mathematics

Many of the deepest theorems in mathematics were not proved within a single domain but by transferring the problem through a sequence of domains, applying tools native to each, and returning.

We reframe the Langlands program as a systematic edge-construction effort within the proof category $\mathbf{Spec}$ introduced in [Nagy, 2026a]. Each proved Langlands-type result — from Artin reciprocity to the modularity theorem — corresponds to a morphism between spectral domains carrying a computable decorrelation gain $\delta \in [0,1]$.

We prove that a single saddle-point formula governs quantitative approximation in every system whose action functional admits a Latent grade decomposition.

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.

We introduce the *decorrelation index* $\delta(P) \in [0,1]$ of a spectral-type mathematical problem $P$, defined as the supremum of the total decorrelation gain over all proof circuits in the proof category $\mathbf{Spec}$ (as formalized in the comp