A Closed Formula for Fibonacci Winding Numbers at Powers of Two

This paper establishes an explicit formula for the Fibonacci winding number modulo powers of two. For every integer k greater than or equal to 5, the Fibonacci winding number modulo 2 to the power of k equals three times 2 to the power of (k minus 2), minus 5. Equivalently, the number of carries arising from the addition of consecutive Fibonacci residues over a complete Pisano period is exactly five less than half the period length. The proof uses a binary lifting recurrence, the 2-adic structure of Lucas numbers, and exact half-period and quarter-period symmetries. These yield a depth-independent carry-count recurrence and explain the appearance of the constant five. Further consequences include exact formulas for mean Fibonacci residues, digit averages, and residue-class bit balances. A complete Python verification script is included in the appendix to provide reproducible checks of finite computations. The Fibonacci winding number itself is a previously studied arithmetic quantity. The contribution of this paper is the explicit formula for powers of two and its symmetry-based proof. No claim of priority over all possible earlier derivations is made. All results concern complete modular periods. They do not establish short-interval equidistribution or normality of Fibonacci digit concatenations.