Core Spectral Pattern Theory

We propose a universal, quantitative definition of "pattern": a pattern is a low-rank spectral component of a data tensor whose eigenvalue exceeds the Marchenko--Pastur noise floor. The smoothness parameter \(\rho > 1\) --- the rate of eigenvalue decay --- is a single number that characterizes the pattern's strength, compressibility, and predictive power across all domains. We show: (1) pattern detection reduces to eigenvalue thresholding (\(\lambda_k > \lambda_{\text{MP}}\)), (2) pattern extraction is optimal truncation (Eckart--Young), (3) pattern quality is measured by \(\rho\) (faster decay = stronger pattern), (4) pattern size is \(N = \Theta(\log(1/\varepsilon)/\log\rho)\) modes (USRT, dimension-free), and (5) overfitting is claiming a pattern when \(\rho < 1.1\). We validate on real data across five domains: chemistry (Wine, \(\rho = 10.1\)), biology (Iris, \(\rho = 4.2\)), medicine (Breast Cancer, \(\rho = 2.1\)), psychology (Many Labs 2, 20 studies \(\times\) 66 labs, \(\rho = 1.3\)), and pure noise (\(\rho = 1.0\)). The ordering is monotone with replication rate (Spearman \(r_s = 0.894\), \(p = 0.041\)): domains with \(\rho > 2\) always replicate; psychology at \(\rho = 1.3\) replicates 54\% of the time; noise never replicates. On the Reproducibility Project (54 studies, raw data from OSF), effect size \(|r|\) predicts replication with AUC \(= 0.697\). The spectral definition resolves long-standing debates: it explains why physics is more predictive than economics (higher \(\rho\)), why deep learning works (data has \(\rho > 1\)), why financial alpha decays (the market learns your \(\rho\)), and why the reproducibility crisis is concentrated in low-\(\rho\) fields. The deepest implication: there is a hierarchy of knowability, and \(\rho\) is its index. The question "how much can we know about this system?" has a quantitative answer.