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.

Constant-kernel aggregation conserves mass and has explicit number decay

Statement

Let $μ_{t}$ be any positive weak solution of the continuous model with constant aggregation kernel $K (x, y) = κ \geq 0$ and no active fragmentation ( $β = 0$ ). If $N_{0} = M_{0} (μ_{0})$ and $M = M_{1} (μ_{0})$ , then $M_{1} (μ_{t}) = M, M_{0} (μ_{t}) = \frac{N_{0}}{1 + κ N_{0} t / 2} .$

Facts & Assumptions

Given: The weak formulation for a positive solution and the choices $φ = 1, x$ .

[F1]

Continuous-size coagulation-fragmentation weak model: coagulation has ordered-pair factor $1 / 2$ , while fragmentation vanishes when $β = 0$ .

Proof

1.1givenF1algebra

With $φ (x) = x$ , the coagulation increment is $(x + y) - x - y = 0$ . The integral equation therefore gives $M_{1} (μ_{t}) = M$ on the solution interval.

2.1givenF1step 1.1algebra∎

With $φ = 1$ , the increment is $1 - 1 - 1 = - 1$ , so $N^{'} (t) = - κ N (t)^{2} / 2$ . If $N_{0} > 0$ , then on every interval where $N > 0$ , differentiating $1 / N (t)$ gives $(1 / N)^{'} = κ / 2$ , hence $N (t) = N_{0} / (1 + κ N_{0} t / 2)$ . This expression is positive at every finite $t \geq 0$ , so continuity precludes a first finite zero and the formula holds throughout the solution interval. If $N_{0} = 0$ , positivity gives $N (t) \geq 0$ and the differential equation gives $N^{'} (t) \leq 0$ , so $N (t) = 0$ , again the formula.

Depends on

Continuous-size coagulation-fragmentation weak model

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

3 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.

Sources

Philippe Laurençot, Weak compactness techniques and coagulation equations, §1 (standard reference, not scraped)