ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedprecheck pendingverified 2026-10-23 (gpt-6.1-sol)

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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.

Resting depth and stratification states

Example

Assume the Axiom of Countable Choice ${A C}_{ω}$ . Let $I \subseteq R$ be a nonempty open time interval. For $d = 2, 3$ , let $T^{d} : = R^{d} / (2 π Z)^{d}$ and represent fields by $2 π$ -periodic lifts, integrating over $Q_{d} = [0, 2 π]^{d}$ with $d x = d λ_{d} (x)$ as defined for the equation family. For any constants $f \in R$ , $N > 0$ , and $ν, κ \geq 0$ , the fields $u = b = π = 0$ on $I \times T^{3}$ solve the rotating linear stratified equations and have $E_{s} = \frac{1}{2} \int_{Q_{3}} (∣ u ∣^{2} + b^{2} / N^{2}) d x = 0$ . For any constants $f \in R$ , $g > 0$ , and $H > 0$ , the fields $h \equiv H$ , $v \equiv 0$ on $I \times T^{2}$ solve the one-layer rotating shallow-water equations and have normalized total energy $E_{h} = \int_{Q_{2}} (h ∣ v ∣^{2} / 2 + g h^{2} / 2) d x = 2 π^{2} g H^{2}$ . Both are stationary smooth periodic states of the stated mathematical systems, not observed fluid states.

Example references

Rotating stratified and shallow-water equations: Rotating stratified and shallow-water equation families.

Countable Choice ${A C}_{ω}$ , assumed for the Lebesgue measure convention: The Axiom of Countable Choice ( ${A C}_{ω}$ ).

Lebesgue measurability and volume of $Q_{d}$ : A box in $R^{n}$ with parameters $a_{i} \leq b_{i}$ is Lebesgue measurable of measure $\prod_{i < n} (b_{i} - a_{i})$ , whichever of its faces are included.

The smooth rotating shallow-water energy balance: Smooth rotating shallow-water energy balance.

Facts & Assumptions

Given: Countable Choice ${A C}_{ω}$ , $f \in R$ , $N > 0$ , $g > 0$ , $H > 0$ , and the periodic equation systems and energy definitions from the stated suppliers.

[A1]

Countable Choice ${A C}_{ω}$ means that every $N$ -indexed family of nonempty sets has a choice function (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

[F1]

Rotating stratified and shallow-water equation families: the rotating, linear, constant-stratification Boussinesq equation family is $u_{t} + f e_{3} \times u + \nabla π = b e_{3} + ν Δ u$ , $\nabla \cdot u = 0$ , and $b_{t} + N^{2} u_{3} = κ Δ b$ .

[F2]

Rotating stratified and shallow-water equation families: the inviscid one-layer shallow-water equation family is $h_{t} + \nabla \cdot (h v) = 0$ and $(h v)_{t} + \nabla \cdot (h v \otimes v) + \nabla (g h^{2} / 2) + f h J v = 0$ .

[F3]

Smooth rotating shallow-water energy balance: for every smooth periodic positive-depth solution of the flat-bottom, inviscid shallow-water system, the energy density $E = h ∣ v ∣^{2} / 2 + g h^{2} / 2$ satisfies $\partial_{t} E + \nabla \cdot ((E + g h^{2} / 2) v) = 0$ and its integral over $Q_{2}$ is conserved.

[F4]

Under ${A C}_{ω}$ , the closed box $Q_{d} = [0, 2 π]^{d}$ is Lebesgue measurable and has measure $(2 π)^{d}$ (A box in $R^{n}$ with parameters $a_{i} \leq b_{i}$ is Lebesgue measurable of measure $\prod_{i < n} (b_{i} - a_{i})$ , whichever of its faces are included).

Verification

1.1givenF1algebra

For $u = b = π = 0$ , every time derivative, spatial derivative, Coriolis term, diffusion term, and buoyancy coupling in the stratified equations vanishes; hence all equations and incompressibility hold pointwise. The stated energy integrand is identically zero.

1.2givenF2F3algebra

For $h \equiv H > 0$ and $v \equiv 0$ , the mass time derivative and flux vanish, and every term in the conservative momentum equation vanishes because $h$ is spatially constant and $v = 0$ . Thus this is a smooth periodic solution to which [F3] applies.

2.1givenA1F3F4step 1.2algebra∎

Substitution into [F3]'s energy density gives $E = g H^{2} / 2$ . By [F4], the cell $Q_{2}$ has Lebesgue area $(2 π)^{2}$ , so its integral is $(2 π)^{2} g H^{2} / 2 = 2 π^{2} g H^{2}$ , independent of time.

Resting depth and stratification states

Example

Example references

Facts & Assumptions

Verification

Depends on

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