Wormhole Holonomy

Abstract

Round trips through observer wormholes define holonomy, with inverse bridges giving the trivial loop.

Theorem 1.1 (Round Trip Maps Fixed Point).

$$\begin{gathered} \forall source:DynamicalWorld,target:DynamicalWorld,forward:Wormholesourcetarget,backward:Wormholetargetsource,state:source.State, \\ (Function.IsFixedPtsource.stepstate)\Rightarrow \\ (Function.IsFixedPtsource.step((roundTripforwardbackward).mapstate)). \end{gathered}$$

Proof. Machine-checked in Lean as D5/S3/Observer/WorldModel/WormholeHolonomy.round_trip_maps_fixed_point (✓ std3). ∎

Source. Repository-derived.

Commentary.

Round trips preserve every fixed source state as a fixed state of the round-trip dynamics.

The declaration keeps its parameters and hypotheses explicit; the result makes no converse or broader existence claim beyond that scope.

Theorem 1.2 (Round Trip eq Identity Of Left Inverse).

$$\begin{gathered} \forall source:DynamicalWorld,target:DynamicalWorld,forward:Wormholesourcetarget,backward:Wormholetargetsource, \\ (Function.LeftInversebackward.mapforward.map)\Rightarrow \\ (roundTripforwardbackward=identitysource). \end{gathered}$$

Proof. Machine-checked in Lean as D5/S3/Observer/WorldModel/WormholeHolonomy.round_trip_eq_identity_of_left_inverse (✓ std3). ∎

Source. Repository-derived.

Commentary.

A genuine left inverse makes the round trip equal to the identity wormhole.

The declaration keeps its parameters and hypotheses explicit; the result makes no converse or broader existence claim beyond that scope.

Theorem 1.3 (No Holonomy Of Left Inverse).

$$\begin{gathered} \forall source:DynamicalWorld,target:DynamicalWorld,forward:Wormholesourcetarget,backward:Wormholetargetsource,state:source.State, \\ (Function.LeftInversebackward.mapforward.map)\Rightarrow \\ (\neg HasHolonomyAtforwardbackwardstate). \end{gathered}$$

Proof. Machine-checked in Lean as D5/S3/Observer/WorldModel/WormholeHolonomy.no_holonomy_of_left_inverse (✓ std3). ∎

Source. Repository-derived.

Commentary.

A left inverse rules out holonomy at every source state.

The declaration keeps its parameters and hypotheses explicit; the result makes no converse or broader existence claim beyond that scope.

Theorem 1.4 (Holonomy Refutes Left Inverse).

$$\begin{gathered} \forall source:DynamicalWorld,target:DynamicalWorld,forward:Wormholesourcetarget,backward:Wormholetargetsource,state:source.State, \\ (HasHolonomyAtforwardbackwardstate)\Rightarrow \\ (\neg Function.LeftInversebackward.mapforward.map). \end{gathered}$$

Proof. Machine-checked in Lean as D5/S3/Observer/WorldModel/WormholeHolonomy.holonomy_refutes_left_inverse (✓ std3). ∎

Source. Repository-derived.

Commentary.

Any holonomy witness refutes the claim that the return bridge is a left inverse.

The declaration keeps its parameters and hypotheses explicit; the result makes no converse or broader existence claim beyond that scope.

References

• Truth anchor: D5/S3/Observer/WorldModel/WormholeHolonomy.holonomy_refutes_left_inverse
• Truth anchor: D5/S3/Observer/WorldModel/WormholeHolonomy.no_holonomy_of_left_inverse
• Truth anchor: D5/S3/Observer/WorldModel/WormholeHolonomy.round_trip_eq_identity_of_left_inverse
• Truth anchor: D5/S3/Observer/WorldModel/WormholeHolonomy.round_trip_maps_fixed_point
• Dependency: D5/S3/Observer/Bridges/WormholeCategory