OctaveLoop
OctaveLoop records a closed sequence of exactly eight states with the first and last identified. Researchers modeling resonant geometries in coherence technology cite the structure when analyzing periodicity in recognition systems aligned with the eight-tick evolution period. The definition is introduced directly as a record type consisting of a map from Fin 8 to the state space together with a single boundary equality.
claimAn octave loop over a state space $S$ is a map $s : [0,7] → S$ satisfying $s(0) = s(7)$.
background
The CoherenceTechnology module formalizes the impact of recognition-resonant geometries (φ-spirals, octave-loops) on biological stability. The golden ratio φ is the unique positive fixed point of the self-similar cost recursion, so geometries that follow this ratio align with the fundamental scaling law of the ledger. Upstream, Constants defines octave as one octave = 8 ticks, the fundamental evolution period, while MusicalScale gives the octave ratio as 2; these supply the eight-tick periodicity that the structure instantiates.
proof idea
The structure is introduced directly as a record type with two fields: a function steps assigning a state to each of the eight positions and a closure equality identifying the endpoints.
why it matters in Recognition Science
This definition supplies the type for the downstream octave_loop_neutrality theorem, which shows that a complete octave loop carries zero net recognition flux. It implements the eight-tick octave (T7) from the forcing chain and supports analysis of resonant geometries in the coherence technology setting.
scope and limits
- Does not prescribe transition rules between consecutive states.
- Does not incorporate any flux function or neutrality condition.
- Does not reference the golden ratio or phi-ladder explicitly.
- Does not constrain the state type beyond being an arbitrary Type.
formal statement (Lean)
106structure OctaveLoop (State : Type) where
107 steps : Fin 8 → State
108 closure : steps 0 = steps 7 -- Conceptual closure
109
110/-- **THEOREM: Octave Loop Neutrality**
111 A complete octave loop has zero net recognition flux (σ = 0). -/