Determine the extremal liminf ratio of maximal term to maximum modulus for entire functions (Erdős #513)

Let $f=\sum_{n=0}^\infty a_nz^n$ be a transcendental entire function. Write $\mu(r)=\max_n \lvert a_n r^n\rvert$ for the maximal term of the series at radius $r$, and $M(r)=\max_{\lvert z\rvert=r}\lvert f(z)\rvert$ for the maximum modulus. What is the greatest possible value of $$\liminf_{r\to \infty} \frac{\mu(r)}{M(r)}\,?$$ Equivalently, determine $B$, the supremum over all transcendental entire $f$ of $\liminf_{r\to\infty}\mu(r)/M(r)$.

Acceptance. FULLY RESOLVES: determine $B$ exactly — a construction (or limiting family) of transcendental entire functions attaining the value in the liminf, plus a matching upper-bound proof valid for all transcendental entire functions; a complete written proof with all steps, machine-checkable (Lean/Coq) preferred. ADVANCES: (a) an explicit entire function, given by its coefficient sequence, together with a rigorous (certified, e.g. interval-arithmetic) proof that its liminf ratio strictly exceeds the best lower bound stated in the background; (b) an upper bound strictly below the best upper bound stated in the background, with proof; or (c) an explicit numeric value for the constant $c$ in the Clunie–Hayman upper bound $2/\pi - c$, with proof. Deliver the proof file, or the construction (coefficient formula) plus certification code and its output.