Universal and Simultaneous Constructions of Perfect Powers with Equal Decimal Digit Frequencies

We construct perfect powers whose decimal representations contain each of the ten digits equally often. A general even-exponent construction is obtained from full-reptend prime-power denominators and a digit-preserving cancellation of signed block carries. The construction is unconditional: a single explicit prime, 1051, suffices, with its higher powers providing the required parameters for arbitrarily large even exponents. Consequently, for each positive integer exponent there are infinitely many such perfect powers. More strongly, for any positive integer $R$ there are infinitely many positive integers $y$ for which $y,y^2,\ldots,y^R$ each contain every decimal digit equally often. We state explicit formulae and relate them to seven OEIS entries: six sequences of square or cube roots and values receive explicit infinite subfamilies, while the least-base sequence A074205 receives existence and upper-bound consequences. We do not determine any of its least terms or completely enumerate any of these sequences.