Sharp state complexity for a quadratic Bernoulli factory

We study exact deterministic finite-state Bernoulli factories for f_m(p) = 1/[2(1 + 2^m(2p - 1)^2)], with positive integer m and 0 < p < 1. Every transient state is counted, including control and randomness extraction; two output absorbers are excluded. An explicit construction uses O(2^(m/2)) states, matching the classical positive-polynomial lower scale. Positive integer combinations of four fixed-length word languages, consisting of monochromatic words and words with two, four or six runs, make language selection and recognition state costs additive. Each parameter is compiled offline, and the machine uses only the unknown-p coin and deterministic heads/tails transitions, terminating almost surely at each interior bias. The results are candidate theorems supported by general arguments and exact finite graph verification. At p = 1/2, this construction has expected toss count exp(Theta(2^(m/2))), doubly exponential in m; this is a limitation of this realization, not a lower bound on optimal factory time. Editable sources, verification code, compact certificates and a reproducibility report accompany the paper.