TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-6.1-sol)

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.

Domain of dependence and local uniqueness

Statement

Assume the Axiom of Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )). Let $n \geq 1$ , $c > 0$ , $x_{0} \in R^{n}$ , $t_{0} > 0$ , and let $u, v \in C^{2}$ solve $□_{c} u = f$ and $□_{c} v = g$ on a neighbourhood of the closed backward cone $K^{-} (x_{0}, t_{0})$ (Forward and backward wave cones, domain of dependence and influence). If $f = g$ on $K^{-} (x_{0}, t_{0})$ and $u (\cdot, 0) = v (\cdot, 0)$ , $u_{t} (\cdot, 0) = v_{t} (\cdot, 0)$ on the base ball $B_{c t_{0}} (x_{0})$ , then $u = v$ on $K^{-} (x_{0}, t_{0})$ .

This is the formal content, required by the design's well-definedness note, of the phrase "the value at $(x_{0}, t_{0})$ depends only on the Cauchy data on the base ball and the source on the cone".

Facts & Assumptions

Given: ${A C}_{ω}$ ; two $C^{2}$ functions $u, v$ solving $□_{c} u = f$ , $□_{c} v = g$ near the cone $K^{-} (x_{0}, t_{0})$ , with $f = g$ on the cone and equal Cauchy data on the base ball $B_{c t_{0}} (x_{0})$ .

[F1]

Finite propagation: if $w$ solves $□_{c} w = h$ near $K^{-} (x^{'}, t^{'})$ with $h = 0$ on the cone and zero Cauchy data on its base ball, then $w = 0$ on $K^{-} (x^{'}, t^{'})$ . (Finite propagation speed for the wave equation)

[F2]

The wave operator is linear, so $u - v$ solves $□_{c} (u - v) = f - g$ and the Cauchy data of the difference are the differences of the data. (Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ , Wave equation, Cauchy data and wave speed)

Proof

1.1givenF2

The difference solves a homogeneous equation on the cone: $w : = u - v$ is $C^{2}$ on a neighbourhood of $K^{-} (x_{0}, t_{0})$ and, by linearity of differentiation [F2], $□_{c} w = f - g$ , which vanishes on the cone because $f = g$ there; moreover $w (\cdot, 0) = 0$ and $w_{t} (\cdot, 0) = 0$ on the base ball $B_{c t_{0}} (x_{0})$ because the Cauchy data of $u$ and $v$ agree there.

2.1step 1.1F1∎

Conclusion: the hypotheses of [F1] are met by $w$ on the cone $K^{-} (x_{0}, t_{0})$ , so $w = 0$ there, that is, $u = v$ on $K^{-} (x_{0}, t_{0})$ ; in particular the value at the vertex is determined by the base data and the source on the cone.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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Sources

Victor Ivrii, Partial Differential Equations (University of Toronto, 2018, CC BY-SA) (standard reference, not scraped)
Gerald Teschl, Partial Differential Equations: From Classical to Modern (2025 archived author manuscript, AMS Graduate Studies in Mathematics) (standard reference, not scraped)
Jared Speck, MIT 18.152 Introduction to Partial Differential Equations, Class Meeting #13-14: Geometric Energy Estimates (Fall 2011) (standard reference, not scraped)