Congruential Completeness for Odd Primes and the Structural Singularity of the Prime Two

We establish a sharp structural dichotomy in indexed congruential frameworks associated with affine maps of the form an + b.For every odd prime p greater than or equal to 3, under the coprimality condition gcd(a,p)=1, we prove a principle of congruential completeness: every coherent p-adic congruence tower on the index axis is automatically realizable by a unique nonnegative integer.In contrast, we show that the prime 2 is structurally singular: coherent 2-adic towers may fail to be Archimedean realizable, and an independent realizability criterion is required.These results isolate the unique arithmetic role of the prime 2 and clarify the logical position of Archimedean arguments in indexed sieve theory.