Self-Knowledge Exceeds Other-Knowledge

When independent random variables with log-concave distributions share an additive budget constraint, the interaction information is non-negative: II (XA; XC; Xⱼ) ≥ 0. The quantitative strengthening II ≥ Var (ε) /log 2 (sharp constant), where ε (s) = H (XA | S = s), is proved, as is the strictly stronger mutual-information monotonicity under inverse temperature for smooth-number staircases. The theorem holds for both discrete and continuous marginals and admits eleven equivalent formulations across probability, spectral theory, information geometry, algebraic combinatorics, analytic number theory, and channel comparison. Seven independent proofs with largely disjoint toolkits reveal a six-level hierarchy of principles, culminating in a meta-principle — low-degree boundary forcing over structured domains — that explains why the multiple proofs exist. Three of the seven proofs establish the main theorem rigorously for all log-concave marginals; the strongest formulation, spectral gap convexity (-log (1-ρ*²) convex in β), is established by a computer-assisted proof for smooth-number staircases with min (|A|, |C|) ≤ 3. Applications include prime factorization under the harmonic measure, where conditioning on any group of prime exponents always increases the mutual information between the remaining groups.