Terminator classification and singularity in the pn+1 family of maps

For an odd prime p, consider the map T_p(n) = pn+1 (n odd), n/2 (n even). An exact terminator is an orbit value that collapses to 1 in a single contraction step. We prove that terminators exist for T_p if and only if the multiplicative order ord_p(2) is even. This yields a complete algebraic classification: primes without terminators begin 7, 23, 31, while p = 3 achieves the maximal terminator density 1/2. Consequently, p = 3 is the unique member of the family satisfying both the algebraic condition (terminators with density 1/2) and the standard heuristic condition (net multiplier p/4 < 1). This provides a structural explanation for why the 3n+1 map is expected to converge universally while maps with p ≥ 5 are not expected to converge. The proofs develop two auxiliary frameworks for the p = 3 case: a base-4 analysis giving exact growth chain lengths via 2-adic valuation, and an m-coordinate system n = 3m ± 1 that decomposes the map into pure growth (×3/2) and explicit contraction phases. All classification results are unconditional; the singularity conclusion for p = 3 depends on the unproved equidistribution of ν₂(3n+1) along individual orbits, which we identify explicitly as the remaining open problem.