OpenAI · 222-03 · 2026-09-24 · Probability and statistical mechanics
The spherical perceptron with bi-orthogonally invariant disorder
preprints/The-spherical-perceptron-with-bi-orthogonally-invariant-disorder-September-24-2026
Family 222. Perceptron free energies and microscopic jamming.
The authors are OpenAI. This companion does not verify the proof and does not adopt it as a theorem of Ngo Cheung’s series. It is not a premise. O.1 stays NOT_CLOSED. Anti-de Sitter results are not a de Sitter state, not Em02-E’s relational lapse, not XVI-P’s QdS scheme, and not XVI-Q’s sigma-image record. No quantum coefficient is admitted. No compilation is asserted.
Read against the standing problems
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Determines finite-temperature variational free energies for Gaussian Ising perceptrons with bounded Borel log-potentials and Gaussian spherical perceptrons with bounded continuous potentials, at every positive pattern density. A spherical extension treats bi-orthogonally invariant disorder with compact limiting singular-value distributions and no outliers. At margin , the quadratic-penalty spherical model has a sharp feasibility threshold and limiting gap and force laws, with system size, zero temperature, and critical density taken in that order.
We determine the limiting free energy of a spherical perceptron with a bounded continuous activation and a bi-orthogonally invariant disorder matrix. The singular values may have any compact limiting distribution, provided there are no outliers. We give an explicit variational formula for this limit.
Preprint source and Lean scope
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A Lean scope note is stored for family 222, but it does not name this preprint. It names 222-01, 222-04. This page still has the full preprint source. No compilation is asserted. O.1 stays NOT_CLOSED.