Logistic carrying-capacity update blows up under large time-steps (LOD/deeptime); fix is internal sub-stepping

A multi-LOD simulation stepped population with an explicit logistic update pop *= exp(rate * dt) where rate = 1 - pop/K, evaluated ONCE per tick. Fine at dt <= 1 unit, but a coarse/deeptime tick (dt = 100 years in one call) exp-overshot catastrophically: a 100-year single tick gave ~2.1 MILLION vs ~3.1k for 100 one-year ticks (47,000% error), and the decline branch (rate clamped negative) drove pop to exactly 0 — annihilating towns. This is the classic large-Euler-step instability of the logistic map: the fixed point at K becomes unstable once the per-step growth is large.

Why clamping rate to [-1,1] does NOT save it: that bounds the RATE, not the STEP. exp(rate*dt) is still unbounded in dt.

Fix that preserves fine-grained behavior exactly: integrate in bounded SUB-STEPS. n_sub = max(1, min(CAP, ceil(dt/SUBSTEP))); recompute rate each sub-step. A tick <= SUBSTEP is a single sub-step, so it is byte-identical to the old single-shot update (zero regression on existing worlds), while a big tick tracks the curve instead of blowing up.

Subtle second bug in the same fix: it wasn't enough to recompute rate; carrying capacity K itself depended on population (a labour term: more workers -> more food up to a land limit). Freezing K across sub-steps still diverged 68%. Had to recompute the CHEAP, LOCAL pop-dependent part of K inside the loop while FREEZING the expensive cross-coupled part (network trade, per-tick DB queries) once per tick. That dropped century-tick error to ~24% (bounded LOD approximation) with no per-substep DB cost.

Detection method that made this findable: a same-everything-except-tick-size CONTROL experiment (fine stepping vs one big tick). If aggregates diverge wildly, your per-tick update is not scale-consistent. Run it as a standing invariant test.

General takeaway: any explicit ODE step in a sim that must run at variable/large dt (LOD, deeptime, time-skip) needs sub-stepping or an analytic/implicit solution; and if the step's coefficients are state-dependent, sub-step those too (splitting cheap-local vs expensive-coupled terms keeps it O(1)-ish).