octave
plain-language theorem explainer
The definition sets the octave period to 2 raised to the spatial dimension D. Researchers deriving coherence energies or mass ladders in Recognition Science cite it when closing the Fibonacci constraint on dimension. The definition is a direct one-line computation that evaluates to 8 once D is fixed at 3 by the forcing chain.
Claim. The octave period is defined as $2^{D}$ where $D=3$ is the spatial dimension.
background
The Coherence Exponent module derives the coherence energy exponent -5 from the requirement that both D and 2^D are Fibonacci numbers. With D fixed at 3 = F_4 and 2^D = 8 = F_6, the identity 8 - 3 = 5 = F_5 fixes the exponent. The sibling definition D : ℕ := 3 records the spatial dimension forced by T8 in the UnifiedForcingChain. Upstream definitions in Constants and MusicTheory identify the octave as the fundamental period of 8 ticks or the frequency ratio 2.
proof idea
One-line definition that computes 2 raised to the power of the dimension D. It directly applies the prior definition of D as 3 and standard natural-number exponentiation, with the attached comment verifying the result equals 8.
why it matters
This supplies the octave value that closes the coherence energy derivation in the module theorem, where the Fibonacci difference directly yields E_coh = φ^{-5}. It is referenced by downstream constructions including OctaveLoop in CoherenceTechnology and layer compositions in RecognitionCategory. The definition instantiates the T7 eight-tick octave and T8 dimension forcing steps, linking the forcing chain to mass and energy formulas without introducing free parameters.
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