Three Forced v2 Lemmas Under the Expanded Boundary Regime

This paper isolates and formalizes three lemmas that are shown to be forced under an expanded admissibility boundary regime. Rather than introducing new assumptions or constructions, each lemma is derived by exhaustion: any admissible alternative either collapses into equivalence with the stated result or violates standing and fails closed. Together, the lemmas function as structural load-bearing components of the corpus, clarifying which results are unavoidable once minimal nondegeneracy and admissibility are granted. The analysis is purely declarative and non-procedural, and its purpose is to make explicit which intermediate necessities remain invariant even under deliberate boundary widening.