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Compatible response forgetting

Abstract

A compatible numbered path certificate constructs an essential finite square graph with both boundaries forgetting after its lag.

Theorem 1.1 (Incoming lifts respect response depth).

Lean statement: D5/S3/ConceptDynamics/Coding/CompatibleResponseForgetting.incoming_response_step

Proof. Machine-checked in Lean as D5/S3/ConceptDynamics/Coding/CompatibleResponseForgetting.incoming_response_step (✓ std3). ∎

Source. Repository-derived.

Commentary.

For two states with the same numbered path responses at depth d plus one, lifting the same incoming base edge places their predecessor states in one depth-d response class. Appending that edge to each depth-d path identifies the resulting lifted paths with the original depth-(d+1) observations.

Numbered compatible squares give incoming and outgoing lifts on the R-edge states. Their finite sweeps identify responses with the certificate path bijections, and counting edge fibers yields the first row and column response matrices.

References