Prime Golden Complex Mode

Abstract

A first golden prime mode splits into prime-faithful heat amplitude and recurrent unit-circle phase.

Theorem 1.1 (Positive amplitude identifies primes while phase recurs).

$$\begin{array}{c}\forall s,0<s\Rightarrow \forall P,\forall e,0<e\Rightarrow \forall b,\forall u,\\ \mathrm{Injective}(\mathrm{firstGoldenComplexMode}(s,u))\wedge \\ \exists t,b<t\wedge \forall p\in P,\mathrm{norm}(\mathrm{firstGoldenComplexMode}(0,t,p)-1)<e.\end{array}$$

Proof. Machine-checked in Lean as D5/S3/Observer/GoldenPrimeCircle/PrimeGoldenComplexMode.complex_mode_amplitude_phase_dichotomy (✓ std3). ∎

Source. Repository-derived.

Commentary.

The real coordinate controls modulus and the imaginary coordinate controls rotation.

This is an analytic-time statement and does not identify the parameter with laboratory time.

References

• Truth anchor: D5/S3/Observer/GoldenPrimeCircle/PrimeGoldenComplexMode.complex_mode_amplitude_phase_dichotomy
• Dependency: D5/S3/ObserverMemory/FourierFibers/PrimeZeckendorfTemporalization