ExampleConstruction: AI-generatedVerification: AI-adaptedPipeline-generatedprecheck passjudge 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.

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Detection-limit assumptions and left-censored readings

Example

Consider the heteroscedastic response law $Y_{c} \sim N (c, σ_{0}^{2} (1 + ρ c)^{2})$ for $c \geq 0$ , with $σ_{0} > 0$ and $ρ \geq 0$ . The blank variance is $σ_{0}^{2}$ , but the standard deviation increases with concentration when $ρ > 0$ . For $0 < α, β < 1 / 2$ , write $a = z_{1 - β}$ . The critical level is $L_{C} = z_{1 - α} σ_{0}$ . A finite detection limit for false-negative target $β$ exists exactly when $a ρ σ_{0} < 1$ , and then $L_{D} = \frac{L_{C} + a σ_{0}}{1 - a ρ σ_{0}} .$ For $k_{Q} > 0$ , the coefficient-of-variation criterion has a finite quantification limit exactly when $k_{Q} > ρ σ_{0}$ , and then $L_{Q} = \frac{σ_{0}}{k_{Q} - ρ σ_{0}} .$ When a reporting procedure records only the category $Y \leq L_{C}$ , its likelihood contribution at concentration $c$ is the probability $P_{c} (Y \leq L_{C}) = Φ ⁣ (\frac{L_{C} - c}{σ_{0} (1 + ρ c)}),$ not a density value at $L_{C}$ .

Facts & Assumptions

Given: AC; the heteroscedastic family and parameter restrictions in the Example; and the positive-decision rule is $Y > L_{C}$ .

[A1]

AC is required by the standard normal probability-law construction. The Axiom of Choice Standard normal and normal laws

[F1]

The standard normal CDF is strictly increasing, has no atoms, and its quantiles obey $z_{q} = - z_{1 - q}$ for every $q \in (0, 1)$ . The standard normal CDF has unique symmetric quantiles

[F2]

A normal response with mean $m$ and scale $σ$ has expectation $m$ and variance $σ^{2}$ . Location-scale normal laws have their stated mean and variance

[F3]

$N (c, σ^{2})$ is the law of $c + σ Z$ for standard normal $Z$ . Standard normal and normal laws

[F4]

A real random variable's CDF is $F (x) = P (Y \leq x)$ . Cumulative distribution function of a real random variable

[F5]

The exact critical level and the false-negative target are defined relative to the specified blank and signal laws. Critical, detection, and quantification levels

[F6]

For a probability measure, $P (A^{c}) = 1 - P (A)$ . Basic identities for a probability measure

Verification

Given: The response model, positive-decision rule, and targets in the Example.

1.1A1F1F3F5F6givenalgebra

At $c = 0$ , the model is $N (0, σ_{0}^{2})$ , so [F3] represents $Y_{0} = σ_{0} Z$ in law. Set $L_{C} = z_{1 - α} σ_{0}$ ; by [F1], [F3], and complement identity [F6], $P_{0} (Y_{0} > L_{C}) = 1 - Φ (z_{1 - α}) = α$ . Also [F1] gives $z_{1 / 2} = - z_{1 / 2}$ , hence $z_{1 / 2} = 0$ and $Φ (0) = 1 / 2$ . Since $1 - α > 1 / 2$ , strict increase gives $z_{1 - α} > 0$ , and [F5] makes this an exact critical level.

1.2F2givenalgebra

By [F2], $CV (c) = σ (c) / c = σ_{0} / c + ρ σ_{0}$ for $c > 0$ . If $k_{Q} > ρ σ_{0}$ , the inequality $CV (c) \leq k_{Q}$ is equivalent to $c \geq σ_{0} / (k_{Q} - ρ σ_{0})$ , giving the displayed least value. If $k_{Q} \leq ρ σ_{0}$ , then $CV (c) > ρ σ_{0} \geq k_{Q}$ for every finite $c > 0$ , so no finite quantification limit exists.

2.1A1F1F3F5givenstep 1.1algebra

For $c > 0$ , the scale is $σ (c) = σ_{0} (1 + ρ c) > 0$ . By [F3], $P_{c} (Y \leq L_{C}) = Φ ((L_{C} - c) / σ (c))$ . Since $a = z_{1 - β} > 0$ , strict increase in [F1] makes this equal to $β$ exactly when $c - L_{C} = a σ_{0} (1 + ρ c), a = z_{1 - β} > 0.$ Rearranging gives $c (1 - a ρ σ_{0}) = L_{C} + a σ_{0}$ , whose right-hand side is positive. If $a ρ σ_{0} < 1$ , the unique solution is the positive displayed $L_{D}$ . If $a ρ σ_{0} = 1$ , the equation would equate zero to a positive number; if $a ρ σ_{0} > 1$ , its only algebraic solution is negative. Thus a nonnegative detection limit exists exactly under the stated condition.

3.1A1F1F3F4step 2.1algebra∎

If the recorded observation is the category $Y \leq L_{C}$ , its probability under the stated law is the CDF at $L_{C}$ , namely the displayed $Φ ((L_{C} - c) / σ (c))$ by [F4]. The affine normal law has no atom at $L_{C}$ by [F1] and [F3], while the CDF gives the probability of the reported half-line. Substituting $L_{C}$ as an observed point is therefore not the likelihood of this censored report.

Source notes

Eurachem §5.3.2 explains that blank correction changes the standard deviation used for LOD/LOQ; §5.3.5 discusses changing blanks across days. The heteroscedastic law and censoring rule here are explicit model assumptions. Their consequences are derived above rather than imported as universal measurement claims.

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Sources

H. Cantwell (ed.), Eurachem Guide: The Fitness for Purpose of Analytical Methods, 3rd ed. (2025), §§5.3.1–5.3.5 and Annex B (standard reference, not scraped)