OctaveLoop
plain-language theorem explainer
OctaveLoop encodes a closed sequence of exactly eight recognition states with the initial and terminal states identified. Researchers in coherence modeling for biological stability cite this structure to represent periodic cycles aligned with the eight-tick evolution period. The definition is introduced directly as a structure with a Fin 8-indexed map and an explicit equality constraint.
Claim. An octave loop over a state type consists of a function $steps : Fin(8) → State$ together with the equality $steps(0) = steps(7)$, forming a closed recognition sequence of length eight.
background
The Coherence Technology 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; geometries that follow this ratio align with the fundamental scaling law of the ledger. Upstream, the octave constant is defined as 8 ticks, the fundamental evolution period, while φ^5 ≈ 11.09 is noted as close to 12. The structure also draws on the discrete 2D Galerkin State type and the completeness predicate for fully determined assignments.
proof idea
As a structure definition with an empty proof body, OctaveLoop is introduced by direct field specification: the steps map over the finite set of eight indices and the closure equality identifying the endpoints. No lemmas or tactics are invoked.
why it matters
OctaveLoop supplies the type used by the downstream theorem octave_loop_neutrality, which proves zero net recognition flux for a complete loop. It instantiates the eight-tick octave (T7) from the forcing chain, supporting analysis of resonant geometries in the Recognition framework where the period 2^3 appears alongside D = 3 and the φ-ladder. The definition closes a representational gap for applied coherence calculations without addressing full state dynamics.
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