Dimension Six Has No Fourth Door

Mutually unbiased bases describe complementary quantum measurements: certainty in one basis reveals nothing about outcomes in another. Dimension six has resisted a complete classification for more than three decades. The new proof treats transition matrices between bases as a topological space. Except for Tao's matrix, the authors show that the order-six complex Hadamard matrices form one connected space. That connectedness reduces the search to finitely many overlapping phase regions that can be certified with interval arithmetic. Every triple then falls either into a two-parameter family or one isolated Fourier-based case, and none admits a fourth basis. The paper says this independently reaches a recently reported impossibility result while adding a full classification. Dimension six becomes the first non-prime-power dimension whose complete sets of mutually unbiased bases are known.