OpenAI · 140-04 · 2026-09-27 · Theoretical computer science
Projection moments, positive cap domination, and Riesz estimates on the sphere
preprints/Projection-moments-positive-cap-domination-and-Riesz-estimates-on-the-sphere-September-27-2026
Family 140. Memory–sample lower bounds for noiseless Gaussian regression.
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
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
For fixed , a one-pass learner with persistent bits needs noiseless Gaussian samples to recover a unit vector to angular error with probability , uniformly in accuracy for large . Computation and randomized updates are unrestricted, but output uses only the terminal state, stopping index and fresh randomness.
We prove moment estimates for exact random projections of finite measures whose mass is controlled on Euclidean balls, and derive positive domination by countable sums of spherical cap measures. For learners with bits of memory, these estimates give three proofs that uniform-sphere average success at least 2/3 at angular accuracy requires noiseless Gaussian observations. The three proofs keep their different stopping and accuracy costs explicit.
Preprint source and Lean scope
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This preprint is named in the Lean scope note for 140: Memory–sample lower bounds for noiseless Gaussian regression. The authors are OpenAI. This companion does not verify the formalization. It is not a premise. The OAI library is not imported. No compilation is asserted. O.1 stays NOT_CLOSED.