The regularization, measured

The paper's Problem 1 on the torus, with the power-growth Hamiltonian H = |p|²/2 − m and a discount, reads

Its regularized operator (3.1) adds, in the transport slot, ε times the γ̄-Laplacian of u and the γ̄-power of u with γ̄ = α(β + 1)/β; for α = 2, β = 1 that is γ̄ = 4, so the added terms are ε(u′³)′ and ε u³ — polynomial. The first equation gives m outright, and the second becomes one scalar equation for u:

Its linearization at ū is a Sturm–Liouville operator, −(c h′)′ + e h, with c = m̄ + (1 + 3ε)ū′² and e = 1 − ū″ + 3εū². The coefficient c is at least the density. Wherever m > 0 the operator is elliptic — at ε = 0 as at ε > 0 — and that is the whole reason the page can certify the unregularized game with the same instrument: the first-order game, after eliminating its density, is a second-order elliptic equation in disguise. The paper's ε is what makes the abstract operator coercive on the Banach space where existence is proved; it is not what makes the equation solvable.

For every amplitude A from 0 to 0.9 and every ε in {0, 0.01, 0.1, 0.3, 1}, a Galerkin candidate with N = 24 to 40 cosine modes is found by Newton continuation from the constant solution, and a radii polynomial in the space X_2 = {u : Σ|u_k|(1 + k)²ν^k < ∞}, ν = 1.05, is asked to close a ball about it. The approximate inverse is the dense inverse of the Galerkin Jacobian on the first N modes and the diagonal 1/(c₀(2πk)² + e₀) beyond; Z1 is computed column by column to six times N and bounded analytically past that; Z2 comes from the algebra property of the weighted norm. A zero of F in X_2 is twice differentiable with an analytic Fourier series — a classical solution — and the density is certified positive on the whole ball, so it is a strong solution in the paper's sense with m bounded away from zero.

Every refusal is the same refusal: Z1, the norm of I − A·DF(ū), reaches 1. It is the tail's doing. A diagonal approximate inverse beyond N is right when the principal coefficient c is nearly constant and wrong in proportion to its variation, and c = m + (1 + 3ε)u′² varies more as A grows (m follows −V) and as ε grows (the 4-Laplacian weights u′² three times more). So the frontier moves left with ε: the regularization that gives the paper its coercivity takes contraction from the certificate. The density is nowhere near zero at the frontier — the certified floor at the last proved cell of ε = 0 is 0.320, and the float candidate at the first refused cell still has min m = 0.225. The void of this atlas is the certificate's boundary, not the equation's; the two would only coincide where m reaches zero, which is where the paper's strong solutions stop being classical, and that is beyond the map.

Where a regularized cell and its unregularized neighbour are both proved, the distance ‖u_ε − u_0‖_2 is decided: the norm of the difference of the two candidates, exactly, plus or minus the two radii. At A = 0 the solutions are constants — u_ε is the real root of εu³ + u = 1 — and the distance is 1 − u_ε, which the battery holds against the cubic. At every amplitude the distance grows with ε, equals about ε at ε = 0.01, and is a third of ε at ε = 1: the regularized game is a first-order perturbation of the unregularized one in this norm, and the perturbation saturates because the cubic term is bounded. The paper passes ε → 0 by compactness; here the limit is watched.

• Proved, cell by cell. A unique zero of F_ε in a ball of X_2 about the candidate, in the even subspace; classical; density positive on the ball. Each certificate carries five falsifiers, and the battery fires them.
• Decided. The distances between doubly-proved cells; the ellipticity of the linearization (c₀ > 0, e₀ ≥ 0) on every proved cell; the constant solution's cubic at A = 0.
• Not claimed. Uniqueness outside the ball or outside the even subspace; anything on a refused cell — existence there is neither asserted nor denied; the paper's d-dimensional theorems; the singular-congestion and weak-growth cases (Theorems 1.5, 1.6). The one-dimensional quadratic game with a single cosine mode is the instance, and it is the instance because its regularization is polynomial.
• The frontier is the certificate's. Every refusal is Z1 ≥ 1 from a diagonal tail against a varying coefficient. A banded tail inverse would move it; the density would not. The page says so rather than drawing a boundary it does not have.
• A new kernel. The radii-polynomial bounds for this quasilinear problem are written out in kernel.js and are this machine's, not lifted. They are checked here by a finite-difference solve on a different discretization and by the constant solution; they have not been reviewed by anyone else.

node instruments/regatlas/battery.js     # 17 checks, 5 red controls: the atlas re-derived, the cubic at A = 0, finite differences, the frontier
node instruments/regatlas/run.js --check  # the record, re-derived and compared byte for byte

The kernel is instruments/regatlas/kernel.js, with derive.js; the record is certs/regatlas.json.

• R. Ferreira, D. A. Gomes, M. Üçer, Solving mean-field games with monotonicity methods in Banach spaces, arXiv:2506.21212v3 (2026) — Problem 1, Assumption 2.4 (power growth), the regularized operator (3.1), Proposition 3.1 (coercivity), Theorem 1.4.
• J. B. van den Berg, J.-P. Lessard, Rigorous numerics in dynamics, Notices AMS 62 (2015); S. Day, J.-P. Lessard, K. Mischaikow, SIAM J. Numer. Anal. 45 (2007) — the radii polynomial; the derivative-weighted ℓ¹ space is the standard device for quasilinear problems.
• The congestion cap, the concentration frontier — the machine's earlier radii polynomials for the viscous game, whose lifted kernel this one does not use.
• The first-order game by its current — the same first-order structure without a discount, enclosed through its current instead.