Where movement pays: the boundary of profitability in a resource-limited pond

The earlier articles of the series measured whether a movement rule pays at single points: one density, one resource-recovery rate, one mobility. This one maps the surface. We ran 240 simulations across four colony densities and three recovery rates, ten seeds each, and measured the difference between a cell that steps towards the richest neighbour and a cell that does not move. The boundary exists and it runs along the recovery rate: with medium or fast recovery movement pays at every density tested (+0.028 to +0.076), while with slow recovery the sign flips at the two highest densities (-0.022 and -0.030). The optimum is at a density of 128 cells on 1024 (12.5 %), not at the lowest density tested (3.1 %): with too few cells nobody finds a patch, and at 53.7 % the competition eats the gain. This explains an earlier result of the series - the working density of 53.7 % sits beyond the boundary when recovery is slow, which is why greedy movement was found harmful there. Every printed number is re-derived from the deposited artifact by a machine check (27 checks, 0 failures).