ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedprecheck passverified 2026-10-22 (gpt-6.1-sol)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Sobolev ellipsoid rates in polynomial and exponential spectra

Example

Assume Countable Choice, take $Q > 0$ and Sobolev-type weights $a_{j} ≍ j$ in the Gaussian sequence inverse model. If the singular values satisfy $s_{j} ≍ j^{- 1}$ , the minimax RMS error is of order $ϵ^{2 / 5}$ as $ϵ ↓ 0$ ; if instead $s_{j} ≍ e^{- j}$ , it is of order $[\log (1 / ϵ)]^{- 1}$ . The comparison constants may depend on the fixed ellipsoid and axis constants.

Example references

Polynomial and exponential minimax estimation rates

Litao Ding and Peter Mathé, Minimax rates for statistical inverse problems under general source conditions, §3 and Table 1

Facts & Assumptions

[F1]

Polynomial and exponential minimax estimation rates: For polynomial singular values and polynomial ellipsoid weights, $R_{*} ≍ ϵ^{κ / (κ + p + 1 / 2)}$ ; for exponential singular values and polynomial weights, $R_{*} ≍ [\log (1 / ϵ)]^{- κ}$ .

[F2]

Gaussian sequence inverse model and Weighted ellipsoid class for sequence inversion: The model uses the positive singular sequence and the parameter class $\sum_{j} a_{j}^{2} θ_{j}^{2} \leq Q^{2}$ .

[A1]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): Countable Choice is inherited from the sequence model.

Given: Countable Choice, $Q > 0$ , $p = κ = 1$ , and either the polynomial or exponential singular-value regime in the Example.

Verification

1.1givenF1algebra

In the polynomial regime, $p = 1$ and $κ = 1$ , so the first formula of [F1] has exponent $1 / (1 + 1 + 1 / 2) = 2 / 5$ .

1.2givenF1algebra

In the exponential regime, $κ = 1$ , so the second formula of [F1] is $[\log (1 / ϵ)]^{- 1}$ .

2.1A1step 1.1step 1.2F2algebra∎

By [A1], the Gaussian sequence model is interpreted with its stated Countable Choice prerequisite. The two substitutions give exactly the two stated RMS orders; $Q$ and the asymptotic comparison constants affect only the multiplicative bounds.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

28 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Litao Ding and Peter Mathé, Minimax rates for statistical inverse problems under general source conditions, §3 and Table 1, PDF pp. 5–6 (standard reference, not scraped)