A starter page, opened 2026-07-30, holding one conjecture together with what is already named, what is genuinely open, and where the conjecture is false. No result, no rung, no occupancy verdict. It exists so the idea can be worked from a corrected position.
For a broad class of multi-population mean-field games, congestion games, market-clearing systems and adaptive network dynamics, the existence and stability of equilibrium are equivalent to the existence of a Riemannian or Hessian-Riemannian geometry in which the collective dynamics become dissipative:
| ż̇ = −gradg E(z) | z is the state of the collective; E is not an obvious energy; g is not Euclidean; and the correct geometry may be hidden in the interaction mechanism itself |
| equilibrium ≃ zero of a geometric gradient | critical points of E |
| stability ≃ curvature | under generalized convexity of E, unique and globally attracting |
Each component of the conjecture corresponds to an established body of work. This is recorded first, and in full, because the alternative is discovering it in a referee report.
| the claim | the existing name |
|---|---|
| population dynamics as gradient flow in a non-Euclidean metric | Otto calculus / JKO — Fokker–Planck is the Wasserstein-2 gradient flow of relative entropy |
| Hessian-Riemannian geometry for Wardrop and congestion | the paper this lab already reproduced. The MWD module is this conjecture, implemented and certified |
| equilibria are critical points of a potential E | potential games — that is the definition, not a consequence |
| replicator dynamics as a gradient flow | the Shahshahani metric, evolutionary game theory |
| Hessian geometry generated by the interaction mechanism | mirror descent / Bregman geometry |
| generalized convexity of E ⇒ unique, globally attracting | Lasry–Lions monotonicity. This lab already calls itself the monotonicity school |
That the intuition reconstructed this independently is a real signal. It is not an open problem. No occupancy verdict is asserted here — no gate has been run, and none of the above may enter a shipping document until each reference is fetched at source.
It is the conjecture's own parenthetical: “possibly after discovering the correct metric”, and “the geometry may be hidden in the interaction mechanism itself.”
For specific systems the metric is known — Wasserstein for Fokker–Planck, Shahshahani for replicator, the Hessian of h = Σ ϑ log ϑ for HRF. Every one was found by hand. There is no general theory that takes an interaction mechanism as input and derives the metric. That gap is the interesting half.
The conjecture claims equivalence. The direction geometry ⇒ equilibrium is standard. The direction equilibrium ⇒ geometry fails, and two modules in this lab already exhibit the failure:
| module | measured property | why it refutes the equivalence |
|---|---|---|
| MST | the explicit toggle stalls by design — a skew-dominated Jacobian | a skew-symmetric part is precisely the obstruction to gradient structure. The system has equilibria and admits no E whose gradient generates it |
| M2D | anti-monotone herding, with a pitchfork traced by repeated solves | multiple equilibria are incompatible with critical points of a generalized-convex E that is unique and globally attracting |
Both are deliberately exposed failure modes of this lab, not bugs. They were built to be honest about where the method stops, and they turn out to bound the conjecture.
The productive question is not when does the geometry exist — that is answered, under monotonicity. It is:
When the geometry does NOT exist, what obstructs it, and can the obstruction be CERTIFIED?
The skew part of the Jacobian is a measurable obstruction. A machine-checked certificate that this system admits no gradient structure would be a new class of certificate — the certify-the-negative direction — and MST is a ready-made first instance.
And the companion, which is the studio's actual opening:
Certify the hypotheses, not the theorem. The theory assumes generalized monotonicity. Almost nobody verifies it rigorously for a given instance. The theorem is taken; the certified instance is empty, and that is what the field lacks.
| a literature gate, first | on the metric-derivation question and on certified obstruction separately. The occupancy of the main conjecture is stated above as a prior, not a verdict. |
| an obstruction has to be decidable | “skew-dominated” must become a quantity with a rigorous bound, not an observation. Until it is enclosed it is a description of a solver's behaviour. |
| the negative needs its own red control | a certificate of non-existence that cannot go red on a system which does admit a gradient structure is not a certificate. |
| never say “equivalent” | this page's own refutation section is the reason. One direction holds; the sentence that claims both is false and this lab measured it. |
Closing note, kept because it is right and because its provenance matters: “the future of computation may not be about simulating millions of agents — it may be about discovering the geometry that makes the millions unnecessary.” That is exactly correct, and it is the founding motivation of mean-field games: replace N agents with a PDE. The instinct is pointed at the right thing; the frontier is one layer further in.