Convolution–Correlation Duality: A Universal Principle for Spectral Damping

We present a universal spectral damping principle that classifies the tractability of problems across number theory, probability, signal processing, combinatorics, PDE, and finance. The core mechanism is elementary: convolution of independent components damps oscillatory Fourier coefficients, while autocorrelation preserves them. We prove 44 machine-verified theorems formalizing this duality. The key insight reduces to a single algebraic fact: for \(0 < a < 1\), we have \(a^2 < a\). This explains why binary Goldbach (sum of two primes, \(k=2\)) sits at the boundary of tractability—the spectral sum \(\sum 1/|\rho|^2\) converges—while twin primes (\(k=1\), correlative) remain intractable with \(\sum 1/|\rho|\) divergent. We formalize the \(k\)-fold damping hierarchy showing that Vinogradov's ternary result (\(k=3\)) is strictly easier than binary Goldbach. The framework extends to CLT variance reduction, heat kernel decay, Cauchy–Davenport sumset growth, and portfolio diversification failure under correlation. All results are verified in the Lean 4 proof kernel with zero axioms.

Keywords: spectral damping, convolution, correlation, Goldbach conjecture, twin primes, CLT, number theory, probability

MSC 2020: 42A85, 11P32, 60F05