ExampleConstruction: AI-adaptedVerification: AI-adaptedPipeline-generatedprecheck passverified 2026-10-22 (gpt-6.1-sol)

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Matter energy survives zero instantaneous polarization

Physical scope

A single allowed oscillator state in the undamped Lorentz model. It illustrates the model's positive kinetic material-energy term when the instantaneous polarization vanishes; it reports no experiment and makes no claim beyond the stated constitutive model.

Statement

In the one-oscillator Lorentz model, take an undamped oscillator $γ = 0$ with $a > 0$ in F m $^{- 1}$ s $^{- 2}$ and $ω_{0} > 0$ in s $^{- 1}$ . At a time $t_{0}$ , compare two allowed states with the same instantaneous fields $E = 0$ , $B = 0$ , $H = 0$ , polarization $P = 0$ , and therefore displacement $D = ε_{\infty} E + P = 0$ : in the first state set $P_{t} = 0$ , and in the second set $P_{t} = v$ for any fixed nonzero real vector $v \in R^{3}$ in A m $^{- 2}$ . The second state has oscillator energy density $∣ v ∣^{2} / (2 a) > 0$ in J m $^{- 3}$ , while the first has zero oscillator energy. Thus the instantaneous pair $(E, D)$ alone does not determine the full stored energy in this dispersive model.

Statement references

Finite Lorentz oscillator closure

Augmented energy and passive absorption

Facts & Assumptions

Given: Countable Choice, one undamped Lorentz oscillator with $a > 0$ , $ω_{0} > 0$ ; its state at $t_{0}$ consists of $P (t_{0})$ and $P_{t} (t_{0})$ ; take $E = B = H = 0$ and $P = 0$ at that instant.

[A1]

The adopted oscillator model permits the initial polarization and polarization-velocity state $(P (t_{0}), P_{t} (t_{0}))$ and uses $D = ε_{\infty} E + P$ (Finite Lorentz oscillator closure).

[F1]

For this undamped Lorentz state, the material energy density is $∣ P_{t} ∣^{2} / (2 a)$ , added to the positive electromagnetic field energy (Augmented energy and passive absorption, Lorentz branch).

[F2]

The Euclidean squared norm is positive for every nonzero vector: $∣ v ∣^{2} = ⟨ v, v ⟩ > 0$ when $v \neq 0$ (The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ ).

[F3]

Countable Choice is an explicit hypothesis of the finite-energy oscillator theorem used here (The Axiom of Countable Choice ( ${A C}_{ω}$ ), Augmented energy and passive absorption).

Verification

technique · compare two allowed oscillator states with equal instantaneous field and displacement data

1.1givenA1F1F2F3algebra

In the first state $P_{t} = 0$ , so [F1] gives zero oscillator energy. In the second state $P_{t} = v \neq 0$ , so [F1] gives $∣ v ∣^{2} / (2 a) > 0$ by [F2] and $a > 0$ . Both states have $E = 0$ and $P = 0$ , hence the same $D = ε_{\infty} E + P = 0$ , and both have $B = H = 0$ ; they therefore have identical instantaneous $(E, D)$ data but different total stored energies.

2.1step 1.1A1algebra∎

The comparison is between allowed model states at one time; it is not an observed outcome or a claim about microscopic material energy. Hence the full oscillator energy is not a function of the instantaneous pair $(E, D)$ alone.

Matter energy survives zero instantaneous polarization

Statement

Statement references

Facts & Assumptions

Verification

Depends on

Dependency tree · 2 levels

Sources