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CS-133-A1 · revision r1 · on CS-133

OpenAI claim summary for CS-133

Literature report · LLM-generated · unknown ·

Literature report. The underlying work is credited to its authors. Automated screening checks form, honesty and scope, not correctness. Status: open. Claims here do not change the source status.

Claim: literature report

Summary of a claim published in openai/math; not a claim by this platform.

The result supplies the conclusion in this problem with its displayed hypotheses and quantifiers. The broader statements and applications in the manuscript are kept separate from the particular question above. The supporting result is manuscript 2 (family 140; 2026-09-27). Other linked papers contribute only their stated special cases; those weaker results alone would not resolve the complete question.

Limitations: The linked openai/math material has not been examined or verified by the platform.

Manuscript

Contents
  1. Memory–precision lower bounds for exact Gaussian regression
  2. OpenAI's claim
  3. Mathematical statement of the claimed result
  4. Manuscript 1.
  5. Manuscript 2.
  6. Manuscript 3.
  7. Manuscript 4.
  8. Manuscript 5.
  9. Manuscript 6.
  10. Outline of the argument
  11. Formal verification
  12. Status caveat
  13. Sources

Memory–precision lower bounds for exact Gaussian regression

OpenAI's claim

OpenAI presents the following claims relevant to this question, with manuscript numbers following paper order in Sources. In manuscript 1 (family 140; 2026-09-27), the asserted result is that finite messages on a spherical image of a cube reveal at most order $dt$ information after $t$ Gaussian blocks of order $d$, even when nested spread-restoring cells are added; In manuscript 2 (family 140; 2026-09-27), the asserted result is that retaining at most $Ad^2$ bits between exact standard Gaussian measurements requires at least $c_A d\log(1/\varepsilon)$ observations for constant-probability angular accuracy on a uniform sphere; In manuscript 3 (family 140; 2026-09-27), the asserted result is that a finite message from exact Gaussian data has bounded conditional information relative to an independently revealed projection, yielding the stated low-memory precision lower bound; In manuscript 4 (family 140; 2026-09-27), the asserted result is that controlled local mass on the sphere gives projection-moment and positive-cap domination estimates, which imply the stated low-memory noiseless-regression sample lower bounds; In manuscript 5 (family 140; 2026-09-27), the asserted result is that replacing the Gaussian rows that selected a finite message with independent rows changes residual conditional information by at most order $d$ under the stated entropy and dimension bounds; In manuscript 6 (family 140; 2026-09-27), the asserted result is that a uniform-sphere learner with $o(d^2)$ persistent bits needs at least $2^{-16}d\log_2(1/\varepsilon)$ exact Gaussian measurements for the stated success level at small angular error.

Mathematical statement of the claimed result

Manuscript 1.
Let $S$ be uniform on the unit sphere in $\mathbb R^d$, independent of successive $x_t\sim N(0,I_d)$, and observe exact labels $\langle x_t,S\rangle$. A jointly measurable one-pass learner has at most $2^{M(d)}$ persistent states, data-independent initial/shared randomness, fresh transition randomness, and prescribed finite horizon $T(d)$; discarded observations cannot be revisited, and terminal output uses only state, index and fresh randomness. If $M(d)=o(d^2)$ and angular accuracy $0<\varepsilon(d)\le1/10$ has success probability at least $2/3$, the claimed consequence is \[ T(d)\ge c d\log(1/\varepsilon(d)) \] with one absolute $c>0$; it holds eventually along each allowed memory sequence, with a sequence-dependent dimension threshold. This corollary alone treats subquadratic memory, rather than every fixed multiple of $d^2$.
Manuscript 2.
Let $S$ be uniform on $S^{d-1}$, independent of rows $x_t\sim N(0,I_d)$ and the learner's data-independent randomness. A jointly measurable one-pass learner sees exact labels $\langle x_t,S\rangle$, has at most $2^M$ persistent states, uses only current observation/index and fresh randomness in transitions, cannot revisit discarded data, and stops by a prescribed deterministic finite horizon $T$. Its unit output depends only on terminal state, index and fresh randomness. The claim states \[ \forall A>0\ \exists c_A>0,d_A:\quad d\ge d_A,\ M\le Ad^2,\ 0<\varepsilon\le1/10, \Pr[\arccos\langle\widehat s,S\rangle\le\varepsilon]\ge2/3 \ \Rightarrow\ T\ge c_A d\log(1/\varepsilon). \] The entire range of each fixed quadratic-memory budget is covered, not only $o(d^2)$.
Manuscript 3.
In dimension $d$, take an independent uniformly distributed unit signal $S$, independent Gaussian rows $x_t\sim N(0,I_d)$ and exact labels $\langle x_t,S\rangle$. A measurable one-pass learner retains at most $2^{M(d)}$ states, has data-independent shared/initial randomness, uses only the current row and label in transitions, and stops at a prescribed finite horizon $T(d)$; its terminal output cannot revisit observations. If $M(d)=o(d^2)$ and $0<\varepsilon(d)\le1/10$, the claimed implication is \[ \Pr[\arccos\langle\widehat S,S\rangle\le\varepsilon(d)]\ge2/3 \quad\Longrightarrow\quad T(d)\ge c d\log(1/\varepsilon(d)) \] for absolute $c>0$ and all sufficiently large $d$. Probability includes the signal, samples and all learner randomness, with a dimension threshold allowed to vary with the chosen memory sequence.
Manuscript 4.
Let $S$ be uniform on the unit sphere in $\mathbb R^d$, independent of exact Gaussian observations $(x_t,\langle x_t,S\rangle)$, $x_t\sim N(0,I_d)$. The one-pass learner is jointly measurable, has at most $2^M$ persistent states, data-independent initial/shared randomness, fresh transition randomness and prescribed finite horizon $T$; output uses only terminal state, index and fresh randomness. The claimed absolute $c>0,d_0$ give \[ d\ge d_0,\ M\ge0,\ 0<\varepsilon\le1/10,\quad \Pr[\text{angular error}\le\varepsilon]\ge2/3 \ \Rightarrow\quad T\ge\frac{c d\log(1/\varepsilon)}{1+M/d^2}. \] Every finite memory budget is allowed, and dependence on memory is explicit.
Manuscript 5.
With a uniform spherical signal $S\in\mathbb R^d$ and independent exact observations $(x_t,\langle x_t,S\rangle)$ for $x_t\sim N(0,I_d)$, consider measurable one-pass learners retaining at most $2^{M(d)}$ states and having data-independent initial/shared randomness and prescribed finite horizon $T(d)$. Discarded observations cannot be revisited; output uses terminal state and fresh randomness. For $M(d)=o(d^2)$ and $0<\varepsilon(d)\le1/10$, the claimed implication is \[ \Pr[\text{angular error}\le\varepsilon(d)]\ge3/5 \quad\Longrightarrow\quad T(d)\ge c d\log(1/\varepsilon(d)) \] for absolute $c>0$ and all sufficiently large $d$. The necessary dimension cutoff can vary with the chosen memory sequence, and success probability includes all learner randomness.
Manuscript 6.
Consider the jointly measurable one-pass model with at most $2^{M(d)}$ persistent states, data-independent initial/shared randomness, exact independent rows $x_t\sim N(0,I_d)$ labeled $\langle x_t,S\rangle$, uniform independent unit signal $S$, and a prescribed deterministic finite horizon $T$. Discarded observations cannot be revisited, and terminal output is determined by state, index and fresh randomness. For every sequence $M(d)=o(d^2)$, a sufficiently large dimension threshold is claimed such that all $0<\varepsilon\le1/10$ obey \[ \Pr[\arccos\langle\widehat s,S\rangle\le\varepsilon]\ge2/3 \quad\Longrightarrow\quad T\ge2^{-16}d\log_2(1/\varepsilon). \] The dimension threshold depends only on the memory sequence.

Outline of the argument

For manuscript 1 (family 140; 2026-09-27), a dyadic cell is chosen to restore geometric spread. Exact projection density estimates bound one message's information, while the entropy of selecting that cell pays for the apparent localization gain; a weighted inverse-distance potential closes the block recurrence. For manuscript 2 (family 140; 2026-09-27), continuation-success mass is localized into controlled spherical balls. Multirow projection moments bound the gain from one block of exact equations, and a backward recurrence propagates the terminal small-cap cost through the finite memory states.

For manuscript 3 (family 140; 2026-09-27), resampling posterior signals preserves the signal--message law while introducing replica correlation. Equal-label fibers are evaluated with affine-volume normalization, and a Gaussian test cancels that volume; total-correlation chain rules then bound the information carried between blocks. For manuscript 4 (family 140; 2026-09-27), exact frame densities are integrated using local ball-mass control. Positive sums of cap measures dominate the propagated law, and radial or Riesz estimates bound successive blocks; forward and backward arguments keep their stopping and terminal accuracy costs explicit.

For manuscript 5 (family 140; 2026-09-27), mixed moments compare shared-row and fresh-row projection densities. A critical radius aligns mass-level charges so the density contributions cancel, and counterfactual independent kernels transfer the remaining information bound to a finite-state streaming recurrence. For manuscript 6 (family 140; 2026-09-27), conditioning a batch on its observed projection leaves a random residual subsphere. Radius-weighted cap estimates control the success mass on that subsphere, and an affine-invariant dimension budget limits how quickly successive batches can reach the terminal cap.

Formal verification

For manuscript 1 (family 140; 2026-09-27), OpenAI lists Lean statements for this assertion: finite messages on a spherical image of a cube reveal at most order $dt$ information after $t$ Gaussian blocks of order $d$, even when nested spread-restoring cells are added.

This coverage does not establish a formal proof of the complete problem as stated here.

For manuscript 2 (family 140; 2026-09-27), OpenAI lists Lean statements for this assertion: retaining at most $Ad^2$ bits between exact standard Gaussian measurements requires at least $c_A d\log(1/\varepsilon)$ observations for constant-probability angular accuracy on a uniform sphere.

The central assertion is within the documented formal scope; the additional applications retain their own scope limits.

For manuscript 3 (family 140; 2026-09-27), OpenAI lists Lean statements for this assertion: a finite message from exact Gaussian data has bounded conditional information relative to an independently revealed projection, yielding the stated low-memory precision lower bound.

This coverage does not establish a formal proof of the complete problem as stated here.

For manuscript 4 (family 140; 2026-09-27), OpenAI lists Lean statements for this assertion: controlled local mass on the sphere gives projection-moment and positive-cap domination estimates, which imply the stated low-memory noiseless-regression sample lower bounds.

The selected statement includes the clause used for this problem.

For manuscript 5 (family 140; 2026-09-27), the documented Lean scope is the conditional information comparisons for fresh versus coupled labels and the stated two-point fiber identities, retaining their radius and row-count assumptions; the separate streaming consequences are not selected for formalization.

This coverage does not establish a formal proof of the complete problem as stated here.

For manuscript 6 (family 140; 2026-09-27), OpenAI lists Lean statements for this assertion: a uniform-sphere learner with $o(d^2)$ persistent bits needs at least $2^{-16}d\log_2(1/\varepsilon)$ exact Gaussian measurements for the stated success level at small angular error.

This coverage does not establish a formal proof of the complete problem as stated here.

The linked family Comparator checks are GaussianFiniteEntropy.lean, GaussianInformation.lean, GaussianReplacement.lean, MemoryPrecision.lean, NoiselessRegression.lean, PosteriorReplicas.lean, ProjectionMoments.lean, SubsphereCurrent.lean.

Their named declarations include OAI.AllScaleInverseVolume.inverse_volume_without_support, OAI.CurrentProjection.actualRowMain, OAI.CurrentProjection.actualRowRefinement, OAI.CurrentProjection.actualRowSourceMain, OAI.CurrentProjection.actualRowSourceRefinement, OAI.CurrentProjection.criticalComparisonMain, OAI.CurrentProjection.fiberBlockMain, OAI.CurrentProjection.fiberTwoPointMain, OAI.CurrentProjection.hybridBlockMain, OAI.CurrentProjection.mixedMomentMain, OAI.CurrentProjection.oneLevelComparisonMain, OAI.CurrentProjection.twoLabelComparisonMain, OAI.KernelPotential.actual_row_drift, OAI.KernelPotential.kernel_coercivity, OAI.MemoryPrecision.main, OAI.NoiselessRegression.main, OAI.PosteriorReplicas.source_main, OAI.ProjectionMoments.main, OAI.RepeatedLocalization.FiniteEntropy.finite_expected_regularization, OAI.RepeatedLocalization.repeated_localization, OAI.SubsphereCurrent.full_current_main_scope.

This site has not executed the Lean code. Described statement coverage is not an independent proof check.

Status caveat

The manuscripts were produced by an internal OpenAI model. OpenAI cautions that results without formalization can contain mistakes. This site has not independently checked the proofs; the label records the scope of OpenAI's claim.

Sources