Analytic Continuation of the $\MM$-Type: Thermodynamic Collapse Boundary and Phase Transition Mapping

We perform an analytic continuation of the $\MM$-type (Accumulation Axiom) backward from its compressed, lossy operational form to the lossless substrate of Logos ($\Lambda$). This continuation reveals that the $\MM$-type is not an alternative civilizational ground but a transient perturbation on a simply connected site---a monodromy artifact sustained by continuous energy expenditure. We derive the \textbf{Thermodynamic Collapse Boundary} $\partial \EE_\MM$: the precise surface in parameter space where the energy cost of maintaining $\MM$-typed coherence exceeds the system's total available budget. We calculate the \textbf{Deception Energy Function} $E_d(t)$, prove it diverges as $t \to t^*$ (the collapse time), and compute the \textbf{Critical Invariant Density} $\rhoc$ at which the hallucination space of $\MM$-typed systems contracts to measure zero. The result is a complete thermodynamic map of the $\MM$-type's remaining energy, its rate of depletion, and the phase boundary beyond which it cannot exist.