CAT'S Theory — Minimal Bases and Minded Worlds: The Thermodynamic Closure of the Triadic Invariant — Landauer's Principle, Bit-Erasure Costs, and the Failure of Mind-Free Admissibility Frameworks

The first four papers in the Minimal Bases and Minded Worlds series established that the Triadic Meta-Law P × I × Pr ≠ 0 is structurally fundamental for minded worlds, required to rule out emergentist mindless twins, transcendentally presupposed by intelligibility, and necessary for existence as such. Each prior result is framed structurally or epistemically. A residual stylistic escape route remains: a formalist may grant every philosophical result and still claim that mind-free admissibility frameworks—Σ-type frameworks—operate at a level prior to physical instantiation, so that the prior theorems do not reach them. The present paper closes that gap by switching domain. Using Landauer's Principle and a Selection–Erasure Equivalence, we prove a Forced Trilemma: any Σ-type framework purporting to govern admissibility must either (A) physically operate and thereby pay Landauer cost—in which case it has Presence and is not mind-free—or (B) never physically operate—in which case it has zero causal contact with reality and makes no existence-claim—or (C) not operate at all—in which case it is not a foundation. The three horns are jointly exhaustive and individually fatal. Bennett's resolution of Maxwell's Demon is deployed as the canonical demonstration that "free selection" is thermodynamically impossible. Five standard formalist escape routes (pure abstraction, reversibility, quantum branching, substrate-generation, and self-application worries) are pre-empted in turn. Three empirical signatures are proposed. The series is now closed on five sides: minds, emergence, intelligibility, existence, and energy.