ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-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.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Perfect correlation can coexist with a method offset

Example

Let $Ω = {0, 1}$ have probability $1 / 2$ at each point, and let two methods on the same target report $A (ω) = ω$ and $B (ω) = ω + c$ for a fixed $c > 0$ . Their correlation is $1$ , but their difference is the nonzero constant $- c$ and their conventional limits of agreement both equal $- c$ .

Facts & Assumptions

Given: $Ω = {0, 1}$ with weights $1 / 2, 1 / 2$ , $A (ω) = ω$ , $B (ω) = ω + c$ , and $c > 0$ . For this example define $Corr (A, B) : = \frac{Cov (A, B)}{\sqrt{Var (A) Var (B)}} .$

[F1]

For $D = A - B$ , the bias is $E [D]$ , the spread is $\sqrt{Var (D)}$ , and conventional limits are their mean plus or minus $1.96$ times that spread. Method agreement and limits of agreement

[F2]

Variance and covariance are expectations of the centered square and centered product. Moments, variance, and covariance on a probability space

[F3]

On a finite probability space, expectation is the weighted sum of values over its atoms. Expectation agrees with the published finite weighted sum

[F4]

Covariance is symmetric and bilinear on finite linear combinations, including constant summands. Covariance is symmetric and bilinear in finite linear combinations

Verification

Given: The two-point probability model and method values in the Example. Verify the correlation and limits.

1.1F2F3F4givenalgebra

By [F3], $E [A] = 1 / 2$ and $E [B] = 1 / 2 + c$ . Directly summing the two centered squares gives $Var (A) = Var (B) = 1 / 4$ . By [F4], $Cov (A, B) = Cov (A, A + c) = Var (A) + Cov (A, c) = 1 / 4$ , since covariance with a constant is zero. Thus the denominator in the definition of correlation is $1 / 4$ and $Corr (A, B) = 1$ .

1.2F1F2F3algebra

At both points $D = A - B = - c$ , so by [F2] and [F3], $E [D] = - c$ and $Var (D) = 0$ . Hence its conventional limits are $- c \pm 1.96 \cdot 0 = (- c, - c)$ , while its bias is nonzero.

2.1step 1.1step 1.2given∎

Therefore perfect correlation does not imply zero bias or interchangeable measurements; any practical tolerance must be supplied for the intended use.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

19 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.