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.

A constant-speed foot can cross many cells in one stable update

Example

Take the one-dimensional torus of length $1$ , $16$ uniform cells of width $h = 1 / 16$ , constant velocity $b = 1$ , one step $Δ t = 3 / 10$ , and arrival node $x_{0} = 0$ . The Courant displacement is $b Δ t / h = 4.8$ , so the backtracked foot crosses more than four cells. Give the two grid nodes bracketing that foot values $2$ and $10$ .

Facts & Assumptions

Given: The grid and velocity in the Example and the explicit Euler/multilinear update of Explicit Euler characteristic backtracking with multilinear interpolation.

[F1]

Periodic tensor grids and multilinear interpolation: with $h = 1 / 16$ , a lifted query $y$ has cell index $q = ⌊ y / h ⌋$ , fraction $θ = y / h - q$ , and weights $1 - θ, θ$ at indices $q, q + 1$ modulo $16$ .

[F2]

Periodic multilinear interpolation is convex and L-infinity contractive: the two weights are nonnegative and sum to one.

Proof

1.1givenF1algebra

Since $b = 1$ , the explicit Euler foot at $x_{0} = 0$ is $- 3 / 10$ , which wraps to $7 / 10$ on the unit torus. On the lift, $(- 3 / 10) / h = - 24 / 5 = - 5 + 1 / 5$ , so the lower index is $- 5 \equiv 11 (m o d 16)$ , the upper index is $12$ , and $θ = 1 / 5$ .

2.1givenF2step 1.1algebra∎

The update is therefore $(1 - θ) 2 + θ 10 = (4 / 5) 2 + (1 / 5) 10 = 18 / 5$ . By [F2] this value lies in $[2, 10]$ , despite the 4.8-cell displacement. The example uses no displacement CFL restriction.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

5 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

European Centre for Medium-Range Weather Forecasts, Numerical Methods, Meteorological Training Course Lecture Series, 2002 (standard reference, not scraped)