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Hamilton–Jacobi conjecture for the bipartite enriched free energy

From papers

Let Q\mathcal Q be the space of paths used to parameterize overlaps, let q=(q1,q2)∈Q2q=(q_1,q_2)\in\mathcal Q^2, and let ψ2:Q2→R\psi_2:\mathcal Q^2\to\mathbb R be the initial free-energy function defined by ψ2(q)=λ1ψ1(q1)+λ2ψ1(q2)\psi_2(q)=\lambda_1\psi_1(q_1)+\lambda_2\psi_1(q_2). For each NN, let FN:R+×Q2→RF_N:\mathbb R_+\times\mathcal Q^2\to\mathbb R be the enriched free energy of the bipartite model. Bipartite enriched free-energy conjecture. The functions FNF_N converge to the function f:R+×Q2→Rf:\mathbb R_+\times\mathcal Q^2\to\mathbb R solving

{∂tf−∫01∂q1f ∂q2f=0on R+×Q2,f(0,⋅)=ψ2on Q2.\begin{cases} \partial_t f-\displaystyle\int_0^1\partial_{q_1}f\,\partial_{q_2}f=0 & \text{on }\mathbb R_+\times\mathcal Q^2,\\ f(0,\cdot)=\psi_2 & \text{on }\mathcal Q^2. \end{cases}

This is the proposed Hamilton–Jacobi description of the limiting free energy for the bipartite spin-glass model. The source does not provide evidence that the conjecture has been proved or disproved.

Progress summary

Open

The conjecture remains unproved: researchers have established only one-sided information about the proposed limit, not convergence to it.

Formulated in the 2021 work on nonconvex interactions, the conjecture asserts that the bipartite enriched free energies converge to the solution of the proposed infinite-dimensional Hamilton–Jacobi equation. The nonconvex interaction is identified as the central obstruction.

Known results

  • Any subsequential limit satisfies the Hamilton–Jacobi equation almost everywhere.
  • The viscosity solution supplies a one-sided bound, including lim inf⁡N→∞FN(t,q)≥f(t,q)\liminf_{N\to\infty}F_N(t,q)\ge f(t,q).
  • The nonlinearity (x,y)↦xy(x,y)\mapsto xy is neither convex nor concave, so standard variational methods do not apply.

October 2024 status update

A later treatment states explicitly that Conjecture 4.24.2 is unresolved: the matching upper bound and hence convergence to the viscosity solution remain missing. No proof, disproof, counterexample, or claimed AI solution was found.

Current status (as of August 2026): The one-sided bound and almost-everywhere subsequential-limit statement are known, but convergence of FNF_N to the viscosity solution remains open.

Sources
Sources & referencesView supporting material

Primary source

Jean-Christophe Mourrat, “Spin glasses and the Parisi formula”, arXiv:2510.01054 (2025).

free-energyhamilton-jacobi-equationsspin-glasses

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