Unity Logic: Binding Without Collapse and a Witnessed Calculus of Admissible Integration

Unity requires more than connection: it requires a supported witness. This paper develops Unity Logic (UL), a typed and parameterized calculus for determining when differentiated components may be treated as one admissible case without collapsing their identities or load-bearing distinctions. UL separates granular support from type-specific validation and requires compatible interpretations across support, validation, and heterogeneous construction models. Its central judgments distinguish positive Unity, partition-relative Independence, and unresolved status. The paper formalizes three canonical families: common-nexus evidential Binding, controlled transition Binding, and configurational Binding through joint-state constraints. It introduces atom-level witness provenance, certified type reduction, burden-sensitive transfer, heterogeneous composition through amalgamated indexed families, and lossless or declared-loss decomposition. A modular relative soundness theorem is established over compatible sound support, validation, and construction systems. Countermodels show that conjunction, co-presence, similarity, common classification, and shared law form do not by themselves entail stronger forms of Unity. A running example demonstrates distinct interactional, configurational, and evidential witnesses over related components, as well as a proposed composition that remains non-derivable when its bridge has not been validated for the claimed output type. The result is an initial witnessed calculus of admissible integration. It does not claim that its Binding taxonomy is exhaustive, that all semantically valid Unity judgments are derivable, or that the present architecture is uniquely forced. Its governing discipline is to bind what is supported, preserve what must remain distinct, and refuse to manufacture a whole where the premises establish only plurality.