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.

Normal, oblique, and grazing impedance reflection

Example

Write $q = k \cos θ$ for $0 \leq θ < π / 2$ . The reflected amplitude is $R (θ) = \frac{\cos θ - 1}{\cos θ + 1} .$ It is $0$ at normal incidence, equals $- 1 / 3$ at $θ = π / 3$ , and tends to $- 1$ at grazing incidence.

Facts & Assumptions

Given: $k > 0$ , $0 \leq θ < π / 2$ , and the reflection theorem [F1].

[F1]

Exact plane-wave reflection coefficient of the local impedance condition: for $q = k \cos θ > 0$ , $R = (q - k) / (q + k)$ .

Verification

Given: $k > 0$ , $0 \leq θ < π / 2$ , and [F1].

1.1F1algebra

Divide numerator and denominator of [F1] by $k > 0$ to obtain $R = (\cos θ - 1) / (\cos θ + 1)$ . At $θ = 0$ , this is $0$ ; at $θ = π / 3$ , $\cos θ = 1 / 2$ and it is $(- 1 / 2) / (3 / 2) = - 1 / 3$ .

2.1F1step 1.1algebra∎

As $θ ↑ π / 2$ , continuity of cosine gives $\cos θ \to 0$ and the denominator tends to $1$ , so $R \to - 1$ and $∣ R ∣ \to 1$ . The endpoint is a limit; the propagating theorem assumed $q > 0$ .

Depends on

Exact plane-wave reflection coefficient of the local impedance condition

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

2 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

Galkowski, Lafontaine, and Spence, Local absorbing boundary conditions on fixed domains give order-one errors for high-frequency waves (standard reference, not scraped)