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Pressure formula for the spherical random perceptron

For the spherical random-perceptron model, does finite-temperature pressure converge to the predicted variational formula for every positive pattern density and inverse temperature and every bounded continuous single-pattern potential?

Why it matters and what counts as progress

Why it matters: A rigorous formula would replace replica-symmetry-breaking predictions by a controlled thermodynamic limit.

What counts as progress: Expectation and probability convergence to the stated functional throughout this potential class, or a model parameter disproving the prediction.

How the problem developed

  1. 2000 · Prediction [1]

    Géza Györgyi and Peter Reimann Proposed a continuous replica-symmetry-breaking description for the spherical perceptron beyond storage capacity; this was a physics prediction, not a proof of pressure convergence.

  2. 24 Sept 2026 · Proof claim [2]

    OpenAI Released a manuscript claiming convergence in expectation and probability to a variational pressure formula for every positive pattern density and inverse temperature and every bounded continuous single-pattern potential.

AI contribution

Sources

  1. Györgyi and Reimann, Beyond Storage Capacity in a Single Model Neuron: Continuous Replica Symmetry Breaking
  2. OpenAI, spherical random-perceptron free energy
  3. The free energy of the spherical random perceptron
  4. OpenAI, spherical-perceptron Lean scope
  5. OpenAI, Lean proof of spherical pressure convergence
  6. OpenAI mathematics collection and provenance
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