Cost Ndim Octave

The octave trajectory

Cost ndim octave is the name for the eight-tick recognition cycle when it is drawn as a curve. The module defines one object, the octave trajectory, and establishes one property about it. The object is a vector of eight real numbers, one for each tick of the cycle, each equal to a common amplitude times the cosine of time plus that tick's own phase. The phase of tick i is 2 pi times i divided by 8, so the eight phases are spread evenly around the full circle. This is a definitional choice, a way to display the cycle, not a new law of nature.

The established property is periodicity. The theorem states that shifting time by 2 pi leaves the whole eight-coordinate trajectory unchanged. In plain language, the octave trajectory repeats itself exactly after one full turn of the phase. The proof is direct: each coordinate's cosine is periodic with period 2 pi, and the phase shift distributes over the addition. This is a machine-checked theorem, verified by the kernel.

The consequence is that the octave picture is a closed loop. No matter how far time runs, the trajectory revisits the same eight-point pattern after each full period. This makes the octave a stable geometric object for studying the eight-tick cycle, even though the trajectory itself is only a visualization, not an additional physical claim.

THEOREM octaveTrajectory_periodic · IndisputableMonolith/Cost/Ndim/Octave.lean

What this page does not claim

This answer does not claim the octave trajectory is derived from the forcing chain; it is a definitional visualization. This answer does not claim the octave trajectory establishes any new physical law beyond periodicity.

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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