Structural Existence of Prime Constellations: Asymptotic Spectral Stability in Finite Sieve Windows

TL;DR

This paper proves the existence of prime constellations by analyzing the spectral properties of a deterministic signal model, showing that finite spectral bandwidth enforces the presence of primes in certain patterns.

Contribution

It introduces a novel structural proof of prime constellations using spectral analysis and finite sieve windows, bypassing traditional probabilistic and heuristic methods.

Findings

Variance of the signal scales linearly with the mean, not quadratically.

Finite spectral bandwidth prevents the signal from maintaining a zero count, implying prime constellations must exist.

The proof leverages the Chinese Remainder Theorem and residue class correlations.

Abstract

The distribution of prime constellations, such as Twin Primes ($p, p+2$), is traditionally analyzed via probabilistic models or analytic sieve theory. While heuristic predictions are accurate, rigorous proofs are obstructed by the "Parity Barrier", which prevents classical sieves from distinguishing primes from semi-primes in the asymptotic limit. In this work, we present a structural proof of existence based on deterministic signal processing. We treat the sequence of integers as a signal generated by a rigid Diophantine basis ($N=2n+3m$) and define a fundamental certification window $\mathcal{W} = [P, m_0^2)$ derived from the basis limit $m_0$. We demonstrate that the non-existence of constellations (the "Null Hypothesis") constitutes a low-entropy signal state, a "Prime Desert", that requires infinite spectral resolution to maintain over a quadratic window. Since the sieving basis is finite ($p \le m_0$), the system is band-limited and structurally incapable of synthesizing the destructive interference required to sustain a zero count. By invoking the Chinese Remainder Theorem and analyzing the detailed correlation structure of residue classes, we prove that positive and negative correlations between sieved positions cancel at leading order, constraining the variance of the signal to scale linearly with the mean ($O(\mu)$) rather than the quadratic scaling ($\Omega(\mu^2)$) required to support a Prime Desert. This Variance Gap implies that the signal must strictly oscillate around its mean, rendering the existence of prime constellations a mandatory consequence of the system's finite spectral bandwidth.