The empirical constant N⋆ ≈ 10⁹ heartbeats per lifetime has been known since the work of Lindstedt and Calder, but it remained without a physical interpretation. Neighboring studies in the archive deal with thermoregulation, heat transfer, and quantum effects, yet this work stands apart: it connects macroscopic physiology with nonequilibrium thermodynamics through a unified formalism of biological time. The allometric laws of Kleiber and Calder explain the reduction in mass dependence, but they provide neither a numerical value for the invariant nor a structure for interclade deviations. The author fills this gap by introducing a closure for entropy production through heart rate. This approach turns an empirical regularity into a consequence of the conservation of the dissipative budget.
At its core lies a constitutive closure: the rate of entropy production is proportional to heart rate with a coefficient σ₀ (entropy per beat). Integrating over a lifetime gives N⋆ = Σ/σ₀, where Σ is the total entropy budget. Substituting Kleiber's law (P ∝ $M^{3/4}$) and Calder's law (f ∝ $M^{-1/4}$) at constant body temperature leads to an exact cancellation of exponents: σ₀* ∝ $M^{3/4+1/4-1}$ = M⁰. The numerical value N₀ ≈ 1.52×10⁹ is reproduced without any fitting parameters. For four clades (primates, bats, birds, cetaceans), a multiplicative correction $Φ_C$ is constructed from independently measured physiological quantities: neurometabolic fraction, torpor factor, mitochondrial efficiency, diving bradycardia. Biological proper time θ(t) = ∫ f(t')dt' is introduced as the accumulated number of cycles.
The specific entropy per beat σ₀* ≈ 3.0×10⁻³ J·K⁻¹·kg⁻¹·beat⁻¹ turns out to be independent of body mass from mouse to elephant (coefficient of variation 16%). Predicted multipliers $Φ_C$: primates 2.60, bats 5.39, birds 2.97, cetaceans 1.76 (after correction for bradycardia). For humans, the calculation gives a lifespan of 70.6 years at α = 0.40, and with a correction for external risks, about 81 years. Bats reach $Φ_{bat}$ ≈ 8 through the product of the torpor factor (1.94) and the hypothermic factor (4.10). Birds compensate for unfavorable thermal and kinematic factors with biochemical efficiency $Φ_{mito}$+oxid ≈ 2.33. Cetaceans exhibit a 'near-miss trap': the raw beat count is close to the baseline, but after correction for bradycardia the budget exceeds N₀ by a factor of 2.4.
The work follows the general direction of neighboring studies on thermoregulation and heat transfer, but it stands out by its scale of generalization: it encompasses all clades of warm-blooded vertebrates within a single thermodynamic formalism. Unlike purely allometric explanations, it introduces a physical quantity σ₀* with a clear operational meaning and predicts its mass independence. This is the first attempt to link the empirical invariant N⋆ to entropy production through a constitutive closure that can be tested by direct calorimetric and telemetric measurements. The geometric interpretation of biological time as arc length opens the way to applying information geometry methods in comparative biology.
The topic could develop toward testing the constitutive closure through direct simultaneous measurements of metabolic power, heart rate, temperature, and mass across a wide range of species. If σ₀* proves to be truly mass-independent, the principle will attain the status of a physical law. Further refinement of clade multipliers will require species-specific data on dive fraction in cetaceans and antioxidant activity in birds. Extension to ectotherms and microorganisms remains an open task. Epigenetic clocks could be recalibrated in units of biological proper time.
The results could influence comparative gerontology, evolutionary biology, and biomedical diagnostics of aging. The biological time formalism is applicable to the interpretation of epigenetic clocks and to the development of chronotherapeutic interventions.
Conduct a decisive experiment with simultaneous measurement of P, f, T, and M in at least 20 species of mammals and birds. Compare predicted slopes of epigenetic clocks with cross-species methylation data.
For the thermodynamics of living systems, the origin of the numerical value N⋆ ≈ 10⁹ and the mechanisms that set the constancy of the entropy cost per heartbeat remain unresolved. The work does not derive σ₀* from microscopic dynamics but postulates it as a closure. The connection to Prigogine's nonequilibrium thermodynamics and the principle of minimum entropy production requires further analysis. The universality of the invariant for ectotherms and invertebrates remains an open problem.
A shrew's heart beats 800 times per minute, while an elephant's beats 28, yet over a lifetime both perform roughly the same number of beats. In humans, the brain consumes 20% of the body's energy despite making up only 2% of its mass.
The idea of biological proper time echoes Greg Egan's novel 'Permutation City', where virtual personalities live in accelerated or decelerated subjective time, and their 'lived life' is measured by internal events rather than external seconds.