{
 "what": "the clearing price of the Gomes–Saúde price-formation model (arXiv:1807.07088, §6 linear-quadratic), closed in exact rationals on the source lab's supply scenario, with a ±15 % forecast box on the supply turned into a proved price band; and the lab's finite-difference kernel measured against the exact model",
 "statement": "With Π = ∫u_x m and Ξ = ∫x m the paper's averaged dynamics Ξ̇ = Q, Π̇ = −η(Ξ − κ) close under quadratic terminal data through Π(T) = γ(Ξ(T) − ζ), so ϖ(t) = −cQ(t) − γ(Ξ(T) − ζ) − η∫ₜᵀ(Ξ − κ): the price is affine in the supply path with every coefficient ≤ 0. Over lo ≤ Q ≤ hi the price band is price(hi) ≤ ϖ ≤ price(lo), attained at both edges, decided exactly. On the lab's scenario (η = 0) Θ = -298385140072281459/180143985094819840 ≈ -1.656370 and the widest band is 2.774661 at grid time 120. The lab's kernel clears its market in the agents' controls to 1e−14 while its fleet ends the day at Ξ(T) = 0.90318 against the 0.98803 that conservation requires: 0.0848 of the day's 0.6861 vanishes at the wall. Forbidding the wall-pointing control halves the loss; clearing on the Fokker–Planck flux removes it (Ξ(T) matches to 1e−14) and, on a domain where the walls do not bind, the same scheme approaches the closed form at first order in the mesh (error ratio 1.88 for H halved), while the lab's wall treatment does not converge to it (ratio 0.72).",
 "verdict": "PROVED",
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  "aw": 0.7,
  "adeq": 1.25,
  "sig": 0.1
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 "model": {
  "c": "1",
  "gamma": "12",
  "zeta": "17/20",
  "T": "1",
  "N": 240
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 "wide": {
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  "nu": 0.005
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 "lab": {
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  "res": 8.03704297558383e-10,
  "converged": true,
  "NX": 120,
  "NT": 240,
  "nu": 0.005000000000000001
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     "exact": "298385140072281459/180143985094819840",
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    "certificate": {
     "claim": "Gomes–Saúde LQ price formation with c = 1, γ = 12, ζ = 17/20, η = 0, x̄₀ ≈ 0.301935, N = 240: for every supply path in the box lo ≤ Q ≤ hi the clearing price lies in the corner band price(hi) ≤ ϖ ≤ price(lo) at every grid time, both edges attained",
     "verdict": "PROVED",
     "evidence": {
      "gridTimes": 241,
      "maxWidth": 2.7746610173104944,
      "maxWidthAt": 120,
      "energyLo": 0.5831817771334623,
      "energyHi": 0.7890106396511548,
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     },
     "assumes": [
      "the model is the paper's (33) on the whole line with quadratic terminal data; the price is the multiplier of the clearing identity (Problem 1)",
      "the closed form: Ξ̇ = Q and Π̇ = −η(Ξ − κ) (the paper's §6.2 averaged dynamics, an identity for C² solutions with decaying m) closed by Π(T) = γ(Ξ(T) − ζ)",
      "Q is a step function on the grid: Q_n rules [t_n, t_{n+1}); the price at t_n is its right limit",
      "first order (ε = 0); the box-constrained battery of the lab kernel is a different model and is measured, not enclosed"
     ],
     "falsifier": [
      "η < 0: a sensitivity ∂ϖ_n/∂Q_k turns positive and a path inside the box prices outside the corner band (the battery shows it)",
      "a supply path inside the box whose exact price leaves the band at any grid time",
      "η = 0: Π_n ≠ −Θ at some n, Θ the paper's (30) — the closed form would not be the paper's line",
      "the Riccati route (u = a x² + b x + e, float ODE) disagreeing with the closed form beyond 1e−9",
      "the wide-domain finite-difference solve of the same system disagreeing beyond first order in the mesh"
     ],
     "provenance": {
      "code": "instruments/price/price.js"
     },
     "why": ""
    }
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    "maxWidth": {
     "exact": "6971753714861787212253/2305843009213693952000",
     "value": 3.0235162094748143,
     "at": 106,
     "parts": {
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      "day": 2.469946350212311,
      "set": 0.25626082285171736
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    },
    "widthAt0": {
     "value": 2.8940326566143106,
     "parts": {
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    },
    "Pi0": {
     "exact": "202214989222702261941/115292150460684697600",
     "value": 1.7539354449950932
    },
    "PiT": {
     "exact": "298385140072281459/180143985094819840",
     "value": 1.6563702635712467
    },
    "certificate": {
     "claim": "Gomes–Saúde LQ price formation with c = 1, γ = 12, ζ = 17/20, η = 3, κ = 3/5, x̄₀ ≈ 0.301935, N = 240: for every supply path in the box lo ≤ Q ≤ hi the clearing price lies in the corner band price(hi) ≤ ϖ ≤ price(lo) at every grid time, both edges attained",
     "verdict": "PROVED",
     "evidence": {
      "gridTimes": 241,
      "maxWidth": 3.0235162094748143,
      "maxWidthAt": 106,
      "energyLo": 0.5831817771334623,
      "energyHi": 0.7890106396511548,
      "maxWidthExact": "6971753714861787212253/2305843009213693952000"
     },
     "assumes": [
      "the model is the paper's (33) on the whole line with quadratic terminal data; the price is the multiplier of the clearing identity (Problem 1)",
      "the closed form: Ξ̇ = Q and Π̇ = −η(Ξ − κ) (the paper's §6.2 averaged dynamics, an identity for C² solutions with decaying m) closed by Π(T) = γ(Ξ(T) − ζ)",
      "Q is a step function on the grid: Q_n rules [t_n, t_{n+1}); the price at t_n is its right limit",
      "first order (ε = 0); the box-constrained battery of the lab kernel is a different model and is measured, not enclosed"
     ],
     "falsifier": [
      "η < 0: a sensitivity ∂ϖ_n/∂Q_k turns positive and a path inside the box prices outside the corner band (the battery shows it)",
      "a supply path inside the box whose exact price leaves the band at any grid time",
      "η = 0: Π_n ≠ −Θ at some n, Θ the paper's (30) — the closed form would not be the paper's line",
      "the Riccati route (u = a x² + b x + e, float ODE) disagreeing with the closed form beyond 1e−9",
      "the wide-domain finite-difference solve of the same system disagreeing beyond first order in the mesh"
     ],
     "provenance": {
      "code": "instruments/price/price.js"
     },
     "why": ""
    }
   },
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    "eta": "12",
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     "value": 3.8291664000145507,
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     "parts": {
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     "exact": "58990121988007161621/28823037615171174400",
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     "exact": "298385140072281459/180143985094819840",
     "value": 1.6563702635712467
    },
    "certificate": {
     "claim": "Gomes–Saúde LQ price formation with c = 1, γ = 12, ζ = 17/20, η = 12, κ = 3/5, x̄₀ ≈ 0.301935, N = 240: for every supply path in the box lo ≤ Q ≤ hi the clearing price lies in the corner band price(hi) ≤ ϖ ≤ price(lo) at every grid time, both edges attained",
     "verdict": "PROVED",
     "evidence": {
      "gridTimes": 241,
      "maxWidth": 3.8291664000145507,
      "maxWidthAt": 80,
      "energyLo": 0.5831817771334623,
      "energyHi": 0.7890106396511548,
      "maxWidthExact": "2207364143647379565029/576460752303423488000"
     },
     "assumes": [
      "the model is the paper's (33) on the whole line with quadratic terminal data; the price is the multiplier of the clearing identity (Problem 1)",
      "the closed form: Ξ̇ = Q and Π̇ = −η(Ξ − κ) (the paper's §6.2 averaged dynamics, an identity for C² solutions with decaying m) closed by Π(T) = γ(Ξ(T) − ζ)",
      "Q is a step function on the grid: Q_n rules [t_n, t_{n+1}); the price at t_n is its right limit",
      "first order (ε = 0); the box-constrained battery of the lab kernel is a different model and is measured, not enclosed"
     ],
     "falsifier": [
      "η < 0: a sensitivity ∂ϖ_n/∂Q_k turns positive and a path inside the box prices outside the corner band (the battery shows it)",
      "a supply path inside the box whose exact price leaves the band at any grid time",
      "η = 0: Π_n ≠ −Θ at some n, Θ the paper's (30) — the closed form would not be the paper's line",
      "the Riccati route (u = a x² + b x + e, float ODE) disagreeing with the closed form beyond 1e−9",
      "the wide-domain finite-difference solve of the same system disagreeing beyond first order in the mesh"
     ],
     "provenance": {
      "code": "instruments/price/price.js"
     },
     "why": ""
    }
   }
  ]
 },
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   "unreachable": 0,
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 "provenance": {
  "code": "instruments/price/price.js",
  "sha256": "15852fabc87149cf620112c3dd611d47c70152fac8246b27a10b4e0337f16598",
  "fd": "instruments/price/fd.js",
  "fdSha256": "a4b9094cf1916aeea6261f34e7e4046ee7643ad35218310cb6b9586bbcd10bc6",
  "derive": "instruments/price/derive.js",
  "deriveSha256": "031f03e7f51b32abb7e4ee64254a7fb6d6a8ba78073df25e8639d270b4895e8f",
  "labKernel": "sin-mfg research/mfg-lab/mfg-lab.html, module MPR, kernel sha256 65c8909331322f85… in artifact sha256 37c0e3bd640ae4c7… (274157 bytes) — read-only; the port in fd.js reproduced its price to the last bit on 2026-09-07 (49 iterations, residual 8.037e−10), see FINDINGS_LIT.md",
  "paper": "D. A. Gomes, J. Saúde, A mean-field game approach to price formation in electricity markets, arXiv:1807.07088 (2018); Dyn. Games Appl. 11 (2021)"
 },
 "generatedBy": "node instruments/price/run.js"
}