Wormhole Kernel Transport

Abstract

Wormhole composition records exact observer-kernel loss.

Theorem 1.1 (Kernel Forward Invariant).

$\forall so u rce : Dy nami c a l W or l d, t a r g e t : Dy nami c a l W or l d, b r i d g e : W or mh o l eso u rce t a r g e t, f i rs t : so u rce . St a t e, seco n d : so u rce . St a t e, (S e t o i d . k er b r i d g e . ma p f i rs t seco n d) \Rightarrow (S e t o i d . k er b r i d g e . ma p (so u rce . s t e p f i rs t) (so u rce . s t e p seco n d)) .$

Proof. Machine-checked in Lean as D5/S3/Observer/Bridges/WormholeKernelTransport.kernel_forward_invariant (✓ std3). ∎

Source. Repository-derived.

Commentary.

The observation kernel of a wormhole is forward-invariant under the source dynamics.

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

Theorem 1.2 (Kernel le Composite).

$\forall so u rce : Dy nami c a l W or l d, mi dd l e : Dy nami c a l W or l d, t a r g e t : Dy nami c a l W or l d, f i rs t : W or mh o l eso u rce mi dd l e, seco n d : W or mh o l e mi dd l e t a r g e t, (S e t o i d . k er f i rs t . ma p \leq S e t o i d . k er (co m p oseseco n df i rs t) . ma p) .$

Proof. Machine-checked in Lean as D5/S3/Observer/Bridges/WormholeKernelTransport.kernel_le_composite (✓ std3). ∎

Source. Repository-derived.

Commentary.

Postcomposing a wormhole can only enlarge its source observer kernel.

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

Theorem 1.3 (Kernel eq Composite Of Outer Injective).

$\forall so u rce : Dy nami c a l W or l d, mi dd l e : Dy nami c a l W or l d, t a r g e t : Dy nami c a l W or l d, f i rs t : W or mh o l eso u rce mi dd l e, seco n d : W or mh o l e mi dd l e t a r g e t, (F u n c t i o n . I nj ec t i v eseco n d . ma p) \Rightarrow (S e t o i d . k er (co m p oseseco n df i rs t) . ma p = S e t o i d . k er f i rs t . ma p) .$

Proof. Machine-checked in Lean as D5/S3/Observer/Bridges/WormholeKernelTransport.kernel_eq_composite_of_outer_injective (✓ std3). ∎

Source. Repository-derived.

Commentary.

An injective outer wormhole preserves the source observer kernel exactly.

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

Theorem 1.4 (Strict Kernel Growth Of Outer Collision).

$\forall so u rce : Dy nami c a l W or l d, mi dd l e : Dy nami c a l W or l d, t a r g e t : Dy nami c a l W or l d, f i rs t : W or mh o l eso u rce mi dd l e, seco n d : W or mh o l e mi dd l e t a r g e t, l e f t : so u rce . St a t e, r i g h t : so u rce . St a t e, (f i rs t . ma pl e f t \neq = f i rs t . ma p r i g h t) \land (seco n d . ma p (f i rs t . ma pl e f t) = seco n d . ma p (f i rs t . ma p r i g h t)) \Rightarrow (S e t o i d . k er f i rs t . ma p < S e t o i d . k er (co m p oseseco n df i rs t) . ma p) .$

Proof. Machine-checked in Lean as D5/S3/Observer/Bridges/WormholeKernelTransport.strict_kernel_growth_of_outer_collision (✓ std3). ∎

Source. Repository-derived.

Commentary.

A pair visible after the first bridge but collapsed by the second bridge witnesses strict growth of the composite kernel.

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

Theorem 1.5 (Strict Growth Refutes Outer Injectivity).

$\forall so u rce : Dy nami c a l W or l d, mi dd l e : Dy nami c a l W or l d, t a r g e t : Dy nami c a l W or l d, f i rs t : W or mh o l eso u rce mi dd l e, seco n d : W or mh o l e mi dd l e t a r g e t, (S e t o i d . k er f i rs t . ma p < S e t o i d . k er (co m p oseseco n df i rs t) . ma p) \Rightarrow (\neg F u n c t i o n . I nj ec t i v eseco n d . ma p) .$

Proof. Machine-checked in Lean as D5/S3/Observer/Bridges/WormholeKernelTransport.strict_growth_refutes_outer_injectivity (✓ std3). ∎

Source. Repository-derived.

Commentary.

Strict information loss through a composite refutes injectivity of the outer bridge.

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/Bridges/WormholeKernelTransport.kernel_eq_composite_of_outer_injective
Truth anchor: D5/S3/Observer/Bridges/WormholeKernelTransport.kernel_forward_invariant
Truth anchor: D5/S3/Observer/Bridges/WormholeKernelTransport.kernel_le_composite
Truth anchor: D5/S3/Observer/Bridges/WormholeKernelTransport.strict_growth_refutes_outer_injectivity
Truth anchor: D5/S3/Observer/Bridges/WormholeKernelTransport.strict_kernel_growth_of_outer_collision
Dependency: D5/S3/Observer/Bridges/WormholeCategory

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Wormhole Kernel Transport

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