OctaveEquiv
Two octaves are equivalent when mutually inverse morphisms exist that preserve the strain functional exactly in each direction. Researchers formalizing pattern transfer across scales cite the structure to equate equilibria and well-posedness. The definition directly records the forward and inverse maps together with the two isometric strain-preservation conditions.
claimAn equivalence between octaves $O_1$ and $O_2$ consists of morphisms $f:O_1.State→O_2.State$ and $g:O_2.State→O_1.State$ such that $J_{O_2}(f(x))=J_{O_1}(x)$ for all $x$ and $J_{O_1}(g(y))=J_{O_2}(y)$ for all $y$, where $J$ denotes the strain functional of each octave.
background
An octave is a state space equipped with a strain functional (the J-cost) and a bundle of display channels, required only to be nonempty. A morphism between octaves is a state map that preserves the weak ordering induced by the strain functional. The module treats octaves as abstract data structures, deferring any concrete scaling factor or periodicity to separate hypotheses.
proof idea
The declaration is the structural definition of octave equivalence. It records the forward morphism, the inverse morphism, and the two axioms asserting exact equality of the strain functional under each map. No lemmas or tactics are applied; the structure itself is the claim.
why it matters in Recognition Science
The structure supplies the equivalence relation used by equiv_equilibria_iff and equiv_wellPosed to transfer balanced states and well-posedness between octaves. It also grounds the definition of samePattern, which declares two octaves to manifest the same pattern precisely when an equivalence exists. In the Recognition framework the construction isolates scaling relations among manifestations while keeping physical constants outside the core octave data.
scope and limits
- Does not assert existence of any scaling factor such as phi.
- Does not incorporate eight-tick periodicity.
- Does not impose physical interpretation on the strain functional.
- Does not require equivalent octaves to share identical display channels.
formal statement (Lean)
88structure OctaveEquiv (O₁ O₂ : Octave) where
89 /-- Forward morphism. -/
90 toFun : OctaveMorphism O₁ O₂
91 /-- Backward morphism. -/
92 invFun : OctaveMorphism O₂ O₁
93 /-- Strain is preserved exactly (forward isometry). -/
94 strain_eq : ∀ x, O₂.strain.J (toFun.map x) = O₁.strain.J x
95 /-- Strain is preserved exactly (inverse isometry). -/
96 strain_eq_inv : ∀ y, O₁.strain.J (invFun.map y) = O₂.strain.J y
97
98namespace OctaveEquiv
99
100/-- Reflexivity. -/