= Impossibility of universal identical-basis perfect correlations
For two systems of equal dimension $d\ge2$, no joint <pure state> guarantees equal outcome labels whenever both systems undergo the same arbitrary <basic measurement>. Equal outcomes in the <computational basis> would force $|\Psi\rangle=\sum_jc_j|jj\rangle$. On any two coordinates j,k, changing both bases to the real sum/difference pair makes the different-outcome amplitude $(c_j-c_k)/2$, so all coefficients must agree. Changing both to the imaginary-phase pair makes a different-outcome amplitude $(c_j+c_k)/2$, so those coefficients must also be negatives. They must all vanish, contradicting <normalization>. The one-dimensional case is an exception.
The requirement of all \i[complex] bases matters. A maximally entangled <vector> $d^{-1/2}\sum_j|jj\rangle$ does produce matching labels if one party uses a basis and the other its coordinatewise <complex conjugate> basis. <Conjugate-state steering from a maximally entangled pair> gives this conjugation explicitly. Identical real bases are a special case, not evidence for identical arbitrary complex bases.
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