Self-Similar Recursion Closes the Gleason Gap: Uniqueness of the Born Rule in Dimension Two

Gleason's theorem (1957) establishes that the Born rule is the unique probability measure on Hilbert spaces of dimension d ≥ 3. The restriction to d ≥ 3 is essential: in dimension 2, non-Born frame functions satisfy all of Gleason's constraints. This paper resolves the dimension-2 gap. On a self-similar binary tree — where every node is a two-way split mapping directly to the qubit case — we prove that cos²(πr/2) is the unique correlation function satisfying boundary conditions, monotonicity, positive-semidefinite composition consistency, and continuity. The ingredient closing the gap is self-similar recursion: the same function must satisfy the composition rule at every depth simultaneously, generating an infinite tower of constraints that eliminates all non-Born solutions. The proof uses no quantum mechanical assumptions, no Hilbert space structure, and no representation theory. An h-substitution reveals the composition rule to be the cosine addition identity, reducing the problem to Cauchy's functional equation with fixed endpoints. A universality corollary shows the result is independent of branching factor p for all primes p ≥ 2, establishing the Born rule as a topological consequence of self-similar hierarchical structure. Numerical verification via constraint propagation on finite-depth trees is included.