Live Collatz Triple Coalescences with Mixed Periodic Ancestry and a Finite Fast-Anchor Obstruction

Description We construct explicit infinite arithmetic families of three distinct positive odd integers whose accelerated Collatz trajectories merge after five iterations and share a common first passage at step nineteen. The construction combines synchronized forward coalescence with linearly deep inverse ancestry arising from different negative periodic orbits. Using a common rational $3$-adic anchor, we connect three inverse branches to the negative fixed point, the negative two-cycle, and the negative seven-cycle. For every integer height $L \ge 3000$, three explicit Chinese-remainder classes contain exponentially many such triples, with certified gate-relative hereditary-rank lower bounds $$\rho_L(W-1) \ge \lfloor L/20 \rfloor, \quad \rho_L(W-7) \ge \lfloor 7L/20 \rfloor, \quad \rho_L(W-91) \ge \lfloor L/125 \rfloor.$$ We also establish an alternative mixed-periodic triple construction and a faster two-branch ancestry construction. A complementary computer-assisted theorem proves that, within a precisely defined class of integral-negative-shadow inverse words and for dyadic rational anchors $0 < A \le 3000$, at most two distinct eligible offsets can connect to the specified fast negative periodic orbits. The finite classification is supported by two independent exact-arithmetic implementations and reproducible certificates. The results concern explicit Collatz orbit constructions and first-passage ancestry. They do not establish the Collatz conjecture or determine unrestricted inverse-tree ranks. Creator Lee Byoungwoo Independent Researcher, Daejeon, Republic of Korea Keywords Collatz conjecture; accelerated Collatz map; Collatz dynamics; synchronized coalescence; first-passage dynamics; inverse Collatz trees; hereditary rank; negative periodic orbits; 3-adic anchors; Chinese remainder theorem; Diophantine congruences; arithmetic dynamics