TheoremStatement: AI-adaptedProof: AI-adaptedPipeline-generatedjudge pass (gpt-6-luna)verified 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.

Kernel directions preclude unrestricted recovery

Statement

Let $(X, d_{X})$ , $(Y, d_{Y})$ , and $(Q, d_{Q})$ be metric spaces, let $U \subseteq X$ be a nonempty admissible truth class, let $F : U \to Y$ be a forward map, and let $q : U \to Q$ assign each truth its target class. Let $x_{0}, x_{1} \in U$ , let $δ \geq 0$ , and suppose a single datum $y \in Y$ satisfies

$d_{Y} (y, F (x_{0})) \leq δ and d_{Y} (y, F (x_{1})) \leq δ .$

For every deterministic data-only estimator, meaning every function $\hat{q} : Y \to Q$ , the two target-class losses at that datum satisfy

$\max {d_{Q} (\hat{q} (y), q (x_{0})), d_{Q} (\hat{q} (y), q (x_{1}))} \geq \frac{1}{2} d_{Q} (q (x_{0}), q (x_{1})) .$

In particular, if $F (x_{0}) = F (x_{1})$ but $q (x_{0}) \neq q (x_{1})$ , no data-only estimator exactly recovers both truths. The quantitative bound also holds whenever the radius- $δ$ data balls around the two forward data share the observed datum $y$ .

Statement references

Inverse-problem model and target-class loss

Identifiability and deterministic estimators

Metric triangle inequality

Nonnegative metric distances

Christian Clason, Regularization of Inverse Problems, Chapter 3 and Chapter 4 §4.1

Tristan van Leeuwen and Christoph Brune, 10 Lectures on Inverse Problems and Imaging, Lecture 1 §§1.2–1.3

Facts & Assumptions

Given: Metric spaces $(X, d_{X})$ , $(Y, d_{Y})$ , and $(Q, d_{Q})$ ; a nonempty set $U \subseteq X$ ; a map $F : U \to Y$ ; a target-class map $q : U \to Q$ ; truths $x_{0}, x_{1} \in U$ ; a radius $δ \geq 0$ ; a datum $y \in Y$ within distance $δ$ of both $F (x_{0})$ and $F (x_{1})$ ; and an estimator $\hat{q} : Y \to Q$ .

[F1]

Inverse-problem models and solution classes: the recovery target has a metric $d_{Q}$ , and $q (x)$ is the target class of truth $x$ .

[F2]

Identifiability and stability moduli: a deterministic data-only estimator is a function $Y \to Q$ and exact recovery requires $\hat{q} (F (x)) = q (x)$ for every admissible truth.

[F3]

Metric space: $d (x, y) = 0$ iff $x = y$ , symmetry, and the triangle inequality; pseudometric and ultrametric: the target metric satisfies separation and the triangle inequality.

[F4]

Nonnegativity of a metric is a consequence of the other axioms, not an axiom: every metric distance is nonnegative.

Proof

1.1givenF2

Put $z : = \hat{q} (y) \in Q$ . Since the estimator is a function evaluated at the single datum $y$ , this is the same estimate for both truths.

1.2F1F3

By the triangle inequality for $d_{Q}$ , $d_{Q} (q (x_{0}), q (x_{1})) \leq d_{Q} (q (x_{0}), z) + d_{Q} (z, q (x_{1})) .$

2.1step 1.2F4algebra

Let $M$ be the maximum of the two losses in the statement. Both losses are nonnegative by [F4] and each summand in step 1.2 is at most $M$ , so $d_{Q} (q (x_{0}), q (x_{1})) \leq 2 M$ and $M \geq \frac{1}{2} d_{Q} (q (x_{0}), q (x_{1}))$ .

3.1givenF2F3F4step 1.1step 2.1∎

If $F (x_{0}) = F (x_{1})$ and their target classes differ, metric separation and nonnegativity give $d_{Q} (q (x_{0}), q (x_{1})) > 0$ ; choosing $y = F (x_{0})$ and $δ = 0$ satisfies the hypotheses, so step 2.1 rules out exact recovery of both. For a shared noisy datum the same estimator output from step 1.1 yields the same bound.

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Sources

Christian Clason, Regularization of Inverse Problems, Chapter 3 and Chapter 4 §4.1 (standard reference, not scraped)
Tristan van Leeuwen and Christoph Brune, 10 Lectures on Inverse Problems and Imaging, Lecture 1 §§1.2–1.3 (standard reference, not scraped)