LemmaStatement: AI-adaptedProof: AI-adaptedPipeline-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 Gaussian matrix-depth scale constant exists uniquely

Statement

Let $Φ (t) = N (0, 1) ((- \infty, t])$ . There is a unique $β > 0$ such that $Φ (\sqrt{β}) = 3 / 4$ .

Facts & Assumptions

Given: The standard normal probability law $N (0, 1)$ .

[F1]

Standard normal and normal laws defines $N (0, 1)$ by the positive even density $e^{- x^{2} / 2} / \sqrt{2 π}$ , whose integral over $R$ is one.

[F2]

Continuity from below for measures: measures of increasing measurable sets converge to the measure of their union.

[F3]

Intermediate value theorem, by bisection with a canonical left-half rule: a continuous function on $[a, b]$ takes every value between $f (a)$ and $f (b)$ : a continuous real function on a closed interval takes every value between its endpoint values.

Proof

1.1F1F2algebra

Since the standard-normal density is bounded by $1 / \sqrt{2 π}$ , $∣ Φ (t + h) - Φ (t) ∣ \leq ∣ h ∣ / \sqrt{2 π}$ ; hence $Φ$ is continuous. The density is positive, so its integral over every nonempty interval is positive and $Φ$ is strictly increasing. Evenness and total mass one give $Φ (0) = 1 / 2$ . Applying [F2] to $(- \infty, n]$ as $n \to \infty$ gives $Φ (n) \to 1$ .

2.1step 1.1F3construct∎

Choose an integer $n$ with $Φ (n) > 3 / 4$ . By [F3] on $[0, n]$ , some $q \in (0, n)$ satisfies $Φ (q) = 3 / 4$ ; strict increase from step 1.1 makes $q$ unique. Then $β = q^{2} > 0$ is the unique positive value with $Φ (\sqrt{β}) = 3 / 4$ .

Depends on

Used by

Dependency tree · two levels

32 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

Mengjie Chen, Chao Gao and Zhao Ren, Robust Covariance and Scatter Matrix Estimation under Huber's Contamination Model, Proposition 3.1 (standard reference, not scraped)
Rick Durrett, Probability: Theory and Examples, 5th ed., §1.6 (standard reference, not scraped)