The universe is fixed at U and calibrated for the linear regime (capacity n = ρmin·U, no resizing). Every range has a density threshold — shown on the left and as a heat-map. Each insert banks q = log₂U coins in every range holding it (log²n in all), and one coin buys a constant M = 2/(1−ρmin) element-moves — tuned so a range's own banked coins exactly pay for its own spread. Click a range to see the accounting.
A range at depth d has threshold τ(d) = ρmin + (1−ρmin)·d/L, so the root (depth 0) is capped at ρmin and leaves (depth L) at 1. A range at depth d has threshold τ(d) = ρmin + (1−ρmin)·d/L, so the root (depth 0) is capped at ρmin and leaves (depth L) at 1. Amortization. Each insert banks q = L = log₂U coins in every range containing it (log²n coins total), starting from 0 — the initial elements sit exactly at the spread baseline, so they owe nothing. A coin is worth a constant M = 2/(1−ρmin) moves. An even spread leaves every child a full gap = (1−ρmin)/L below its own threshold, so a range must bank ≥ gap·(capacity)/2 inserts before it can be forced to spread — and then q·(banked)·M ≥ capacity ≥ the moves the spread costs. Every range's own coins pay for its own spread; the spread resets its (and its sub-ranges') coins to 0, while ancestor ranges keep theirs. When n reaches ρmin·U the linear universe is full (a real PMA would double U and rebuild).