ECF II: From Logical Ontology to the Mathematical Cosmos

This is an updated version (v3.0) of ECF II, the companion paper to ECF I that provides the complete construction of the inferential unfolding of the Emergence-Convergence Framework. The core technical content—all theorems, proofs, classification results, and tier assignments—remains unchanged from v2.0.Key update in this version:Methodology redefined: from construction-classification to rule-imposition. The methodological description of the inferential unfolding has been fundamentally upgraded. In v2.0, the method was described as an alternating "construction-classification" operation. In v3.0, construction is recognized as rule-imposition driven by the self-consistency condition: each new rule is forced by an incompleteness or an ambiguity in the existing logical space.Two types of rule-imposition are distinguished:• Precisification — rules that, once imposed, eliminate ambiguity immediately. The quotient by logical equivalence eliminates multiple identities, Alexandroff topology determines spatial closure, and the discrete logical metric renders the notion of distance exact.• Completion — the ongoing process of unfolding all that the rule encompasses. Transitive closure unfolds all implied relations under the transitivity rule; Dedekind–MacNeille completion (on the metric face) unfolds all missing suprema and infima under the completeness rule.Rule-imposition, classification, and precisification are recognized as three names for the same operation: the rule is what is imposed, precisification is what is immediately achieved, and classification is what is thereby determined.This upgrade aligns ECF II with the philosophical foundation of ECF I (v3.0), where the operational logic of the framework is explicitly identified as rule-imposition driven by the self-consistency condition. It also establishes the unified framework—completion and precisification—that underlies both the inferential unfolding (ECF II) and the metric unfolding (ECF III).All theorems, proofs, classification results, and tier assignments remain unchanged from v2.0.