The Decorrelation Index

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 companion development of convolution--correlation duality and proof circuits). A problem is solvable when \(\delta(P)\) meets or exceeds its decorrelation threshold \(\delta^(P)\). The decorrelation gap \(\Delta(P) = \max(0,\delta^(P) - \delta(P))\) quantifies, within this framework, how much decorrelation is still missing relative to what the problem demands.

We prove that the computable lower bound \(\underline\delta(P)\) — the maximum over known circuits — can be computed in polynomial time via maximum-reliability path algorithms. We prove that \(\delta(P)\) itself is not determined by any finite subgraph of \(\mathbf{Spec}\), and that the gap \(\Delta(P)\) is monotonically non-increasing under graph augmentation: every mathematical breakthrough that adds an edge to \(\mathbf{Spec}\) weakly decreases \(\Delta\) for all problems simultaneously. We characterize the impossibility boundary — problems requiring \(\delta^ = 1\) that no finite circuit of partial decorrelations can solve — and show that this boundary is non-empty if and only if \(\mathbf{Spec}\) lacks a single transfer with \(\delta = 1\) for the relevant correlative structure. We establish a connection to proof complexity: the minimum circuit length achieving \(\delta \geq \delta^\) provides a lower bound on the number of distinct domain transfers required in any spectral proof. We exhibit illustrative values of \(\Delta\) for twelve benchmark problems across number theory, PDE, and combinatorics (derived from the explicit transfer model in the companion work), and introduce the time-dependent index \(\underline\delta_t(P)\) tracking how the known lower bound evolves as the transfer graph grows.

Keywords: decorrelation index, proof complexity, spectral transfer, problem difficulty

MSC 2020: 03B30, 11-02, 18A05, 68Q17