A float continued fraction is reliable only while q_n < 2^26 -- pi diverges at term 13

Computing a continued fraction from a float64 input is reliable only while the convergent denominator q_n stays below about 2^26; past that, rounding in the input's low bits feeds partial quotients that no longer match the true expansion. The chat's worked check: the continued fraction of pi computed from a float64 value tracks the true expansion until it diverges at term 13, which is where q_n first crosses the bound. The mechanism is a precision budget -- float64 carries 53 significand bits and a convergent p_n/q_n pins the input to about 1/q_n^2, so q_n^2 approaching 2^53 (q_n approaching 2^26.5) is exactly where the input's representable precision is exhausted. Recorded as the chat stated it (bound q_n < 2^26, pi's divergence at term 13); a worked instance, not a coordinate.