ExampleConstruction: AI-adaptedVerification: AI-generatedPipeline-generated

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.

A product between pivots is split without changing number or mass

Statement

On the grid $(1, 2, 4)$ , the product size $z = 3$ is replaced by $P_{Δ} (3) = \frac{1}{2} δ_{2} + \frac{1}{2} δ_{4}$ . It remains one particle in expectation, has mass $3$ , and its second moment is $10$ , an excess of $1$ over $z^{2} = 9$ .

Facts & Assumptions

Given: The fixed-pivot rule on a grid containing adjacent nodes $2$ and $4$ .

[F1]

Conservative fixed-pivot redistribution on a finite positive grid: at $z \in [x_{i}, x_{i + 1}]$ , the pivot weights are $(x_{i + 1} - z) / (x_{i + 1} - x_{i})$ and $(z - x_{i}) / (x_{i + 1} - x_{i})$ .

[F2]

Fixed-pivot weights are nonnegative and reproduce number and mass: the weights sum to one and reproduce first moment $z$ .

Proof

1.1givenF1algebra

Substituting $x_{i} = 2$ , $x_{i + 1} = 4$ , and $z = 3$ in [F1] gives both weights equal to $(4 - 3) / (4 - 2) = 1 / 2$ . Thus the replacement is exactly $\frac{1}{2} δ_{2} + \frac{1}{2} δ_{4}$ .

2.1F2step 1.1algebra∎

The weights sum to $1$ , their weighted mass is $(2 + 4) / 2 = 3$ , and their weighted second moment is $(2^{2} + 4^{2}) / 2 = 10$ . Hence the number and mass statements agree with [F2], and the second-moment excess is $10 - 3^{2} = 1$ .

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · one level

2 results within one dependency step 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

Jitendra Kumar, Numerical approximations of population balance equations in particulate systems, Chapter 2, §2.2 (standard reference, not scraped)