{ "what": "the maximal value function of Gomes–Üçer (Theorem 1.8) on an explicit first-order time-dependent MFG with a vacuum, built here: the parabolic bump of shrinking support on the circle of length 6 with H = ½p² − m", "statement": "On the circle of length 6 with H = ½p² − m and the parabolic bump of shrinking support r(t) ∈ [3/2, 2]: (m, u = aφ + b) is an MFG solution (Definition 1.1) with a vacuum — HJ and transport residuals enclose 0 on every support cell, the strict subsolution inequality holds on every vacuum cell, u(T) = u_T. The maximal value function u* of Theorem 1.8 equals u on the 1842 support cells (by the theorem), equals the Hopf–Lax value on 1150 vacuum cells where every minimising path is proved clear of the support, and is REFUSED on 80 cells; on every decided vacuum cell u* ≥ u, with a gap up to 0.7309849116.", "verdict": "VERIFIED", "instance": { "L": 6, "rho0": 2.25, "r0": 2, "rT": 1.5, "T": [ 1.5621400012944524, 1.5621400012944613 ], "thetaT": "π/3", "H": "½p² − m", "formulas": { "t": "t(θ) = √(2/3)(π − 2θ + sin 2θ)", "r": "r = 2 sin²θ", "a": "a = √(3/2) cos θ/(4 sin³θ)", "A": "A = 3/(8 sin²θ)", "B": "B = 3/(32 sin⁶θ)", "b": "b = √(3/2)(θ − π/3)", "phi": "φ = x² on |x| ≤ 9/4; ρ₀² + 2ρ₀s − ρ₀s²/w on s = |x| − ρ₀ ∈ [0, 3/4]", "uT": "u_T = a(T) φ" } }, "options": { "NT": 32, "NX": 96, "profileRows": [ 0, 8, 16, 24, 31 ] }, "counts": { "SUPPORT": 1842, "DECIDED": 1150, "REFUSED": 80, "tight": 388, "gapProven": 786 }, "pairOK": true, "gapOK": true, "terminalOK": true, "maxGap": 0.7309849116440201, 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is an MFG solution (Definition 1.1) with a vacuum — HJ and transport residuals enclose 0 on every support cell, the strict subsolution inequality holds on every vacuum cell, u(T) = u_T. The maximal value function u* of Theorem 1.8 equals u on the 1842 support cells (by the theorem), equals the Hopf–Lax value on 1150 vacuum cells where every minimising path is proved clear of the support, and is REFUSED on 80 cells; on every decided vacuum cell u* ≥ u, with a gap up to 0.7309849116.", "verdict": "PROVED", "evidence": { "cells": 3072, "support": 1842, "decided": 1150, "refused": 80, "maxGap": 0.7309849116440201, "T": [ 1.5621400012944524, 1.5621400012944613 ] }, "assumes": [ "the maximal subsolution u* of Theorem 1.8 is the value of the control problem inf ∫(½|ẋ|² + m) + u_T(x(T)) — the classical identification for continuous bounded m (Bardi–Capuzzo-Dolcetta), cited, not re-proved", "the instance is built here, not taken from the paper: the paper has no worked example; the derivation of the parabolic bump is on the page and its residuals are enclosed as a check on the algebra", "mass 1 by the algebra ∫m = (4/3) A^{3/2} B^{−1/2} = (4/3)(3/(4r))^{3/2}(3/(4r³))^{−1/2} = 1" ], "falsifier": [ "the HJ residual with the coupling sign flipped (−u_t + ½u_x² + m) must NOT enclose 0 on the support", "the transport residual with the velocity sign flipped must NOT enclose 0", "a vacuum cell whose minimising path crosses the support must be REFUSED, and the float shortcut (the centre path only) must be shown to decide it", "the Hopf–Lax lower bound evaluated with the branch-and-bound pruning disabled to one round must be wider, never tighter", "a decided cell with u* forged below u must fail the gap check" ], "provenance": { "code": "instruments/maxval/maxval.js", "sha256": "09ebda9daab5d8c29e7a4466d41360a585c73ee9b2d9c76c063a6198e0011db2" }, "why": "" }, "provenance": { "code": "instruments/maxval/maxval.js", "sha256": "09ebda9daab5d8c29e7a4466d41360a585c73ee9b2d9c76c063a6198e0011db2", "paper": "Diogo Gomes, Melih Üçer, \"Existence and Structure for First-Order Time-Dependent Mean-Field Games with Local Couplings\", arXiv:2606.28378 (19 Jun 2026): Definition 1.1, Definition 1.7, Theorem 1.8, Corollary 1.9, Example 2.7", "identification": "the maximal subsolution is the viscosity solution of −u_t + ½u_x² = m with u(T) = u_T, the value of inf ∫(½|ẋ|² + m) + u_T(x(T)) — Bardi & Capuzzo-Dolcetta, Optimal Control and Viscosity Solutions of HJB Equations (1997), classical; assumed" }, "generatedBy": "node instruments/maxval/run.js" }