Zero-Target Decision-State Separation in Collatz Digit Towers: A Myhill–Nerode Lower Bound from Mixed 2-adic and 3-adic Swap Geometry

We study the exact decision problem associated with the correctiondigits q_k of Collatz valuation prefixes. A companion paper("Multiple-Swap Superposition and Ultrametric Memory Separation inCollatz Correction Coordinates", DOI: 10.5281/zenodo.21502490)establishes unbounded state complexity for computing the fullcorrection coordinate B_m exactly; that result does not by itselfimply hardness of the binary predicate q_k=0. We close this gapunconditionally. Using disjoint, non-overlapping swaps on adjacent unequal symbols in avaluation word, we construct, for every r, a cumulative chain of r+1prefixes sharing a common terminal height -- an explicit, self-containedconstruction on the elementary alternating word (1,2)^r. The pairwisecorrection differences satisfy two complementary ultrametric laws: the2-adic valuation of a pairwise difference is controlled by the firstdiffering swap, while its 3-adic valuation is controlled by the last.Under any common continuation, the residue difference reduces to asingle fixed 2-adic integer, independent of the continuation and of theposition; the 3-adic law shows this fixed integer is never an ordinaryinteger. Along the common valuation-one continuation, thisnon-integrality forces an explicit separation of the correspondingzero-target predicates. Consequently the zero-target language has infinite Myhill-Nerode index:no deterministic finite-state streaming automaton can decide the exactpredicate q_k=0 on all finite valuation words. We verify theconstruction exhaustively for an explicit r=20 family (all 210 pairs).This result does not address termination, undecidability of theclassical Collatz map, or the eventual condition "q_k=0 eventually"; itisolates a specific unbounded-memory obstruction in exact, single-timecarry coherence decisions. This is a companion paper to "Prefix Recovery from IRA CorrectionCoordinates" (DOI: 10.5281/zenodo.21486899) and "Multiple-SwapSuperposition and Ultrametric Memory Separation in Collatz CorrectionCoordinates" (DOI: 10.5281/zenodo.21502490), together forming theIRA-RCS (Recover-Compose-Separate) research program on the statecomplexity of Collatz correction coordinates.