The coupling plane of a two-population mean-field game, cut into 21,567 cells and decided uniformly over each one. The condition everyone cites for uniqueness stops at s = 1. The first certificate proving two equilibria does not appear until s = 11.24 — a factor of 11.2. Everything between is a corridor where one equilibrium is enclosed and nothing rules out a second.
Two populations of agents share a torus. Each solves its own ergodic control problem, and each pays for crowding according to where the other one is:
−σ ui″ + ½ (ui′)² + ρi = Σj cij mj + Vi(x) and −σ mi″ − (mi ui′)′ = 0, with ∫mi = 1, ∫ui = 0, mi > 0.
The coupling matrix C = (c_ij) is the whole story. Lasry–Lions monotonicity — the hypothesis that buys a unique equilibrium — asks that the interaction be monotone, which for a constant matrix is exactly that its symmetric part C + Cᵀ be positive semidefinite. Writing the cross-coupling as c₁₂ = s + d and c₂₁ = s − d, with the self-coupling fixed at c₁₁ = c₂₂ = 1, that condition reads |s| ≤ 1 — and it does not mention d.
That is worth sitting with. The parameter d is precisely the attack–defense asymmetry: one population attracted to where the other is, the other repelled from where the first is. Pursuit and evasion. The standard uniqueness condition is blind to it. This map asks whether it should be.
The monotone region is real and it is certified: 24 cells, covering 5.3% of the rectangle, lie where |s| ≤ 1 over the whole cell and carry an enclosure of the unique equilibrium. Uniqueness there is the cited theorem; the enclosure is ours.
Multiplicity is also real, and it starts a long way away. The first cell carrying two provably disjoint certified balls sits at s = 11.238. Between s = 1 and s = 11.24 — a factor of 11.2 — the map is a corridor of 5,285 cells in which exactly one equilibrium is enclosed and no theorem available to us rules out a second. That corridor is not a gap in the computation. It is the honest width of what is unknown.
Whether d bounds the multiplicity region is NOT settled by this map. The certified multiplicity runs to d = 1.50, which is the top edge of the rectangle we swept, so the boundary visible at the top of the figure is the edge of the picture and nothing more. Locating the fold in d means sweeping further, and until that is done the honest statement is that d does not appear in the cited condition and we have not yet measured whether it should.
| MULTIPLE | 11,628 | 24.3% | two exact equilibria for EVERY parameter in the cell, in provably disjoint balls, both densities positive |
| UNIQUE | 24 | 5.3% | Lasry–Lions monotone over the whole cell [cited] and the enclosure certified here |
| UNDECIDED | 9,915 | 70.4% | 5,285 of them enclose one equilibrium; the reason is kept verbatim on every cell |
The multiplicity witnesses are not marginal. Across every MULTIPLE cell the separation between the two certified balls exceeds the sum of their radii by a factor of at least 246.3 and at most 7958; the worst contraction factor anywhere on the map is 0.9975; and the smallest certified density is 1.72e-1, so positivity — a hypothesis of the model, never an assumption here — is proved and not assumed on every decided cell.
At d = 0 the two populations are interchangeable, and the second equilibrium appears through a symmetry-breaking pitchfork: the Jacobian is singular at the branch point, and any continuation method that walks in s will feel it. Switch d away from zero and the symmetry is gone. The pitchfork unfolds. The primary branch becomes perfectly smooth — no singular Jacobian, no warning, nothing to detect — while the second equilibrium survives on a branch that is no longer connected to it.
That is the part worth stating plainly. In the attack–defense regime, which is the regime with d ≠ 0 by definition, a method that starts from a natural initialisation and descends — a continuation, a homotopy, a trained network — lands on the primary branch and converges beautifully. Its residual goes to zero. Its residual is not lying. It has simply solved for one equilibrium out of two, and nothing in its own output can tell it so.
A box certificate has no such blind spot, because it does not walk anywhere. It takes a candidate and a radius and asks whether the Newton–Kantorovich operator contracts. Two candidates, two radii, one separation bound, and the multiplicity is proved without either branch ever having to be reachable from the other. That is the structural advantage, and it is the reason this lab exists.
The map puts a number on it. 11,463 of the 11,628 MULTIPLE cells — 98.6% of them, covering 23.9% of the swept rectangle — lie entirely at d > 0. In every one of those cells the symmetry is broken throughout, so the primary branch is smooth and a continuation feels nothing; and in every one of them two distinct equilibria are certified for every parameter in the cell. That is the whole argument in one count: the second equilibrium is provably there, and it is provably not where a solver following the obvious branch is going to look.
Every cell on this map is one certificate away from being a mistake, so the gates are built to catch the ways it could be. The two-population certifier is a second implementation of an argument the one-population lab already runs, and at zero cross-coupling the system decouples into two independent copies of that lab. The battery therefore demands that the two agree: the solvers to 1.78e-15 in every coefficient, and the box verdicts cell for cell — 24 agree, 0 disagree, refusals included.
A certificate that claims to hold over a whole rectangle is checked against the rectangle: 4 corners of a cell are independently re-solved and each must land inside the certified enclosure, which they do with a worst excess of 3.49e-6 against a radius of 3.67e-4. The tangent predictor that makes wide cells possible is switched off on demand and the same cell must then fail — it does, with Y₀ rising 178×. The cross-coupling really is charged in the tail bound rather than quietly dropped: a large off-diagonal entry moves Z₁ from 5.576e-2 to 1.020e+0, which is exactly the term that would vanish if the coupling had been omitted.
And the record is not trusted either, though checking it takes more care than it first appears. Finding a branch is search; certifying it is proof, and only the second is what a gate should test — because, as §3 says, a cold seed genuinely cannot find the second equilibrium once d ≠ 0. So the 8 cells drawn deterministically from the map are checked two ways. Every MULTIPLE cell has its segregated branch RE-REACHED at build by continuing in from the pitchfork at d = 0 along a different path than the sweep used, and the whole decision is then recomputed: verdict, radii, positivity, separation. Every UNIQUE cell is re-decided cold with no seed at all. And every sampled cell is re-decided cold as well, where it may return UNDECIDED — the branch went unfound, which is the point — but may never return the other decided verdict. 5 of the 8 reproduced cold outright.
What is NOT claimed: nothing here proves uniqueness outside the monotone region, and the corridor is labelled undecided precisely because we cannot decide it. The map is one two-dimensional slice — σ = 0.5, c₁₁ = c₂₂ = 1, A = 1, 16 Fourier modes — of a seven-parameter family, and the viscosity axis is expensive here for a reason the lab states in the open: the certifier's operator is built at one σ, so a σ-box is charged a defect that does not decay with frequency. Widening that axis is the next piece of work, not a footnote.
Four files, MIT, no dependencies: labs/mfg2p/mfg2p.js is the model and its Newton solver; box2p.js is the certifier, uniform over a rectangle of coupling matrices; regime2p.js walks the plane and refines what refuses; battery.js is the gate that this page re-runs before it will build. The map record itself is certs/mfg2p-regime-map.json, and every number on this page is read from it or recomputed here.
One design decision is worth naming because it is the difference between a sweep that finishes and one that does not. A cell that is undecided because a certificate refused can be repaired by making the cell smaller, so it is refined. A cell that is undecided because one equilibrium is enclosed and monotonicity is unavailable cannot: what is missing there is a theorem, not resolution, and a smaller cell returns the same sentence. Refining those would have spent the entire budget re-proving the corridor at finer and finer scales and called it progress.
The instruments next door: the one-population regime observatory, which this generalises and which the battery checks against; certified multiplicity at a point; a congestion MFG enclosed; and the methods note.
If you work on mean-field games and there is a statement your group keeps defending by hand — a uniqueness regime, a numerical equilibrium, a bound someone doubts — send it. If it is decidable, it gets decided here, with the witness, and the answer is yours whether or not it is the one you wanted.