Golden Sampling Disk Atom

Abstract

Golden negative-time sampling sends positive-height modes inside the unit disk.

Theorem 1.1 (Positive-height golden samples are disk atoms).

$$\forall omega\in \mathbb{R},hObserver\in \mathbb{R},hMode\in \mathbb{R},hObserver<hMode\Rightarrow leth:=hMode-hObserver;letq:=\mathrm{goldenSamplingAtom}\left(omega,h\right);\left(\left(\left(q={\phi }^{-2\cdot h}\cdot \exp \left(-i\cdot T_{\phi }\cdot omega\right)\wedge \Vert q\Vert ={\phi }^{-2\cdot h}\right)\wedge \Vert q\Vert <1\right)\wedge \Vert \mathrm{goldenSamplingAtom}\left(omega,0\right)\Vert =1\right)\wedge 1<\Vert q^{-1}\Vert$$

Proof. Machine-checked in Lean as D5/S3/Observer/GoldenCoding/GoldenSamplingDiskAtom.golden_sampling_disk_atom (✓ std3). ∎

Source. Repository-derived.

Commentary.

The height h is the mode height minus the observer height, and T_phi is the repository’s positive golden scale period. The displayed multiplier separates into a radial golden-ratio power and a unit-norm complex phase.

Strictly positive height makes the radial exponential less than one. The same norm calculation gives unit norm at height zero and places the reciprocal of every positive-height atom outside the unit disk.

The inverse-Fourier residue formula in the source depends on a transform convention not defined by the atom. This theorem records the self-contained pointwise consequence for its displayed multiplier.

References

• Truth anchor: D5/S3/Observer/GoldenCoding/GoldenSamplingDiskAtom.golden_sampling_disk_atom
• Dependency: D5/S3/CompletionDynamics/GoldenMobius/GoldenScaleHelix