Balanced Two-Sided Hereditary-Rank Lower Bounds for Live Collatz Coalescences: A Common Rational Anchor for Long Inverse Valuation-One Tails

Description We establish explicit arithmetic families of live Collatz coalescences exhibiting balanced, linearly growing hereditary-rank lower bounds on both branches of the accelerated odd Collatz dynamics. For every integer height $L \ge 200$, we construct exponentially many pairs of positive odd integers $X = W - 1$ and $Y = W - 7$ whose accelerated Collatz trajectories coalesce after exactly three iterations and share the same live first-passage state after eleven iterations. Both starting points possess gate-relative hereditary ranks satisfying $$\rho_L(X) \ge D_L + 1, \qquad \rho_L(Y) \ge D_L \ge \lfloor 0.29L \rfloor.$$ The central construction uses a common rational anchor to synchronize two distinct inverse valuation words. After finite initial inverse segments, both ancestral branches admit arbitrarily long valuation-one tails, yielding comparable linear growth of their hereditary-rank lower bounds. More generally, we establish a quantitative trade-off between simultaneous ancestral depth and the exponential abundance of admissible coalescing pairs. For suitable parameters $0 < q < \log_3 2$, the two rank lower bounds grow with asymptotic coefficient $q \log_2(3/2)$, while the number of constructed pairs grows as $$2^{(1 - q \log_2 3)L + O_q(1)}.$$ Thus the common lower-bound coefficient can approach $\log_3(3/2) \approx 0.36907$ while retaining a positive exponential population growth rate. The results provide a quantitative structural description of synchronized coalescence and inverse ancestry in Collatz dynamics. Complete proofs and reproducible exact-integer verification programs are provided. This work is an independent sequel to Live Collatz Coalescence with Two-Sided Linear Hereditary-Rank Lower Bounds (DOI: 10.5281/zenodo.23790756). 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; inverse valuation words; Diophantine congruences; arithmetic dynamics