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 ≥ 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 ρL(X) ≥ DL + 1, ρL(Y) ≥ DL ≥ 0.29L . 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 < log₃ 2, the two rank lower bounds grow with asymptotic coefficient q log₂(3/2), while the number of constructed pairs grows as 2(1 - q log₂ 3)L + Oq(1). Thus the common lower-bound coefficient can approach log₃(3/2) ≈ 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