twoStep
twoStep constructs a minimal Chain over recognition structure M with exactly one recognition step between unit elements. Researchers verifying flux cancellation on balanced ledgers cite it as the base case before invoking chainFlux_zero_of_balanced. The definition is a direct structure literal whose single proof obligation is discharged by the trivial tactic.
claimOver the recognition structure $M$ with universe $U$ and relation recog, define the chain $C$ by length parameter $n=1$, sequence function constantly equal to the unit element, and recognition condition holding by triviality for the single index.
background
The Chain structure consists of a natural number $n$, a map from Fin(n+1) into the universe $U$ of a RecognitionStructure, and a proof that each consecutive pair satisfies the recognition relation R. Here M is the concrete RecognitionStructure whose universe is U and whose relation is recog. The ledger L is defined with constant debit and credit functions both equal to 1, hence balanced. Upstream, chainFlux is defined as phi of the last element minus phi of the head, and the lemma chainFlux_zero_of_balanced states that this difference vanishes whenever debit equals credit everywhere.
proof idea
Direct structure construction: n is set to 1, f is the constant function returning the unit element, and the ok field is proved by intro i; trivial. The attached example first establishes the balance hypothesis by reflexivity, then applies chainFlux_zero_of_balanced via simpa.
why it matters in Recognition Science
Supplies the simplest concrete witness for the Chain structure inside the Recognition module, enabling immediate verification that flux is zero on balanced ledgers. This supports the T1 (MP) claim that nothing cannot recognize itself by exhibiting a trivial recognition step. It precedes any appeal to the Recognition Composition Law or longer chains in the forcing sequence.
scope and limits
- Does not construct chains with n greater than 1.
- Does not compute nonzero flux values.
- Does not address unbalanced ledgers or non-unit elements.
- Does not invoke the Recognition Composition Law.
Lean usage
example : chainFlux L twoStep = 0 := by have hbal : ∀ u, L.debit u = L.credit u := by intro _; rfl; simpa [chainFlux_zero_of_balanced (M:=M) L twoStep hbal]formal statement (Lean)
105def twoStep : Chain M :=
proof body
Definition body.
106 { n := 1
107 , f := fun _ => ⟨()⟩
108 , ok := by intro i; trivial }
109
110example : chainFlux L twoStep = 0 := by
111 have hbal : ∀ u, L.debit u = L.credit u := by intro _; rfl
112 simpa [chainFlux_zero_of_balanced (M:=M) L twoStep hbal]
113
114end Demo
115
116-- Moved advanced Recognition sections (ClassicalBridge, Potential uniqueness) to Recognition/WIP.lean during modularization.