A Note on Left-Zero Padding in Reverse-and-Add Dynamics: Representation Dependence and a One-Step Palindromic Closure

The classical reverse-and-add process reverses the ordinary digit expansion of an integer and adds the result. This note studies a representation-sensitive variant in which leading zeros are retained before the first reversal. A k-zero launch changes the first update to n + 10^k R(n). More generally, in base b, if n has d digits and a nonzero units digit, prepending d zeros before reversal yields n + b^d R_b(n), whose digit string is rev(s)s and is therefore a palindrome. For 196, a single leading zero gives 0196 + 6910 = 7106, followed by 13123 and then the palindrome 45254; three leading zeros give the one-step palindrome 691196. The note explicitly does not claim to solve the standard 196 problem, which uses ordinary integer reversal and discards leading zeros. It documents the representation dependence, a general padded-closure proposition, and a small reproducible computation on the 13 three-digit decimal candidate starting values.