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

TL;DR

This work reframes prime constellations as deterministic signals generated by a finite Diophantine basis, introducing a certification window to circumvent the parity barrier. By analyzing residues through a CRT-guided, band-limited system, it proves a variance gap and a correlation-cancellation mechanism that force the existence of constellations rather than relying on probabilistic models. The authors derive asymptotic dominance results showing the mean count grows and variance remains linear, enabling second-moment arguments to establish infinitude for twin primes, sums in Goldbach, and general k-tuple configurations. Numerical evidence supports sub-Poissonian fluctuations and the predicted convergence rates, suggesting a new, structurally grounded route to long-standing conjectures. The framework links spectral rigidity with combinatorial sieving and offers a universal approach potentially applicable to broader questions in additive number theory and equidistribution.

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 ($O(μ)$) rather than the quadratic scaling ($Ω(μ^2)$) 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.

Figures (3)

• Figure 1: Variance-to-Mean Ratio Convergence. The ratio $\sigma^2/\mu$ increases toward the Poisson limit of 1.0 as $m_0$ grows. For finite $m_0$, the system exhibits sub-Poissonian behavior ($F < 1$) due to the finite sieving basis. This empirical evidence supports the correlation cancellation theorem, confirming that the system lacks the "noise" energy required to generate large prime-free gaps.
• Figure 2: Twin Prime Count vs Theory. The actual count of certified twin primes (gray squares) tracks the theoretical prediction (solid line) with high precision across logarithmic scales. The absence of deviation corroborates the Asymptotic Dominance Theorem.
• Figure 3: Convergence of Error Rate. The relative error (Coefficient of Variation) between the signal model count and the theoretical density decays as $O(L^{-1/2})$. This vanishing error rate is the signature of a deterministic system stabilizing as the energy scale increases.

Theorems & Definitions (89)

• Theorem 2.1: Canonical Seed Generation
• proof
• Proposition 2.2: Parity and Coprimality Constraints
• proof
• Definition 2.3: The Gear System
• Theorem 2.4: Structural Primality Test
• proof
• Lemma 2.5: Factor Coverage
• proof
• Remark 2.6