Fixed-Weight Hamming Distance Distributions of Fibonacci Strings

A Fibonacci string is a binary string with no two consecutive 1s. This work studies the Hamming-distance distribution inside each fixed-weight layer of the Fibonacci cube. It derives an explicit finite formula for the number of ordered equal-weight pairs at a prescribed distance using an alternating-block decomposition, and introduces an endpoint-weight-refined distance enumerator with a rational generating function obtained from a four-state transfer matrix. The known ordered Hosoya polynomial is recovered as a specialization. Consequences include a closed distance-two formula, polynomiality in the string length for fixed parameters, and exact near-extremal values. The archive includes the manuscript source, exact verification code, tests, and selected machine-readable outputs.