二桥纽结符号差的精确方差与精细极限律

The signature counts of two-bridge knots admit exact binomial formulas, and their normalized distributions satisfy a central limit theorem. We use these formulas to determine the finite-crossing-number mean and variance in the partially double-counted word model. In particular, the variance minus the crossing number converges to -14/3, giving an exact answer to Conjecture 5.1, the numerical variance question in A central limit theorem for the signatures of 2-bridge knots by Cohen, Baker, Dam, Felber, Madras, Saha and Thackrah (arXiv:2604.21107). A single analytic factor determines every fixed-order cumulant correction and a uniform local expansion to any fixed order. We give its first correction explicitly, together with a relative binomial approximation on the full support, sharp interior large deviations and endpoint asymptotics. For uniform mirror classes, an independent fair choice of sign gives a canonical symmetric distribution. We prove its parity-dependent variance and local corrections, with the sampling normalization retained exactly, and obtain the same large-deviation rate for both models.