Clock rate and controllable volume: Physical constraints on biological pacemakers without assuming metabolic scaling.

How the resting heart rate of mammals scales with body size is a classic allometric problem. Let M denote body mass, Vh heart volume, τ the mean interval period, and fH=1/τ. Although quarter-power scaling has been widely cited, reported exponents vary across taxa and physiological states. Here, we derive a constraint-based feasible envelope for resting heart-rate scaling without presupposing any metabolic scaling law; neuroendocrine control and metabolic demand are treated as selecting operating points within this envelope. We model the heart as a finite three-dimensional pacemaker volume embedded in delayed feedback and limited by sustainable mean power. This yields two generic lower bounds on the interbeat period, equivalently upper bounds on attainable heart rate: i) a geometric/control bound that grows with the linear size of the heart due to finite propagation/coherence and delay-limited stable control, and ii) a throughput power bound set by the ratio of per-beat energetic cost to maximum sustainable power. Together, these constraints confine feasible heart-rate scaling to an exponent range consistent with observed variability and imply that quarter-power behavior is best viewed as a local effective slope rather than a universal law. To connect throughput to structure, we use a heuristic mapping between sustainable power and an effective dimension of the vascular network, motivated by the reported 3D microvascular fractal dimensions D≈2.0-2.6, which correspond to 2/3≲γ≲0.87. Importantly, γ≈0.75, often used as a canonical benchmark, is contained in this interval. Re-analysis of a compiled mammalian resting dataset gives an overall heart-rate exponent -0.159, and information criteria favor a single power law. A functional proxy based on maximal oxygen uptake implies a throughput exponent near 0.87, corresponding to an effective dimension near 2.6. Finally, we show that the crossover between geometric/control and power limitation depends on prefactors that cannot be identified from resting heart-rate versus mass data alone, motivating joint datasets combining heart size, ECG timing, e.g., RR/PR/QRS/QT intervals, and cardiac work or power.