The Inaccessibility of Actual Infinity for Energy-Bounded Subsystems

Aristotle, in Physics III.5–8, argued that nature admits only potential infinity — endless process, never completed totality. An actual infinity, he held, could possess no determinate place or operation within the cosmos. Two millennia later, continuous spacetime and uncountably dense truth-value structures became the default background of mathematical physics, and Aristotle's argument was set aside. The question he raised, however, has never been answered. It has only been bracketed. This paper removes the bracket. We present a reductio showing that any embedded subsystem with bounded energy and the capacity to acquire approximately stable knowledge cannot operationally distinguish a continuous truth-value structure from a discrete one of finite cardinality. The argument rests on two empirical premises — that energy within any bounded region is finite, and that subsystems do in fact form stable records — together with the derived consequence, via Landauer's principle, that each registration carries a non-zero energetic cost. The continuum, if it exists at all, is thereby operationally inaccessible: a substrate posit with no consequence for any energy-bounded epistemic system. Four available counter-strategies — denying the operational claim, denying registration cost, denying bounded energy, or asserting stable knowledge in a continuous regime — are enumerated and foreclosed. Each is either physically untenable, metaphysically unobservable, or structurally equivalent to the discreteness conclusion under different terminology. The argument proceeds independently of quantum mechanics, general relativity, or any specific commitment to spacetime structure. It uses only what no working physical theory currently denies. Where modern physics treats the continuum as a calculational background while leaving its operational status unexamined, we argue that the operational status is structurally determined: any subsystem that actually knows anything must operate on a finite quotient. Aristotle's rejection of actual infinity in nature is recovered, not as metaphysical dogma, but as a theorem about the necessary conditions for stable knowledge under finite resources. The path to actual infinity, like the path to the Ding an sich, leads back through the very resource constraints that make knowledge possible — and stops there.