Cocycle accounting does not determine which legal action a policy selects.
Theorem 1.1 (Cocycle composition leaves policy choice nonunique).
∀ U , S i g ma , C a se , P u b l i c F a c t , O u t co m e : Type , [ Nonempty ( U ) ] , [ AddCommGroup ( S i g ma ) ] , p ro j ec t i o n : AddMonoidHom ( S i g ma , AddCircle ( 1 ) ) , h P ro j ec t i o n S u r j ec t i v e : Surjective ( p ro j ec t i o n ) , hi dd e n Eq u i v : AdditiveEquiv ( H i dd e n J u m pC oor d ina t es , ker ( p ro j ec t i o n ) ) , sec t i o n A lp ha , sec t i o n B e t a , sec t i o n G amma : U → S i g ma , j u m p A lp ha B e t a , j u m pB e t a G amma , j u m p A lp ha G amma : U → H i dd e n J u m pC oor d ina t es , h A lp ha B e t a : ∀ u , sec t i o n B e t a ( u ) = sec t i o n A lp ha ( u ) + hi dd e n Eq u i v ( j u m p A lp ha B e t a ( u )) , h B e t a G amma : ∀ u , sec t i o n G amma ( u ) = sec t i o n B e t a ( u ) + hi dd e n Eq u i v ( j u m pB e t a G amma ( u )) , p u b l i c L a w : C a se → P u b l i c F a c t , a d mi ss ib l e : C a se → Prop , p er mi tt e d : C a se → O u t co m e → Prop , b : P u b l i c F a c t , ∃ l e f tO u t co m e , r i g h tO u t co m e : O u t co m e , l e f tO u t co m e = r i g h tO u t co m e ∧ ( ∃ x : C a se , a d mi ss ib l e ( x ) ∧ p u b l i c L a w ( x ) = b ∧ p er mi tt e d ( x ) ( l e f tO u t co m e )) ∧ ∃ x : C a se , a d mi ss ib l e ( x ) ∧ p u b l i c L a w ( x ) = b ∧ p er mi tt e d ( x ) ( r i g h tO u t co m e ) ⇒ (( ∀ u , p ro j ec t i o n ( sec t i o n A lp ha ( u )) = p ro j ec t i o n ( sec t i o n B e t a ( u )) ∧ p ro j ec t i o n ( sec t i o n B e t a ( u )) = p ro j ec t i o n ( sec t i o n G amma ( u ))) ∧ ( ∀ u , sec t i o n G amma ( u ) = sec t i o n A lp ha ( u ) + hi dd e n Eq u i v ( j u m p A lp ha G amma ( u )) ⟺ ∀ u , j u m p A lp ha G amma ( u ) = j u m p A lp ha B e t a ( u ) + j u m pB e t a G amma ( u )) ∧ ( ∃ u , j u m p A lp ha G amma ( u ) = j u m p A lp ha B e t a ( u ) + j u m pB e t a G amma ( u ) ⇒ ∃ u , sec t i o n G amma ( u ) = sec t i o n A lp ha ( u ) + hi dd e n Eq u i v ( j u m p A lp ha G amma ( u )))) ∧ ¬ ( ∃ ! c an d i d a t e : O u t co m e , ∃ x : C a se , a d mi ss ib l e ( x ) ∧ p u b l i c L a w ( x ) = b ∧ p er mi tt e d ( x ) ( c an d i d a t e )) .
Proof. Machine-checked in Lean as D5/S3/ConceptDynamics/Policy/CocyclePolicySeparation.cocycle_does_not_select_policy✓ std3). ∎
Source. Repository-derived.
Commentary.
The hidden-jump construction records visible agreement, the endpoint/cocycle equivalence, and an explicit endpoint residual when the selected jumps disagree.
The same public model carries a permitted-outcome relation with two distinct outcomes in one public-law fiber. Consequently the cocycle law supplies composition and accounting, but no unique policy choice.