= Solution
Let $p^a$ be the future-directed causal <four-momentum> of the fragment captured by the <Kerr black hole>. In signature $(-+++)$ its conserved <Killing energy> and axial <angular momentum> are
$$
E=-p_ak^a,\qquad L=p_am^a.
$$
The future-directed null generator of the <Killing horizon> is $\chi^a=k^a+\Omega_Hm^a$, where $\Omega_H$ is the <Kerr horizon angular velocity>. At a crossing of the future <event horizon>, the scalar product of a future-directed causal vector with a future-directed <null vector> is nonpositive. For example, in a local orthonormal frame, $\chi=\alpha(1,\mathbf n)$ with $|\mathbf n|=1$ and $\alpha>0$, while $p^0\ge|\mathbf p|$; hence $p\cdot\chi=\alpha(-p^0+\mathbf p\cdot\mathbf n)\le0$. Therefore
$$
\boxed{E-\Omega_HL=-p\cdot\chi\ge0,\qquad E\ge\Omega_HL.}
$$
This is a local causal restriction, so it applies also to an extremal <Kerr black hole>. A massive fragment crossing a regular future <event horizon> has a strict inequality; equality is the reversible limiting case, with causal <four-momentum> approaching the generator direction.
The <Penrose process> exploits the fact that $k^a$ becomes spacelike in the <Kerr ergoregion>, allowing future-directed fragments with negative <Killing energy>. For a positive choice of <Kerr horizon angular velocity>, a captured negative-<energy> fragment must have sufficiently negative <angular momentum> to satisfy the inequality. Conservation of <energy> in the splitting then gives $E_{\mathrm{escape}}=E_{\mathrm{initial}}-E_{\mathrm{capture}}>E_{\mathrm{initial}}$: the escaping fragment extracts rotational <energy> and the captured fragment reduces the <black hole>'s mass and <angular momentum>.
For the independent thermodynamic argument, take a subextremal <Kerr black hole> and a small capture process covered by the <Physical-process first law of black-hole mechanics>. To first order, $\delta M=E$, $\delta J=L$, so
$$
E-\Omega_HL=\frac{\kappa}{8\pi}\delta A.
$$
The <second law of black-hole mechanics>, under its classical <null energy condition> and regularity hypotheses, gives $\delta A\ge0$; since $\kappa>0$, this reproduces $E\ge\Omega_HL$. The ideal reversible limit has $\delta A\to0$. One should not infer a general finite-capture equality for an extremal <black hole> by simply substituting $\kappa=0$: the linearized nonextremal process proof and its differentiability assumptions must first be controlled. The causal proof already establishes the required inequality there without this issue.
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