Biological proper time and entropy-cost invariance in cardiac and respiratory lifespan scaling

Abstract: Warm-blooded vertebrates accumulate approximately conserved numbers of physiological cycles over a natural lifetime: of order $10^9$ heartbeats and $10^8$--3×10<sup>83\times10<sup>8 breaths. These regularities are not exact constants, but their persistence across orders-of-magnitude variation in body mass, metabolic power, physiological frequency, and lifespan suggests that biological time is not measured by chronological duration alone. We develop the Principle of Biological Time Equivalence (PBTE), a thermodynamic framework in which lifetime cycle count is determined by the ratio between total lifetime entropy production and the entropy cost of one physiological cycle. Starting from the open-system entropy balance $\dot S=\dot e_p-\dot h_d$, we define the entropy cost per cycle as $σ<em>0=dΣ/dN$, where $dΣ$ is the entropy produced as the physiological clock advances by $dN$ cycles. For an adult homeostatic regime, this gives the cycle-count relation $N</em>\star=Σ/\langleσ<em>0\rangle$, with $Σ=\int_0^L \dot e_p(t)\,dt$, where $N</em>\star$ is the lifetime cycle count, $Σ$ is total lifetime entropy production, and $\langleσ_0\rangle$ is the lifetime-averaged entropy cost per cycle. In the homeostatic limit, $\dot e_p\simeq P/T$, so direct measurement of metabolic power $P$, body temperature $T$, and physiological frequency $f$ gives $σ_0\simeq P/(Tf)$. PBTE converts the empirical lifetime-cycle invariants into entropy-cost invariants. Under Kleiber metabolic scaling and quarter-power physiological-frequency scaling, the mass-specific entropy cost satisfies $\barσ_0=P/(TfM)\propto M<sup>{3/4+1/4-1}=M<sup>0$, providing a thermodynamic interpretation of allometric mass cancellation.

• The paper develops the Principle of Biological Time Equivalence, expressing lifetime heartbeats or breaths as total entropy production divided by mean entropy cost per cycle.
• The cardiac entropy cost remains approximately mass-independent across 46 placental mammals, averaging 3.0 × 10⁻³ J K⁻¹ beat⁻¹ kg⁻¹ with 16% variation, while measured respiratory data show a nonsignificant positive mass trend and roughly 100% scatter.
• The framework predicts clade-specific longevity effects from torpor, thermal regulation, neural investment, oxidative efficiency, and dive bradycardia, and proposes cardiac entropy-normalized time as a testable biological-aging coordinate.

Overview and central claim

This paper develops the Principle of Biological Time Equivalence (PBTE), a thermodynamic framework in which the well-known lifetime cycle invariants of comparative physiology—of order $10^9$ heartbeats and $10^8$–$3\times10^8$ breaths per lifespan—are reinterpreted as entropy-cost invariants. The core construction defines the entropy cost per physiological cycle as $\sigma_0 = d\Sigma/dN = \dot e_p/f$, where $\dot e_p \simeq P/T$ is the irreversible entropy-production rate in an adult homeostatic regime, $P$ is metabolic power, $T$ is body temperature, and $f$ is the physiological frequency. The resulting identity, $N_\star = \Sigma/\langle\sigma_0\rangle$, states that the lifetime cycle count equals total lifetime entropy production divided by the mean entropy cost per cycle. Under Kleiber scaling ($P\propto M^{3/4}$) and quarter-power frequency scaling ($10^8$0), the mass-specific cost $10^8$1 scales as $10^8$2, giving allometric mass cancellation a thermodynamic interpretation.

The paper is explicit about its logical structure, separating three levels: (1) an exact algebraic identity that cannot be falsified; (2) a constitutive closure, $10^8$3, which is an approximation; and (3) the falsifiable empirical claim that the mass-specific entropy cost is approximately invariant within defined physiological regimes. This stratification matters because the paper's most substantive empirical finding is a partial refutation at Level 3.

The cardiac clock

Applied to heart rate, PBTE predicts that the mass-specific entropy cost per beat, $10^8$4, is approximately mass independent. On representative mammals spanning house mouse to African elephant, the absolute cost per beat varies by orders of magnitude while the mass-specific cost remains narrowly distributed around $10^8$5, with a coefficient of variation of only 16% across nearly six orders of magnitude in body mass. The a priori reference cardiac budget from canonical allometries is $10^8$6 beats; for clade-level predictions the paper uses a fitted empirical anchor $10^8$7 beats from $10^8$8 non-primate placental species. The two differ by about 50% (0.18 dex), a discrepancy the authors attribute to idealized exponents versus residual scatter in real allometries.

The respiratory clock and the decisive negative result

The respiratory analysis is the paper's most important empirical contribution because it confronts circularity directly. When Kleiber-derived metabolic power is imposed, the respiratory cancellation $10^8$9 holds by construction and carries no evidential weight. As a non-circular test, the authors recompute $3\times10^8$0 using measured species-level basal metabolic rates on $3\times10^8$1 mammals with both measured BMR and measured resting breath rate. The mass cancellation does not survive: the fitted slope of $3\times10^8$2 versus $3\times10^8$3 shifts from approximately zero (+0.005) under Kleiber power to +0.21 under measured BMR, though this slope is not statistically resolved ($3\times10^8$4), and scatter rises to a coefficient of variation near 100%. The positive trend is driven by aquatic mammals, whose low resting breath rates relative to metabolic rate sharply inflate the entropy cost per breath at large body size.

The authors treat this as an informative asymmetry rather than a failure: breathing is a strongly regulated control variable—shaped by tidal volume, ventilation strategy, thermoregulation, and the aquatic–terrestrial dichotomy—rather than a passive metabolic tick. Consequently, only the cardiac clock supplies a non-circular reference entropy cost, and this asymmetry governs which rhythm can serve as the clock for aging. A limitation should be noted plainly here: the positive slope rests on 29 species and is statistically unresolved, so the respiratory result establishes absence of demonstrated cancellation rather than demonstrated positive scaling.

Clade multipliers as structured departures

Clade deviations are modeled not as residual scatter but as renormalizations of the effective cycle budget, $3\times10^8$5, with $3\times10^8$6. Two channels are derived from closed-form expressions given measured inputs (duty-cycle state occupancy; Arrhenius thermal kinetics), while two are phenomenological imports from comparative biochemistry and demography—the latter carrying the larger share of residual uncertainty. Once $3\times10^8$7 is fixed, lifespan follows from frequency via $3\times10^8$8.

The clade applications assign distinct thermodynamic strategies to each lineage:

Clade	Dominant mechanism	Representative prediction
Non-primate placentals	Reference class, $3\times10^8$9	Mouse 3.0 yr vs. 3.5 observed; elephant 67 vs. 65
Primates	Budget expansion via neural investment, $\sigma_0 = d\Sigma/dN = \dot e_p/f$0	Human 69.8 yr vs. 70–85 observed
Bats	Duty-cycle torpor × thermal suppression	Myotis lucifugus 33.8 yr vs. 34 observed
Birds	Entropy-cost reduction despite adverse thermal/duty factors	Wandering albatross 64.1 yr vs. ~70 observed
Cetaceans	Dive-bradycardia time dilation	Bowhead whale 203 yr vs. 150–200 observed

The paper concedes that the baseline placental table is an internal consistency check, not an independent prediction, since the anchor is the mean over precisely those species; genuine predictive content lies in the other clades, whose multipliers are fixed independently of the lifespans they predict. It also warns against double counting: duty-cycled organisms must use active or surface heart rate as the reference frequency, since suppression is already encoded in $\sigma_0 = d\Sigma/dN = \dot e_p/f$1.

Biological proper time and aging

Under the PBTE closure, biological proper time $\sigma_0 = d\Sigma/dN = \dot e_p/f$2 is shown to be the unique intrinsic coordinate (up to affine rescaling) in which entropy accumulates uniformly, since any monotone reparametrization preserving uniform accumulation must satisfy $\sigma_0 = d\Sigma/dN = \dot e_p/f$3. The framework then promotes biological age to an entropy-normalized coordinate:

$\sigma_0 = d\Sigma/dN = \dot e_p/f$4

the fraction of a reference entropy–cycle budget consumed, with aging velocity $\sigma_0 = d\Sigma/dN = \dot e_p/f$5. This organizes longevity mechanisms into three classes: time dilation (caloric restriction, torpor, cetacean bradycardia), entropy-cost reduction and budget expansion (avian and primate maintenance efficiency), and hypertemporal pathologies (inflammation, metabolic syndrome, neurodegeneration, cancer), which accelerate the velocity of biological time. A falsifiable consequence is proposed: DNA-methylation clocks should correlate more tightly with accumulated internal time than with chronological age, and damage trajectories should collapse more cleanly against the cardiac $\sigma_0 = d\Sigma/dN = \dot e_p/f$6 than against chronological time or the respiratory clock.

Limitations and open questions

Several constraints bound the results. The homeostatic closure $\sigma_0 = d\Sigma/dN = \dot e_p/f$7 neglects the difference between basal, field, and total metabolic throughput, which can be substantial. Mass cancellation is exact only when the metabolic and frequency exponents satisfy $\sigma_0 = d\Sigma/dN = \dot e_p/f$8; measured metabolic exponents range from roughly 0.67 to 0.75 depending on taxon and state, so residual slopes of order ±0.05 are expected even where cancellation holds. The respiratory slope of +0.21 exceeds this regime substantially, but it is unresolved at $\sigma_0 = d\Sigma/dN = \dot e_p/f$9 on 29 species. The $\dot e_p \simeq P/T$0 and $\dot e_p \simeq P/T$1 channels are phenomenological and clade-level rather than species-resolved, so clade predictions inherit their uncertainty. The decisive test remains unperformed: simultaneous calorimetric, cardiac, respiratory, temperature, and body-mass measurement across species to determine whether $\dot e_p \simeq P/T$2 is genuinely mass-independent within defined physiological regimes. Open questions include whether tidal-volume-based ventilatory coordinates restore respiratory invariance, and whether the primate exponent $\dot e_p \simeq P/T$3 can be derived rather than calibrated.

Conclusion

The paper recasts lifetime heartbeat and breath counts as ratios between total entropy production and per-cycle entropy cost, deriving the cardiac invariant thermodynamically and demonstrating empirically that it survives a non-circular test with measured metabolic data, while the respiratory invariant does not. The cardiac–respiratory asymmetry is presented as the central empirical finding: one rhythm provides a validated reference cost for an entropy-normalized biological-age coordinate, the other does not. The framework is positioned as a macroscopic thermodynamic organization of comparative longevity that complements Gompertz demography and molecular aging clocks, with its falsifiability hinging on direct simultaneous measurement of $\dot e_p \simeq P/T$4, $\dot e_p \simeq P/T$5, $\dot e_p \simeq P/T$6, and $\dot e_p \simeq P/T$7 across species.