400390363838 ], "mu": [ -19.795347943178818, 17.038116253659148 ] } ], "pairs": [ { "shift": [ -30.258001301212797, 10.077400390363838 ], "muFloat": [ -19.795347943178818, 17.038116253659148 ], "ok": true, "mu": [ [ -19.795347943234418, -19.795347943123218 ], [ 17.038116253596055, 17.03811625372224 ] ], "muWidth": [ 1.1119993814645569e-10, 1.2618528444363622e-10 ], "nonReal": true, "rounds": 1 } ], "certified": [ { "mu": [ [ -19.795347943234418, -19.795347943123218 ], [ 17.038116253596055, 17.03811625372224 ] ], "muWidth": [ 1.1119993814645569e-10, 1.2618528444363622e-10 ] } ], "control": null, "verdict": "NOT A GRADIENT FLOW", "certificate": { "claim": "the monotone flow of the Galerkin MFG at (σ, c, A, N) = (0.5, -12, 0.5, 16) herding branch is NOT a gradient flow under any Riemannian metric near its certified equilibrium: an eigenvalue μ of the flow Jacobian is enclosed with Re μ ∈ [-19.79534794, -19.79534794], Im μ ∈ [17.03811625, 17.03811625] (Im μ ≠ 0); the linearised Lasry–Lions form is indefinite (c < 0, witness value c/2 = -6): not monotone", "verdict": "PROVED", "evidence": { "galerkinRadius": 1.861621967691463e-12, "pdeRadius": 4.377915128205962e-12, "muRe": [ -19.795347943234418, -19.795347943123218 ], "muIm": [ 17.038116253596055, 17.03811625372224 ], "formValue": [ -6.000000000000033, -5.999999999999967 ] }, "assumes": [ "the flow is the L²-paired monotone flow (ṁ, u̇) = −A(m, u) restricted to the even Galerkin space of order N with mass and mean held; the theorem is about that finite-dimensional ODE", "a gradient flow ż = −G⁻¹∇E has a Jacobian similar to a symmetric matrix at an equilibrium, hence real spectrum (linear algebra, not re-proved)" ], "falsifier": [ "the Lasry–Lions form evaluated from the assembled Jacobian at δm = cos 2πx must return c/2 — a sign error in the assembly breaks it", "a symmetric matrix handed to the eigenpair certifier with a complex shift must NOT return a non-real pair", "the eigenpair certifier started 0.5 away from the eigenvalue must return no contraction", "at A = 0 the certified pair must contain the closed-form mode eigenvalue, and the closed form with σ replaced by 1 − σ must be excluded", "the Galerkin equilibrium certifier started from a candidate perturbed by 1e-2 must return no contraction" ], "provenance": { "code": "instruments/monoflow/monoflow.js", "sha256": "4f54364e4932e6ee5c8ed184e44e3e88c84f160cbb35ebe0430fd2bb238d3b4c" }, "why": "" }, "x": [ -15.681790485240482, -0.30530490555466294, -0.0011184098790654841, -0.002229189427369755, -0.000022039451524474385, -0.00002992808775263553, -4.907134590143237e-7, -4.796657697829863e-7, -1.096813467809388e-8, -8.387701404544928e-9, -2.4540043805275206e-10, -1.5458204354665272e-10, -5.494490456481003e-12, -2.951692496329487e-12, -1.2310252804669006e-13, -5.782852557490138e-14, -2.7776140939455358e-15, 0.52074188546056, 0.15396390851543124, 0.03515880343801783, 0.007154968647264166, 0.0013655189364208883, 0.0002502009473562253, 0.000044571200343153855, 0.000007778000807242456, 0.0000013361140605143919, 2.2668599043661402e-7, 3.8075125837795396e-8, 6.342406465968258e-9, 1.0491568397978051e-9, 1.725246332096383e-10, 2.8224057237464794e-11, 4.624366639978177e-12 ] }, { "id": "H0", "words": "the constant solution at c = −12: real spectrum by the closed form, one eigenvalue positive — the flow leaves it", "sigma": 0.5, "c": -12, "A": 0, "N": 16, "branch": "aligned", "galerkinRadius": 1.2132517213103713e-12, "pde": { "r": 7.216737409379427e-15, "Z1": 0.22468933142386083, "kappa": 0.22468933142396344, "minM": 0.9999999999999908 }, "form": { "value": [ -6.000000000000033, -5.999999999999967 ], "expected": -6, "contains": true }, "monotone": false, "modes": [ { "k": 1, "mu1": 2.779583696855365, "mu2": -30.258001301212797, "complex": false }, { "k": 2, "mu1": -41.59631827253255, "mu2": -104.31735214489717, "complex": false }, { "k": 3, "mu1": -125.09286901089712, "mu2": -218.21288942831978, "complex": false }, { "k": 4, "mu1": -247.97325374258796, "mu2": -371.6814279271309, "complex": false } ], "tried": [ { "shift": [ 2.779583696855365, 1.8338751090566094 ], "mu": [ 2.779583696855362, 3.469446951953614e-18 ] }, { "shift": [ -30.258001301212797, 10.077400390363838 ], "mu": [ -30.258001301212797, 2.220446049250313e-16 ] }, { "shift": [ -104.31735214489717, 32.295205643469146 ], "mu": [ -125.09286901089712, -2.6645352591003757e-15 ] }, { "shift": [ -218.21288942831978, 66.46386682849594 ], "mu": [ -218.21288942831973, -7.105427357601002e-15 ] }, { "shift": [ -247.97325374258796, 75.39197612277638 ], "mu": [ -247.97325374258804, 1.4210854715202004e-14 ] }, { "shift": [ -371.6814279271309, 112.50442837813928 ], "mu": [ -371.68142792713115, 3.552713678800501e-15 ] } ], "pairs": [], "certified": [], "control": [ { "k": 1, "complex": false, "re": [ -13.739208802178734, -13.739208802178698 ], "disc": [ 1091.4820225045708, 1091.482022504586 ] }, { "k": 2, "complex": false, "re": [ -72.95683520871495, -72.95683520871478 ], "disc": [ 3933.9280900182052, 3933.928090018417 ] }, { "k": 3, "complex": false, "re": [ -171.65287921960865, -171.65287921960822 ], "disc": [ 8671.3382025406, 8671.338202541783 ] }, { "k": 4, "complex": false, "re": [ -309.8273408348598, -309.8273408348591 ], "disc": [ 15303.712360071599, 15303.712360074862 ] } ], "verdict": "NOT DECIDED", "certificate": { "claim": "the monotone flow at (σ, c, A, N) = (0.5, -12, 0, 16) is not a gradient flow under any metric — NOT DECIDED: no non-real eigenpair was certified (every float eigenvalue reached was real)", "verdict": "NOT_CHECKED", "evidence": {}, "assumes": [], "falsifier": [ "the Lasry–Lions form evaluated from the assembled Jacobian at δm = cos 2πx must return c/2 — a sign error in the assembly breaks it", "a symmetric matrix handed to the eigenpair certifier with a complex shift must NOT return a non-real pair", "the eigenpair certifier started 0.5 away from the eigenvalue must return no contraction", "at A = 0 the certified pair must contain the closed-form mode eigenvalue, and the closed form with σ replaced by 1 − σ must be excluded", "the Galerkin equilibrium certifier started from a candidate perturbed by 1e-2 must return no contraction" ], "provenance": { "code": "instruments/monoflow/monoflow.js", "sha256": "4f54364e4932e6ee5c8ed184e44e3e88c84f160cbb35ebe0430fd2bb238d3b4c" }, "why": "" }, "x": [ -12, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ] } ], "provenance": { "code": "instruments/monoflow/monoflow.js", "sha256": "4f54364e4932e6ee5c8ed184e44e3e88c84f160cbb35ebe0430fd2bb238d3b4c", "kernel": "legacy/core/mfg/{mfg1d,validate}.js (the lifted mfg-cap kernel; Jacobian rows dRow, interval)", "flow": "Almulla–Ferreira–Gomes, DGA 7(4) 2017, §2.6 (2.7)–(2.8): the monotone flow; Ferreira–Gomes–Tada, arXiv:2502.20091: monotonicity methods", "argument": "a gradient flow ż = −G⁻¹∇E has Jacobian −G⁻¹ Hess E ~ symmetric at an equilibrium, hence real spectrum; a certified non-real pair excludes every (G, E)" }, "generatedBy": "node instruments/monoflow/run.js" }