Live Collatz Coalescence with Two-Sided Linear Hereditary-Rank Lower Bounds: An Explicit First-Passage Arithmetic Family

Description For the accelerated odd $3x+1$ map we study a gate-relative hereditary rank defined by genuine first-passage predecessor gates. For every prescribed dyadic height $L \ge 200$, we construct an explicit arithmetic family of exponentially many pairs of distinct positive odd states $X = W - 1$ and $Y = W - 7$. Each pair first coalesces after exactly three accelerated steps and has a common live first passage after eleven steps. Both initial states simultaneously have hereditary-rank lower bounds linear in $L$, with coefficients $0.2753979\dots$ and $\frac{1}{25}$, respectively. Beyond this existence theorem, we establish a quantitative trade-off between the two depth bounds and the exponential number of pairs. A global cardinality bound for the predecessor gates then gives explicit windows for the actual integer ranks of exponentially many pairs and forces an exponentially repeated (though not individually specified) ordered rank profile. The arguments are constructive and the finite arithmetic certificates are reproducible using Python's standard library. These are statements about gate-relative inverse ancestry; they neither determine individual exact ranks nor prove the Collatz conjecture. Creator Lee Byoungwoo Independent Researcher, Daejeon, Republic of Korea Keywords Collatz conjecture; accelerated $3x+1$ map; hereditary rank; predecessor gates; first-passage gates; dyadic height; arithmetic family; inverse ancestry; integer ranks; computational complexity; finite arithmetic certificates