Exact Tail-Error Formulas for Continued Fractions and a Certified Approximation to π

A finite simple continued-fraction prefix acts on its remaining tail by a fractional-linear map. This mathematical note gives a self-contained derivation of the classical determinant identity and its exact finite tail-error formula, distinguishing finite error propagation from infinitesimal sensitivity. For the pi prefix [3;7,15,1,292], substitution of tau = log(3)/log(2) gives pi_hat = (103993*tau+355)/(33102*tau+113). Exact rational bounds for arctangent and logarithm series certify 3.32696344545485e-12 < pi - pi_hat < 3.32696344545486e-12. The two numbers share exactly eleven initial digits after the decimal point. The identities are standard; this note supplies an explicit treatment and a reproducible certificate for the particular example. The coefficients already encode a known prefix of pi. No structural relation, statistical significance, or computational advantage is claimed. The supplement contains the editable LaTeX source and bibliography, exact-rational verifier, regression tests, certificate, build instructions, and licensing information. This is an unreviewed preprint, not a journal-accepted article.