Cost Ndim Octave Octave Trajectory Periodic

The octave cycle

A cycle is a loop: after one full turn, the pattern returns to where it started. The octave trajectory is a concrete example of that idea, built from eight cosine waves whose starting points are evenly spaced around a circle. Each wave is a smooth back-and-forth motion, and the octave trajectory is the eight of them taken together as a single object in eight-dimensional space.

The trajectory is defined by a simple formula: for each of the eight positions, the value is the amplitude times the cosine of time plus a phase shift. The phase shifts are the eight evenly spaced fractions of a full circle, so the eight waves are staggered like the petals of a flower. This is the standard way to represent a cycle in harmonic analysis: a set of oscillators that are identical except for their timing.

The key fact, proved in the framework's machine-checked library of formal theorems, is that this trajectory is periodic: if you add one full turn of time, which is 2π, to any moment, the entire eight-dimensional object comes back exactly to its previous state. The proof is a direct calculation using the cosine addition formula, and it holds for any amplitude and any starting time. This is not a physical claim about the world; it is a mathematical property of the chosen definition, verified step by step.

In Recognition Science, this periodic trajectory is used as a visualization aid for the framework's eight-tick recognition cycle, the discrete sequence of steps that the framework derives from its cost function. The trajectory gives a smooth, continuous picture of that discrete cycle: as time runs, the eight coordinates trace out a closed loop, and the loop closes exactly when a full cycle of eight ticks has passed. The periodicity theorem is what guarantees the picture is a genuine loop rather than a spiral that never returns.

What the theorem does not claim is just as important. It says nothing about the physical meaning of the trajectory, nothing about whether the eight ticks are real events in the world, and nothing about the cost function that the framework uses to derive the cycle. It is a purely formal statement about a cosine construction, proved from the definition alone. The bridge from this mathematical picture to any physical recognition process is a separate question that the framework leaves open.

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]

/-- 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)

What this page does not claim

The theorem does not claim that the octave trajectory has any physical existence or that the eight ticks are real events. The theorem does not establish that the cost function J forces this particular trajectory; the trajectory is a definitional choice, not a derived consequence. The theorem does not prove that a full cycle takes exactly 2π units of time in any physical sense.

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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Derived articles

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