ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedprecheck pass

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.

Perfectly correlated error components

Example

Let $U$ equal $- 1$ or $1$ with probability $1 / 2$ each, and set $V = U$ . Both component errors are centered and have variance one, but they are perfectly correlated.

Facts & Assumptions

Given: A two-point probability space with $U \in {- 1, 1}$ equally likely and $V = U$ , interpreted as two successive signed stage errors. Compute the mean-squared sum and compare it with the variance-only sum.

[F1]

The mean-squared sum of square-integrable components includes twice each pairwise covariance. (MSE of a sum of error components includes covariance terms)

[F2]

An error ledger records each stage error as the difference of adjacent scalar references. (Successive statistical and computational error targets)

Verification

1.1givenF2construct

Set the first reference to $0$ , the next to $U$ , and the returned value to $U + V$ ; by [F2] the two signed stage errors are exactly $U$ and $V$ .

1.2givenalgebra

Since $U$ equals $- 1$ or $1$ with equal probability and $V = U$ , $E U = E V = 0$ and $Var (U) = Var (V) = 1$ .

2.1givenstep 1.2algebra

Because $V = U$ , $Cov (U, V) = E [U V] = 1$ and $E [(U + V)^{2}] = 4$ .

3.1F1step 1.2step 2.1algebra∎

[F1] gives the same value, $1 + 1 + 2 (1) = 4$ , whereas adding only the two component variances gives $2$ ; the omitted covariance term changes the result.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

8 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.