universalInstantiationCert
Any nonempty carrier K with at least one distinction admits a LogicRealization on exactly that carrier, using two-valued equality cost and a constant step map. Foundation researchers cite this to justify applying Universal Forcing to arbitrary non-singleton types before reaching the continuous J layer. The proof is a direct term assignment that packages two existence lemmas into the required structure.
claimFor any nonempty type $K$, if there exist distinct $x,y$ in $K$, then there is a nonempty LogicRealization whose carrier is $K$, whose comparison is the two-valued equality cost, whose identity point is $x$, and whose step map is the constant map to $y$; moreover a named version exists that retains the explicit choice of $x,y$ and the inequality witness.
background
The module constructs a minimal LogicRealization on any carrier that is not a singleton. The realization uses the two-valued equality cost as comparison function, picks one point as identity, and maps every step to the other point; its internal orbit is the free LogicNat orbit. This supplies the Law-of-Logic interface without assuming a native smooth real-valued J-cost on the carrier.
proof idea
The proof is a one-line term-mode wrapper. It supplies the instantiate field of the structure by direct reference to the existence theorem that builds a LogicRealization from any distinction, and supplies the named field by reference to the version that retains the chosen points and witness.
why it matters in Recognition Science
This declaration removes the skeptical objection that a bare distinction cannot instantiate the Law-of-Logic interface on its own carrier. It therefore lets Universal Forcing apply to every non-singleton type, yielding the same forced arithmetic object as the canonical recognition realization. The result is the first universal step before realization-invariance reaches the continuous J/spacetime layer.
scope and limits
- Does not assert that an arbitrary carrier K carries a native smooth real-valued J-cost.
- Does not prove continuity or differentiability of any map on K.
- Does not claim uniqueness of the constructed LogicRealization up to isomorphism.
- Does not apply to singleton carriers.
formal statement (Lean)
198theorem universalInstantiationCert
199 (K : Type u) [Nonempty K] :
200 UniversalInstantiationCert K where
201 instantiate := exists_logicRealization_of_distinction K
proof body
Term-mode proof.
202 named := exists_named_logicRealization_of_distinction K
203
204end UniversalInstantiationFromDistinction
205end Foundation
206end IndisputableMonolith