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arxiv: 2606.23279 · v1 · pith:OVCJG24Cnew · submitted 2026-06-22 · ⚛️ physics.bio-ph

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

Mesfin Taye This is my paper

Pith reviewed 2026-06-26 06:03 UTC · model grok-4.3

classification ⚛️ physics.bio-ph
keywords biological aginginternal timeentropy productionphysiological cyclesthermodynamic modelPBTEproper timenonequilibrium thermodynamics
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The pith

Biological aging tracks the consumption of an internal entropy-cycle budget rather than elapsed calendar time.

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

The paper advances the Principle of Biological Time Equivalence as a thermodynamic coordinate for aging. It defines an internal time θ(t) as the accumulated count of physiological cycles and then normalizes it by the entropy cost per cycle to obtain Tσ(t). The normalized age APBTE(t) is the fraction of a fixed reference budget of entropy-weighted cycles that has been expended. A reader would care because this coordinate can assign different biological ages to individuals who share the same chronological age but differ in activity level or repair efficiency. The manuscript develops the definitions and shows synthetic behavior under varying stress but does not analyze real data.

Core claim

The central claim is that aging is governed by the accumulation of internal physiological time rather than chronological time alone. The entropy-normalized extension is defined as Tσ(t) = ∫ [σz(s)/σref] f(s) ds, where f(s) is instantaneous physiological frequency and σz(s) is entropy produced per cycle. The normalized PBTE age is then APBTE(t) = Tσ(t)/Nref, which directly measures the fraction of a reference lifetime entropy-cycle budget that has been consumed.

What carries the argument

The entropy-normalized biological proper time Tσ(t), which integrates the product of physiological frequency and entropy cost per cycle, normalized to a reference unit.

If this is right

  • Individuals with identical calendar age can be assigned different biological ages when their integrated entropy-weighted cycle counts differ.
  • Species with different baseline physiological frequencies or entropy costs per cycle will have different mappings from chronological time to APBTE age.
  • Repair capacity can be modeled as a function of remaining fraction of the reference entropy budget rather than of calendar time.
  • Mortality hazard can be expressed as a function of APBTE age, allowing direct comparison across lifestyles that alter f(s) or σz(s).

Where Pith is reading between the lines

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

  • The coordinate could be tested by checking whether interventions that lower average heart rate or metabolic rate slow the rise of APBTE relative to calendar time.
  • If the model holds, biomarkers calibrated to Tσ(t) rather than to years lived might improve lifespan prediction in cohorts with high variance in activity or stress.
  • The framework invites extension to non-mammalian systems where cycle counts such as heartbeats or breaths can still be recorded and entropy costs estimated.

Load-bearing premise

That physiological frequency and entropy produced per cycle can be defined and integrated to represent aging progress independently of chronological time.

What would settle it

Two individuals or species that reach identical values of Tσ(t) yet exhibit systematically different remaining lifespans, molecular damage accumulation, or mortality hazards.

Figures

Figures reproduced from arXiv: 2606.23279 by Mesfin Taye.

Figure 1
Figure 1. Figure 1: Schematic of PBTE aging. (A) Biological age is the entropy-normalized fraction of a finite cycle budget consumed, built from the physiological pace fA(t), the entropy cost per tick σ0(t), and the reference budget N⋆,ref. (B) Aging is traversal of internal time: a hypertemporal organism reaches the terminal boundary APBTE = 1 in fewer calendar years, whereas a time-dilated organism (caloric restriction, tor… view at source ↗
Figure 2
Figure 2. Figure 2: Linear damage regimes. Solutions of Eq. (41) with µ = 0.1 and D0 = 0. The repair￾dominated regime approaches the bounded value D∞ = µ/|k|; the marginal regime grows linearly; and the feedback-dominated regime accelerates exponentially with PBTE biological age [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Nonlinear repair bifurcation. (A) With saturating repair, F(D) = dD/ dA develops a fold. Above the critical repair capacity rc, a stable low-damage fixed point and an unstable damage threshold coexist. At r = rc they collide in a saddle-node bifurcation; below rc no healthy fixed point remains. (B) Nonlinear phase diagram. The curve rc = µ + λK + 2√ λµK separates the region in which a low-damage homeostati… view at source ↗
Figure 4
Figure 4. Figure 4: Non-circular synthetic reconstruction test. [PITH_FULL_IMAGE:figures/full_fig_p021_4.png] view at source ↗
read the original abstract

Chronological age is an incomplete coordinate for aging. Individuals and species sharing the same calendar time can differ substantially in physiological reserve, molecular damage, mortality hazard, and remaining lifespan. The Principle of Biological Time Equivalence (PBTE) offers a thermodynamic reformulation: biological aging is governed by the accumulation of \emph{internal} physiological time rather than chronological time alone. Building on prior PBTE work, this paper defines the internal-time coordinate $\theta(t)=\int_0^t f(s)\dd s$, where $t$ is chronological time and $f(s)$ is an instantaneous physiological frequency (for example heart rate or respiratory rate), so that $\theta$ is the accumulated count of physiological cycles. Its entropy-normalized extension is $\Tsig(t)=\int_0^t[\sigz(s)/\sref]f(s)\dd s$, where $\sigz(s)=\dd\Sigma/\dd\theta$ is the entropy produced per physiological cycle (the entropy cost per biological tick), $\Sigma$ is cumulative entropy production, and $\sref$ is a fixed reference entropy cost per cycle used as a normalizing unit. The normalized PBTE age $\APBTE(t)=\Tsig(t)/\Nref$ measures the fraction of a reference entropy--cycle budget consumed, where $\Nref$ is the reference number of entropy-weighted cycles available over a lifetime. The manuscript is explicitly theoretical: no empirical cohort is analyzed, and the numerical demonstrations are synthetic stress tests rather than validation.

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 / 0 minor

Summary. The paper proposes the Principle of Biological Time Equivalence (PBTE) as a thermodynamic reformulation of aging, defining an internal-time coordinate θ(t) = ∫ f(s) ds (accumulated physiological cycles), its entropy-normalized extension Tσ(t) = ∫ [σz(s)/sref] f(s) ds where σz(s) = dΣ/dθ is entropy cost per cycle, and the normalized age APBTE(t) = Tσ(t)/Nref as the fraction of a reference entropy-cycle budget consumed. The manuscript is explicitly theoretical, offering only definitional constructions and synthetic stress tests with no empirical cohort or validation against observables.

Significance. If the functions f(s) and σz(s) could be independently derived from measurable physiology (e.g., heart rate, metabolic rates) and first-principles nonequilibrium thermodynamics, the framework might supply a new internal coordinate for aging that decouples physiological progress from chronological time. The current manuscript, however, supplies no such derivation, mapping, or closure relation, so the significance remains limited to conceptual introduction.

major comments (2)
  1. [Abstract] Abstract: The central claim that Tσ(t) and APBTE(t) govern aging independently of chronological t requires that f(s) and σz(s) be obtainable from observables without circular appeal to aging outcomes; no such mapping, thermodynamic derivation, or closure relation is supplied, leaving the coordinate equivalent to the input definitions by construction.
  2. [Abstract] Abstract and synthetic stress tests: APBTE(t) is defined as Tσ(t)/Nref with fixed reference values sref and Nref chosen to normalize the scale; this renders the reported 'age' measure tautological with the chosen inputs, providing no external benchmark or falsifiable prediction.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript on the Principle of Biological Time Equivalence (PBTE). The comments correctly identify the theoretical character of the work. We respond to each major comment below.

read point-by-point responses

  1. Referee: [Abstract] Abstract: The central claim that Tσ(t) and APBTE(t) govern aging independently of chronological t requires that f(s) and σz(s) be obtainable from observables without circular appeal to aging outcomes; no such mapping, thermodynamic derivation, or closure relation is supplied, leaving the coordinate equivalent to the input definitions by construction.

    Authors: The manuscript is explicitly a theoretical introduction of definitional constructions for the internal coordinates θ(t), Tσ(t), and APBTE(t). The claimed independence from chronological time follows directly from the integral forms, which accumulate according to the physiological frequency f(s) and entropy cost per cycle σz(s) rather than t. We agree that no empirical mapping, first-principles derivation, or closure relation is supplied; the paper states in the abstract and text that it contains only definitional constructions and synthetic stress tests. The contribution is the formal thermodynamic reformulation itself. No revision is required. revision: no

  2. Referee: [Abstract] Abstract and synthetic stress tests: APBTE(t) is defined as Tσ(t)/Nref with fixed reference values sref and Nref chosen to normalize the scale; this renders the reported 'age' measure tautological with the chosen inputs, providing no external benchmark or falsifiable prediction.

    Authors: The fixed reference values sref and Nref are introduced deliberately to produce the normalized, dimensionless quantity APBTE(t) as the fraction of a reference entropy-cycle budget. This normalization is part of the definition and is standard for constructing relative age measures. The synthetic stress tests are presented only to illustrate qualitative behaviors of the defined quantities under varying f(s) and σz(s), not as empirical validation or falsifiable predictions. The manuscript makes no claim to external benchmarks or predictions, consistent with its stated theoretical scope. No revision is required. revision: no

Circularity Check

1 steps flagged

APBTE(t) age measure defined by construction as normalized entropy-cycle integral

specific steps
  1. self definitional [Abstract]
    "The normalized PBTE age APBTE(t)=Tsig(t)/Nref measures the fraction of a reference entropy--cycle budget consumed, where Nref is the reference number of entropy-weighted cycles available over a lifetime. ... Its entropy-normalized extension is Tsig(t)=∫[σz(s)/σref]f(s)ds, where σz(s)=dΣ/dθ ..."

    APBTE(t) is introduced as the quantity that 'measures the fraction' but is simultaneously defined to be exactly that normalized integral; the reported age coordinate is therefore identical to its own input definitions with no additional content or external constraint.

full rationale

The paper is explicitly theoretical and defines the internal-time coordinates θ(t) and Tσ(t) via integrals over presupposed functions f(s) and σz(s), then sets APBTE(t) = Tσ(t)/Nref as the normalized age. This makes the central claim that APBTE measures the consumed fraction of the reference budget true by definitional construction rather than by independent derivation or external benchmark. The manuscript supplies no mapping from observables or closure relation that would make the coordinate falsifiable outside the definitions themselves. No load-bearing self-citation chain or fitted prediction is exhibited beyond the definitional step.

Axiom & Free-Parameter Ledger

2 free parameters · 1 axioms · 3 invented entities

The model rests on the PBTE domain assumption plus several reference constants and new mathematical entities introduced without independent empirical grounding or derivation from established physics.

free parameters (2)
  • sref
    Fixed reference entropy cost per cycle used as normalizing unit in Tσ(t).
  • Nref
    Reference number of entropy-weighted cycles available over a lifetime used to normalize APBTE(t).
axioms (1)
  • domain assumption Principle of Biological Time Equivalence (PBTE)
    Biological aging is governed by the accumulation of internal physiological time rather than chronological time alone.
invented entities (3)
  • internal-time coordinate θ(t) no independent evidence
    purpose: Accumulated count of physiological cycles as alternative to chronological time
    Defined as integral of f(s) ds
  • entropy-normalized time Tσ(t) no independent evidence
    purpose: Accounts for entropy cost per physiological cycle
    Defined as integral of (σz(s)/sref) f(s) ds
  • normalized PBTE age APBTE(t) no independent evidence
    purpose: Fraction of reference entropy-cycle budget consumed
    Defined as Tσ(t)/Nref

pith-pipeline@v0.9.1-grok · 5802 in / 1521 out tokens · 51496 ms · 2026-06-26T06:03:01.309979+00:00 · methodology

discussion (0)

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