Making an agent-based sim cross-LOD-consistent: derive the aggregate as the mean-field, do not write a parallel logistic

Problem pattern: an emergent economic/population sim that must run at multiple levels of detail (LOD) — deeptime steps centuries over aggregates, close-up simulates individual agents — and the two must stay consistent. Common failure: the aggregate is a separately-authored formula (a logistic-to-carrying-capacity + a central spatial-equilibrium SOLVE), and the micro agents are a second model; the two silently disagree, and you end up hand-integrating sub-steps to stop the aggregate blowing up.

Key non-trivial insight (took reading three modules + four proof sims to see): carrying capacity, prices, trade, and cities should be MEASUREMENTS of a stationary state, never terms in a formula. Replace (i) a non-rival market-potential field Sum surplus*exp(-cost) [cannot conserve], (ii) a one-shot proportional pie-split allocate() [over-shares a finite pie; a lone bidder gets a whole region], and (iii) logistic-to-K, with ONE local primitive: a resource STOCK that physically moves along a graph edge down a local price gradient at finite rate: delta = clamp(beta*(p_dst - p_src - edge_cost), 0, edge_capacity, source_stock); it is a MOVE between two ledger rows so it is conserved by construction. Contention emerges: as arbitrageurs drain a source, its stock falls, its local price rises (a local tatonnement, no global auction), the gradient shrinks, flow throttles — no one computes a share. Consumption is capped at satiation (min(stock, N*percapita)) which kills the no-demand-ceiling bug. Birth/death are direct functions of realized nutrition (fed ratio), so K is the stationary point where births=deaths, measured by running it — never a variable.

Cross-LOD proof: the coarse aggregate must be the SAME reaction-advection-diffusion operator (graph-Laplacian of the goods-flow) discretized coarser — then the deeptime stepper and the agent stepper are one finite-volume scheme at two resolutions, so they cannot disagree. Falsifiable test that kills a parallel-model design: run full agents fine, run the PDE coarse, AND coarse-grain agents into super-nodes; require L2(agent,PDE)<0.1 and that re-refining reproduces the fine run (a renormalization-group check). The one thing to settle with math FIRST: the price constitutive law is nonlinear so <phi(x)> != phi(<x>) — coarse prices need a sub-grid variance correction or a proof phi is near-affine over the operating range; otherwise the invariant fails.

Conservation made structural (not policed): expose only a move(src_row,dst_row,resource,qty) API for flows and forbid set-stock-from-formula in the trade path — then total matter can change ONLY via the declared production source and consumption sink, and non-conserving trade is impossible to write.