Further remarks on the inverse Littlewood conjecture

For a finite set A, write L(A)=∫₀¹|∑a∈ Ae2π iat|\,dt. We prove the uniform lower bound L(A)≥(c_*-o(1))log|A|, where c_*>0.2459209 is explicit. We also establish an almost-covering inverse theorem for an explicit class of integer sets generated from dense separated blocks and long arithmetic block inflations by cyclic Sidon-fibre decompositions of arbitrary depth. If A belongs to this class and L(A)≤ Klog|A|, then, for every ε>0, all but ε|A| points are covered by boundedly many disjoint, linear-sized subsets of bounded doubling. The constants depend only on K, ε, and the fixed terminal-geometry parameters, and are independent of the decomposition depth, branching, moduli, and diameter. The same conclusion holds under high-order Freiman modelling. An accompanying Lean 4 formalization is available at: https://github.com/hxypqr/littlewood-constant-inverse