specifiabilityClosureCert
The declaration certifies that for any nonempty carrier K the existence of distinct elements is equivalent to the presence of nontrivial specifications on K. Workers on the absolute-floor program cite this to close Route B without extra hypotheses. The proof is a direct term construction that supplies the equivalence field of the certificate structure from the prior distinguishability theorem.
claimFor any nonempty type $K$, the specifiability closure certificate holds: the existence of distinct elements $x,y$ in $K$ is equivalent to the nonemptiness of the type of nontrivial specifications on $K$.
background
Route B of the absolute-floor program equates specifiability with distinguishability at the foundational level. The structure asserts that a non-singleton universe of discourse is precisely the condition for nontrivial specifiability. The upstream theorem establishes the biconditional between the existence of distinct elements and the nonemptiness of the nontrivial specification type on an inhabited carrier.
proof idea
The proof is a one-line term wrapper that constructs the certificate structure by assigning the equivalence field directly to the distinguishability theorem.
why it matters in Recognition Science
This theorem completes the certificate for Route B, confirming that specifiability sits at the same floor as a non-singleton carrier. It supports the Law-of-Logic chain by providing a theorem-backed equivalence without additional assumptions. No downstream uses are recorded, but it closes the foundational distinction needed for later derivations.
scope and limits
- Does not establish existence of any particular ontology or physical model.
- Does not extend to empty carriers or non-inhabited types.
- Does not address computability or higher-order properties of specifications.
formal statement (Lean)
96theorem specifiabilityClosureCert (K : Type*) [Nonempty K] :
97 SpecifiabilityClosureCert K where
98 equivalence := distinguishability_iff_nontrivial_specifiability
proof body
Term-mode proof.
99
100end SpecifiabilityClosure
101end Foundation
102end IndisputableMonolith