{
 "what": "the first-order stationary MFG of Almulla–Ferreira–Gomes (1.1) on the torus, V = sin 2πx, enclosed by the current: two scalars (j, H̄) fixed by two integral conditions, a Krawczyk box, rigorous quadrature throughout",
 "statement": "CASE B (b = cos² 2πx, ∫b = 1/2, the case the paper says has no known closed form): a classical solution (u, m, H̄) of (1.1) EXISTS, with current j and ergodic constant H̄ enclosed in the recorded box, locally unique there, and density enclosed pointwise and certified strictly positive. CASE A (b = 0, the paper's closed form u = 0, m = e^V/∫e^V, H̄ = ln ∫e^V): the same certifier, run as its own control — the enclosure of H̄ contains the closed form enclosed independently, and j is enclosed at 0 to 1e-13. Global uniqueness is AFG Lemma 2.3 (monotone operator), cited and assumed, not re-proved. Every number below is an interval endpoint computed in outward-rounded arithmetic; the only floats trusted are the Newton candidate and the approximate inverse, and they are trusted for nothing.",
 "verdict": "VERIFIED",
 "grid": {
  "KF": 16384,
  "KD": 2048,
  "KP": 256
 },
 "cases": {
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   "label": "b = 0",
   "words": "their closed form: u = 0, m = e^V / ∫e^V, H̄ = ln ∫e^V — the control",
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   "rounds": 1,
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   ],
   "certificate": {
    "claim": "AFG (1.1) with V = sin 2πx, b = 0: a classical solution (u, m, H̄) exists with current j ∈ [-1.000e-13, 1.000e-13] and H̄ ∈ [0.2359143495, 0.2359143675], locally unique in that box, density in [0.2905689541, 2.147030341] > 0",
    "verdict": "PROVED",
    "evidence": {
     "j": [
      -1.0000000000000002e-13,
      1.0000000000000002e-13
     ],
     "H": [
      0.23591434950775014,
      0.23591436750660538
     ],
     "jWidth": 2.0000000000000006e-13,
     "HWidth": 1.7998855239564907e-8,
     "mMin": 0.2905689540531119,
     "mMax": 2.147030340750151,
     "krawczykRounds": 1,
     "thinCells": 16384,
     "jacobianCells": 2048
    },
    "assumes": [
     "global uniqueness of the solution: AFG Lemma 2.3 (the operator is monotone in L²×L²) with Lasry–Lions; cited, not re-proved here — what is proved is uniqueness of (j, H̄) in the box",
     "classical solutions of (1.1) carry a constant current m(u_x + b) = j — the one-dimensional current formulation (Gomes–Nurbekyan–Prazeres), which is an identity for C¹ solutions on the torus",
     "∫ b is entered exactly (0 for b = 0; 1/2 for cos² 2πx)"
    ],
    "falsifier": [
     "the quadrature with its remainder term deleted (the interval then misses a value it must contain)",
     "the scalar inverse with the sign check disabled (a bracket that does not bracket is refused)",
     "Krawczyk started from a candidate 0.05 away from the root (must return no contraction)",
     "case A: the closed-form H̄ shifted by 1e-7 must lie OUTSIDE the enclosure",
     "a certified density floor forged to ≤ 0 must refuse"
    ],
    "provenance": {
     "code": "instruments/afg/afg.js",
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   ],
   "certificate": {
    "claim": "AFG (1.1) with V = sin 2πx, b = cos²(2πx): a classical solution (u, m, H̄) exists with current j ∈ [0.3461135133, 0.3461135308] and H̄ ∈ [0.1711645503, 0.1711645816], locally unique in that box, density in [0.429153308, 2.316364134] > 0",
    "verdict": "PROVED",
    "evidence": {
     "j": [
      0.3461135133116429,
      0.3461135307748013
     ],
     "H": [
      0.17116455025732513,
      0.1711645816326927
     ],
     "jWidth": 1.7463158363817403e-8,
     "HWidth": 3.137536758179849e-8,
     "mMin": 0.42915330799471596,
     "mMax": 2.316364134313385,
     "krawczykRounds": 1,
     "thinCells": 16384,
     "jacobianCells": 2048
    },
    "assumes": [
     "global uniqueness of the solution: AFG Lemma 2.3 (the operator is monotone in L²×L²) with Lasry–Lions; cited, not re-proved here — what is proved is uniqueness of (j, H̄) in the box",
     "classical solutions of (1.1) carry a constant current m(u_x + b) = j — the one-dimensional current formulation (Gomes–Nurbekyan–Prazeres), which is an identity for C¹ solutions on the torus",
     "∫ b is entered exactly (0 for b = 0; 1/2 for cos² 2πx)"
    ],
    "falsifier": [
     "the quadrature with its remainder term deleted (the interval then misses a value it must contain)",
     "the scalar inverse with the sign check disabled (a bracket that does not bracket is refused)",
     "Krawczyk started from a candidate 0.05 away from the root (must return no contraction)",
     "case A: the closed-form H̄ shifted by 1e-7 must lie OUTSIDE the enclosure",
     "a certified density floor forged to ≤ 0 must refuse"
    ],
    "provenance": {
     "code": "instruments/afg/afg.js",
     "sha256": "ce5f0d0e5a18b1053055cb45cc1c2fbc9fb106b0f760bea098218ebeb0eb07bb"
    },
    "why": ""
   }
  }
 },
 "provenance": {
  "code": "instruments/afg/afg.js",
  "sha256": "ce5f0d0e5a18b1053055cb45cc1c2fbc9fb106b0f760bea098218ebeb0eb07bb",
  "quadrature": "instruments/interval/quadrature.js",
  "quadratureSha256": "2b2f723e6dce48b7ba136dd580597f59910c8756f48927dca6c006f039a3b577",
  "paper": "Noha Almulla, Rita Ferreira, Diogo Gomes, \"Two Numerical Approaches to Stationary Mean-Field Games\", Dynamic Games and Applications 7(4) 657–682 (2017), DOI 10.1007/s13235-016-0203-5, arXiv:1511.06576 (v1, 20 Nov 2015)",
  "reduction": "the one-dimensional current formulation: Gomes, Nurbekyan, Prazeres, \"One-dimensional stationary mean-field games with local coupling\", Dynamic Games and Applications 8 (2018), arXiv:1611.08161; and \"Explicit solutions of one-dimensional, first-order, stationary mean-field games with congestion\", CDC 2016"
 },
 "generatedBy": "node instruments/afg/run.js"
}