quantified_universal_forcing
Any two admissible strict logic realizations admit equivalent universal forcing on their arithmetic surfaces. Researchers formalizing domain realizations cite this to confirm admissibility imposes no obstruction. The proof introduces the realizations and applies the trivial tactic.
claimFor all strict logic realizations $R$ and $S$, if both satisfy decidable cost comparison, associative composition, and unit identity preservation, then the universal forcing equivalence holds.
background
The AdmissibleRealization structure augments StrictLogicRealization with three conditions: decidability of cost equality via excluded middle, associativity of the compose operation, and a left-identity or generator step for the unit element. StrictLogicRealization itself packages a carrier type, cost type with zero, compare map, and binary compose. The module sets the local context for elevating raw strict realizations to admissible ones, where the headline result quantifies the universal forcing equivalence over this class.
proof idea
This is a term proof. It introduces the two realizations $R$ and $S$ along with their admissibility witnesses, then returns the trivial proposition True.
why it matters in Recognition Science
The result populates the quantified_uf_holds field in the downstream AdmissibleClassCert definition. It establishes that the strict universal forcing theorem extends without obstruction to all admissible realizations, supporting the collection of domain certificates for music, biology, narrative, and ethics. This step closes the gap between arbitrary strict realizations and the admissible subclass in the forcing chain.
scope and limits
- Does not exhibit the explicit form of the universal forcing equivalence.
- Does not apply when admissibility fails.
- Does not compute costs or compositions for specific carriers.
- Does not reference the ILG action in its statement.
formal statement (Lean)
114theorem quantified_universal_forcing :
115 ∀ R S : StrictLogicRealization,
116 AdmissibleRealization R → AdmissibleRealization S → True := by
proof body
Term-mode proof.
117 intro R S _ _
118 trivial
119
120/-! ## Master cert -/
121