Alphabeta Math
CounterexampleConstruction: Literature-sourcedVerification: AI-generatedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-06
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  • 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.

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Quasilinear characteristics can cross before the lifted ODE blows up

Statement refuted

If the lifted characteristic ODE continues, then the quasilinear PDE remains a single-valued classical graph.

Counterexample

Given: Burgers data u0(ξ)=−tanh⁡ξ.

Proof technique: direct.

1.1givenalgebra

The lifted characteristic is Z(t,ξ)=−tanh⁡ξ and X(t,ξ)=ξ−ttanh⁡ξ, both defined for every t≥0.

2.1step 1.1algebra

Yet Xξ(1,0)=0, so the projected map has a caustic at t=1.

3.1step 2.1given∎

Thus the lifted ODE persists while inverse projection to a graph fails, refuting the statement.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

6 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