Agency Enrichment

Abstract

Agency enrichment pairs current state and strategy, isolates the strategy residual inside current fibers, and becomes agency completion only after controlled behavior closure.

Theorem 1.1 (Current Kernel Strategy Residual Partition).

$\forall H : T y p e, B : T y p e, P : T y p e, c u rre n t : H \to B, s t r a t e g y : H \to P, x : H, y : H, (c u rre n t x = c u rre n t y \Leftrightarrow a g e n cy E n r i c hm e n t c u rre n t s t r a t e g y x = a g e n cy E n r i c hm e n t c u rre n t s t r a t e g yy \lor St r a t e g y R es i d u a l c u rre n t s t r a t e g y x y) .$

Proof. Machine-checked in Lean as D5/S3/Observer/AgencySelf/AgencyEnrichment.current_kernel_strategy_residual_partition (✓ std3). ∎

Source. Repository-derived.

Commentary.

Inside a current-state fiber, a pair either agrees under the enriched readout or is a strategy residual.

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

Theorem 1.2 (Agency Kernel Disjoint Strategy Residual).

$\forall H : T y p e, B : T y p e, P : T y p e, c u rre n t : H \to B, s t r a t e g y : H \to P, x : H, y : H, (\neg (a g e n cy E n r i c hm e n t c u rre n t s t r a t e g y x = a g e n cy E n r i c hm e n t c u rre n t s t r a t e g yy \land St r a t e g y R es i d u a l c u rre n t s t r a t e g y x y)) .$

Proof. Machine-checked in Lean as D5/S3/Observer/AgencySelf/AgencyEnrichment.agency_kernel_disjoint_strategy_residual (✓ std3). ∎

Source. Repository-derived.

Commentary.

The enriched kernel and the strategy residual are disjoint.

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

Theorem 1.3 (No Strategy Residual iff Kernel Inclusion).

$\forall H : T y p e, B : T y p e, P : T y p e, c u rre n t : H \to B, s t r a t e g y : H \to P, ((\forall x y, \neg St r a t e g y R es i d u a l c u rre n t s t r a t e g y x y) \Leftrightarrow \forall x y, c u rre n t x = c u rre n t y \Rightarrow s t r a t e g y x = s t r a t e g yy) .$

Proof. Machine-checked in Lean as D5/S3/Observer/AgencySelf/AgencyEnrichment.no_strategy_residual_iff_kernel_inclusion (✓ std3). ∎

Source. Repository-derived.

Commentary.

There is no strategy residual exactly when strategy is constant on every current-state fiber.

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

Theorem 1.4 (Strategy Factorization iff No Residual).

$\forall H : T y p e, B : T y p e, P : T y p e, c u rre n t : H \to B, s t r a t e g y : H \to P, ((\exists! f a c t or : S e t . r an g ec u rre n t \to S e t . r an g es t r a t e g y, \forall x, f a c t or (S e t . r an g e F a c t or i z a t i o n c u rre n t x) = S e t . r an g e F a c t or i z a t i o n s t r a t e g y x) \Leftrightarrow \forall x y, \neg St r a t e g y R es i d u a l c u rre n t s t r a t e g y x y) .$

Proof. Machine-checked in Lean as D5/S3/Observer/AgencySelf/AgencyEnrichment.strategy_factorization_iff_no_residual (✓ std3). ∎

Source. Repository-derived.

Commentary.

Vanishing strategy residual is equivalent to a unique factor from the realized current-state image to the realized strategy image.

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

Theorem 1.5 (Agency Enrichment Kernel eq Current iff No Residual).

$\forall H : T y p e, B : T y p e, P : T y p e, c u rre n t : H \to B, s t r a t e g y : H \to P, (S e t o i d . k er (a g e n cy E n r i c hm e n t c u rre n t s t r a t e g y) = S e t o i d . k erc u rre n t \Leftrightarrow \forall x y, \neg St r a t e g y R es i d u a l c u rre n t s t r a t e g y x y) .$

Proof. Machine-checked in Lean as D5/S3/Observer/AgencySelf/AgencyEnrichment.agency_enrichment_kernel_eq_current_iff_no_residual (✓ std3). ∎

Source. Repository-derived.

Commentary.

Pairing strategy adds no new distinction exactly when the strategy residual vanishes.

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/AgencySelf/AgencyEnrichment.agency_enrichment_kernel_eq_current_iff_no_residual
Truth anchor: D5/S3/Observer/AgencySelf/AgencyEnrichment.agency_kernel_disjoint_strategy_residual
Truth anchor: D5/S3/Observer/AgencySelf/AgencyEnrichment.current_kernel_strategy_residual_partition
Truth anchor: D5/S3/Observer/AgencySelf/AgencyEnrichment.no_strategy_residual_iff_kernel_inclusion
Truth anchor: D5/S3/Observer/AgencySelf/AgencyEnrichment.strategy_factorization_iff_no_residual
Dependency: D5/S3/ObserverMemory/Prediction/ControlledBehaviorUniversality
Dependency: D5/S3/ObserverMemory/Refinement/EffectiveImageKernelCriterion
Dependency: D5/S3/ObserverMemory/Refinement/JointReadoutSupremum

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Agency Enrichment

Abstract

References