The Mounssif Transform (V2): A Terminal-Digit Modular Inverse and Digit-Block Polynomial Congruence Criteria for Odd Integers

We present a structured V2 exposition of the Mounssif Transform for odd integers in base 10. To each odd integer N, the transform assigns an integer m = T(N), determined by the terminal digit of N, such that 10m ≡ 1 (mod N). This inverse relation yields systematic congruences derived from decimal block splitting. Writing N = A_i 10^{n−i} + B_i, we introduce the digit-block polynomials E_i(x) = A_i + B_i x^{n−i} and prove that E_i(m) ≡ 0 (mod N) for all i. Moreover, for any divisor P | N with m_P = T(P), we obtain factor-sensitive criteria E_i(m_P) ≡ 0 (mod P). We also derive a quadratic cancellation identity by combining adjacent criteria and illustrate the method with the complete example 1919 = 19·101. An appendix collects further perspectives, including CRT-based simultaneity, iterations, base-b variants, and a possible primality viewpoint.