Phangara's conjecture: every order-six S-Hadamard matrix is equivalent to Tao's cubic matrix

Proposition 3.1: if $H$ and its entrywise square are both complex Hadamard matrices of order six, then $H$ is equivalent to Tao's cubic matrix; in particular all its dephased entries are cube roots of unity. The manuscript notes that uniqueness of the cube-root class was already known from Butson classifications (Lampio-Ostergard-Szollosi; Liang-Chen-Long-Qiu); the new step derives cube-root entries from the two Hadamard conditions for arbitrary phases. It says nothing about other orders.

Proposition 3.1: if $H$ and its entrywise square are both complex Hadamard matrices of order six, then $H$ is equivalent to Tao's cubic matrix; in particular all its dephased entries are cube roots of unity. The manuscript notes that uniqueness of the cube-root class was already known from Butson classifications (Lampio-Ostergard-Szollosi; Liang-Chen-Long-Qiu); the new step derives cube-root entries from the two Hadamard conditions for arbitrary phases. It says nothing about other orders.