Cost Ndim Octave Octave Phase

The octave phase

The octave phase is a number that spaces eight cosine waves evenly around a full circle. If you mark eight points on a clock face at 12, 3, 6, and 9, then add the half-hour marks between them, you have the same idea: each point sits 45 degrees, or one eighth of a turn, from its neighbor. The definition sets the phase for step i as 2πi/8, where i runs from 0 to 7. This is not a discovery about nature; it is a choice of how to arrange a set of curves for display.

The curves themselves are cosine waves, each starting at its own phase. The trajectory for step i is amp times the cosine of (time plus the phase for i). So at time zero, the eight waves are not aligned; they are staggered like runners in adjacent lanes. The one proved fact about this arrangement is that it repeats: shifting time by a full 2π brings every wave back to its starting value. That is the periodicity of cosine, not a new claim.

In Recognition Science, this octave arrangement appears in the n=8 visualization, where the framework's eight-tick recognition cycle is shown as eight phase-shifted waves. The framework models a cycle of eight recognition events, and the octave phase gives each event a distinct starting point on the circle. The label 'octave' borrows from music: eight notes, evenly spaced, repeating. The visualization is a way to see the cycle, not evidence that the cycle is real.

The declaration does not claim that the phase values are derived from anything deeper. It does not claim that eight is forced by the phase choice itself, nor that the cosine form has physical content. The phase is a coordinate choice for a picture. The periodicity theorem is about the mathematics of cosine, not about recognition. What the octave phase establishes is modest: a clean, repeating way to draw eight related waves, useful for showing a cycle in the framework's own terms.

/-- Uniform phase shift for an octave index. -/
noncomputable def octavePhase (i : Fin 8) : ℝ := 2 * Real.pi * (i : ℝ) / 8

/-- Phase-shifted cosine trajectory used in the `n=8` visualization. -/
noncomputable def octaveTrajectory (amp t : ℝ) : Vec 8 :=
  fun i => amp * Real.cos (t + octavePhase i)

theorem octaveTrajectory_periodic (amp t : ℝ) :
    octaveTrajectory amp (t + 2 * Real.pi) = octaveTrajectory amp t := by
  ext i
  unfold octaveTrajectory
  have hrew : t + 2 * Real.pi + octavePhase i = (t + octavePhase i) + 2 * Real.pi := by ring
  calc
    amp * Real.cos (t + 2 * Real.pi + octavePhase i)
        = amp * Real.cos ((t + octavePhase i) + 2 * Real.pi) := by rw [hrew]
    _ = amp * Real.cos (t + octavePhase i) := by
          rw [Real.cos_add_two_pi]

What this page does not claim

The phase values are not derived from the cost function; they are chosen for the display. The periodicity theorem does not imply that recognition events repeat in time. The octave label does not connect to musical tuning or acoustics.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Cost/Ndim/Octave.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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