Physics theoremStatement: AI-adaptedProof: 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.

Local positive-density polytropic model evolution

Physical scope

Unforced, smooth, positive-density barotropic Euler model on a periodic cell, with one fixed polytropic parameter and exponent; dimensions one and two are transversely invariant sectors of the three-dimensional model. It adopts specific internal energy $e_{p h y s} (ρ) = κ_{p h y s} ρ^{γ - 1} / (γ - 1)$ for the model energy and asserts local smooth evolution and its mean-energy conservation. It does not cover vacuum, shocks, thermal coupling, spatially varying $κ_{p h y s}$ or $γ$ , or a molecular derivation of either constitutive law.

Statement

Assume the Axiom of Countable Choice. Let $d \in {1, 2, 3}$ , $L > 0$ , reference density $ρ_{*} > 0$ in kilograms per cubic metre, reference speed $U_{*} > 0$ in metres per second, physical polytropic coefficient $κ_{p h y s} > 0$ in pascals times $(m^{3} / k g)^{γ}$ , and dimensionless exponent $γ > 1$ . Let $I \subset R$ be an open time interval containing $0$ and $T_{L}^{d} = (R / (2 π L Z))^{d}$ , with position $X$ in metres and time $t$ in seconds; this is the physical cell for $d = 3$ and the reduced coordinate torus for $d = 1, 2$ , whose physical cell is $T_{L}^{3}$ . The smooth model fields are $ρ_{p h y s} \in C^{\infty} (I \times T_{L}^{d}; (0, \infty))$ and $u_{p h y s} \in C^{\infty} (I \times T_{L}^{d}; R^{d})$ . Torus fields are understood as smooth periodic lifts to Euclidean coordinates when applying the local continuum equations. The physical density $ρ_{p h y s}$ is measured in kilograms per cubic metre, velocity $u_{p h y s}$ in metres per second, and pressure $p_{p h y s}$ in pascals. Adopt the unforced, positive-density barotropic Euler model $\partial_{t} ρ_{p h y s} + \nabla_{X} \cdot (ρ_{p h y s} u_{p h y s}) = 0, \partial_{t} u_{p h y s} + (u_{p h y s} \cdot \nabla_{X}) u_{p h y s} + ρ_{p h y s}^{- 1} \nabla_{X} ⁣ (κ_{p h y s} ρ_{p h y s}^{γ}) = 0,$ where the same fixed $κ_{p h y s}$ and $γ$ apply throughout the flow. Also adopt the specific internal-energy closure $e_{p h y s} (ρ) = κ_{p h y s} ρ^{γ - 1} / (γ - 1)$ in joules per kilogram, so the model internal-energy density is $ρ e_{p h y s} = κ_{p h y s} ρ^{γ} / (γ - 1)$ in pascals and $ρ^{2} e_{p h y s}' (ρ) = κ_{p h y s} ρ^{γ} = p_{p h y s} (ρ)$ for $ρ > 0$ . This is an additional constitutive assumption for the model energy; neither it nor the pressure law is derived from molecular mechanics or from an empirical measurement here.

For $d = 3$ , these are the direct three-dimensional equations. For $d = 1, 2$ , define $J_{d} : T_{L}^{3} \to T_{L}^{d}$ by $J_{d} (X_{1}, X_{2}, X_{3}) = (X_{1}, \dots, X_{d})$ , with coordinates modulo $2 π L$ , and lift reduced fields by ${\tilde{ρ}}_{p h y s} (t, X) = ρ_{p h y s} (t, J_{d} X), {\tilde{u}}_{p h y s} (t, X) = ((u_{p h y s})_{1} (t, J_{d} X), \dots, (u_{p h y s})_{d} (t, J_{d} X), 0, \dots, 0) .$ The three-dimensional pressure and Cauchy stress are ${\tilde{p}}_{p h y s} = κ_{p h y s} {\tilde{ρ}}_{p h y s}^{γ}$ and $\tilde{σ} = - {\tilde{p}}_{p h y s} I_{3}$ . Direct differentiation gives $\partial_{t} {\tilde{ρ}}_{p h y s} + \nabla_{3} \cdot ({\tilde{ρ}}_{p h y s} {\tilde{u}}_{p h y s}) = (\partial_{t} ρ_{p h y s} + \nabla_{d} \cdot (ρ_{p h y s} u_{p h y s})) \circ J_{d},$ $\partial_{t} {\tilde{u}}_{p h y s} + ({\tilde{u}}_{p h y s} \cdot \nabla_{3}) {\tilde{u}}_{p h y s} + {\tilde{ρ}}_{p h y s}^{- 1} \nabla_{3} (κ_{p h y s} {\tilde{ρ}}_{p h y s}^{γ}) = (\partial_{t} u_{p h y s} + (u_{p h y s} \cdot \nabla_{d}) u_{p h y s} + ρ_{p h y s}^{- 1} \nabla_{d} (κ_{p h y s} ρ_{p h y s}^{γ}), 0, \dots, 0) \circ J_{d} .$ Thus the lower-dimensional equations are exactly the mass and momentum reduction of the adopted three-dimensional barotropic Euler model for fields independent of the omitted coordinates and with zero omitted velocity components. Conversely, every smooth three-dimensional solution in this invariant class reduces to the stated equations. For a scalar field $f$ on $T_{L}^{d}$ and its lift $\tilde{f} = f \circ J_{d}$ , the physical-cell means satisfy $\frac{1}{(2 π L)^{3}} \int_{T_{L}^{3}} \tilde{f} (X) d X = \frac{1}{(2 π L)^{d}} \int_{T_{L}^{d}} f (X_{1}, \dots, X_{d}) d X_{1} \dots d X_{d};$ this follows by integrating the constant-in-transverse-coordinate lift over the remaining $3 - d$ periodic coordinates.

Set $x = X / L \in T^{d} = (R / (2 π Z))^{d}$ , $τ = t U_{*} / L$ , $R = ρ_{p h y s} / ρ_{*}$ , $V = u_{p h y s} / U_{*}$ , $P = p_{p h y s} / (ρ_{*} U_{*}^{2})$ , and $\overset{ˉ}{κ} = κ_{p h y s} ρ_{*}^{γ - 1} / U_{*}^{2} > 0$ . Put $a = (γ - 1) / 2$ and $r (R) = \sqrt{\overset{ˉ}{κ} γ} R^{a} / a$ . For smooth periodic $f$ , use normalized measure $d μ_{x} = (2 π)^{- d} d x$ , Fourier coefficients $\hat{f} (k) = \int f (x) e^{- i k \cdot x} d μ_{x}$ , and $∥ f ∥_{H^{m}}^{2} = \sum_{k \in Z^{d}} (1 + ∣ k ∣^{2})^{m} ∣ \hat{f} (k) ∣^{2}$ . Let $ρ_{0} : T_{L}^{d} \to (0, \infty)$ and $u_{0} : T_{L}^{d} \to R^{d}$ be smooth real periodic initial data, and define $R_{0} (x) = ρ_{0} (L x) / ρ_{*}$ and $V_{0} (x) = u_{0} (L x) / U_{*}$ . Then for every integer $s \geq 4$ there is $T > 0$ , depending only on $d, s, \overset{ˉ}{κ}, γ, ∥ (r (R_{0}), V_{0}) ∥_{H^{s}}$ , and $\min r (R_{0})$ , such that the normalized system $\partial_{τ} R + \nabla_{x} \cdot (R V) = 0, \partial_{τ} V + (V \cdot \nabla_{x}) V + R^{- 1} \nabla_{x} (\overset{ˉ}{κ} R^{γ}) = 0, R (0, \cdot) = R_{0}, V (0, \cdot) = V_{0}$ has a unique smooth real solution on $[0, T] \times T^{d}$ with $R (τ, x) > 0$ and every integer Sobolev order $m \geq s$ bounded on that same interval. The dimensionless mean total-energy density $\overline{E} (τ) = \int_{T^{d}} (\frac{1}{2} R ∣ V ∣^{2} + \frac{\overset{ˉ}{κ}}{γ - 1} R^{γ}) d μ_{x}$ is constant. Equivalently, $\frac{1}{(2 π L)^{3}} \int_{T_{L}^{3}} (\frac{1}{2} {\tilde{ρ}}_{p h y s} ∣ {\tilde{u}}_{p h y s} ∣^{2} + κ_{p h y s} {\tilde{ρ}}_{p h y s}^{γ} / (γ - 1)) d X$ is constant and has units of pascals; for $d = 3$ the tildes are omitted, and for $d = 1, 2$ this mean equals the reduced $d$ -dimensional cell mean by the formula above. The polytropic pressure law is an adopted constitutive closure, not a molecular derivation. No temperature-coupled equation, vacuum or shock continuation, spatially varying constitutive parameters, or empirical validity claim is included.

Statement references

François Golse, The Boltzmann Equation and Its Hydrodynamic Limits, §7.4.

Thomas C. Sideris, “Formation of Singularities in Three-Dimensional Compressible Fluids”.

David Tong, Fluid Mechanics, Chapter 4: Waves.

Facts & Assumptions

Given: Countable Choice; the SI units, positive reference scales, periodic cell, smooth data, fixed $κ_{p h y s} > 0$ , fixed $γ > 1$ , and unforced model in the Statement.

[A1]

Single-component nonpolar continuum and material balances: the adopted smooth continuum supplies mass and linear-momentum balances; this item uses the unforced, periodic, positive-density branch and asserts no solution of the full thermal continuum balance system.

[F1]

Smooth positive-density barotropic Euler model: the current definition specifies three-dimensional $C^{2}$ fields $ρ > 0$ and $u$ , a $C^{1}$ pressure law $P : (0, \infty) \to R$ , continuity and momentum equations, and Cauchy stress $- P (ρ) I_{3}$ ; it does not assume a sign for $P^{'} (ρ)$ . This use sets $P (ρ) = κ_{p h y s} ρ^{γ}$ ; the Statement gives the exact lower-dimensional mass and momentum reduction. The current batch-11 manifest registers this carrier, and step1Decision reports a current ready record; this records authoring readiness, not independent proof approval.

[F2]

Local smooth positive-density polytropic Euler: assuming Countable Choice, $d \in {1, 2, 3}$ , $\overset{ˉ}{κ} > 0$ , $γ > 1$ , smooth periodic data with strictly positive minimum density, and integer $s \geq 4$ , the normalized system with fixed pressure law $P (R) = \overset{ˉ}{κ} R^{γ}$ has a unique smooth positive-density solution on a parameter-dependent local interval with all higher Sobolev norms bounded. Its actual complete proof and current Step-1 ready record were inspected; independent acceptance remains a later review duty.

[F3]

Smooth polytropic energy conservation: the current theorem assumes Countable Choice, $d \in {1, 2, 3}$ , $\overset{ˉ}{κ} > 0$ , $γ > 1$ , integer $s \geq 4$ , and smooth periodic initial data with positive minimum density; the local Euler theorem supplies the unique smooth positive-density solution. It proves the local law $\partial_{t} E + div ((E + p) u) = 0$ and constancy of the torus mean for $E = \frac{1}{2} ρ ∣ u ∣^{2} + \overset{ˉ}{κ} ρ^{γ} / (γ - 1)$ . Its full current carrier proof was inspected; the batch-19 manifest registers this item.

[A2]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): “For every family $(X_{n})_{n \in N}$ of nonempty sets indexed by $N$ there is a function $f$ with domain $N$ such that $f (n) \in X_{n}$ for every $n$ .”

[A3]

The Statement adopts $e_{p h y s} (ρ) = κ_{p h y s} ρ^{γ - 1} / (γ - 1)$ as the barotropic model specific internal energy. The total-energy density in the claim is therefore the energy of this model closure, not an energy-temperature law derived from the continuum postulate or measured material data.

Proof

Given: The Statement hypotheses and exact conditional interfaces [A1], [A2], and [F1]–[F3].

Proof technique: Scale the adopted physical balance system to the normalized torus, apply the current local positive-density mathematical theorem and the current mathematical energy supplier, then scale the conserved mean back to SI variables.

1.1givenA1F1algebra

For $d = 3$ , the mass and momentum equations in the Statement are the adopted three-dimensional balance and barotropic-model equations [A1, F1]. For $d = 1, 2$ , the lifts defined in the Statement have density independent of the omitted coordinates and zero velocity components there; the displayed lift identities show that the 3D mass equation is exactly the reduced mass equation and the first $d$ momentum components are exactly the reduced momentum equation, with all omitted components zero. The definitions of $R, V, P, x, τ$ give $\partial_{t} ρ_{p h y s} = (ρ_{*} U_{*} / L) \partial_{τ} R$ and $\nabla_{d} \cdot (ρ_{p h y s} u_{p h y s}) = (ρ_{*} U_{*} / L) \nabla_{x} \cdot (R V)$ , so the mass equation becomes $\partial_{τ} R + \nabla_{x} \cdot (R V) = 0$ . Also $\partial_{t} u_{p h y s} + (u_{p h y s} \cdot \nabla_{d}) u_{p h y s} = (U_{*}^{2} / L) (\partial_{τ} V + (V \cdot \nabla_{x}) V)$ and $ρ_{p h y s}^{- 1} \nabla_{d} p_{p h y s} = (U_{*}^{2} / L) R^{- 1} \nabla_{x} P$ . Since $P = κ_{p h y s} ρ_{*}^{γ - 1} U_{*}^{- 2} R^{γ} = \overset{ˉ}{κ} R^{γ}$ , the velocity equation becomes exactly the normalized equation in the Statement.

2.1givenA2F2step 1.1

The hypotheses of [F2] hold for the normalized data and parameters, including Countable Choice [A2], $d \leq 3$ , integer $s \geq 4$ , $\overset{ˉ}{κ} > 0$ , $γ > 1$ , and $\min R_{0} > 0$ . Thus the normalized system has a unique smooth solution on the claimed interval, remains positive there, and has bounded Sobolev norms at all integer orders $m \geq s$ .

3.1givenF3step 2.1

Applying the exact conditional energy interface [F3] to that positive smooth periodic solution gives constancy of $\overline{E} (τ)$ throughout $[0, T]$ .

4.1givenA3step 1.1step 3.1algebra

The physical energy density equals $ρ_{*} U_{*}^{2} (\frac{1}{2} R ∣ V ∣^{2} + \overset{ˉ}{κ} R^{γ} / (γ - 1))$ . The change of variables $X = L x$ gives the normalized physical-cell average in dimension $d$ ; for $d = 1, 2$ , the full-cell average of its 3D lift equals this reduced average because the lifted density and velocity are independent of the omitted coordinates. Thus the physical mean energy density is $ρ_{*} U_{*}^{2} \overline{E} (τ)$ and is constant.

5.1givenA1F1step 2.1step 4.1∎

These deductions require the stated positive-density local branch and fixed polytropic closure; they assert no continuation through vacuum or shocks and no thermal or empirical conclusion.

Local positive-density polytropic model evolution

Statement

Statement references

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources