Research with AI

AI-generated results

These proofs were generated almost entirely by AI. My role was primarily to prompt the models and request audits and revisions; I barely contributed mathematical insights or proof ideas. I take zero credit and ε responsibility for these results, as I have not had time to carefully check the proofs myself.

Several rounds of AI review found no obvious unresolved substantive errors or gaps within their scope. Due to limited computational resources, only parts have been verified in Lean, where available, conditional on stated assumptions.

• Initial full-RSB interval for SK at positive temperature [GitHub] [PDF]

  An interval starting at zero in the support of the zero-field SK Parisi measure for every finite inverse temperature β > 1, establishing full replica symmetry breaking near the origin.

• Mézard–Parisi formula for the Viana–Bray model [GitHub] [PDF]

  A proposed proof of the variational formula for the limiting free energy at finite connectivity and temperature, with bounded symmetric couplings. Complete verification remains pending.

• Cutoff for SK Glauber dynamics [GitHub] [PDF]

  A sharp transition to equilibrium for zero-field SK Glauber dynamics at each fixed inverse temperature 0 < β < 1/3, with a mixing window bounded independently of system size.

• Critical free-energy fluctuations of the pure Ising p-spin model [GitHub] [PDF]

  At the static critical temperature, for every fixed integer p ≥ 3, the zero-field Gaussian model has an annealed-normalized partition function tending to 1/2 and centered fluctuations with an index-one stable component. Additional Gaussian terms depend on p and the covariance convention. I have not yet personally checked this manuscript carefully.

AI-assisted results

These works build, to varying degrees, on my previous research. I contributed mathematical ideas and insights to varying extents across projects. AI played a significant role in developing both the ideas and the proofs. I checked the underlying proof ideas and am more confident about the correctness of these results, though potential minor errors or gaps may remain.

• Inverse freezing in the Ghatak–Sherrington model [GitHub] [PDF]

  For a nonempty open set of crystal fields, the zero-field spin-one model is paramagnetic at high and low temperatures and replica-symmetry breaking on an intermediate interval, with each conclusion holding at every global maximizing density. Heejune and I began this project in February 2026; AI subsequently played a significant role in developing and revising the manuscript.

• Mixed p-spin free-energy fluctuations throughout the high-temperature regime [GitHub] [PDF]

  A quadratic approximation and central limit theorem for finite, zero-field Gaussian mixed p-spin models without a two-spin component, at every fixed inverse temperature in the strict high-temperature regime.

• Joint MPLE fluctuations in the SK model [GitHub] [PDF]

  Conditional Gaussian fluctuations of joint pseudolikelihood estimates of inverse temperature and external field from one SK configuration with observed couplings, at every fixed positive inverse temperature and zero true field.

• Approximate counting for pure Ising spin glasses [GitHub] [PDF]

  Partition-function approximation to arbitrary requested accuracy for pure zero-field models at a fixed margin below the graphical second-moment threshold, with exact-arithmetic operation bounds and a Lean companion. Proposed extensions to finite mixtures and vanishing fields are also outlined.