logicNatToLattice_zero
plain-language theorem explainer
The interpretation of logic-natural numbers into the recognition lattice sends the identity element to the base cell of any primitive interface. Researchers establishing the pre-spatial quotient structure of observers cite this when fixing the base case of the recursive map. The equality holds immediately by reflexivity on the definition of the lattice interpretation at the identity constructor.
Claim. Let $I$ be a primitive interface on carrier $K$, with base configuration $b$ and successor map $s:K→K$. The interpretation of the logic-natural numbers into the lattice satisfies $f(I,b,s)(id)=cell(I,b)$, where $id$ denotes the identity element of the logic naturals and $cell(I,b)$ is the equivalence class of the base under the kernel of $I$.
background
The module converts the structural claim of Recognition Geometry into a theorem: kernel-equivalence classes of a recognizer form the first recognition lattice. This lattice is the quotient of the carrier by the indistinguishability kernel induced by a PrimitiveInterface, whose observe map lands in Fin n and therefore carries finite resolution. LogicNat is the inductive type with constructors identity (the zero-cost element at J-cost minimum) and step, mirroring the multiplicative orbit generated by the forcing constant.
proof idea
One-line wrapper that applies reflexivity to the definition of logicNatToLattice at the identity constructor of LogicNat.
why it matters
The declaration supplies the base case for mapping the universal iteration object LogicNat into the recognition lattice, completing the interpretation step described in the module documentation. It anchors the pre-spatial quotient before metric or dimensional structure appears, consistent with the forcing chain landmarks that begin from the J-uniqueness relation and the identity event. No downstream uses are recorded, indicating its role as a foundational lemma for lattice properties.
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