The Convolution–Correlation Duality

We identify a structural dichotomy — the convolution–correlation duality — that governs whether problems involving oscillatory spectral expansions are tractable. The principle is this: when a quantity is formed by convolving independent components, each independent integration contributes a damping factor to the spectral coefficients. When the same quantity is formed by correlating locked components (measuring a single object against a shifted copy of itself), no such damping occurs.

In number theory, this explains why Goldbach's conjecture (a convolution: \(p + q = n\)) has a convergent zero sum \(\sum 1/|\rho|^2 \approx 0.046\), while the twin prime conjecture (a correlation: \(p\) and \(p+2\) locked) retains the divergent sum \(\sum 1/|\rho|\). We prove that the same mechanism operates in five other domains: the central limit theorem versus correlated-sum fluctuations in probability, matched filtering versus autocorrelation noise in signal processing, sumset versus difference-set growth in additive combinatorics, heat kernel smoothing versus self-similar blowup in PDE, and the pricing of correlated lognormal portfolios in quantitative finance.

Beyond classification, we identify a constructive principle: cross-domain transfer. A problem that is correlative in one domain may become partially convolutive in another, because different domains have different native independence structures. The most powerful proofs in mathematics — Vinogradov's ternary theorem, Roth's theorem on arithmetic progressions, the Green–Tao theorem on primes in AP — are circuits through multiple domains, where each transfer either decorrelates the problem or provides tools unavailable in the previous domain. We formalize this as a category \(\mathbf{Spec}\) whose objects are spectral domains and whose morphisms are spectral transfers carrying a decorrelation gain \(\delta \in [0,1]\). We prove that \(\mathbf{Spec}\) is a dagger category enriched over \(([0,1], \geq, \cdot)\), compute \(\delta\) for the principal known transfers, draw the explicit transfer graph, identify the missing edges (including Langlands as the highest-value candidate), and characterize the density increment paradigm as iterated circuit composition with compounding gain. The decorrelation gap \(\delta^* - \delta(P)\) — the difference between the required and achievable gain — gives a single number measuring how far we are from a proof of any spectral-type problem.

Keywords: convolution–correlation duality, spectral damping, additive number theory, central limit theorem, signal processing, additive combinatorics

MSC 2020: 42A85, 11P32, 60F05, 94A12, 11B13