Physics theoremStatement: AI-adaptedProof: AI-generatedPipeline-generatedjudge pass (gpt-6-luna)

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Causal stable perfect-material linear response

Physical scope

Adiabatic perfect-material first variations about the specified uniform Minkowski rest state with h0>0 and 0<a_s<=1; linear global evolution only, not nonlinear or curved-spacetime fluid existence or empirical applicability.

Statement

Assume the adiabatic perfect-material linear branch of Specified small-perturbation material dynamics on its prescribed Minkowski background, with $c > 0$ , $h_{0} > 0$ , and $0 < a_{s} \leq 1$ , and assume the Axiom of Countable Choice ${A C}_{ω}$ . For every initial datum $U_{0} = (r_{0}, w_{0}) \in S (R^{3}; R^{4})$ or $U_{0} \in C^{\infty} (T_{L}^{3}; R^{4})$ with $L > 0$ , the Cauchy problem $U (0, \cdot) = U_{0}$ for the system

$r_{t} + c h_{0} div w = 0, w_{t} + \frac{c a_{s}}{h_{0}} \nabla r = 0$

has a unique global smooth solution on the corresponding space-time domain, contractive in the positive quadratic energy and its weighted Sobolev norms. Its energy density

$E_{p f} = \frac{1}{2} [\frac{a_{s}}{h_{0}} r^{2} + h_{0} ∣ w ∣^{2}]$

satisfies $\partial_{t} E_{p f} + div (c a_{s} r w) = 0$ . For each Sobolev exponent $q \geq 0$ , its weighted Sobolev norm is nonincreasing for $t \geq 0$ . The characteristic speeds, meaning eigenvalues of the principal matrix in a unit spatial direction, are $0$ and $\pm c \sqrt{a_{s}}$ . Finite propagation means that if initial data vanish on a ball of radius $R$ , then at time $t$ they vanish on the concentric ball of radius $R - c \sqrt{a_{s}} t$ whenever $0 \leq t < R / (c \sqrt{a_{s}})$ ; on $T_{L}^{3}$ use geodesic balls. Thus the exact linear propagation speed is $c \sqrt{a_{s}} \leq c$ . Equality is permitted at $a_{s} = 1$ . This is a conditional model deduction and makes no claim about nonlinear or curved-spacetime evolution, or about nature beyond the adopted model.

Facts & Assumptions

Given: The physical time and spatial coordinates $(t, x)$ , the adiabatic perfect branch, and its state parameters and units as specified in Specified small-perturbation material dynamics.

[A1]

$c > 0$ , $h_{0} > 0$ , and $0 < a_{s} \leq 1$ . The strict lower bound is required for the positive definite energy; the upper bound is the additional causal-branch hypothesis.

[A2]

Assume ${A C}_{ω}$ : for every sequence $(X_{n})_{n \in N}$ of nonempty sets there is a function $f$ with $f (n) \in X_{n}$ for every $n$ (The Axiom of Countable Choice ( ${A C}_{ω}$ )). This is the exact choice strength used by the mathematical supplier proof; no stronger choice principle is assumed.

[F1]

Specified small-perturbation material dynamics: the perfect branch uses $e \in C^{2} (X)$ , $n_{0}, T_{0}, h_{0} > 0$ , $a_{s} = e_{v v} / (n_{0} h_{0}) > 0$ , and $δ σ_{p} = 0$ , and adopts the displayed perfect system. In particular $e_{v v} > 0$ follows from the response identity and these strict inequalities; [A1] supplies the additional upper bound $a_{s} \leq 1$ . Put $U = (r, w) : R \times R^{3} \to R^{4}$ .

[F2]

Perfect sound characteristic and actual stable causal linear branch (current batch-56 mathematical draft): under its exact state and adiabatic hypotheses, $c > 0$ , $a_{s} = e_{v v} / (n_{0} h_{0}) \in (0, 1]$ , and [A2], the complete conditional proof supplies the same constant-coefficient system's positive energy, unique global smooth solutions for $U_{0} \in S (R^{3}; R^{4})$ or $C^{\infty} (T_{L}^{3}; R^{4})$ , and contraction $∥ U (t) ∥_{H_{M}^{q}} \leq ∥ U_{0} ∥_{H_{M}^{q}}$ for each $q \geq 0$ . On $R^{3}$ , $M = diag (a_{s} / h_{0}, h_{0} I_{3})$ (where $I_{3}$ is the $3 \times 3$ identity), $\hat{U} (ξ) = \int_{R^{3}} e^{- 2 π i x \cdot ξ} U (x) d x$ , and $∥ U ∥_{H_{M}^{q}}^{2} = \int_{R^{3}} (1 + ∣ ξ ∣^{2})^{q} \hat{U} (ξ)^{*} M \hat{U} (ξ) d ξ$ . On the torus, ${\hat{U}}_{k} = L^{- 3} \int_{[0, L]^{3}} U (x) e^{- 2 π i k \cdot x / L} d x$ and $∥ U ∥_{H_{M}^{q}}^{2} = \sum_{k \in Z^{3}} (1 + ∣ k / L ∣^{2})^{q} {\hat{U}}_{k}^{*} M {\hat{U}}_{k}$ . It gives speeds $0, \pm c \sqrt{a_{s}}$ and the ball-shrinking finite-propagation property stated above. The state, response, data-class and countable-choice hypotheses here match that exact mathematical interface.

Proof

1.1A1F1givenalgebra

By [F1] the physical model's perfect branch is exactly the displayed constant-coefficient system, with $U = (r, w)$ , physical time $t$ , and the same constants $c, h_{0}, a_{s}$ . Hypothesis [A1] supplies the parameter restrictions required for the positive-energy causal branch.

2.1A1A2F2step 1.1

Apply [F2] under [A2] to this identical system and matching state, parameter, and data hypotheses. It yields the stated energy density and flux identity, unique global smooth solutions for both specified data classes, contraction in the stated norms, characteristic speeds $0, \pm c \sqrt{a_{s}}$ , and finite propagation at $c \sqrt{a_{s}}$ .

3.1A1A2F2step 2.1algebra∎

Since $0 < a_{s} \leq 1$ and $c > 0$ , multiplication by $c$ and monotonicity of the nonnegative square root give $c \sqrt{a_{s}} \leq c$ , with equality when $a_{s} = 1$ . The conclusion is therefore causal relative to the specified Minkowski reference cone and remains conditional on the adopted model, [A2], and current mathematical supplier.

Causal stable perfect-material linear response

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources