phiRing_id_right
The right identity law for morphisms in the PhiRing category states that composing any morphism f from A to B with the identity on A recovers the underlying carrier map of f. Algebraists verifying that PhiRing forms a category within Recognition Science would cite this result when checking the unit laws. The proof is a one-line term applying function extensionality followed by reflexivity after unfolding composition and identity.
claimLet $A$ and $B$ be phi-ring objects and let $f$ be a phi-ring morphism from $A$ to $B$. Then the composition of $f$ with the identity morphism on $A$ satisfies $(f ∘ id_A).map = f.map$.
background
PhiRingObj is a structure consisting of a carrier type with commutative unital ring operations (zero, one, add, neg, mul) together with a distinguished golden element phi. PhiRingHom consists of maps between carriers that preserve all ring operations and send phi to phi. The module RecognitionCategory assembles these data into a category whose morphisms model algebraic recognition flows. Upstream results include phiRing_id, which supplies the identity map as the identity function on the carrier, and phiRing_comp, which composes the underlying maps pointwise.
proof idea
The proof is a one-line term that applies funext to reduce map equality to pointwise equality on the carrier, after which rfl closes the goal by the definitions of phiRing_comp and phiRing_id.
why it matters in Recognition Science
This identity law completes the unit axioms needed to show that PhiRing forms a category, paralleling the recAlg_id_right sibling and supporting the algebraic layer of Recognition Science. It sits downstream of the phi-ring definitions and upstream of any functorial constructions that rely on PhiRing as a model for the J-cost and phi-ladder structures. No open scaffolding questions are directly addressed.
scope and limits
- Does not establish associativity of composition.
- Does not provide a physical interpretation of the morphisms.
- Does not extend the result to non-commutative or infinite carriers.
formal statement (Lean)
569theorem phiRing_id_right {A B : PhiRingObj} (f : PhiRingHom A B) :
570 (phiRing_comp f (phiRing_id A)).map = f.map := by
proof body
Term-mode proof.
571 funext x
572 rfl
573
574/-- The ambient real flow space on a finite node set. -/