Canonical Causal Response Signatures

Abstract

A finite total causal order yields finite predecessor-indexed deterministic response signatures whose event probabilities are exact linear objectives.

A total order assigns every endogenous coordinate a unique position. At position j, a canonical response table maps assignments to the j predecessor positions into the current variable’s value.

For finite value spaces, the dependent product of all such response tables is finite. Its probability masses therefore form a finite response-type vector.

Every Boolean observational or counterfactual event on signatures has an indicator coefficient. Summing those coefficients against signature masses is exactly a rational linear objective. Pushing a finite exogenous law through its deterministic signature map preserves total mass and event probabilities.

Theorem 1.1 (Every node occupies a unique total-order position).

Lean statement: D5/S3/ConceptDynamics/PartialIdentification/CanonicalResponseSignature.node_has_unique_position

Proof. Machine-checked in Lean as D5/S3/ConceptDynamics/PartialIdentification/CanonicalResponseSignature.node_has_unique_position (✓ std3). ∎

Source. Repository-derived.

Commentary.

The inverse permutation gives the position witness, and injectivity gives uniqueness.

Theorem 1.2 (A signature event probability is an exact linear objective).

Lean statement: D5/S3/ConceptDynamics/PartialIdentification/CanonicalResponseSignature.signature_event_mass_eq_linearObjective

Proof. Machine-checked in Lean as D5/S3/ConceptDynamics/PartialIdentification/CanonicalResponseSignature.signature_event_mass_eq_linearObjective (✓ std3). ∎

Source. Repository-derived.

Commentary.

The event indicator is zero or one on each response signature, so the finite event sum coincides term by term with linear-objective evaluation.

Theorem 1.3 (Every signature mass has an identity exogenous realization).

Lean statement: D5/S3/ConceptDynamics/PartialIdentification/CanonicalResponseSignature.pushforwardSignatureMass_id

Proof. Machine-checked in Lean as D5/S3/ConceptDynamics/PartialIdentification/CanonicalResponseSignature.pushforwardSignatureMass_id (✓ std3). ∎

Source. Repository-derived.

Commentary.

Taking the signature carrier itself as the exogenous state space and the identity as the signature map reproduces every mass exactly.

References

• Truth anchor: D5/S3/ConceptDynamics/PartialIdentification/CanonicalResponseSignature.node_has_unique_position
• Truth anchor: D5/S3/ConceptDynamics/PartialIdentification/CanonicalResponseSignature.pushforwardSignatureMass_id
• Truth anchor: D5/S3/ConceptDynamics/PartialIdentification/CanonicalResponseSignature.signature_event_mass_eq_linearObjective
• Dependency: D5/S0/Certificates/LinearObjectiveDual