samePattern_refl
plain-language theorem explainer
Reflexivity of the same-pattern relation follows directly from the reflexivity of octave equivalence. Researchers establishing that the same-pattern predicate forms an equivalence relation cite this result when verifying basic properties of octave transfer. The proof is a one-line wrapper applying the reflexivity constructor of OctaveEquiv.
Claim. For every octave $O$, the octave $O$ manifests the same pattern as itself.
background
An octave is a structure consisting of a state space together with a strain functional that records the J-cost of configurations. Octave equivalence is the structure containing a pair of mutually inverse morphisms that preserve strain exactly in each direction. The same-pattern predicate between two octaves holds exactly when an octave equivalence exists between them, i.e., when the type of such equivalences is nonempty.
proof idea
The proof is a one-line wrapper that applies the reflexivity constructor of OctaveEquiv to the input octave.
why it matters
This result supplies the reflexive case for the same-pattern relation used throughout the octave transfer theorems. It aligns with the abstract treatment of the eight-tick octave before physical constants are introduced. No downstream uses are recorded in the current module.
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