samePattern_refl
For any octave O the samePattern relation holds between O and itself because OctaveEquiv is reflexive. Workers on RRF scale-transfer arguments cite this when they need to treat pattern equivalence as a relation on the space of octaves. The proof is a direct term that packages the reflexivity constructor of OctaveEquiv.
claimFor every octave $O$, there exist mutually inverse morphisms from $O$ to $O$ that preserve the strain functional exactly, so $O$ manifests the same pattern as itself.
background
An octave is a state space equipped with a strain functional (the J-cost) and a set of display channels. Two octaves manifest the same pattern precisely when they are equivalent, i.e., when there exists a pair of mutually inverse morphisms that preserve strain isometrically in both directions. The module develops transfer theorems that move equilibrium structures and ordering relations across such equivalent octaves.
proof idea
The proof is a one-line term wrapper that applies the reflexivity constructor of OctaveEquiv to the input octave O, thereby producing a witness for Nonempty (OctaveEquiv O O).
why it matters in Recognition Science
Reflexivity of samePattern is required before the relation can be used as an equivalence in octave-transfer arguments. It directly supports the module's main results on equilibria transfer and order preservation under morphisms. Within the Recognition framework it supplies the base case for showing that the same pattern can appear at different scales while preserving strain structure.
scope and limits
- Does not prove symmetry or transitivity of samePattern.
- Does not incorporate the phi-ladder or eight-tick structure.
- Does not address physical constants or mass formulas.
- Does not transfer equilibria or ordering relations.
formal statement (Lean)
90theorem samePattern_refl (O : Octave) : samePattern O O :=
proof body
Term-mode proof.
91 ⟨OctaveEquiv.refl O⟩
92