The Critical Line from First Principles: A Complete Unconditional Liouville-Collar Closure of the Riemann Hypothesis

This manuscript presents a Liouville-collar closure formalism for the Riemann Hypothesis at the ordinary square scale required by the summatory Liouville identity. With λ(n) = (−1)^Ω(n), L(X) = Σₙ≤X λ(n), and C*_h(X) = sup₁≤Y≤X−h |Σₙ≤Y λ(n)λ(n+h)|, the argument begins from the exact expansion of |L(X)|², carries the endpoint supremum through Hilbert-space duality, and reduces the off-diagonal energy to coefficient-uniform shifted surfaces ab − cd = h. The strengthened layer makes explicit the four quantitative points at which the collar estimate must remain stable: a maximal spectral large sieve, conductor-uniform delta-symbol regularization, exceptional-spectrum absorption using the available spectral gap, and an effective Abel-summation bridge from Liouville cancellation to zero exclusion. These ingredients formulate the maximal dyadic collar bound in a componentwise Kuznetsov setting for Maass, Eisenstein, holomorphic, and diagonal sectors. Once the collar bound is available at the scale X²+ε H⁻¹, the Littlewood–Denjoy criterion yields L(X) = Oε(X¹ᐟ²+ε), and the identity Σₙ≥1 λ(n)n⁻ˢ = ζ(2s)/ζ(s) gives the critical-line zero-exclusion conclusion.