ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedjudge pass (gpt-6-luna)

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.

One corrupted block is rejected by the geometric median

Example

Take five real block means $0, 0, 0, 0, M$ with $M > 0$ . Their geometric median is exactly $0$ , regardless of how large $M$ is.

Facts & Assumptions

Given: Countable Choice and the five block means $0, 0, 0, 0, M$ with $M > 0$ . Countable Choice is a prerequisite of the minimum-norm selector used by Geometric median of block means, through Measurable minimum-norm geometric-median selection.

[F1]

A geometric median minimizes the sum of distances to the listed block means (Geometric median of block means).

[F2]

With inner product $⟨ x, y ⟩ = x y$ , $R$ is a real inner-product space: ordered-field arithmetic gives bilinearity, symmetry, and $x^{2} \geq 0$ with equality only for $x = 0$ (The reals form a totally ordered field, Real and complex inner-product spaces and their induced length). Its induced norm is $\sqrt{x^{2}} = ∣ x ∣$ (Square roots exist: a unique $\sqrt{a} \geq 0$ with $(\sqrt{a})^{2} = a$ ; the positives are ${x^{2} : x \neq 0}$ , Basic properties of the absolute value), so completeness of the real line makes it a Hilbert space (The reals are complete, Hilbert space). The embedded rationals are countable and dense, so this Hilbert space is separable ( $Q$ is countably infinite, The rationals embed densely in the reals, Separability: the existence of an at most countable dense subset).

Verification

technique · direct

Given: $M > 0$ and the objective $F (z) = 4 ∣ z ∣ + ∣ z - M ∣$ .

1.1F1F2algebra

If $z \leq 0$ , then $F (z) = M - 5 z \geq M$ , with equality only at $z = 0$ . If $0 \leq z \leq M$ , then $F (z) = M + 3 z \geq M$ , again with equality only at $0$ .

2.1F1step 1.1algebra∎

If $z \geq M$ , then $F (z) = 5 z - M \geq 4 M > M = F (0)$ . Therefore $0$ is the unique minimizer, and the minimum-norm selection rule also returns $0$ . Four of five centers are exact while the fifth can be arbitrarily far away.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

67 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

Stanislav Minsker, Geometric Median and Robust Estimation in Banach Spaces, Lemma 2.1 (standard reference, not scraped)