Observer-Mode Confinement

Abstract

Proper joint growth of the completed-zeta Archimedean mode confines every bounded-prime strict sublevel in both observer mode and frequency.

Theorem 1.1 (Finite dangerous modes and a common frequency window).

$$\begin{array}{r}\forall primeMultiplier\in \mathbb{R}\to \left(\mathbb{Z}\to \left(\mathbb{R}\to \mathbb{R}\right)\right),L\in \mathbb{R},A\in \mathbb{R},B\in \mathbb{R},\left(\forall n\in \mathbb{Z},t\in \mathbb{R},\Vert primeMultiplier\left(L,n,t\right)\Vert \le B\right)\Rightarrow \mathrm{let}modeShift:\mathbb{Z}\to \mathbb{R}=n:\mathbb{Z}\mapsto n\cdot goldenAngularFrequency,\mathrm{let}archimedeanMode:\mathbb{Z}\times \mathbb{R}\to \mathbb{R}=nt:\mathbb{Z}\times \mathbb{R}\mapsto \frac{1}{2}\cdot \left(2\cdot \pi \cdot \mathrm{mu}\left(\mathrm{snd}\left(nt\right)+modeShift\left(\mathrm{fst}\left(nt\right)\right)\right)+2\cdot \pi \cdot \mathrm{mu}\left(\mathrm{snd}\left(nt\right)-modeShift\left(\mathrm{fst}\left(nt\right)\right)\right)\right),\mathrm{Tendsto}\left(archimedeanMode,\mathrm{cocompact}\left(\mathbb{Z}\times \mathbb{R}\right),atTop\right)\Rightarrow \mathrm{let}jointMultiplier:\mathbb{Z}\times \mathbb{R}\to \mathbb{R}=nt:\mathbb{Z}\times \mathbb{R}\mapsto archimedeanMode\left(nt\right)-primeMultiplier\left(L,\mathrm{fst}\left(nt\right),\mathrm{snd}\left(nt\right)\right),\mathrm{let}dangerousSet:\mathrm{Set}\left(\mathbb{Z}\times \mathbb{R}\right)=\{nt\in \mathbb{Z}\times \mathbb{R}\mid jointMultiplier\left(nt\right)<A\},\mathrm{Finite}\left(\{n\in \mathbb{Z}\mid \exists t\in \mathbb{R},(n,t)\in dangerousSet\}\right)\wedge \left(\left(\forall n\in \mathbb{Z},\mathrm{IsBounded}\left(\{t\in \mathbb{R}\mid (n,t)\in dangerousSet\}\right)\right)\wedge \left(\exists modes\in \mathrm{Finset}\left(\mathbb{Z}\right),radius\in \mathbb{R},0\le radius\wedge \left(\forall n\in \mathbb{Z},t\in \mathbb{R},jointMultiplier\left((n,t)\right)<0\Rightarrow \left(n\in modes\wedge \Vert t\Vert \le radius\right)\right)\right)\right)\end{array}$$

Proof. Machine-checked in Lean as D5/S3/Weil/ZetaGamma/ObserverModeConfinement.two_direction_archimedean_confinement (✓ std3). ∎

Source. Repository-derived.

Commentary.

The observer shift is the canonical integer multiple of the golden angular frequency. The Archimedean term is the symmetric average of the existing completed-zeta digamma multiplier 2 pi mu.

The public hypotheses state proper growth on the full integer-by-real carrier and a uniform bound for the fixed-support prime multiplier. No mode or frequency is specialized.

A compact complement of a sufficiently high Archimedean superlevel contains both the threshold danger set and the negativity set. Its integer projection is finite, while its real projection is bounded.

References

• Truth anchor: D5/S3/Weil/ZetaGamma/ObserverModeConfinement.two_direction_archimedean_confinement
• Dependency: D5/S3/Observer/GoldenPrimeCircle/GoldenVerticalSampling