{
 "what": "the one-dimensional first-order MFG with mixed boundary conditions of Alharbi–Ashrafyan–Gomes (AMO 93:40, 2026; arXiv:2305.15952) §3.2: the empty region decided cell by cell with refusal at the free boundary, the two value functions enclosed, the boundary complementarity evaluated; case 2 certified over a γ box",
 "statement": "AAG §3.2, V = γ + ½ sin(3π(x + ¼)): CASE 1 (j₀ = 0, γ = 0) the density vanishes exactly on the empty region {V < 0} — on 2 runs decided EMPTY, 2 runs decided OCCUPIED with m = V enclosed, and 4 of 256 cells REFUSED, each containing one of the three exact roots 1/12, 5/12, 3/4; the two value functions u± are enclosed and differ, so u is not unique off the support; x = 1 is in the contact set with exit flux 0. CASE 2 (j₀ = 1/10, every γ in [-0.5,-0.3]): m > 0.06840945657 on every cell, u enclosed, x = 1 in the contact set with exit flux −j₀ exactly",
 "verdict": "VERIFIED",
 "options": {
  "K": 256,
  "ladder": [
   16,
   64,
   256,
   1024
  ],
  "gamma": [
   -0.5,
   -0.3
  ]
 },
 "roots": [
  0.08333333333333333,
  0.4166666666666667,
  0.75
 ],
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    "x1": 0.08203125,
    "standing": "OCCUPIED"
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    "x1": 0.0859375,
    "standing": "REFUSED"
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    "x1": 0.4140625,
    "standing": "EMPTY"
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    "x0": 0.4140625,
    "x1": 0.41796875,
    "standing": "REFUSED"
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    "standing": "OCCUPIED"
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    "standing": "REFUSED"
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   {
    "x0": 0.75390625,
    "x1": 1,
    "standing": "EMPTY"
   }
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  "refused": 4,
  "refusedLength": 0.015625,
  "refusedOK": true,
  "rootsOK": true,
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    0.004739223825020247,
    0.0053539493305148405
   ],
   [
    1,
    0,
    0
   ]
  ],
  "boundary": {
   "inflow": {
    "value": [
     0.09999999999999999,
     0.10000000000000002
    ],
    "required": [
     0.09999999999999999,
     0.10000000000000002
    ],
    "ok": true,
    "words": "m u_x ≡ −j₀ by the current, so −m(0)u_x(0) = j₀ identically"
   },
   "relaxed": {
    "u1": [
     0,
     0
    ],
    "psi": [
     0,
     0
    ],
    "ok": true
   },
   "noEntry": {
    "flux": [
     -0.10000000000000002,
     -0.09999999999999999
    ],
    "ok": true
   },
   "contact": {
    "product": [
     -5e-324,
     5e-324
    ],
    "ok": true,
    "inContactSet": true,
    "exit": true
   },
   "words": "x = 1 is in the contact set (u(1) = ψ = 0) and carries exit flux −j₀: contact with exit"
  }
 },
 "certificate": {
  "claim": "AAG §3.2, V = γ + ½ sin(3π(x + ¼)): CASE 1 (j₀ = 0, γ = 0) the density vanishes exactly on the empty region {V < 0} — on 2 runs decided EMPTY, 2 runs decided OCCUPIED with m = V enclosed, and 4 of 256 cells REFUSED, each containing one of the three exact roots 1/12, 5/12, 3/4; the two value functions u± are enclosed and differ, so u is not unique off the support; x = 1 is in the contact set with exit flux 0. CASE 2 (j₀ = 1/10, every γ in [-0.5,-0.3]): m > 0.06840945657 on every cell, u enclosed, x = 1 in the contact set with exit flux −j₀ exactly",
  "verdict": "PROVED",
  "evidence": {
   "K": 256,
   "refusedCells": 4,
   "refusedLength": 0.015625,
   "roots": [
    0.08333333333333333,
    0.4166666666666667,
    0.75
   ],
   "case2DensityFloor": 0.06840945657036854,
   "case2Gamma": [
    -0.5,
    -0.3
   ]
  },
  "assumes": [
   "the explicit solutions are the paper's own (§3.2, Cases 1 and 2); what is certified is that they satisfy Definition 2.11 and (3.6) cell by cell, in intervals, and where the density vanishes",
   "γ in Figure 2 is not printed; case 2 is certified over the box [-0.5,-0.3] that contains the value read off the figure",
   "the value functions are Riemann brackets, O(h) wide: rigorous, and wider than the density's"
  ],
  "falsifier": [
   "the float rule (sign at the cell centre) paints every root cell — the interval rule must REFUSE those same cells",
   "a root removed from the exact list makes a refused cell contain no root: the consistency check must fail",
   "the exit cost ψ raised to 0.1 with j₀ = 0.1: contact is lost and the complementarity product becomes 0.1 × (−0.1) ≠ 0 — must refuse",
   "the cubic bracket shifted off the root fails the sign check",
   "the cubic with j₀ forged to 0 at a cell where V < 0 has no positive root: the bracket must be refused"
  ],
  "provenance": {
   "code": "instruments/aag/aag.js",
   "sha256": "0448eb66afae8091995d219536dbe25b9773b5faa304724767a7308dce010df0"
  },
  "why": ""
 },
 "provenance": {
  "code": "instruments/aag/aag.js",
  "sha256": "0448eb66afae8091995d219536dbe25b9773b5faa304724767a7308dce010df0",
  "paper": "AbdulRahman M. Alharbi, Yuri Ashrafyan, Diogo Gomes, \"A First-Order Mean-Field Game on a Bounded Domain with Mixed Boundary Conditions\", Applied Mathematics & Optimization 93, Art. 40 (2026); arXiv:2305.15952v4 (2 Mar 2026). System (3.5), boundary conditions (3.6), Cases 1 and 2 of §3.2, Figures 1 and 2, Definition 2.11, Theorem 1.3.",
  "gammaNote": "γ of Figure 2 is not printed in the text; read from the figure as about −0.4; case 2 is certified for every γ in the recorded box"
 },
 "generatedBy": "node instruments/aag/run.js"
}