distinctionInterpret
distinctionInterpret supplies the canonical embedding of the lifted LogicNat orbit into an arbitrary carrier K by sending the identity constructor to base point x and every step constructor to distinct point y. Researchers building universal forcing from a single distinction cite it as the minimal map that turns any two-point set into a LogicRealization. The definition is a direct pattern match on the two constructors after ULift.
claimLet $K$ be any type equipped with distinguished elements $x,y$. The map $i_{x,y}:$ ULift(LogicNat) $→ K$ is defined by $i_{x,y}(↑identity)=x$ and $i_{x,y}(↑(step n))=y$ for every $n$.
background
LogicNat is the inductive type whose constructors identity and step generate the forced arithmetic orbit under the Law of Logic; identity is the zero-cost multiplicative unit and step iterates the generator. ULift lifts this orbit into the same universe as the arbitrary carrier $K$. The module shows that any $K$ with $x≠y$ carries a LogicRealization whose carrier is exactly $K$, using the two-valued equality cost, identity at $x$, and the constant step map to $y$.
proof idea
The declaration is a direct pattern-matching definition on the lifted LogicNat value. It routes the identity constructor to the supplied base point $x$ and routes every step constructor to the supplied distinct point $y$, without inspecting the argument of step.
why it matters in Recognition Science
This definition is the interpretation function required by logicRealizationOfDistinction, the universal instantiation theorem that equips every non-singleton carrier with a LogicRealization. It thereby lets Universal Forcing apply to $K$ and yields the same forced arithmetic object as the canonical recognition realization. The construction is deliberately minimal and recovers the continuous J/spacetime layer only through realization-invariance.
scope and limits
- Does not assert that an arbitrary carrier $K$ carries a native smooth real-valued J-cost.
- Does not construct spacetime geometry or continuous structure on $K$.
- Does not claim uniqueness of the induced LogicRealization.
- Does not extend beyond the minimal two-point distinction to richer internal structure.
formal statement (Lean)
70def distinctionInterpret {K : Type u} (x y : K) : ULift.{u} LogicNat → K
71 | ⟨LogicNat.identity⟩ => x
72 | ⟨LogicNat.step _⟩ => y
73
74/-- The step map induced by a single distinction: every state advances to the
75distinguished second point. This gives a total endomap on `K`. -/