Adjacent-Digit Distinct Squares from Repeated 27 and 72 Blocks

OEIS A135140 consists of the nonnegative integers \(n\) for which both \(n\) and \(n^2\) have no pair of equal adjacent decimal digits. We study two natural repeated-block families: \(X_m\), obtained by repeating the block \(27\) exactly \(m\) times, and \(Y_m\), obtained by repeating the block \(72\) exactly \(m\) times. Using base-\(100\) carry recurrences and the fact that \(100\) has multiplicative order \(11\) modulo \(121\), we prove the exact classifications \[X_m^2 is adjacent-digit distinct m≡ 3{11},\] and \[Y_m^2 is adjacent-digit distinct m≡ 2,6,7{11}.\] Since the roots \(X_m\) and \(Y_m\) themselves have alternating digits, these classifications give explicit infinite subfamilies of A135140 and in particular prove that A135140 is infinite. Exact integer computations through \(m=1000\) are included only as an independent consistency check and are not used in the proof.