Alphabeta Math
ExampleConstruction: Literature-sourcedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-05
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-velocity transport translates the initial profile

Example

For a constant vector b∈Rn and a datum u0∈C1(Rn), the equation

ut+b⋅∇xu=0,u(x,0)=u0(x),

has solution

u(x,t)=u0(x−bt).

Facts & Assumptions

Given: A constant velocity b, a datum u0∈C1(Rn), and the homogeneous transport equation above.

[L1]

Homogeneous transport is represented by the inverse characteristic flow (Homogeneous linear transport is solved by the inverse characteristic flow).

Verification

technique · direct
1.1

The characteristic ODE is X′(s)=b, so the characteristic through x at time t is X(s;t,x)=x+(s−t)b, and in particular X(0;t,x)=x−bt.

given
2.1

Apply [L1] with the characteristic from step 1.1; the transport solution is therefore u(x,t)=u0(X(0;t,x))=u0(x−bt).

L1step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

4 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