LemmaStatement: Literature-sourcedProof: 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.

The printed all-scale probability cell-mass bound is inconsistent

Statement

Let $X$ be a nonempty measurable space, let $d \geq 1$ , let $c > 0$ , and let $μ$ be a probability measure on $X$ . There is no family of finite measurable partitions $P_{ε}$ of $X$ , one for every $ε > 0$ , for which every cell $E \in P_{ε}$ satisfies $μ (E) \geq c ε^{d}$ .

Statement references

Shin et al., Assumption 3.1, clause 3: the all- $ϵ > 0$ lower cell-mass condition

Facts & Assumptions

Given: A probability measure $μ$ on nonempty $X$ , an integer $d \geq 1$ , and a constant $c > 0$ .

[F1]

A probability measure satisfies $μ (X) = 1$ (Probability measures and probability spaces).

[F2]

If $E \subseteq X$ is measurable, then $μ (E) \leq μ (X)$ (Measures are monotone).

Proof

technique · choose one scale at which the asserted lower bound exceeds total mass

1.1givenF1F2choose

Choose $ε > c^{- 1 / d}$ . Since $X$ is nonempty, any partition of $X$ has at least one cell $E$ . The set $E$ is measurable and $E \subseteq X$ , so [F1] and [F2] give $μ (E) \leq 1$ .

2.1givenstep 1.1algebracontradiction∎

The asserted lower bound gives $μ (E) \geq c ε^{d} > 1$ , contradicting step 1.1. Hence no such all-scale family of partitions exists.

Depends on

Used by

Regularized PINN minimizers have vanishing residuals under sample refinement Theorem

Dependency tree · two levels

4 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

Shin, Darbon, and Karniadakis, On the Convergence of Physics-Informed Neural Networks for Linear Second-Order Elliptic and Parabolic Type PDEs (standard reference, not scraped)