Closure Activation as Partition-Warrant: A Two-Axis Resolution to Metric-Preserving Identity Underdetermination

Paper 1 establishes that persistence-based frameworks — the Free Energy Principle, Global Workspace Theory, and adjacent approaches — rely on batteries of dynamical observables insufficient, on at least one substrate where the frameworks would naturally apply them, to individuate the systems they characterize. The supplementary structural condition Paper 1 §7 calls a partition-warrant must be sensitive to substrate properties not invariant under the admissible transformation group of the dynamics. The present paper develops closure activation as a candidate filling for that slot. The formal predicate W_CA is partition-evaluative: for each subsystem under a partition P, it returns a verdict structure (τ₀, μ*, κ) recording whether the activation event T₀ has been crossed, the maintenance margin μ* under graded perturbation, and the resulting classification (closed, closure-competent, subthreshold). The activation conditions — state-referential constraint, internally governed corrective response, internally viable maintenance budget — are evaluated relative to the partition's imposed boundary. The predicate's two-axis structure (binary T₀, scalar T*) is phase-structured in a way that distinguishes closure activation from classical autopoietic operational closure. W_CA is non-G-invariant at the predicate level: the causal support graph relative to the imposed boundary is partition-relative and not preserved under admissible transformations. The predicate operates in three structural modes — rejection, selection, and stratification — depending on the substrate's closure-bearing properties under G-related partitions. Evaluation on the Paper 1 witness yields a rejection-mode verdict: neither partition is closure-bearing, and the uniform negative output is the predicate's correctness behavior on a substrate where the activation conditions cannot be satisfied under any admissible labeling. Distinctions from optimization-based (Integrated Information Theory), structural-definitional (Global Workspace Theory), statistical-structural (Free Energy Principle in nonlinear regimes), and process-based (autopoietic operational closure) alternatives are developed structurally and by operational-mode coverage. The route × mode cross-tabulation supplies a meta-taxonomy for partition-warrants more generally. Falsification conditions are stated at predicate, verdict-structure, and discriminative-scope levels. In conjunction with Paper 1, the present paper separates three questions persistence-based frameworks have often conflated: what persists dynamically, what individuates structurally, and what sustains recursive closure. The class of closure-bearing systems is a proper subset of the class of persistence-bearing systems under the battery characterization, and the structural rarity of closure-bearing organization is a derivable consequence of the activation conditions' joint stringency.