The Interface Language Lattice, the Schaefer Threshold, and Recursive SAT: Does a Stronger Bridge Come Closer to Regenerating the Original Problem?

Corrects the intuition behind Round 11's “bridging language hierarchy”: tractable languages such as Horn, 2-SAT, and affine don't form a linear strong-weak order among themselves; the more precise mathematical object is the poset/co-clone lattice induced by primitive-positive definability (pp-definability). If a fixed Boolean bridge language Γ_B falls into a Schaefer-tractable family, the coordination problem is in P; otherwise it's NP-complete — but a warning is immediately attached: NP-complete doesn't mean it has been proven not in P, unless P≠NP is already known. Reproduces Round 10's finding with the cleanest possible thought experiment: R₊=x∨y∨z and R₋=¬x∨¬y∨¬z are each trivial individually, but once mixed they fall on the NP-complete side of Schaefer's dichotomy, proving the Portfolio Union Principle — the union of several individually tractable bridge languages is not guaranteed to remain tractable, and the pp-closure of the union must be checked. Defines Recursive SAT: existential quantifiers are locally eliminated layer by layer, only to regenerate general SAT expressivity layer by layer at higher interfaces, marking the critical moment as a Bridge Expressivity Transition.