Lasse Hyyrynen Independent researcher
July 23, 2026
This paper proposes an accounting framework for information-persisting systems (IPS): driven open systems whose selected macroscopic identity is actively maintained during non-equilibrium steady or quasi-steady operation. The proposal is deliberately narrower than a universal theory of persistence. Its core quantity is the persistence ratio
Here \(P_{\mathrm{in}}\) is power income expressed as an exergy or other explicitly defined work-capacity rate, \(\eta\) is coupling efficiency, and the denominator contains measured or estimated power assigned to routine maintenance, protocol-dependent model mismatch, and repair. We call this identity and its accompanying measurement procedure the Fractal Persistence Equation (FPE) framework. The label does not denote a new fundamental physical law. The threshold \(\mathcal{R}\ge 1\) is a bookkeeping threshold: current usable income covers the required power as those terms have been operationally defined. A system with \(\mathcal{R}<1\) may continue operating by drawing on a finite reserve \(B\).
The framework separates four epistemic categories. Energy conservation, non-negative entropy production, Landauer’s erasure bound, and established mismatch-cost identities are prior physical results. IPS, persistence ratio, delusion divergence, structural fatigue, substrate integrity, and shelter coefficient are definitions. Decompositions of measured power into these terms are modelling assumptions. Cross-domain relations between those quantities and survival are empirical hypotheses. Under a transparent reserve balance, we prove only a reserve-depletion proposition: if the power deficit is bounded below by \(d>0\), a reserve initially equal to \(B_0\) is exhausted no later than \(B_0/d\). Reserve exhaustion is not itself a theorem of dissolution; that conclusion requires a system-specific dependence of identity maintenance on supplied power. Likewise, known mismatch-dissipation results do not imply a universal linear or exponential lifetime penalty.
Substrate and shelter graphs are retained as optional, domain-specific ansatze requiring explicit cut sets and noise channels. Stable nuclei are excluded from the IPS evidence base because ordinary nuclear stability concerns bound quantum states, not driven open-system maintenance. Biological, chemical, ecological, organisational, and machine-learning examples are presented as candidate operationalisations rather than independent confirmations of a common law. A fractal mixture-of-experts network is treated as an architectural analogue: hard routing gives inactive expert parameters zero task gradient, while exact parameter preservation additionally requires that no shared, optimiser-state, or regularisation update reaches them. The framework therefore remains a proposal requiring independent operationalisation, accounting closure tests, and empirical validation.
Keywords: open systems, non-equilibrium steady state, stochastic thermodynamics, Landauer principle, mismatch cost, persistence, information processing, reserve depletion.
Long-lived organised patterns need not conflict with the second law of thermodynamics. A driven open system can maintain a non-equilibrium distribution while exchanging energy, matter, and entropy with its environment. The local organisation is sustained by throughput and dissipation; total entropy production for system plus environment remains non-negative. This standard resolution underlies the study of dissipative structures, biochemical networks, control systems, and stochastic thermodynamics (Schrödinger, 1944; Prigogine, 1967; de Groot & Mazur, 1962; Seifert, 2012).
What is less standard is the attempt to compare, in one ledger, the usable power available to maintain a selected macroscopic identity with the power required to do so. Such a comparison can be useful, but only if it avoids converting definitions into laws. A ratio called “persistence” cannot by itself explain persistence. Terms such as maintenance, mismatch, damage, boundary, model, and lifetime must be independently operationalised, and any causal claim connecting the ratio to loss of identity must be tested rather than inferred from the ratio’s name.
This paper therefore develops a cautious open-system accounting framework. It defines a class of information-persisting systems, specifies a measurable power balance, proves a limited reserve-depletion result, and states the additional assumptions needed to connect depleted reserves to dissolution. It also identifies where established information thermodynamics can constrain particular entries in the ledger. The purpose is not to derive a new force, a universal lifespan law, or a thermodynamic theory of every stable object. The purpose is to make proposed persistence budgets explicit enough to audit and falsify.
The central accounting definition is simple. Over a stated observation window, usable power is \(P_{\mathrm{usable}}=\eta P_{\mathrm{in}}\). Required power is the sum of non-overlapping, operationally assigned maintenance, mismatch, and repair powers. Their ratio is \(\mathcal{R}\). If \(\mathcal{R}<1\), the ledger records a deficit. A finite reserve can cover that deficit temporarily. This is compatible with the second law and with quasi-steady operation; it is not instantaneous failure. The only general time bound follows from integrating the deficit against the reserve.
To prevent category errors, claims are marked conceptually as follows.
Established results are imported from prior physics: the first and second laws for open systems; Landauer’s lower bound for logically irreversible erasure under its physical assumptions; fluctuation relations; and protocol-specific mismatch-cost relations. This paper does not rederive or strengthen those results.
Definitions fix terminology and bookkeeping: IPS, coarse-grained identity, power income, coupling efficiency, complexity, internal noise floor, internal model, delusion divergence, structural fatigue, substrate integrity, shelter coefficient, reserve, and persistence ratio. Definitions can be useful or unhelpful, but they are not empirical discoveries.
Modelling assumptions connect observables to ledger entries. Examples include an assumed separation between routine maintenance and repair, a proxy for delusion divergence, a graph cut set for substrate integrity, or a noise-transfer model for shelter. These choices may differ by domain.
Empirical hypotheses claim reproducible relations not guaranteed by the accounting identity—for example, that increased mismatch cost predicts higher reserve draw under controlled conditions, or that a specified cut-set failure raises maintenance power. Such claims require independent data.
The original contribution is thus an auditable assembly and research programme, not a first-principles derivation of lifespan.
The scope is driven open systems operating in a non-equilibrium steady state (NESS) or in a quasi-steady regime over a declared time window. A strict NESS has a stationary probability distribution for the variables included in its state description while non-zero currents may persist. A quasi-steady system has slowly drifting macroscopic variables relative to its relaxation time. If a reserve \(B\) is declining, the enlarged state that includes \(B\) is not in an exact stationary state, although faster operating variables may remain close to a NESS. This distinction is important.
An IPS in this paper is not every object that lasts. In particular, a stable nucleus is ordinarily described as a bound quantum state, or a metastable state with a decay width. It does not require continuous environmental power income to “re-establish” its binding. Stable nuclei are therefore outside the driven-open-system scope and are not evidence for the framework. Rocks and equilibrium crystals are excluded for the same accounting reason unless a particular driven maintenance process is specified.
The FPE is not a fundamental force or a fifth interaction. Thermodynamics is force-agnostic: it constrains transformations and statistical balances regardless of which microscopic interactions implement them. Electromagnetic, strong, weak, and gravitational interactions enter system-specific Hamiltonians, transition rates, transport coefficients, boundary fluxes, and noise spectra. The Standard Model supplies the accepted nongravitational microscopic interactions within its tested domain, and general relativity supplies the classical description of gravitation. This framework neither unifies them nor extends them into a theory of quantum gravity. It presumes that an appropriate open-system thermodynamic description is available at the scale under study.
The framework does not claim that:
The system is \(\Sigma\), its environment is \(\mathcal{E}\), and an operational boundary is \(\partial\Sigma\). The coarse-graining map is
where \(\mathcal{X}_\Sigma\) is a microstate or fine-state space and \(\mathcal{M}\) is a declared macrostate space. The symbol \(\Phi\) is reserved for substrate integrity and is never used for coarse-graining.
Temperature is \(T\), Boltzmann’s constant is \(k_B\), and entropy is measured in nats unless noted. For distributions \(p\) and \(q\) on a common measurable space,
when \(p\) is absolutely continuous with respect to \(q\). Powers are non-negative rates unless a signed balance is explicitly shown. Time-averaged quantities over a window \(W=[t,t+\tau_W]\) carry angle brackets, for example \(\langle P\rangle_W\).
An open-system boundary is a declared interface across which energy, matter, entropy, and information-bearing physical states may pass. The boundary must be specific enough for fluxes to be measured or inferred. A Markov blanket is one possible statistical boundary. If internal states \(\sigma\), blanket states \(b\), and external states \(e\) satisfy
then internal and external states are conditionally independent given \(b\) (Pearl, 1988; Friston, 2013). Equation (2.1) is a property to test for the chosen variables and time resolution, not a licence to label any legal, geographic, or conceptual perimeter a thermodynamic blanket.
A coarse-grained identity is an admissible set \(\mathcal{I}\subseteq\mathcal{M}\). The system persists over \([t_0,t_1]\) when
subject to a stated tolerance and sampling rule. The identity criterion must be fixed independently of the observed failure time. Otherwise the claim that the system persisted is circular.
Definition 2.1 (IPS). An information-persisting system is a model of a physical subsystem for which all of the following are specified:
“Internal model” is used functionally and does not imply consciousness, explicit Bayesian inference, or a particular neural representation. A biochemical controller, an adaptive estimator, and a stored predictive distribution may each qualify if an intervention can show that the state affects maintenance-relevant action. Purely descriptive variables assigned by an observer do not automatically qualify.
Requiring an internal model makes IPS narrower than the class of all dissipative structures. A Bénard cell, flame, or vortex may be an open non-equilibrium pattern without possessing a separately identifiable control-relevant model. Such systems remain valid subjects of non-equilibrium thermodynamics but need not be IPS under Definition 2.1.
The reserve \(B(t)\ge0\) is the amount of stored work capacity that the chosen boundary and protocol allow to be converted into maintenance-relevant work. Depending on the domain it might be chemical free energy, a charged storage device, or another physically justified stock. Nominal assets, mass, or energy that cannot be converted on the relevant timescale are not automatically part of \(B\).
The reserve definition includes a conversion convention so that its units are energy. If storage conversion has a separate efficiency, that loss belongs either in the definition of usable reserve or in the power terms, but not both. The maximum capacity may be finite even when the system is connected to a continuing external drive.
The accounting window \(\tau_W\) must separate fast fluctuations from sustained imbalance. An instantaneous value of \(\mathcal{R}\) can be noisy or undefined. For a cyclic controller, a period average may be appropriate; for a stochastic system, an ensemble or long-time average with uncertainty intervals may be needed. Temporary deficits can occur during otherwise stable operation and do not violate the second law. They indicate reserve draw or another compensating term in the energy balance.
When an internal model supplies a probability distribution \(q_\mu(y\mid z)\) for outcomes \(y\) conditional on observations \(z\), and a reference distribution \(p(y\mid z)\) can be independently estimated, define the delusion divergence
The ordering in (2.3) is a convention chosen for scoring predictions under outcomes generated by \(p\). Another ordering may be appropriate for a specific mismatch theorem, but it must be stated rather than silently substituted. The reference distribution, support, conditioning information, and estimator are part of the operational definition. “Distance from reality” without these items is not a measurable KL divergence.
The free-energy principle literature often introduces a variational density and an identity in which variational free energy equals surprise plus a non-negative KL divergence (Friston, 2010). That mathematical decomposition is relevant when its variables and generative model are identified. It does not establish that every macroscopic organisation literally performs gradient descent, nor does it by itself provide a power in watts. Converting a variational objective into a thermodynamic ledger entry requires a physical implementation and protocol.
Power income \(P_{\mathrm{in}}\) is the gross rate of imported work capacity before conversion losses. Whenever possible it should be an exergy rate relative to a stated environment, not raw energy throughput.
Coupling efficiency \(\eta\) is the fraction of \(P_{\mathrm{in}}\) delivered as maintenance-relevant usable power under the selected boundary and accounting convention. If all inputs and outputs are consistently expressed as exergy and no reserve discharge is hidden in the numerator, \(0\le\eta\le1\). No universal dependence \(\eta(\mathcal{D}_{\mathrm{KL}})\) is assumed.
Complexity \(\omega\) is a declared measure of maintenance-relevant degrees of freedom or constraints. It is not automatically a thermodynamic state variable.
The internal noise floor \(\mathcal{E}_\Sigma\) is a measured spectral or effective disturbance scale for information-carrying or control-relevant modes. Its units must be stated. A relation such as \(P_{\mathrm{maint}}\approx\omega\mathcal{E}_\Sigma\) is an optional model requiring assumptions about independence, refresh rates, coding, and control; it is not a consequence of Landauer’s principle.
Structural fatigue \(\Gamma\) is a measured state or index of accumulated damage and deferred repair that may affect \(P_{\mathrm{repair}}\) or \(P_{\mathrm{maint}}\). It is not itself a power unless calibrated as one.
Substrate integrity \(\Phi\) summarises the condition of specified constituent nodes or components. The shelter coefficient \(\Psi\) summarises transmission through specified environmental disturbance channels. Section 4.5 states the restrictions on these graph quantities.
For a chosen control volume, the first law is an energy balance. One schematic convention is
with enthalpy and chemical-work terms included in the appropriate fluxes for systems exchanging matter. The exact balance depends on the boundary and sign convention. Equation (3.1), not the persistence ratio, enforces energy conservation.
The entropy balance may be written
where \(\dot S_{\mathrm{prod}}\) is internal entropy production and \(\dot S_{\mathrm{flow}}\) is net entropy flow into the system. In a strict steady state for the included variables, \(dS_\Sigma/dt=0\), so an outward entropy flow balances internal production. In a periodic or stochastic steady state, the corresponding equality holds after the appropriate cycle or ensemble average.
Neither (3.1) nor (3.2) says that a selected macrostate disappears as soon as maintenance demand exceeds current income. Stored free energy can bridge a deficit, control can tolerate excursions, and some identity variables may relax slowly. Conversely, non-negative total entropy production does not guarantee persistence of any particular macrostate.
Landauer’s principle concerns logically irreversible manipulation of physically encoded information. For an ideal one-bit memory erased quasistatically while coupled to a heat bath at temperature \(T\), the minimal average heat delivered to the bath is
for erasure of an unbiased bit under the standard assumptions (Landauer, 1961; Bennett, 1982; Parrondo, Horowitz & Sagawa, 2015). General formulations depend on the memory distribution, nonequilibrium free energy, error tolerance, correlations, and protocol. Finite-time operations normally dissipate more.
Equation (3.3) is not a standing rent per stored bit. A stable memory can retain a bit without repeated erasure, and reversible operations can in principle approach different limits. To turn (3.3) into a power, one must count actual logically irreversible operations and divide by their execution time. If \(\dot N_{\mathrm{erase}}\) unbiased erasures per unit time satisfy the standard assumptions, then
Whether this power belongs in routine maintenance, mismatch correction, or repair depends on the protocol. It must not be counted in more than one category.
Relative entropy appears in several established nonequilibrium identities, but the distributions have specific meanings. The Jarzynski equality and Crooks fluctuation relation constrain work distributions for specified forward and reverse protocols (Jarzynski, 1997; Crooks, 1999). Kawai, Parrondo, and Van den Broeck (2007) relate dissipated work to relative entropy between forward and reverse phase-space distributions under a driven process. Still et al. (2012) relate nonpredictive information in a driven system to dissipation. Kolchinsky and Wolpert (2017) analyse the additional dissipation incurred when a process designed for one input distribution is run on another.
A useful special case is a reset or computational protocol optimised for a design distribution \(q\) but operated on an actual distribution \(p\). In the mismatch-cost formulation, excess dissipation can involve a contraction of relative entropy,
where \(p'\) and \(q'\) are the corresponding output distributions under the same channel. Details and equality conditions depend on the implementation. If the protocol maps all relevant inputs to a common reset state so that the output-divergence term vanishes, and if the operational delusion divergence is exactly \(D_{\mathrm{KL}}(p\|q)\) for those input distributions, then a protocol repeated every \(\tau_{\mathrm{model}}\) obeys the conditional bound
Equation (3.6) is therefore not universal. It applies only when the cited protocol assumptions hold: a physically specified operation optimised for \(q\), actual inputs distributed as \(p\), a thermal environment at \(T\), a reset-like output that removes the relevant distinction (or an explicit correction for the output KL term), and repeated operation on the stated timescale. For a general adaptive controller, organisational forecast, or learned world-model, a KL score is not automatically heat and (3.6) may not apply.
Nothing in (3.5) or (3.6) imposes a second factor such as \(1+\mathcal{D}_{\mathrm{KL}}\) on total maintenance. Nothing implies
Coupling efficiency can depend on model quality in a particular engine or controller, but that dependence must be measured or derived from that system’s dynamics. Adding both an assumed efficiency penalty and a mismatch-power penalty without separate measurements risks double counting.
Thermodynamic accounting does not identify a new microscopic interaction. The forces and fields relevant to a particular IPS determine its energy levels, reaction pathways, barriers, transport laws, and stochastic transition rates. At ordinary molecular scales these are primarily electromagnetic interactions constrained by quantum mechanics; nuclear reactions require strong and weak interactions; gravitating systems may require general relativity. Coarse-grained thermodynamic descriptions can be compatible with these theories without being derivable from a single universal power formula.
Accordingly, the internal noise floor may include thermal fluctuations, shot noise, chemical stochasticity, externally transmitted disturbances, or quantum measurement noise when relevant. Vacuum fluctuations are not a generic extractable power source or maintenance tax. Each contribution must be connected to a physical channel and measurement bandwidth.
For a fixed boundary, identity criterion, and observation window, define
and, when \(P_{\mathrm{required}}>0\),
This is the persistence ratio. The name Fractal Persistence Equation (FPE) is retained for (4.1)–(4.3), the reserve balance below, and optional graph-indexed extensions. FPE names an accounting framework and identity, not a fundamental physical law.
\(P_{\mathrm{maint}}\) is the measured power needed for routine operation within the declared identity set under the reference disturbance distribution. \(P_{\mathrm{mismatch}}\) is additional power causally attributable to running a specified operation under a mismatch between its design and actual distributions. \(P_{\mathrm{repair}}\) is power used to reverse or compensate identified damage. The categories must be defined so they do not overlap. If a device reports only total required power, using that total is safer than forcing an unidentifiable decomposition.
The framework does not require every IPS to have non-zero values in all three categories. A controller with no independently measurable mismatch term can set \(P_{\mathrm{mismatch}}=0\) in the ledger while reporting delusion divergence separately. A repair-free observation window can have \(P_{\mathrm{repair}}=0\). These zeros mean “not assigned under this protocol,” not “physically absent in all descriptions.”
By definition,
Equation (4.4) is a bookkeeping threshold. It is not an additional law of thermodynamics and does not prove that a system persists. If the required-power estimate is wrong, the ratio is wrong. Even with accurate powers, a system can fail through a rare transition, an unmodelled shock, loss of a critical constraint, or a change in the identity criterion while \(\mathcal{R}\ge1\). Conversely, it can operate for a finite period with \(\mathcal{R}<1\) by drawing its reserve.
For fluctuating systems, a windowed ratio can be defined as
provided the denominator is positive. The ratio of averages is generally not the average of instantaneous ratios. Reporting the integrated form, window length, variance, and reserve trajectory avoids false threshold precision.
In the simplest closed ledger for maintenance-relevant work capacity,
where
is the signed deficit. If storage saturates, surplus may be exported or dissipated and (4.6) needs a capacity term. If there are unmeasured reserve inflows or losses, they must be added explicitly. The proposition in Section 5 applies only when (4.6) or its stated generalisation closes empirically.
An operating subsystem can remain near a stationary distribution during a temporary period with \(D(t)>0\) because its fast variables are buffered by \(B\). There is no second-law violation: reserve conversion and exported heat contribute to the full entropy balance. A persistent mean deficit cannot coexist indefinitely with a finite reserve and an exactly stationary enlarged state, but finite excursions are routine in driven systems.
The vocabulary terms complexity \(\omega\), internal noise floor \(\mathcal{E}_\Sigma\), and structural fatigue \(\Gamma\) can help parameterise the measured powers. For example, a domain model might propose
with parameters \(\theta\) and \(\xi\) estimated from interventions. A linear ansatz \(P_{\mathrm{maint}}=\omega\mathcal{E}_\Sigma\) is defensible only if the units agree and the maintained channels contribute approximately independently with comparable refresh demands. Correlation, redundancy, coding, nonlinear control, and shared infrastructure can make the relation sublinear, superlinear, or nonseparable.
Similarly, structural fatigue can raise routine maintenance, trigger episodic repair, lower coupling efficiency, or alter the admissible identity set. Those are competing system-specific models. The accounting identity does not choose among them, and fatigue is not required to increase monotonically.
Suppose a system is represented by a component graph \(\mathcal{G}_S\) and an environmental-channel graph \(\mathcal{G}_E\). Define substrate integrity \(\Phi\) only after specifying:
A series system, parallel redundant system, flow network, and adaptive network have different composition laws. No universal minimum, mean, or product follows from the IPS definition.
Define shelter coefficient \(\Psi_j\) for a disturbance channel \(j\) as a measured transfer quantity, such as output disturbance power divided by incident disturbance power over a specified frequency band. Smaller \(\Psi_j\) may represent better attenuation under that convention. Distinct independent linear channels may sometimes be composed multiplicatively; overlapping or correlated channels generally cannot. A fully exposed channel with \(\Psi_j=1\) still leaves the system’s own maintenance capacity and other channels intact. Therefore \(\Psi\to1\) does not imply \(\mathcal{R}\to0\).
Substrate and shelter may enter the ledger through measured functions such as
but multiplying the core ratio by universal prefactors \(\Phi\) or \(\Psi\) is not justified. Equations such as (4.9) are domain-specific ansatze to compare by predictive performance and intervention, not derived factors of (4.3).
Proposition 5.1 (reserve depletion under a sustained deficit). Let \(B:[0,\tau)\rightarrow[0,\infty)\) be absolutely continuous with \(B(0)=B_0<\infty\). Assume that, while \(B(t)>0\), the closed reserve balance
holds almost everywhere. If there is a constant \(d>0\) such that
almost everywhere until exhaustion, then the exhaustion time
satisfies
Proof. For every \(t<t_{\mathrm{exh}}\), absolute continuity and (5.1)–(5.2) give
The right-hand side is non-positive at \(t=B_0/d\). Since \(B\) is non-negative, it must reach zero no later than that time. \(\square\)
An integral version needs no constant lower bound: if cumulative deficit first reaches \(B_0\) at \(t^\star\),
then exhaustion occurs by \(t^\star\), assuming no omitted reserve flows.
Proposition 5.1 proves exhaustion of the defined reserve, not dissolution of the defined identity. To infer dissolution, one needs an additional relation between available maintenance power and macrostate dynamics. One possible system-specific assumption is:
There exists a required delivered maintenance power \(P_{\min}(x,t)\) such that, after \(B=0\), a continuing shortfall causes the state to leave \(\mathcal{I}\) within a separately established failure time \(\tau_{\mathrm{fail}}\).
With that assumption and a persistent shortfall, one can conclude exit by \(t_{\mathrm{exh}}+\tau_{\mathrm{fail}}\). But \(\tau_{\mathrm{fail}}\) must come from mechanics, kinetics, control theory, reliability data, or experiment.
Kramers’ escape-rate theory is not a generic bridge from reserve exhaustion to dissolution. It applies to specified stochastic dynamics near a metastable basin, commonly an overdamped or underdamped Markov diffusion with a defined potential barrier, noise intensity, and timescale separation (Kramers, 1940; Hänggi, Talkner & Borkovec, 1990). An IPS may satisfy those assumptions, in which case a Kramers-type model can be added. The accounting framework alone supplies neither the barrier nor the rate.
The established content is conditional equation (3.5), and its reset-like special case (3.6). There is no general theorem that increasing delusion divergence shortens lifetime linearly or exponentially. Lifetime depends on how mismatch changes usable power, required power, reserve, transition barriers, repair, and external feedback.
The framework instead proposes the following empirical hypothesis:
H1 (controlled mismatch cost). For a fixed physical protocol, boundary, input rate, temperature, and output task, interventions that increase the protocol-relevant input mismatch \(D_{\mathrm{KL}}(p\|q)\) increase measured dissipation by the contraction predicted in (3.5), within experimental uncertainty.
This hypothesis is close to the established mismatch-cost result but still requires correct operational identification in each apparatus. A separate survival hypothesis may be tested:
H2 (mismatch-mediated reserve draw). Holding gross power income, reserve, disturbance exposure, and maintenance target fixed, an intervention that raises independently measured \(P_{\mathrm{mismatch}}\) increases cumulative reserve draw by the same added energy, up to measured compensatory changes.
Neither H1 nor H2 asserts a universal lifespan curve. Any exponential survival relation would require an independently justified hazard or barrier model.
For substrate graphs, a defensible hypothesis must name the service and cut set. For example:
H3 (critical cut-set cost). In a specified flow or reliability network, disabling a component belonging to every currently feasible service path causes a measurable increase in \(P_{\mathrm{required}}\), decrease in \(\eta\), or exit from \(\mathcal{I}\), as predicted by the fitted network model.
For shelter:
H4 (channel-specific shelter). Increasing attenuation on a specified disturbance channel lowers the portion of \(P_{\mathrm{maint}}\) causally attributable to that channel, with the transfer function measured before the persistence outcome.
These hypotheses do not establish a numeric universal composition theorem. A node can survive loss of an enclosure if its own control capacity or alternative channels suffice. Likewise, a component with low local persistence ratio need not be critical if it is redundant or replaceable.
A serious test of the framework should predefine at least the following.
Boundary and fluxes. State what is inside \(\Sigma\), what is outside, and how energy and matter fluxes are measured. Use exergy or another work-capacity convention consistently.
Identity. Specify \(G\), \(\mathcal{I}\), tolerance, and sampling frequency before observing failure. Report sensitivity to alternative reasonable coarse-grainings.
Operating regime. Establish whether the system is in a NESS, periodic steady state, or quasi-steady regime, and estimate relaxation times relative to \(\tau_W\).
Income and efficiency. Measure \(P_{\mathrm{in}}\) and useful delivered power separately. Do not infer \(\eta\) from the outcome that the framework is meant to predict.
Required power. Estimate total required power by controlled interruption, load variation, calorimetry, or a validated process model. Decompose it only when interventions identify maintenance, mismatch, and repair contributions without double counting.
Reserve. Measure \(B_0\), conversion losses, capacity limits, and omitted inflows. Test accounting closure by comparing observed \(\Delta B\) with integrated power imbalance.
Internal model and mismatch. Identify the physically instantiated model, prediction target, reference distribution, and KL orientation. If applying (3.6), verify its reset-like protocol assumptions.
Graph quantities. For \(\Phi\), publish components, capacities, dependencies, and cut sets. For \(\Psi\), publish channels, spectra, transfer functions, and correlation assumptions.
Outcomes and uncertainty. Report both reserve exhaustion and macrostate exit. Include uncertainty propagation, temporal autocorrelation, and out-of-sample validation.
Independent operationalisation is essential. If the same failure observations are used to tune \(P_{\mathrm{required}}\), choose \(\mathcal{I}\), estimate \(B\), and validate \(\mathcal{R}\), apparent success is tautological.
Driven chemical reaction networks are natural initial candidates because input fluxes, chemical potentials, dissipation, concentrations, and reserve species can often be controlled. An autocatalytic set may maintain a concentration pattern under continuous feed. The FPE ledger could compare imported chemical exergy with measured maintenance and correction costs. Eigen’s error threshold and autocatalytic-set theory motivate experiments, but they are not direct instances of the persistence ratio until the ledger quantities are independently mapped (Eigen & Schuster, 1977; Kauffman, 1986).
Cells are candidate IPS because they are driven open systems with membranes, metabolic throughput, regulatory states, repair, and chemical reserves. The framework does not claim a universal bit count or that minimum cellular metabolism equals a Landauer estimate. Cellular maintenance power is dominated by system-specific biochemical processes far above ideal erasure limits. A useful experiment would manipulate one identified repair load or prediction-dependent control loop while measuring metabolic power, reserve depletion, and a preregistered viability macrostate. Different cell types and environmental regimes should be expected to yield different functions.
Ageing can be discussed through structural fatigue only as a modelling hypothesis. Molecular damage, repair capacity, and physiological reserve are heterogeneous and sometimes reversible. The framework does not derive Gompertz mortality, monotonic \(\Gamma\), or a universal mapping from damage to hazard. It can instead organise measurements of repair power and reserve under a stated reliability model (López-Otín et al., 2013).
Ecosystems are open and driven, but their boundaries, identity variables, and exergy flows are difficult to define. Regime shifts and early-warning indicators motivate a quasi-steady analysis (Scheffer et al., 2001, 2009). A defensible study would define an ecosystem service or composition basin, measure external energy and material fluxes, and compare explicit network models. Biodiversity is not automatically substrate integrity, and climate is not automatically a scalar shelter coefficient.
These examples are illustrations and candidate test beds. They are not independent evidence for a common law merely because each can be described using income and cost metaphors.
Firms and institutions consume physical power, use information, hold reserves, and maintain boundaries, but financial variables are not automatically thermodynamic powers. Revenue can purchase energy and labour, yet dollars per unit time cannot be inserted into (4.3) beside watts without a declared conversion and boundary. A management forecast KL score can be operationally meaningful, but equation (3.6) does not turn forecast error directly into heat dissipated by the organisation.
Organisational applications should therefore be described as control or resource-accounting analogues unless a physical exergy ledger is actually constructed. A useful empirical model might test whether calibrated forecast error predicts additional resource expenditure after controlling for scale, income, reserves, and shocks. Such a result would remain a domain-level statistical relation, not confirmation of a universal thermodynamic lifetime penalty. Existing firm-mortality results do not by themselves identify delusion divergence or FPE terms (Daepp et al., 2015).
Nuclear binding illustrates the breadth of physical stability mechanisms, not the IPS framework. A stable nucleus occupies a bound quantum state determined by strong, electromagnetic, and weak interactions. It does not normally import continuous usable power to maintain a Markov blanket or internal model. A radioactive nucleus is treated through quantum decay and coupling to available channels, not through depletion of a maintenance reserve. Calling binding energy “power income” confuses an energy difference with a rate. Nuclei should therefore serve as an explicit boundary case showing why persistence in ordinary language is broader than IPS.
A sparsely routed mixture-of-experts (MoE) network can represent a graph of components and conditional computation. This resemblance makes it a useful architectural analogue and follow-up experiment, not a literal thermodynamic realisation of the FPE. Gradient signal is not power income, cross-entropy loss is not heat, parameter count is not complexity in thermodynamic units, and a router is not automatically a Markov blanket.
There is, however, one clean structural result. Let a hard router select an active expert set \(A(x)\) for input \(x\), and suppose the task output and task loss do not depend on the parameters \(\theta_j\) of an inactive expert \(j\notin A(x)\). Then, wherever the task loss is differentiable,
This follows directly from functional independence and the chain rule. Zero task gradient does not guarantee unchanged parameters. Exact preservation additionally requires that the inactive expert receive no weight decay, regularisation gradient, shared-parameter update, optimiser-state update such as momentum applied without a current gradient, parameter averaging, quantisation rewrite, or other mutation. Router changes can also alter future access to the expert even if its own parameters remain fixed.
The proposed follow-up experiment is continual learning on tasks \(A\) then \(B\), comparing a hard-routed hierarchical MoE with FLOP- and parameter-matched baselines. The experiment should log route overlap, expert-specific task gradients, all non-task updates, and forgetting on held-out \(A\). Ablations should vary depth, top-\(k\), router stability, weight decay, shared layers, and optimiser state.
An exploratory relation such as
where \(\bar o_{AB}\) is mean route overlap and \(N\) is depth, is assumption-dependent. It would require independence or factorisation of routing events across levels, comparable expert sensitivities, stable routers, and negligible shared updates. Equation (6.2) is not implied by (6.1), is not a test of thermodynamic FPE, and should be fitted against alternatives rather than treated as a prediction of a physical law.
The framework retains a useful intuition: a driven system that actively preserves a selected identity has an auditable relation among imported work capacity, conversion efficiency, required operation, repair, and reserves. The persistence ratio provides a dimensionless summary of that ledger. Delusion divergence names an operational model mismatch when two distributions can be specified. Structural fatigue names a damage variable when it can be measured. Substrate integrity and shelter coefficient name graph and channel summaries when their construction is explicit.
The resulting framework is compatible with established open-system thermodynamics but does not “reduce” to the first or second law. Equation (4.3) is a definition layered on top of those laws. Landauer’s bound constrains eligible erasure operations rather than all maintenance. Mismatch-cost identities constrain eligible protocols rather than all inaccurate models. The free-energy principle supplies related variational language but is neither proved nor generally converted to watts here.
The reserve proposition is intentionally modest. Its usefulness depends on whether reserve and deficit can be measured well enough for (5.1) to close. If they can, the exhaustion bound is exact under its assumptions and does not need a stochastic escape argument. What happens after exhaustion is a separate dynamical question.
Because the core ratio is an identity, tests must target closure, transportability, or causal submodels rather than ask whether income divided by cost equals a ratio. The following predictions are deliberately conditional.
P1: reserve closure. In a preregistered system with measured \(B\), the observed reserve change should match the integrated usable-power surplus or deficit within stated measurement uncertainty. Persistent residuals falsify the proposed boundary or reveal omitted flows.
P2: depletion timing. When an intervention maintains a deficit at least \(d\) and the assumptions of Proposition 5.1 hold, reserve exhaustion should occur no later than \(B_0/d\). Failure of the bound falsifies at least one measurement or closure assumption; it does not falsify the second law.
P3: protocol-specific mismatch. For a physical operation satisfying the assumptions of (3.5), changes in input mismatch should produce the corresponding change in excess dissipation, including the output-divergence correction. Applying only the input KL term when the output term is non-zero should fail.
P4: maintenance dependence. If a system-specific model identifies a delivered-power threshold and failure time, then reserve exhaustion followed by a continued shortfall should predict macrostate exit under that model. Different dynamics may yield abrupt, delayed, or probabilistic exit; no universal hazard shape is predicted.
P5: graph interventions. A preregistered cut-set or transfer-channel model should predict changes in measured ledger entries under component or shelter interventions better than simpler size-only models. Neither discontinuous collapse nor a particular critical exponent is presumed.
P6: hard-routing isolation. In the MoE follow-up, inactive expert parameters should have zero task gradient. Any parameter drift must be attributable to logged non-task updates or a violation of hard functional isolation. The overlap-depth relation (6.2) remains exploratory and may be rejected without bearing on thermodynamic accounting.
Information is physical when encoded in physical degrees of freedom, but informational measures and energetic quantities retain different units and roles. Multiplication by \(k_B T\) converts a dimensionless information quantity into an energy scale only under a theorem whose assumptions identify the relevant process. The framework therefore rejects free substitution among cross-entropy, KL divergence, task loss, money, and heat.
Control theory offers a natural language for many IPS applications. A regulator uses internal states to keep outputs in an admissible region under disturbances. Power constraints, actuator saturation, stored energy, model mismatch, and component reliability are familiar. In that setting the persistence ratio may be a useful diagnostic summary, while state-space and stochastic-control equations provide the actual dynamics. The FPE should complement, not replace, those models.
The most important limitation is operational non-uniqueness. Boundaries, coarse-grainings, exergy reference states, reserve definitions, and accounting windows can change numerical values. Sensitivity analysis is mandatory. A credible application should show that conclusions survive a reasonable family of choices or explain why one choice is physically privileged.
A second limitation is causal identifiability. Maintenance, mismatch, and repair may share machinery and cannot always be additively separated. The framework permits a total required-power measurement in that case, but then more detailed claims about delusion divergence or structural fatigue remain unsupported.
A third limitation is scale. Coarse-grained thermodynamics may work well at one level while graph terms at another introduce hidden feedback. “Fractal” in FPE should therefore be read as an invitation to explicit multiscale accounting, not a claim of exact self-similarity or a universal recursive equation.
Priority experiments should use systems with controllable protocols and direct calorimetry or chemical-work measurements. Synthetic memories, biochemical reaction networks, and autonomous devices with measured batteries are more decisive first tests than broad comparisons among cells, ecosystems, firms, and states. Cross-domain work becomes informative only after each domain independently closes its ledger.
The framework currently needs independent operationalisation and empirical testing. Its value will be determined by measurement reliability, out-of-sample prediction, and comparison with simpler alternatives, not by the breadth of its vocabulary.
This paper has recast information-persisting systems as a cautious open-system accounting framework. An IPS is a driven open system in NESS or quasi-steady operation with a specified boundary, identity coarse-graining, control-relevant internal model, and finite usable reserve where deficits are buffered. The FPE is the identity
not a new fundamental law or force. The threshold \(\mathcal{R}\ge1\) says only that measured usable income covers measured required power over the chosen window.
Established physics constrains selected entries: the first and second laws govern the full balances, Landauer constrains specified irreversible erasures, and mismatch-cost results constrain specified mismatched protocols. None establishes a universal efficiency penalty or a linear or exponential relation between model error and lifetime.
The sole general mathematical result is reserve depletion under a sustained bounded deficit: \(t_{\mathrm{exh}}\le B_0/d\). Dissolution requires additional system-specific dynamics. Substrate integrity and shelter coefficient remain optional measured ansatze based on explicit cut sets and noise channels. Biological, chemical, ecological, and organisational examples are research candidates, stable nuclei are outside scope, and fractal MoE is an architectural analogue whose robust result is zero task gradient for hard-routed inactive experts under the stated isolation conditions.
The proposal is therefore falsifiable in a practical sense: ledgers may fail to close, terms may prove non-identifiable, and domain-specific models may not transport. Until independent experiments succeed, FPE should be treated as a disciplined accounting vocabulary and hypothesis generator, not an established common law of persistence.
The author thanks researchers in non-equilibrium thermodynamics, information thermodynamics, control, and reliability whose work motivates the distinctions made here.
LH conceived and wrote the manuscript with the assistance of LLM-based AI systems.
The author declares no competing interests. No external funding supported this work.