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 binary split preserves first moment and increases number by one

Statement

For a parent of size $5 a$ , let its expected daughters be one cluster of size $2 a$ and one of size $3 a$ . The event removes one parent and adds two daughters, so number increases by one and mass is unchanged.

Facts & Assumptions

Given: The finite fragmentation event convention with daughter counts $b_{5} (2) = b_{5} (3) = 1$ and all other $b_{5} (k) = 0$ .

[F1]

Finite discrete coagulation and fragmentation with a mass reservoir: one fragmentation event has increment $- e_{i} + \sum_{k < i} b_{i} (k) e_{k}$ and requires $\sum_{k < i} k b_{i} (k) = i$ .

[F2]

Stoichiometric number and mass changes for finite coagulation and fragmentation events: a fragmentation event changes number by $\sum b_{i} (k) - 1$ and mass by zero.

Proof

1.1givenF1algebra

The daughter mass sum is $2 a + 3 a = 5 a$ , so this daughter law satisfies the required parent-mass normalization in [F1].

2.1givenF2step 1.1algebra∎

Its number increment is $1 + 1 - 1 = 1$ . Its mass increment is $- 5 a + 2 a + 3 a = 0$ , agreeing with [F2].

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · one level

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Sources

Philippe Laurençot and Stéphane Mischler, From the discrete to the continuous coagulation-fragmentation equations, §1 (standard reference, not scraped)