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arxiv: 2606.15310 · v2 · pith:KJ6YQN5Fnew · submitted 2026-06-13 · 🧬 q-bio.OT · cond-mat.stat-mech

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

Mesfin Taye This is my paper

Pith reviewed 2026-06-27 04:40 UTC · model grok-4.3

classification 🧬 q-bio.OT cond-mat.stat-mech
keywords allometric scalingentropy productionbiological timeKleiber's lawphysiological cycleslifespan scalingopen-system thermodynamics
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The pith

Mass-specific entropy cost per physiological cycle is independent of body mass under Kleiber and quarter-power scalings.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces the Principle of Biological Time Equivalence to explain why warm-blooded vertebrates accumulate roughly fixed numbers of heartbeats and breaths over a lifetime despite huge differences in size and lifespan. It starts from the open-system entropy balance and defines an entropy cost per cycle that equals metabolic power divided by temperature and frequency. When Kleiber scaling for power and the observed quarter-power scaling for frequency are inserted, the mass-specific version of this cost becomes independent of mass. This supplies a thermodynamic reason for the observed cycle-count regularities instead of treating them as chronological coincidences.

Core claim

Under the Principle of Biological Time Equivalence, lifetime cycle count equals total lifetime entropy production divided by the average entropy cost per cycle. In the homeostatic adult regime this cost is approximately metabolic power over temperature times frequency, and the mass-specific form of the cost is invariant because the M^{3/4} metabolic scaling and M^{-1/4} frequency scaling cancel exactly.

What carries the argument

Entropy cost per cycle σ₀ ≈ P/(T f), whose mass-specific version is rendered constant by the combination of Kleiber metabolic scaling and quarter-power frequency scaling.

If this is right

  • The roughly 10^9 heartbeats and 10^8–3×10^8 breaths are entropy-cost invariants rather than fixed chronological durations.
  • Biological proper time is set by cumulative entropy production rather than by clock time alone.
  • Allometric mass cancellation receives a direct thermodynamic interpretation through invariance of the mass-specific entropy cost.
  • The same invariance should appear in any physiological process whose power and frequency follow the same pair of scaling relations.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the invariance holds, then species that deviate from Kleiber scaling should also deviate from the canonical cycle counts.
  • The framework suggests that interventions altering metabolic efficiency or frequency without changing the ratio P/(T f) would leave lifetime cycle count largely unchanged.
  • Extension to non-homeostatic regimes would require integrating the time-dependent entropy production rate rather than using the simple adult approximation.

Load-bearing premise

The adult homeostatic regime lets the entropy production rate be approximated as metabolic power divided by temperature without large deviations from the open-system balance over the lifetime.

What would settle it

Measurements of metabolic power, body temperature, and heart or breathing rate across at least two orders of magnitude in body mass that show whether P/(T f M) is constant or varies systematically with mass.

Figures

Figures reproduced from arXiv: 2606.15310 by Mesfin Taye.

Figure 1
Figure 1. Figure 1: Schematic summary of the PBTE framework. (A) The lifetime number of physiological cycles is determined by the ratio of total lifetime entropy production Σ to the mean entropy cost per cycle ⟨σ0⟩. (B) Biological proper time θ(t) is the accumulated number of physiological cycles. Small mammals, with high cardiac frequency, traverse the reference cycle budget rapidly in chronological time, whereas large mamma… view at source ↗
Figure 2
Figure 2. Figure 2: Mass-specific cardiac entropy cost ¯σ (M) H versus body mass. The dashed line shows the PBTE prediction ¯σ (M) H ∝ M0 . we instead use the fitted empirical anchor N (emp) H,0 = 10 8.995 ≃ 9.9 × 108 beats, (34) the measured mean of log10 N⋆ H over the n = 46 non-primate placental species (Eq. Supplementary Eq. (S7), Supplementary Table S8). The two differ by about 50% (0.18 dex, where one dex is a factor of… view at source ↗
Figure 3
Figure 3. Figure 3: Lifetime cardiac cycle count ℓ = log10(N⋆ H) versus body mass. PBTE predicts flat within-clade profiles with clade-dependent vertical offsets ∆ℓ = log10 ΦC . denote the accumulated number of cardiac cycles up to chronological time t. The corresponding normalized cardiac age is ˆθH(t) = θH(t) N⋆ H . (36) Species with high heart rates advance rapidly in cardiac biological time, whereas species with low heart… view at source ↗
Figure 4
Figure 4. Figure 4: Mass-specific respiratory entropy cost ¯σ (M) R estimated with Kleiber metabolic power. The leading allometric exponents cancel, yielding an approximately scale-invariant respiratory entropy cost per breath per unit mass, although the scatter is larger than in the cardiac clock. The respiratory clock is a more stringent and more protocol-sensitive test of PBTE than the cardiac clock. This is because the ca… view at source ↗
Figure 5
Figure 5. Figure 5: The non-circular test of the respiratory clock: mass cancellation does not survive [PITH_FULL_IMAGE:figures/full_fig_p029_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Respiratory lifetime cycle count ℓ = log10(N⋆ R) versus body mass. The dominant pattern is clade-dependent vertical displacement rather than continuous mass dependence. 30 [PITH_FULL_IMAGE:figures/full_fig_p030_6.png] view at source ↗
read the original abstract

Warm-blooded vertebrates accumulate approximately conserved numbers of physiological cycles over a natural lifetime: of order $10^9$ heartbeats and $10^8$--$3\times10^8$ 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 $\sigma_0=d\Sigma/dN$, where $d\Sigma$ is the entropy produced as the physiological clock advances by $dN$ cycles. For an adult homeostatic regime, this gives the cycle-count relation $N_\star=\Sigma/\langle\sigma_0\rangle$, with $\Sigma=\int_0^L \dot e_p(t)\,dt$, where $N_\star$ is the lifetime cycle count, $\Sigma$ is total lifetime entropy production, and $\langle\sigma_0\rangle$ 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 $\sigma_0\simeq P/(Tf)$. 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 $\bar\sigma_0=P/(TfM)\propto M^{3/4+1/4-1}=M^0$, providing a thermodynamic interpretation of allometric mass cancellation.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The manuscript proposes the Principle of Biological Time Equivalence (PBTE), a thermodynamic framework in which lifetime physiological cycle counts N_★ are given by the ratio of total lifetime entropy production Σ to the average entropy cost per cycle ⟨σ_0⟩. Starting from the open-system balance \dot S = \dot e_p - \dot h_d, it defines σ_0 = dΣ/dN and approximates ar σ_0 ≃ P/(T f) in the adult homeostatic regime. Under Kleiber scaling P ∝ M^{3/4} and quarter-power frequency scaling f ∝ M^{-1/4}, the mass-specific entropy cost satisfies ar σ_0 = P/(T f M) ∝ M^0, which is presented as a thermodynamic interpretation of the approximate invariance of total heartbeats and breaths across body mass.

Significance. If the homeostatic approximation and negligible lifetime deviations from entropy balance can be justified, the framework supplies a thermodynamic reading of allometric cancellation in physiological lifespans. The derivation itself is parameter-free once the two empirical exponents are inserted, but this also means the mass invariance is an algebraic identity rather than a new falsifiable prediction.

major comments (2)
  1. [Abstract] Abstract: the claim that ar σ_0 ∝ M^0 supplies an independent thermodynamic interpretation is undermined because the result follows by direct substitution of the input exponents 3/4 and -1/4 into the definitional expression ar σ_0 = P/(T f M); the mass cancellation is therefore an algebraic consequence rather than a derived thermodynamic prediction.
  2. [Abstract] Abstract: the cycle-count relation N_★ = Σ / ⟨σ_0⟩ and the identification ar σ_0 ≃ P/(T f) rest on the homeostatic approximation \dot e_p ≃ P/T together with the assumption that integrated deviations from \dot S = ar e_p - ar h_d (ontogeny, senescence, or non-homeostatic intervals) remain negligible over the full lifetime; no quantitative bound or estimate of these deviations is supplied, yet this assumption is load-bearing for the validity of the entropy-cost invariance.
minor comments (1)
  1. [Abstract] Abstract: the manuscript states the scaling relations and the cancellation but supplies neither data, error analysis, nor direct comparison against measured lifetime entropy production.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive report. The comments highlight important aspects of the derivation and assumptions in the Principle of Biological Time Equivalence (PBTE). We respond to each major comment below and indicate the revisions we will make to strengthen the manuscript.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the claim that bar σ_0 ∝ M^0 supplies an independent thermodynamic interpretation is undermined because the result follows by direct substitution of the input exponents 3/4 and -1/4 into the definitional expression bar σ_0 = P/(T f M); the mass cancellation is therefore an algebraic consequence rather than a derived thermodynamic prediction.

    Authors: We agree that the mass invariance follows algebraically from substituting the empirical exponents into the definition of bar σ_0. The PBTE framework supplies a thermodynamic reading of this cancellation by connecting cycle count to entropy production but does not derive the scaling exponents. We will revise the abstract to describe the result as a thermodynamic consequence within the PBTE framework rather than an 'independent' interpretation, removing any implication of a new prediction. revision: partial

  2. Referee: [Abstract] Abstract: the cycle-count relation N_★ = Σ / ⟨σ_0⟩ and the identification bar σ_0 ≃ P/(T f) rest on the homeostatic approximation ė_p ≃ P/T together with the assumption that integrated deviations from ė_S = bar e_p - bar h_d (ontogeny, senescence, or non-homeostatic intervals) remain negligible over the full lifetime; no quantitative bound or estimate of these deviations is supplied, yet this assumption is load-bearing for the validity of the entropy-cost invariance.

    Authors: The referee correctly notes that the homeostatic approximation and negligibility of lifetime deviations are central and load-bearing. The manuscript already restricts the relations to the 'adult homeostatic regime' and 'homeostatic limit'. We acknowledge that no quantitative bounds on deviations are given; this is a limitation of the present analysis. We will revise the abstract to state explicitly that the entropy-cost invariance is derived under the assumption of dominant homeostatic contributions over the lifetime. revision: partial

Circularity Check

1 steps flagged

Mass-specific entropy cost invariance is algebraic consequence of substituting empirical allometric exponents into the definition

specific steps
  1. fitted input called prediction [Abstract]
    "Under Kleiber metabolic scaling and quarter-power physiological-frequency scaling, the mass-specific entropy cost satisfies bar sigma_0 = P/(T f M) proportional to M^{3/4+1/4-1}=M^0, providing a thermodynamic interpretation of allometric mass cancellation."

    bar sigma_0 is defined as P/(T f M); substituting the empirical inputs P ∝ M^{3/4} and f ∝ M^{-1/4} produces exact cancellation to M^0 by algebra alone. The invariance is therefore forced by the definition and the input scalings rather than derived from the open-system entropy balance or PBTE.

full rationale

The paper defines the mass-specific entropy cost as bar sigma_0 = P/(T f M) and states that under the known Kleiber scaling P ∝ M^{3/4} and quarter-power frequency scaling f ∝ M^{-1/4} this quantity is proportional to M^0. This M^0 result follows immediately by algebraic cancellation of the exponents (3/4 + 1/4 - 1 = 0) with no additional thermodynamic content or independent derivation supplied. The homeostatic approximation dot e_p ≃ P/T is used to reach sigma_0 ≃ P/(T f), but the claimed invariance itself reduces directly to the input scalings by construction. This is a clear instance of a fitted/empirical input being presented as a derived thermodynamic result.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 2 invented entities

The framework rests on the open-system entropy balance, the homeostatic approximation linking entropy production to metabolic power, and the empirical Kleiber and frequency scaling exponents; it introduces the PBTE principle and biological proper time without independent falsifiable handles.

free parameters (2)
  • Kleiber exponent = 3/4
    Empirical metabolic scaling exponent 3/4 inserted to obtain mass cancellation
  • frequency scaling exponent = -1/4
    Quarter-power physiological frequency scaling exponent -1/4 inserted to obtain mass cancellation
axioms (2)
  • standard math Open-system entropy balance dot S = dot e_p - dot h_d
    Starting equation invoked for entropy production in living systems
  • domain assumption Homeostatic regime approximation dot e_p approximately P/T
    Used to connect entropy production rate to measurable metabolic power and temperature
invented entities (2)
  • Principle of Biological Time Equivalence (PBTE) no independent evidence
    purpose: Thermodynamic framework equating lifetime cycle count to total entropy production divided by entropy cost per cycle
    New principle introduced to reinterpret cycle invariants; no independent evidence supplied
  • Biological proper time no independent evidence
    purpose: Entropy-based measure of biological time distinct from chronological duration
    Conceptual entity proposed to explain why cycle counts are conserved; no external validation

pith-pipeline@v0.9.1-grok · 5876 in / 1656 out tokens · 53019 ms · 2026-06-27T04:40:34.136107+00:00 · methodology

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. A Nonequilibrium Internal-Time Model of Aging: Entropy-Normalized Biological Proper Time and Repair Bifurcations

    physics.bio-ph 2026-06 unverdicted novelty 3.0

    The paper introduces entropy-normalized internal time θ and Tσ as measures of accumulated physiological cycles and entropy cost to define a normalized PBTE age APBTE as the fraction of a reference lifetime budget consumed.

Reference graph

Works this paper leans on

48 extracted references · cited by 1 Pith paper

  1. [1]

    Einstein A.Ann Phys322:891 (1905)

    1905

  2. [2]

    Oldenbourg, 1908

    Rubner M.Das Problem der Lebensdauer. Oldenbourg, 1908

    1908

  3. [3]

    Lindstedt SL, Calder WA.Q Rev Biol56:1 (1981)

    1981

  4. [4]

    Harvard, 1984

    Calder WA.Size, Function, and Life History. Harvard, 1984

    1984

  5. [5]

    Livingstone SD, Kuehn LA.Aviat Space Environ Med50:592 (1979)

    1979

  6. [6]

    Levine HJ.J Am Coll Cardiol30:1104 (1997)

    1997

  7. [7]

    Stahl WR.J Appl Physiol22:453 (1967)

    1967

  8. [8]

    Escala A.Sci Rep12:2407 (2022)

    2022

  9. [9]

    Knopf, 1928

    Pearl R.The Rate of Living. Knopf, 1928

    1928

  10. [10]

    Speakman JR.J Exp Biol208:1717 (2005)

    2005

  11. [11]

    Glazier DS.Biology11:1106 (2022). 53

    2022

  12. [12]

    Schr¨ odinger E.What Is Life?Cambridge, 1944

    1944

  13. [13]

    Prigogine I.Introduction to Thermodynamics of Irreversible Processes, 3rd ed., 1967

    1967

  14. [14]

    Kleiber M.Hilgardia6:315 (1932)

    1932

  15. [15]

    West GB, Brown JH, Enquist BJ.Science276:122 (1997)

    1997

  16. [16]

    Mortola JP, Limoges M-J.Respir Physiol Neurobiol154:500 (2006)

    2006

  17. [17]

    McKechnie AE, Wolf BO.Physiol Biochem Zool77:502 (2004)

    2004

  18. [18]

    de Magalh˜ aes JP, Costa J.J Evol Biol22:1770 (2009)

    2009

  19. [19]

    Hulbert AJ, et al.Physiol Rev87:1175 (2007)

    2007

  20. [20]

    McNab BK.Comp Biochem Physiol A151:5 (2008)

    2008

  21. [21]

    White CR, Seymour RS.PNAS100:4046 (2003)

    2003

  22. [22]

    Genoud M, Isler K, Martin RD.Biol Rev93:404 (2018)

    2018

  23. [23]

    Pontzer H, et al.PNAS111:1433 (2014)

    2014

  24. [24]

    Fahlman A, et al.Exp Physiol110:1349 (2025)

    2025

  25. [25]

    Gompertz B.Phil Trans R Soc Lond115:513 (1825)

  26. [26]

    Herculano-Houzel S.PLOS ONE6:e17514 (2011)

    2011

  27. [27]

    Friston K.Nat Rev Neurosci11:127 (2010)

    2010

  28. [28]

    Wilkinson GS, South JM.Aging Cell1:124 (2002)

    2002

  29. [29]

    Barja G, Herrero A.J Bioenerg Biomembr30:235 (1998)

    1998

  30. [30]

    Brand MD, et al.Biochem J392:353 (2000)

    2000

  31. [31]

    Ogburn CE, et al.J Gerontol A56:B468 (2001)

    2001

  32. [32]

    Goldbogen JA, et al.PNAS116:25329 (2019)

    2019

  33. [33]

    Williams TM, et al.Nat Commun6:6055 (2015)

    2015

  34. [34]

    Noren SR, Williams TM.J Exp Biol203:3601 (2000)

    2000

  35. [35]

    Human Ageing Genomic Resources.AnAge build 15(2023).https://genomics.senescence.info/ species/

    2023

  36. [36]

    Jones KE, et al.Ecology90:2648 (2009)

    2009

  37. [37]

    Bininda-Emonds ORP, et al.Nature446:507 (2007)

    2007

  38. [38]

    Christian KA, Weavers BW.Copeia1999:688 (1999)

    1999

  39. [39]

    Gillooly JF, et al.Science293:2248 (2001)

    2001

  40. [40]

    Felsenstein J.Am Nat125:1 (1985)

    1985

  41. [41]

    Horvath S.Genome Biol14:R115 (2013)

    2013

  42. [42]

    Colman RJ, et al.Nat Commun5:3557 (2014)

    2014

  43. [43]

    Brown JH, et al.Ecology85:1771 (2004)

    2004

  44. [44]

    Yegian AK, et al.PNAS121:e2313703121 (2024)

    2024

  45. [45]

    Academic Press, 1982

    Lyman CP, Willis JS, Malan A, Wang LCH.Hibernation and Torpor in Mammals and Birds. Academic Press, 1982

    1982

  46. [46]

    Prinzinger R, Preßmar A, Schleucher E.Comp Biochem Physiol A99:499 (1991)

    1991

  47. [47]

    Clarke A, Rothery P.Funct Ecol22:58 (2008)

    2008

  48. [48]

    Cambridge, 2015

    Ponganis PJ.Diving Physiology of Marine Mammals and Seabirds. Cambridge, 2015. 54 Table S10:Measured resting breathing frequencyf R for mammals (M≥10 kg) in the He et al. (2023) dataset.Min kg;f R in breaths min−1. Habitat: land or aqua (aquatic/semi-aquatic).∗denotes species with a matched measured BMR, used in the entropy-cost analysis (Appendix S3). Sp...

    2015