Population rising fast, then leveling off at the ceiling the environment allows. The continuous logistic ODE dN/dt = r·N·(1−N/K) bends every positive start into one S-curve and parks it at the carrying capacity K. Rendered, not quoted — the sigmoid is computed live, the fixed points proven, the inflection located.
source Verhulst, P.-F., “Notice sur la loi que la population suit dans son accroissement,” Correspondance mathématique et physique, vol. X (1838), pp. 113–121 — reproduction PDF · AMBER: 1838 memoir, no single canonical stable link; cited by year/volume/pages.
One state variable N(t), a population. Growth is proportional to N but throttled by crowding:
dN/dt = r·N·(1−N/K)
r is the intrinsic per-capita rate; K the carrying capacity. Two fixed points where dN/dt = 0: N=0 and N=K. The bracket (1−N/K) is the brake: near 1 when N is small (near-exponential), near 0 as N→K (stall).
Assumptions (named, not hidden): a single well-mixed population, constant r and K, no age structure, no time delay, continuous time. A model of growth with a ceiling — not advice about any real population.
Growth with a ceiling. Verhulst named the logistique in 1838 to answer Malthus’ runaway exponential: real populations meet a wall.
This is the continuous, monotone cousin — the tame ODE. Discretize it with a big step and you get the-logistic-map x→μx(1−x), whose period-doubling cascade is chaos. Same brake, opposite temperament: here the sigmoid never overshoots; there it can oscillate and never settle. One law, two worlds.
Live re-check of the engine’s core claims — stability of the fixed points, sigmoid closed form, the K/2 inflection, and no overshoot. Confirms green; flips red the instant the Tamper (window 6) corrupts the law.
The witness recomputes from the same functions the panel runs — not a stored verdict.
Rate r, carrying capacity K, and a starting population N₀. Feed any positive N₀ — below K or above it.
RK4 integration, fixed step dt=0.01, over t∈[0,T]. Green curve = numerical trajectory; the faint marks are the closed-form sigmoid sampled independently — they coincide within tolerance. Dashed line = K. Dot = the K/2 inflection.
“Constant r and K” is the wall this model cannot see past. Real carrying capacity moves — seasons, disease, competition, its own waste. Add a time delay to the brake (crowding felt late) and the continuous logistic can overshoot and oscillate too: Hutchinson’s delayed logistic. The smooth S-curve is a property of the assumptions, not a law of nature.
The engine is honest within its stated model. The adversary attacks the model’s reach, not its arithmetic.
The disclosed planted void. Flip one sign — dN/dt = r·N·(1+N/K) — and the brake becomes an accelerator: super-exponential blow-up, no ceiling. The witness (window 7) catches it because N no longer approaches K.
Known-bad on purpose. selfcheck() plants and reverts this same flip at boot to prove the witness fires.