ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedprecheck pendingverified 2026-10-23 (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.

A four-axis parameterized family of planar velocity fields

Physical scope

A constructed nonrelativistic Euclidean data-domain example for a family of planar velocity fields; it asserts no numerical result, experiment, or physical prediction.

Example

Let $Ω \subset R^{2}$ be a nonempty open planar spatial region with coordinates $x, y$ in metres, let $I \subset R$ be a nonempty open time interval measured in seconds, and let $Θ \subset (0, \infty)$ be a nonempty open interval of dimensionless Reynolds numbers. Fix a characteristic speed $U > 0$ in metres per second and characteristic length $L > 0$ in metres. For each $θ \in Θ$ , set the kinematic viscosity to $ν (θ) : = U L / θ$ in square metres per second, so that the Reynolds number $U L / ν (θ)$ equals $θ$ . A parameterized velocity family is a continuously differentiable map $u : Ω \times I \times Θ \to R^{2}$ whose value is a planar velocity vector in metres per second; no equation or computed value is stipulated. Sampling this family on grids in $x, y, t, θ$ gives a data array with four independent input axes. For fixed $θ$ , its spatial domain remains $Ω \subset R^{2}$ and its physical space-time coordinates are $x, y, t$ : Reynolds number indexes different simulation settings and is not an additional spatial or time coordinate, and the two output components of $u$ are not input axes.

Facts & Assumptions

Given: The stated nonempty open sets $Ω$ , $I$ , and $Θ$ ; the fixed positive scales $U$ and $L$ ; the map $u$ ; SI units; and the stated interpretation of $θ$ as Reynolds number.

[A1]

The input domain is the product $Ω \times I \times Θ \subset R^{2} \times R \times R$ , with coordinates $(x, y, t, θ)$ ; $u$ maps this four-coordinate domain to $R^{2}$ .

Verification

1.1givenA1algebra

Since $Ω$ , $I$ , and $Θ$ are nonempty open subsets of $R^{2}$ , $R$ , and $R$ , respectively, their product is open in $R^{4}$ and its independent coordinates are $x, y, t, θ$ . Thus a sample grid that varies all four factors has four input axes.

2.1givenA1step 1.1algebra∎

Fixing $θ$ leaves $Ω \times I$ with coordinates $x, y, t$ , and fixing both $t$ and $θ$ leaves the spatial slice $Ω \subset R^{2}$ . The codomain $R^{2}$ gives two velocity components but adds no input coordinate. Hence the family has two spatial coordinates, one time coordinate, and one extra dimensionless parameter axis, not four spatial coordinates or four space-time coordinates.

Dependency tree · 0 levels

Nothing. This result depends on no other item in the library.