refl
plain-language theorem explainer
Reflexivity for octave equivalence is supplied by the identity morphism in both directions. Researchers formalizing equivalence classes of manifestation scales cite this when they need every octave to be equivalent to itself under exact strain preservation. The definition reuses the identity morphism already constructed for OctaveMorphism and discharges the two strain equations by reflexivity of equality.
Claim. For any octave $O$, the reflexive equivalence from $O$ to itself consists of the identity morphism on states in both directions together with the two equations asserting that the strain functional $J$ is preserved exactly under these identities.
background
An octave is an abstract structure consisting of a state space, a strain functional (the J-cost), a bundle of display channels, and a non-emptiness witness. Two octaves are equivalent when there exist mutually inverse morphisms that preserve the strain ordering exactly in both directions. The identity morphism on a single octave is the map that sends every state to itself while preserving the weak-better relation by reflexivity.
proof idea
One-line wrapper that applies the identity morphism from OctaveMorphism in both the forward and inverse slots and uses the reflexivity tactic to discharge the two strain-preservation equations.
why it matters
This definition supplies the reflexivity leg of the equivalence relation on octaves inside the RRF core layer. It sits directly beneath any later construction of equivalence classes or quotient structures on manifestation scales. The module treats octaves as abstract without committing to the phi-ladder or eight-tick octave, leaving those as separate hypotheses.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.