An Information-Theoretic Approach to the Adjacent Chowla Conjecture via the Möbius Entropy Deficit

We introduce an information-theoretic framework for the adjacent two-point Chowla problem C₁(N) = (N−1)⁻¹ Σ μ(n)μ(n+1) based on the Kullback–Leibler divergence between empirical block laws of the Möbius function and a support-balanced reference law. The paper gives an exact fibre decomposition of the two-symbol entropy deficit showing that Δ₂(N) → 0 if and only if C₁(N) → 0, a conductor decomposition of the adjacency Dirichlet series through inverse Dirichlet L-functions, and a two-parameter family D(s,w). New analytic results include dyadic coefficient energy bounds, a phase-measure formulation of the backbone harmonic, an averaged first-harmonic decay theorem, an all-τ de la Vallée Poussin bound, and a bridge theorem yielding summability of dyadic backbone blocks for every fixed Möbius twist. The diagnostic section identifies a Kronecker resonance obstruction to uniform backbone decay, proves that resonant prime phases lie beyond polynomial Perron height, and isolates the remaining gap as a short-range conductor-block problem for sums of μ in arithmetic progressions at Re s < 1. This paper does not prove the Riemann Hypothesis and does not prove Chowla's conjecture.