Pipeline-generated

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.

✓ 1 result · all verified · 1 also independently AI-judged

Every result on this page is machine-checked by a proof checker and read in full by a delegated reviewing agent on the owner's instruction; the judge is an additional, independent cross-model AI review of the proofs; all 1 also cleared it.

Parameter rules and the noise-free veto

1 · Prerequisites

2 · Logical flowchart

3 · Definitions, theorems and proofs

DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-6-luna)verified 2026-10-22 (gpt-6.1-sol)Open item page →

Noise-level-free parameter-choice rule

Definition

Fix a compact Hilbert inverse model $K : X \to Y$ and a family of reconstruction maps $(R_{α})_{α > 0}$ , with each $R_{α} : Y \to X$ , both held fixed. A deterministic parameter-choice rule is a function $α : Y \to (0, \infty)$ . It is noise-level-free (also called heuristic) when the value $α (z)$ for an observed datum $z \in Y$ uses only $K$ , the fixed family, and $z$ ; it does not take the exact datum or a noise bound $δ$ as input. Here $δ \geq 0$ is a valid noise bound for $z$ and an exact datum $y$ when $∥ z - y ∥_{Y} \leq δ$ . The resulting reconstruction is $R_{α (z)} z$ .

Definition references

Compact Hilbert inverse model

Clason, Regularization of Inverse Problems, Definition 4.4

Sherry, Inverse Problems: Lecture Notes 2024, Section 3.2.3

TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (gpt-6-luna)verified 2026-10-22 (gpt-6.1-sol)Open item page →

Bakushinskii veto for discontinuous pseudoinverses

Statement

Assume Countable Choice. Let $K : X \to Y$ be a compact linear operator between real or complex Hilbert spaces over one scalar field, with Hilbert-space Moore–Penrose inverse $K^{†} : D (K^{†}) \to X$ . Suppose $K^{†}$ is discontinuous on $D (K^{†})$ with its norm inherited from $Y$ . Let $(R_{α})_{α > 0}$ be maps $Y \to X$ such that $R_{α} y \to K^{†} y$ for every $y \in D (K^{†})$ as $α ↓ 0$ , and let $α : Y \to (0, \infty)$ be a deterministic noise-level-free rule. For each exact datum $y \in D (K^{†})$ and $δ > 0$ , set $E_{y} (δ) : = \sup {∥ R_{α (z)} z - K^{†} y ∥_{X} : z \in Y, ∥ z - y ∥_{Y} \leq δ}$ . If $E_{y} (δ) \to 0$ as $δ ↓ 0$ for every $y \in D (K^{†})$ , then $K^{†}$ is continuous on $D (K^{†}) .$ Thus, when $K^{†}$ is discontinuous, no such noise-level-free rule yields worst-case convergence over each exact datum's full admissible error balls. The limit is required separately for each exact datum, not uniformly over $y$ . This veto does not exclude convergence on restricted solution or error classes, stochastic noise models, or finite-dimensional problems.

Statement references

Compact Hilbert inverse model

Hilbert-space Moore–Penrose inverse and its domain

Noise-level-free parameter-choice rule

Clason, Regularization of Inverse Problems, Theorem 4.8

Sherry, Inverse Problems: Lecture Notes 2024, Theorem 3.12

Facts & Assumptions

Given: Countable Choice, a compact linear map $K : X \to Y$ between real or complex Hilbert spaces over one scalar field, its Moore–Penrose inverse $K^{†}$ discontinuous on $D (K^{†})$ , a family $(R_{α})_{α > 0}$ that converges pointwise to $K^{†}$ on this domain, a noise-level-free rule $α : Y \to (0, \infty)$ , and the convergence condition $E_{y} (δ) \to 0$ for each $y \in D (K^{†})$ .

[A1]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): For every family $(X_{n})_{n \in N}$ of nonempty sets indexed by $N$ , there is a function $f$ with domain $N$ such that $f (n) \in X_{n}$ for every $n$ .

[F1]

Compact linear inverse model between Hilbert spaces: The compact inverse model has real or complex Hilbert parameter and data spaces over one scalar field and a compact linear forward map $K : X \to Y$ .

[F2]

Hilbert-space Moore–Penrose inverse: The inverse has domain $D (K^{†}) = R (K) \oplus R (K)^{⊥}$ and maps $r + z$ to the unique minimum-norm solution of $K x = r$ .

[F3]

Noise-level-free parameter-choice rule: For the fixed operator and reconstruction family, $α$ is a function of the observed datum alone; its value does not take the exact datum or noise bound as input.

Proof

1.1givenA1F1F2F3algebra

Fix $y \in D (K^{†})$ and put $T (y) : = R_{α (y)} y$ . By [F3], this vector is independent of $δ$ . Since $y$ itself satisfies $∥ y - y ∥_{Y} = 0 \leq δ$ for every $δ > 0$ , the supremum defining $E_{y} (δ)$ gives $∥ T (y) - K^{†} y ∥_{X} \leq E_{y} (δ)$ for every $δ > 0$ . Letting $δ ↓ 0$ proves $T (y) = K^{†} y$ .

2.1givenstep 1.1F2F3choosealgebra∎

Fix $y \in D (K^{†})$ and $ε > 0$ . By the assumed limit choose $r > 0$ such that $E_{y} (r) < ε$ . For any $z \in D (K^{†})$ with $∥ z - y ∥_{Y} \leq r$ , step 1.1 applied to $z$ gives $R_{α (z)} z = K^{†} z$ , while $z$ belongs to the ball in $E_{y} (r)$ . Hence $∥ K^{†} z - K^{†} y ∥_{X} \leq E_{y} (r) < ε$ . This is continuity of $K^{†}$ at the arbitrary point $y$ in its subspace domain, contradicting the hypothesis.

4 · Examples, counterexamples and false statements

None yet.

Parameter rules and the noise-free veto

2 · Logical flowchart

3 · Definitions, theorems and proofs

Noise-level-free parameter-choice rule

Definition

Definition references

Bakushinskii veto for discontinuous pseudoinverses

Statement

Statement references

Facts & Assumptions

Proof

4 · Examples, counterexamples and false statements

Sources