LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedverified 2026-10-15 (gpt-6-luna, gpt-6.1-sol, and delegated Codex review/repair agents; see completed run evidence)

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 two-point lever rule on a scalar interval

Statement

Let $v_{a}, v_{b} \in R$ with $v_{a} < v_{b}$ . A fraction $λ \in [0, 1]$ satisfies $v = (1 - λ) v_{a} + λ v_{b}$ exactly when $v \in [v_{a}, v_{b}]$ and

$λ = \frac{v - v_{a}}{v_{b} - v_{a}} .$

This fraction is unique. At $v = v_{a}$ it is $0$ , and at $v = v_{b}$ it is $1$ ; no admissible fraction gives a value outside the endpoint interval.

Facts & Assumptions

Given: Ordered real numbers $v_{a} < v_{b}$ , an interval point $v \in R$ , and a mixing fraction $λ \in [0, 1]$ .

[F1]

Complete ordered field (least-upper-bound property): $R$ is an ordered field, so division by the positive number $v_{b} - v_{a}$ is valid and preserves inequalities.

[F2]

Intervals of $R$ : the nine order-convex forms, nondegeneracy, and length: $[0, 1] = {λ \in R : 0 \leq λ \leq 1}$ .

Proof

technique · direct

1.1F1F2givenalgebra

From $v = (1 - λ) v_{a} + λ v_{b}$ one obtains $v - v_{a} = λ (v_{b} - v_{a})$ . Since $v_{b} - v_{a} > 0$ and $0 \leq λ \leq 1$ , this gives $v_{a} \leq v \leq v_{b}$ and forces $λ = (v - v_{a}) / (v_{b} - v_{a})$ .

2.1F1F2step 1.1algebra∎

Conversely, if $v_{a} \leq v \leq v_{b}$ , the quotient $λ = (v - v_{a}) / (v_{b} - v_{a})$ lies in $[0, 1]$ and substitution gives $v = (1 - λ) v_{a} + λ v_{b}$ . The same forced quotient proves uniqueness and yields the two endpoint values.

The two-point lever rule on a scalar interval

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources