Further remarks on the inverse Littlewood conjecture

For a finite set $A\subset\mathbb{Z}$, write $\mathcal{L}(A)=\int_0^1|\sum_{a\in A}e^{2\pi iat}|\,dt$. We prove the uniform lower bound $\mathcal{L}(A)\ge(c_*-o(1))\log|A|$, 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 $\mathcal{L}(A)\le K\log|A|$, then, for every $\varepsilon>0$, all but $\varepsilon|A|$ points are covered by boundedly many disjoint, linear-sized subsets of bounded doubling. The constants depend only on $K$, $\varepsilon$, 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