The Sharan-Sidford-Valiant precision question: do learners with $o(d^2)$ memory need $\Omega(d\log(1/\epsilon))$ samples for linear regression?

Main theorem: for fixed $A>0$ there are $c_A,d_A$ such that any learner with $M\le Ad^2$ bits, $d\ge d_A$ , $0<\epsilon\le1/10$ and uniform-prior success at least $2/3$ in reaching angular error $\epsilon$ needs horizon $T\ge c_A d\log(1/\epsilon)$ ; for $M=o(d^2)$ the constant is absolute, and the bound also holds under an every-signal guarantee. Transitions may use unbounded computation and randomness; the output uses only the terminal state, stopping index and fresh randomness, with a deterministic horizon. It implies an $\Omega_A(dr\log d)$ bound in the noisy experiment of Sharan-Sidford-Valiant at Euclidean accuracy $d^{-r}$ in their range. No matching finite-memory upper bound is proved, and memory well above $d^2$ is not treated.

Exact FrontierDelta

Prior state unknown→proved

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Occurred: Sep 27, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: lean-checked. Publication: announcement. AI contribution: ai-discovered. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

Canonical aliases: The Sharan-Sidford-Valiant precision question: do learners with $o(d^2)$ memory need $\Omega(d\log(1/\epsilon))$ samples for linear regression? · Memory-precision tradeoff for linear regression

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VibeMathed record: The Sharan-Sidford-Valiant precision question: do learners with $o(d^2)$ memory need $\Omega(d\log(1/\epsilon))$ samples for linear regression?

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No independent mathematician has checked this yet. Checked here: the main theorem, its two corollaries and the predecessors section of the principal manuscript were read against the Sharan-Sidford-Valiant question as the manuscript cites it. Lean: lean/docs/140.md lists eight challenges. ComparatorChallenges/NoiselessRegression (theorem OAI.NoiselessRegression.main, solution module OAI.Probability.GaussianRegression.Main, file exists at the pinned commit) is not in formalization.yaml; its statement was read here: for $M=o(d^2)$ an absolute $c$ with $T\ge c\,d\log(1/\epsilon)$ under uniform-prior or every-signal success $2/3$ , and for $M\le Ad^2$ a constant $c_A$ , for finite-state randomized learners with exact Gaussian observations and angular error. That is the headline. MemoryPrecision.main (in formalization.yaml) covers supporting estimates and the $M\le d^2$ case. Not rebuilt here. The model is exact (noiseless) observations; the noisy consequence is derived in the paper, not formalized.