The Architecture of Outer Admissibility: Unification, Maximality, and Residual Structure in Certificate Worlds

Outer certificates — compressed witnesses, abstraction layers, quotient representations, transport claims, upgrade witnesses — are routinely used as if they defined distinct admissibility worlds. This paper proves that, in a broad and disciplined certificate world, they do not: apparently different outer certificate forms collapse to one unified gate, force a derived higher layer above the collapsing certificates, and leave a residual heterogeneous structure that cannot itself be collapsed. The unified gate theorem, the heterogeneous outer admissibility theorem, and the strict three-level architecture theorem are all machine-checked in Lean 4 as part of the Adequacy Architecture program.