ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-6-luna)

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.

Continuous and discrete heat maximum bounds on sampled data

Example

On $Ω = (0, 1)^{2}$ , the function

u (x, y, t) = \sin (π x) \sin (π y) e^{- 2 π^{2} t}

solves the unit-diffusivity heat equation with zero Dirichlet data and initial values in $[0, 1]$ . It remains in $[0, 1]$ continuously. On an equal-spacing grid, forward Euler also remains in $[0, 1]$ provided $2 Δ t / h^{2} \leq 1 / 2$ .

Verification

Given: $T > 0$ , $N \geq 2$ , $h = 1 / N$ , $M \geq 1$ , $Δ t = T / M$ with $2 Δ t / h^{2} \leq 1 / 2$ , and the exact sampled initial and zero Dirichlet boundary data for the forward-Euler heat scheme with $κ_{x} = κ_{y} = 1$ .

[F1] Fully discrete θ heat finite-difference scheme: the centered Dirichlet update imposes the specified boundary rows at every time level.

[F2] A boundary-aware discrete maximum principle for the θ scheme: for the zero-source Dirichlet θ scheme, if $Δ t (1 - θ) (κ_{x} / h_{x}^{2} + κ_{y} / h_{y}^{2}) \leq 1 / 2$ , all grid values lie between the initial and boundary extrema.

1.1givenalgebra

Direct differentiation gives $u_{t} = - 2 π^{2} u$ and $Δ u = - 2 π^{2} u$ , so $u_{t} - Δ u = 0$ . Also $u = 0$ on the four sides and $u (x, y, 0) = \sin (π x) \sin (π y) \in [0, 1]$ .

2.1givenstep 1.1algebra

On $[0, 1]$ , $0 \leq \sin (π x), \sin (π y) \leq 1$ , and for $t \geq 0$ the factor $e^{- 2 π^{2} t}$ lies in $(0, 1]$ . The explicit formula therefore gives $0 \leq u \leq 1$ throughout the closed cylinder.

3.1F1F2givenalgebra∎

The sampled initial values lie in $[0, 1]$ and the boundary rows are zero. With $θ = 0$ and $κ_{x} = κ_{y} = 1$ , the condition in [F2] is exactly $2 Δ t / h^{2} \leq 1 / 2$ . Therefore [F2] gives $0 \leq U_{i j}^{n} \leq 1$ at every grid node and time level.

Remarks

The continuous supplier is used only for unit diffusivity on a bounded rectangle with smooth exact data. The discrete conclusion follows from the separately proved finite-grid boundary condition.

Depends on

Used by

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