OpenAI · 140-06 · 2026-09-27 · Theoretical computer science
Subsphere methods for memory-sample lower bounds in noiseless Gaussian regression
preprints/Subsphere-methods-for-memory-sample-lower-bounds-in-noiseless-Gaussian-regression-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.
Let a finite-state streaming learner estimate a uniformly random unit vector from independent exact Gaussian linear measurements. We prove that bits of persistent memory and angular success probability at least 2/3 require at least samples for all sufficiently large d, uniformly for . The proof conditions each batch on its observed projection and controls the resulting random residual subsphere.
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.