Paper 0D — The Bath Theorem: Axiomatic Construction of the Unique Recovery Initiator for Systems with Memory (ZLT Impossibility Result)

Paper 0C establishes that monolithic recovery (V⁻₁) is thermodynamically insufficient for systems under CRCS (Critical Regime of Compromised Self-certification). This creates a structural gap: a degraded system cannot self-certify, cannot tolerate external forcing, yet must recover. Paper 0D resolves this gap by constructing the Bath Operator — a boundary-conditioned recovery initiator defined by four canonical axioms (B0–B3): Informational Orthogonality, Scalar Magnitude Limit, Boundary Variance Collapse, and Endogenous Non-Interference. We prove that these axioms are individually necessary (Cases A–D) and jointly sufficient (Case E) for recovery initiation without violating CRCS constraints. The Uniqueness Theorem (3.3) proves by exhaustive elimination from the impossibility architecture (Papers 0A–0C) that B0–B3 is the only admissible operator class under CRCS. The Sustainability Theorem establishes a calculable minimum bath frequency f_min = α/ε per channel. The Universality Theorem demonstrates that in every system with memory, at least one B0–B3 operator exists (3.1a, logical); moreover, in every domain examined, non-trivial baths with ε > 0 are documented (3.1b, empirical). Thirteen corollaries derive consequences for anti-bath detection, load leakage, cascade dynamics, and cross-channel recovery. An economic case study (FullContact, Appendix F) instantiates the framework in the organizational domain with currency-neutral analysis: bath cost ~10–15% of annual compensation vs replacement cost at 4×–16× ratio across US, European, Latin American, and Southeast Asian markets. A computational proof engine (Appendix H, 71/71 steps passed) proves all theorems and corollaries in the algorithmic domain (per Unification Theorem, Paper 50). The bath is not luxury — it is the unique admissible recovery initiator for systems with memory under CRCS.