Why an elephant burns far less energy per kilogram than a mouse. Metabolic rate follows a power law across roughly 20 orders of magnitude of body mass — a straight line of slope 3/4 on a log-log plot. Rendered, not quoted.
SOURCE Kleiber, M. — Body size and metabolism, Hilgardia 6(11):315–353 (1932). doi:10.3733/hilg.v06n11p315 · AMBER: coefficients illustrative; the 3/4 exponent and its origin remain debated. No medical advice.
The allometric law: Y = a·M^b. For whole-organism metabolic rate Kleiber found B = a·M^(3/4).
Take logs: log B = log a + b·log M — a straight line of slope b. Recover b from any two masses: b = Δ(log B) / Δ(log M).
Because b < 1, the rate per unit mass is B/M = a·M^(b−1) = a·M^(−1/4) — it falls as animals get bigger.
Life's power law — Kleiber 1932. B ~ M^(3/4) makes per-mass metabolism fall as M^(−1/4); paired with lifespan ~M^(1/4) and heart rate ~M^(−1/4), heartbeats-per-life are roughly invariant.
This is the whole-organism scaling standing above the single cell of the-hodgkin-huxley — one spike, then a whole zoo of bodies obeying one line.
Live re-check of the engine's invariants. Green = the slope recovers to 3/4 and per-mass metabolism declines. If window 6 tampers the exponent, this flips red.
WITNESS: booting…
Two reference bodies plus a coefficient. All parameters AMBER (illustrative).
| exponent b (Kleiber) | 3/4 |
| coefficient a | 3.4 kcal·d⁻¹ |
| mouse M₁ | 0.02 kg |
| elephant M₂ | 5000 kg |
Live log-log line B = a·M^b across 6 decades of mass. Slope drawn from the pure functions; the dot grid is the per-mass rate B/M collapsing as M grows.
Proven result: the recovered slope equals 3/4 exactly; the mouse burns far more per gram than the elephant; heartbeats-per-life are constant.
WALL "The exponent is obviously 2/3 — metabolism just tracks surface area (M^(2/3)) shedding heat." A clean geometric argument, and wrong at the population level: broad datasets sit nearer 3/4. The West–Brown–Enquist fractal-network model derives 3/4 from space-filling transport, but it is contested (AMBER — debated).
"Bigger animals have faster metabolism per kilogram."
Correction: total B rises with M, but B/M ~ M^(−1/4) falls — a shrew must eat near-constantly; an elephant idles per gram.
"One clean exponent (exactly 3/4) fits every taxon."
Correction: fitted exponents vary by clade and method (~0.66–0.85); 3/4 is a central tendency, not a law of nature (AMBER).
"Allometry lets you predict drug/energy doses for humans exactly."
Correction: scaling gives an order-of-magnitude guide only — not medical advice.
Planted void (disclosed): force b = 1 (isometric) so metabolism is proportional to mass and per-gram cost becomes constant — the mouse/elephant gap vanishes. The Witness (7) must catch it.
state: untampered