Remark: Interpretation boundary

For arbitrary interior dual endpoints, $\mathcal{M}(a,b)$ is a declared optimization of local entanglement costs. An open interior path is not automatically the entropy of one global vertex bipartition.

The theorem concerns a post-quench cost model. It does not derive the time-dependent entropy of a generic many-body system, the usual spacetime membrane, or its finite-temperature free energy. The exponential law was engineered through independent controls; it was not inferred from chaotic quantum dynamics.