ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedprecheck passverified 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.

Exact Reynolds and Mach nondimensionalization — worked interpretation

Physical scope

Formal periodic smooth-model example with explicitly selected SI parameters and fields. The values are mathematical inputs, not measurements or representative data for a named fluid. No approximation, limit, turbulence, or empirical conclusion is asserted.

Example

Use SI units and select the formal inputs $ρ_{0} = 10^{3} k g m^{- 3}$ , $μ = 10^{- 3} P a s$ , $U = 1 m s^{- 1}$ , $L = 10^{- 1} m$ , and $c = 10^{2} m s^{- 1}$ . These are exact illustrative parameters, not measurements. Then $ν = μ / ρ_{0} = 10^{- 6} m^{2} s^{- 1}$ , $Re = U L / ν = 10^{5}$ , $Ma = U / c = 10^{- 2}$ , and ${Ma}^{- 2} = 10^{4}$ .

Let $J = (- 1, 1)$ be a dimensionless open time interval containing $Θ = 0$ and let the physical time interval be $I = (L / U) J = (- L / U, L / U)$ seconds. For a side length $ℓ > 0$ , write $T_{ℓ}^{3} : = R^{3} / (ℓ Z^{3})$ and let $[x]_{ℓ}$ denote the class of $x \in R^{3}$ modulo $ℓ Z^{3}$ . Define the dimensionless flat torus $T_{2 π}^{3}$ and the physical flat torus $T_{2 π L}^{3}$ . The map $D_{L} : T_{2 π L}^{3} \to T_{2 π}^{3}, D_{L} ([x]_{2 π L}) = [x / L]_{2 π},$ is a well-defined smooth bijection with inverse $[X]_{2 π} \mapsto [L X]_{2 π L}$ . The quotient, not the open cube, is the periodic spatial domain; $[0, 2 π)^{3}$ is only a representative cell.

Define the periodic lift ${\tilde{u}}^{*} : J \times R^{3} \to R^{3}$ and pressure field ${\tilde{p}}_{i n c}^{*} : J \times R^{3} \to R$ by ${\tilde{u}}^{*} (Θ, X) = (e^{- Θ / Re} \sin X_{2}, 0, 0), {\tilde{p}}_{i n c}^{*} (Θ, X) = 0.$ For every $k \in Z^{3}$ , ${\tilde{u}}^{*} (Θ, X + 2 π k) = {\tilde{u}}^{*} (Θ, X)$ , so these smooth fields descend to $u^{*} : J \times T_{2 π}^{3} \to R^{3}$ and $p_{i n c}^{*} : J \times T_{2 π}^{3} \to R$ . Spatial derivatives on this flat torus are defined by differentiating the periodic lifts. They satisfy the scaled incompressible equations $\partial_{Θ} u^{*} + (u^{*} \cdot \nabla_{X}) u^{*} = - \nabla_{X} p_{i n c}^{*} + {Re}^{- 1} Δ_{X} u^{*}, {div}_{X} u^{*} = 0.$ Under the spatial dilation $D_{L}$ and the time map $Θ = U t / L$ , define the corresponding physical fields on $I \times T_{2 π L}^{3}$ by $u (t, [x]_{2 π L}) = U u^{*} (U t / L, D_{L} ([x]_{2 π L})), p (t, [x]_{2 π L}) = p_{r e f} + ρ_{0} U^{2} p_{i n c}^{*} (U t / L, D_{L} ([x]_{2 π L})) = p_{r e f} .$ Equivalently, the physical velocity lift is $\tilde{u} (t, x) = U {\tilde{u}}^{*} (U t / L, x / L)$ , which is periodic under $x \mapsto x + 2 π L k$ for every $k \in Z^{3}$ ; physical spatial derivatives on the quotient are defined by this lift. A representative physical cell has side length $2 π L$ . This is an exact solution of the selected incompressible equations.

On the same dimensionless spacetime domain $J \times T_{2 π}^{3}$ , for the separate barotropic branch choose the pressure law $P : (0, \infty) \to R$ given by $P (ρ) = 10^{5} P a + c^{2} (ρ - ρ_{0})$ and the constant fields $ρ^{*} : J \times T_{2 π}^{3} \to (0, \infty)$ , $u^{*} : J \times T_{2 π}^{3} \to R^{3}$ , and $p_{b a r}^{*} : J \times T_{2 π}^{3} \to R$ , with values $ρ^{*} = 1$ , $u^{*} = 0$ , and $p_{b a r}^{*} = 0$ . For these fields the scaled equations $\partial_{Θ} ρ^{*} + {div}_{X} (ρ^{*} u^{*}) = 0, \partial_{Θ} u^{*} + (u^{*} \cdot \nabla_{X}) u^{*} = - {Ma}^{- 2} (ρ^{*})^{- 1} \nabla_{X} p_{b a r}^{*}$ hold identically. They are smooth, $P^{'} (ρ_{0}) = c^{2} > 0$ , and their corresponding physical fields $ρ : I \times T_{2 π L}^{3} \to (0, \infty)$ and $u : I \times T_{2 π L}^{3} \to R^{3}$ have the constant values $ρ (t, [x]_{2 π L}) = ρ_{0}$ and $u (t, [x]_{2 π L}) = 0$ . This is an exact equilibrium of the unforced barotropic model. The computed coefficients and fields illustrate algebra in the adopted equations only; they report no observation and imply no low-Mach or high-Reynolds approximation.

Example references

Exact conditional Reynolds and Mach model prediction

Facts & Assumptions

Given: The exact parameter selection, periodic cell, smooth fields, and barotropic pressure law in the Example.

[F1]

Exact Reynolds and Mach nondimensionalization — conditional physical prediction: for positive scales, the unforced constant-density incompressible Newtonian system and the unforced positive-density barotropic Euler system have the exact scaled equations stated there. This example is conditional on those adopted model formulations and is not empirical evidence.

Verification

Given: The formal SI inputs and the two selected model fields in the Example. Use $Θ$ for dimensionless time, distinct from any physical temperature.

1.1givenalgebra

The definitions give $ν = μ / ρ_{0} = 10^{- 3} / 10^{3} = 10^{- 6} m^{2} s^{- 1}$ . Therefore $Re = U L / ν = (1) (10^{- 1}) / 10^{- 6} = 10^{5}$ and $Ma = U / c = 1 / 10^{2} = 10^{- 2}$ , so ${Ma}^{- 2} = 10^{4}$ .

2.1F1givenstep 1.1algebra

For the selected shear field, ${div}_{X} u^{*} = 0$ because its first component is independent of $X_{1}$ , and $(u^{*} \cdot \nabla_{X}) u^{*} = 0$ for the same reason. Direct differentiation gives $\partial_{Θ} u^{*} = - {Re}^{- 1} u^{*}$ and $Δ_{X} u^{*} = - u^{*}$ . With $p_{i n c}^{*} = 0$ , the momentum equation is thus satisfied componentwise. Apply [F1] to the smooth periodic lift on the open domain $R^{3}$ ; its physical lift satisfies the dimensional incompressible equations and is $2 π L$ -periodic in each spatial coordinate, so it descends to the stated physical torus.

2.2F1givenstep 1.1algebra

For the linear pressure law, $P^{'} (ρ_{0}) = c^{2} > 0$ . At $ρ^{*} = 1$ , $p_{b a r}^{*} = 0$ ; the selected constant density and zero velocity make both the mass equation and the barotropic momentum equation vanish on each side. The coefficient $10^{4}$ multiplies a zero pressure gradient in this equilibrium and is not an approximation parameter conclusion.

3.1F1step 1.1step 2.1step 2.2∎

Every selected value and field is a formal input or exact solution of the stated model equations; none is a measured result. The calculation supplies no limit or empirical claim.

Depends on

Exact Reynolds and Mach nondimensionalization — conditional physical prediction