{ "what": "the p-Laplacian regularisation of Ferreira–Gomes–Üçer (arXiv:2506.21212, operator (3.1)) on the one-dimensional stationary first-order MFG with H = |p|²/2 − m and V = A cos 2πx: every cell of a 10 × 5 atlas in (A, ε) certified as a unique classical even solution with positive density by a radii polynomial in a derivative-weighted ℓ¹ space, or refused by name; the distance between the regularised and the unregularised solution decided as a bracket", "statement": "34 of 50 cells PROVED (16 refused: 15 by Z1 ≥ 1, 1 by Z1 near 1, no radius closes). The unregularised game (ε = 0) certifies as readily as the regularised one: after eliminating m the linearisation is −(c h′)′ + e h with c = m + (1 + 3ε)u′² ≥ m > 0, elliptic wherever the density is positive, so the certifier needs no ε. Along A at fixed ε every row is PROVED then REFUSED; the frontier is ε = 0: proved to A = 0.7, refused from 0.8; ε = 0.01: proved to A = 0.7, refused from 0.8; ε = 0.1: proved to A = 0.6, refused from 0.7; ε = 0.3: proved to A = 0.5, refused from 0.6; ε = 1: proved to A = 0.4, refused from 0.5. Where both cells are proved, ‖u_ε − u_0‖_2 is bracketed to the sum of the two radii; at A = 0 it equals ε exactly (u ≡ 1 − ε).", "verdict": "PROVED (cell by cell)", "amps": [ 0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9 ], "eps": [ 0, 0.01, 0.1, 0.3, 1 ], "nu": 1.05, "cells": [ { "A": 0, "eps": 0, "N": 24, "ok": true, "verdict": "PROVED", "why": null, "r": 9.32587340686168e-16, "kappa": 2.2691748667015117e-12, "Y0": 8.881784197001452e-16, "Z1": 1.5321077739827826e-14, "Z1tail": 9.725e-320, "Z1worst": 82, "Anorm": 1.000000000000006, "Z2": 2416.775020024794, "c0": [ 0.9999999999999997, 1.0000000000000007 ], "e0": [ 0.9999999999999999, 1.0000000000000002 ], "mMin": 0.9999999999999944, "mLow": 0.9999999999999936, "mMinFloat": 1, "newton": { "res": 0, "it": 0 }, "u": [ 1, 0, 0, 0, 0, 0 ], "uAt0": 1, "uAtHalf": 1, "mAt0": 1, "mAtHalf": 1 }, { "A": 0, "eps": 0.01, "N": 24, "ok": true, "verdict": "PROVED", "why": null, "r": 9.059346166839256e-16, "kappa": 2.1829707313303045e-12, "Y0": 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approximate inverse is not one; no contraction at any radius", "r": null, "kappa": null, "Y0": 0.0858167045748274, "Z1": 2.03080939495526, "Z1tail": 1.1962148965252273, "Z1worst": 33, "Anorm": 4.293569389693662, "Z2": null, "c0": [ 0.786566734177116, 0.786566734177118 ], "e0": [ 2.375645918433778, 2.375645918433791 ], "mMin": null, "mLow": null, "mMinFloat": 0.02053689998868813, "newton": { "res": 2.220446049250313e-16, "it": 0 }, "u": [ 0.6768252487203805, 0.014466723854798271, 0.0039956334610076714, 0.0014787123464736261, 0.0006667237809111398, 0.00035946656559899593 ], "uAt0": 0.7205368999886881, "uAtHalf": 0.6538088840169101, "mAt0": 0.02053689998868813, "mAtHalf": 1.35380888401691 }, { "A": 0.8, "eps": 0, "N": 40, "ok": false, "verdict": "REFUSED", "why": "Z1 >= 1 — the approximate inverse is not one; no contraction at any radius", "r": null, "kappa": null, "Y0": 6.527320023624313e-8, "Z1": 1.2190135729111287, "Z1tail": 0.8070854639908027, "Z1worst": 41, "Anorm": 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approximate inverse is not one; no contraction at any radius", "r": null, "kappa": null, "Y0": 1.6856885076696664, "Z1": 2.872368057743764, "Z1tail": 2.1411243121461703, "Z1worst": 41, "Anorm": 6.6181040358065175, "Z2": null, "c0": [ 0.8478343975228627, 0.8478343975228665 ], "e0": [ 2.363297713926717, 2.363297713926733 ], "mMin": null, "mLow": null, "mMinFloat": -0.07056094381501676, "newton": { "res": 1.2620113287731272e-16, "it": 0 }, "u": [ 0.6736314738922943, 0.017105657382153465, 0.005285200761932041, 0.0021027350180245373, 0.001000326276415538, 0.0005692525095225768 ], "uAt0": 0.7294390561849833, "uAtHalf": 0.647088506075117, "mAt0": -0.07056094381501676, "mAtHalf": 1.447088506075117 }, { "A": 0.9, "eps": 0, "N": 40, "ok": false, "verdict": "REFUSED", "why": "Z1 >= 1 — the approximate inverse is not one; no contraction at any radius", "r": null, "kappa": null, "Y0": 0.00010262434740001185, "Z1": 1.642382843976409, "Z1tail": 0.9242986294749932, "Z1worst": 41, "Anorm": 3.850706461639599, "Z2": null, "c0": [ 1.023269615823252, 1.0232696158232537 ], "e0": [ 0.9999999999999999, 1.0000000000000002 ], "mMin": null, "mLow": null, "mMinFloat": 0.13023786950005978, "newton": { "res": 5.551115123125783e-17, "it": 0 }, "u": [ 0.9883651920883738, 0.014287522718009172, 0.003953882430488302, 0.0014474026000548153, 0.0006060246088009207, 0.0002813300338383897 ], "uAt0": 1.0302378695000598, "uAtHalf": 0.9656387630528884, "mAt0": 0.13023786950005978, "mAtHalf": 1.8656387630528886 }, { "A": 0.9, "eps": 0.01, "N": 40, "ok": false, "verdict": "REFUSED", "why": "Z1 >= 1 — the approximate inverse is not one; no contraction at any radius", "r": null, "kappa": null, "Y0": 0.00019672185582680937, "Z1": 1.6887868024087969, "Z1tail": 0.9364164833396151, "Z1worst": 41, "Anorm": 4.198690288544842, "Z2": null, "c0": [ 1.015525733645986, 1.0155257336459875 ], "e0": [ 1.028739095383608, 1.0287390953836086 ], "mMin": null, "mLow": null, "mMinFloat": 0.1212880972430802, "newton": { 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