Zero-Density Reciprocal Prime Evidence

Abstract

A zero-relative-density prime set can carry divergent reciprocal mass and, under the product-law criterion, statistical completion.

Theorem 1.1 (Sparse primes can retain divergent reciprocal evidence).

$$\exists S\subseteq Primes,\underset{n\to \infty }{\lim }\mathrm{relativePrimeCountingRatio}\left(S,n\right)=0\wedge \left(\neg \mathrm{Summable}\left((p:S\mapsto \frac{1}{p})\right)\wedge \forall e,mu,nu,\left(mu\perp nu\iff \neg \mathrm{Summable}\left(e\right)\right)\Rightarrow \left(\mathrm{IsTheta}\left((p:S\mapsto e(p)),cofinite,(p:S\mapsto \frac{1}{p})\right)\Rightarrow mu\perp nu\right)\right)$$

Proof. Machine-checked in Lean as D5/S3/Analytic/ZetaEntropyPlane/ZeroDensityReciprocalPrimeSet.zero_density_divergent_reciprocal_prime_set (✓ std3). ∎

Source. Repository-derived.

Commentary.

There is one subset of the primes whose relative counting ratio tends to zero while its reciprocal-prime family is not summable.

The Lean witness is a classical, noncomputable selection: its block endpoints use Nat.find on analytic existence of enough reciprocal mass. It is not a computable enumeration of the subset.

For the same subset, evidence asymptotic to one over p yields mutually singular transcript laws under the singularity-versus-divergent-energy equivalence from the source’s Theorem 233.1. No separate absolute-continuity-versus-summability premise is assumed.

References

• Truth anchor: D5/S3/Analytic/ZetaEntropyPlane/ZeroDensityReciprocalPrimeSet.zero_density_divergent_reciprocal_prime_set
• Dependency: D5/S3/Analytic/ZetaEntropyPlane/PrimeRelativeDensityEvidenceDivergence