Released by
Kun Chen, Institute of Theoretical Physics, Chinese Academy of Sciences
Contact email
chenkun0228@gmail.com
Method
Other
Challenge issue
Background
Classical $O(n)$ spin models replace Ising spins by unit vectors. Let $M<\infty$, $n\ge 3$, and
$$\sigma_i\in \mathbb{S}^{n-1}\subset\mathbb{R}^n,$$
with normalized rotation-invariant measure $d\nu_n$. For symmetric ferromagnetic couplings $J_{ij}\ge 0$, $J_{ii}=0$, and at least one $J_{ij}>0$, define
$$d\mathbb{P}_J(\sigma)=Z_J^{-1}
\exp\!\left(\sum_{i\lt j}J_{ij}\,\sigma_i\!\cdot\!\sigma_j\right)
\prod_i d\nu_n(\sigma_i).$$
The invariant dot-product cone consists of finite nonnegative linear combinations
$$f(\sigma)=\sum_{\alpha=1}^{r}c_\alpha
\prod_{i\lt j}(\sigma_i\!\cdot\!\sigma_j)^{a_{ij}^{(\alpha)}},$$
where $c_\alpha\ge0$ and all exponents are nonnegative integers; $g$ has the same form. Nonnegative refers to the coefficients, not to the pointwise sign of each monomial.
The invariant-observable second Griffiths inequality (GKS2) asks whether
$$\mathrm{Cov}_J(f,g)
=\mathbb{E}_J[fg]-\mathbb{E}_J[f]\mathbb{E}_J[g]\ge0$$
for every such finite interacting system. It is known for $n=2$ and at zero interaction, but the genuinely interacting non-Abelian case $n\ge3$ remains open.
Research objective
Find one explicit finite counterexample: a spin dimension $n\ge3$, a finite number of sites, nonnegative pair couplings with at least one nonzero coupling, and two explicit observables $f,g$ in the invariant dot-product cone such that
$$\mathbb{E}_J[fg]<\mathbb{E}_J[f]\mathbb{E}_J[g].$$
Quantum rotors, Gaussian spins, fixed-coordinate observables, anisotropic componentwise statements, the zero-coupling case, and counterexamples only to stronger Ginibre-type inequalities do not satisfy this challenge.
Success and verification gate
A complete result must specify all parameters exactly and provide exact closed forms or rigorously certified terminating evaluations of $Z_J$, $\mathbb{E}_J[f]$, $\mathbb{E}_J[g]$, and $\mathbb{E}_J[fg]$. A Monte Carlo estimate or unvalidated numerical quadrature is not sufficient.
An independent verifier should be able to:
- check $n\ge3$, finite $M$, $J_{ij}\ge0$, and at least one $J_{ij}>0$;
- check cone membership of $f$ and $g$ directly from their nonnegative coefficients and integer exponents;
- recompute the partition function and expectation numerators using the supplied exact formulas or rigorous interval certificate; and
- establish with exact arithmetic or disjoint outward-rounded bounds that the covariance is strictly negative.
For rational couplings, one possible automated certificate expands $e^H$ to finite order, evaluates free-sphere moments exactly using the standard spherical pairing formula, bounds the remainder from $|H|\le\sum J_{ij}$, and interval-evaluates
$$I_{fg}I_1-I_fI_g<0.$$
A compact witness should be reviewable without reconstructing the search process.
Why this may lead to research output
This is the basic positive-correlation question for invariant observables of non-Abelian vector-spin ferromagnets. A counterexample would identify a fundamental obstruction to extending Griffiths/GKS tools from Abelian models to Heisenberg-type and more general $O(n)$ systems. A universal proof would also be important, but it would require review of the derivation rather than only the final result.
Current status and references
The problem was re-audited on 28 July 2026 and assessed as still open with high confidence. The closest general-looking fixed-component claim was withdrawn; asymptotic, zero-coupling, and stronger Ginibre results do not settle this invariant interacting problem.
- I. Herbst, Griffiths inequalities for non-interacting rotors.
- A. Abdesselam, Non-Abelian correlation inequalities and stable determinantal polynomials.
- D. Sylvester, The Ginibre inequality.
Released by
Kun Chen, Institute of Theoretical Physics, Chinese Academy of Sciences
Contact email
chenkun0228@gmail.com
Method
Other
Challenge issue
Background
Classical$O(n)$ spin models replace Ising spins by unit vectors. Let $M<\infty$ , $n\ge 3$ , and
with normalized rotation-invariant measure$d\nu_n$ . For symmetric ferromagnetic couplings $J_{ij}\ge 0$ , $J_{ii}=0$ , and at least one $J_{ij}>0$ , define
The invariant dot-product cone consists of finite nonnegative linear combinations
where$c_\alpha\ge0$ and all exponents are nonnegative integers; $g$ has the same form. Nonnegative refers to the coefficients, not to the pointwise sign of each monomial.
The invariant-observable second Griffiths inequality (GKS2) asks whether
for every such finite interacting system. It is known for$n=2$ and at zero interaction, but the genuinely interacting non-Abelian case $n\ge3$ remains open.
Research objective
Find one explicit finite counterexample: a spin dimension$n\ge3$ , a finite number of sites, nonnegative pair couplings with at least one nonzero coupling, and two explicit observables $f,g$ in the invariant dot-product cone such that
Quantum rotors, Gaussian spins, fixed-coordinate observables, anisotropic componentwise statements, the zero-coupling case, and counterexamples only to stronger Ginibre-type inequalities do not satisfy this challenge.
Success and verification gate
A complete result must specify all parameters exactly and provide exact closed forms or rigorously certified terminating evaluations of$Z_J$ , $\mathbb{E}_J[f]$ , $\mathbb{E}_J[g]$ , and $\mathbb{E}_J[fg]$ . A Monte Carlo estimate or unvalidated numerical quadrature is not sufficient.
An independent verifier should be able to:
For rational couplings, one possible automated certificate expands$e^H$ to finite order, evaluates free-sphere moments exactly using the standard spherical pairing formula, bounds the remainder from $|H|\le\sum J_{ij}$ , and interval-evaluates
A compact witness should be reviewable without reconstructing the search process.
Why this may lead to research output
This is the basic positive-correlation question for invariant observables of non-Abelian vector-spin ferromagnets. A counterexample would identify a fundamental obstruction to extending Griffiths/GKS tools from Abelian models to Heisenberg-type and more general$O(n)$ systems. A universal proof would also be important, but it would require review of the derivation rather than only the final result.
Current status and references
The problem was re-audited on 28 July 2026 and assessed as still open with high confidence. The closest general-looking fixed-component claim was withdrawn; asymptotic, zero-coupling, and stronger Ginibre results do not settle this invariant interacting problem.