More than planetary-scale feedback self-regulation : 1 A Biological-centred approach to the Gaia Hypothesis 2 3

12 Recent appraisals of the Gaia theory tend to focus on the claim that planetary life is a cybernetic 13 regulator that would self-regulate Earth’s chemistry composition and climate dynamics, following either 14 a weak (biotic and physical processes create feedback loops), or a strong (biological activity control and 15 regulates the physical processes) interpretation of the Gaia hypothesis. Here, we contrast with the 16 regulator interpretation and return to the initial motivation of the Gaia hypothesis: extending 17 Schrödinger’s question about the nature of life at the planetary scale. To this end, we propose a relational 18 and systemic biological approach using autopoiesis as the realization of the living and the (M,R)-system as 19 the formal theory of biological systems. By applying a minimum of key categories to a set of interacting 20 causal processes operating on a wide range of spatial time scales through the atmosphere, lithosphere, 21 hydrosphere, and biosphere of the Earth system, we suggest a one-to-one realization map between the 22 Gaia phenomenon and (M,R)-Autopoiesis. We show that metabolic molecular self-production by closure 23 to efficient causation on a planetary scale is plausible. This suggests that the Gaia phenomenon may be the 24 embodiment of Life itself in the planetary domain, a sui-generis biological unity and thus more 25 fundamental than self-regulation by feedback mechanisms. Formulating the Gaia theory in biological 26 terms provides a formal basis for the claim that planetary biology elsewhere in the universe must involve 27 and have a formal equivalence to a self-referential physical process which cannot be implemented by a 28 Turing machine and, therefore, has a non-computable character. 29


Introduction
Lovelock's hypothesis that Life is a planetary-scale phenomenon (Lovelock, 1979) followed attempts to detect life on other planets by inspecting absorption spectra of their atmospheres (Lovelock, 1965). The underlying idea was that chemical disequilibrium would be the signature of life on a planetary scale (Hitchcock and Lovelock, 1967;Giffin, 1969, Lovelock, 1972), thereby extending Schrodinger's (1945) characterization of life as maintaining 'negative entropy' 1 at this planetary scale. The land-based measures of Mars's atmosphere spectra then revealed that Mars's atmosphere is in chemical equilibrium, hence, lifeless (Lovelock, 1975(Lovelock, , 1980. In contrast, Earth's atmosphere is in chemical disequilibrium because of the continuous biological production of a molecular mixture of highly reactive gases . The claim that life is a planetary phenomenon was later coined the Gaia hypothesis . The Gaia hypothesis has motivated a 'research programme' (Margulis and Sagan, 1997) on which experimental evidence show how biological activity affects Earth's dynamics (Lovelock 2003a). Today, it is not much disputed that Earth's climate, water, and trace elements dynamics involve biological activity on a wide range of time and spatial scales. Examples include i) metabolic-enhanced rock weathering, ii) the existence of a cloud albedo feedback to algal gas emission, iii) geological evidence that the Archaean atmospheric chemistry dominated by methane, iv) the metabolic production and balanced levels of oxygen in the atmosphere, v) impact of boreal forest and biodiversity on local and global climates, and vi) ocean to land transfer of elements by biogenic gases. These empirical evidences have framed the present-day Gaia theory (Thompson, 1991(Thompson, , 1987Bunyard and Goldsmith, 1989;Barlow, 1992;Bunyard, 1996;Schneider and Boston, 1992;Schneider et al., 2004;Crist et al., 2009).
However, as we argue in Section 2, much of the theoretical work associated with the Gaia hypothesis addressed the question of whether life on Earth 'regulates' Earth's climate, and therefore appeals to the notion of self-regulation by 'feedback mechanisms' which are a legacy of the development of cybernetic systems (Wiener 1948;Von Foerster et al., 1951;Ashby 1956). Lovelock originally, and later with Margulis, advocates the notion that Gaia is a "a biological cybernetic system able to homeostat the planet for an optimum physical and chemical state appropriate to its current biosphere" such that "life moderates the global environment to make it more favorable for life" (Lovelock, 1972, p. 579). That is, the physiological conditions of Earth were regulated by and for the biosphere . However, initially this conception has been viewed as teleological and implausible to any neo-Darwinist mechanism of evolution (Doolittle 1981). Lovelock and Margulis later used slightly different heuristic explanatory notions to represent and maintain the Gaia hypothesis strictly in the scientific field, and to dissociate teleology from it. Over time, advocates and opponents have used or proposed other multiple explanatory notions, and different catalogs and dictionaries to address the Gaia hypothesis. It is therefore no wonder that the epistemological status of the Gaia hypothesis -as Life as planetary scale phenomenon-grew unclear and controversial (Schneider 1986;Kirchner, 2002;Lenton and Wilkinson, 2003;Lovelock, 2003b;Margulis, 1993;Kleidon, 2002, Volk, 2003, Margulis 2004, Lenton and Latour, 2018. We argue therefore that a clear-cut mathematical formulation of the Gaia hypothesis is necessary.
It could focus, reinvigorate, and consolidate the Gaian research programme towards the recognition of central features associated to biological phenomenon such as cognition and learning across scales of organization.
The present article attempts to place some key biological formal theory and clear-cut characterizations of living systems in the hope to identify a common and unified grounding to the Gaia hypothesis. We propose a biological-centred formal framework of the Gaia hypothesis based on the language of relational and systemic biology. In Section 2, we distinguish the Gaia phenomenon, the Gaia hypothesis and the Gaia theory. Section 3 outlines the steps leading towards a theoretical biology framework of the Gaia hypothesis with reference to Autopoiesis and the (M,R)-system. Section 4 establishes the Gaian Autopoiesis through the modelling relation between Autopoiesis and Gaia. In Section 5 we establish rigorously the theoretical ground of the Gaia hypothesis in terms of the (M,R)-system. Section 6 outlines briefly some implications of biologizing Gaia theory in Daisyworld mechanisms. Finally, section 7 summarizes the paper.

Gaia hypothesis: Asking Schrödinger's question in the planetary scale
The Gaia phenomenon is the perception, distinction and recognition of the phenomenon of life on a planetary scale from the differential observables of planets like Earth, Mars or Venus. As Lovelock stated: "At present most biologists can be convinced that a creature is alive by arguments drawn from phenomenological evidence" (Lovelock, 1972, p. 579). The observation that Earth is qualitatively distinct from Mars and Venus, therefore, provides the recognition of the Gaia phenomenon, and thus the expression of the Gaia hypothesis: Life as planetary scale phenomenon Lovelock, 1979) (Fig. 1).
The first prediction of the Gaia hypothesis as Schrödinger's question in the planetary scale was confirmed: at the planetary scale, Mars is a lifeless planet. Yet, paradoxically, the present-day Gaia theory sidesteps the Gaia hypothesis. Either their supporters or opponents implicitly assume that Life must be, on a planetary scale, a stabilizing or a destabilizing regulator. The regulator thesis refers to cybernetic systems, which are systems with a specific organization to self-regulate by feedback mechanisms (Wiener 1948;Von Foerster et al., 1951;Ashby 1956). It has been stated that every good regulator of a system must be a model of that system (Conant and Ashby, 1970). Following the cybernetic paradigm and around the notion of the regulator, the Gaia hypothesis has been taxonomized, reinterpreted (weak and strong interpretation), and then refuted (Kirchner, 1989(Kirchner, , 1990. In the weak interpretation of the Gaia hypothesis, both the biotic and physical processes create feedback loops and are coupled regulators (Kirchner, 1989). In the strong interpretation, the biological activity is the regulator, and it controls the Earth chemistry and physical processes (Kirchner, 1989).
That is, the Gaia hypothesis, hence the phenomenon of life at the planetary scale, is reduced to an epiphenomenon of chemical and climatic regulation by negative or positive feedback mechanisms in the Earth system, as can be deduced from Kirchner's (1989, p. 225) own argument: " Margulis and Lovelock (1974) (Kirchner, 1989(Kirchner, , 1990. In summary, the regulator thesis along with the neo-Darwinism confines the present-day Gaia theory to the weak and strong interpretations of the Gaia hypothesis. Indeed, much of the present-day Gaia theory addressed the question on what time-scale natural selection 'regulates' Earth's chemistry and climate. Whether it is at local, regional or at the planetary scale. Under the weak interpretations, the adaptationist programme (neo-Darwinism) (Gould and Lewontin, 1979;Nielsen, 2009) of natural selection (Darwin, 1859) provides the evolutionary background for explaining whether self-regulation can be explained by mechanisms involving and units of selection, competition, cheating and selfishness (Lenton, 1998, Lenton andLovelock, 2000;Staley, 2002;Lenton, 2008, Doolittle, 2017). It also features explicative notions such as selection by survival alone, and progress by accumulation (Doolittle, 2017;Lenton et al., 2018). On the other hand, under the strong interpretations, homeostasis and optimization (Cannon, 1929) provide the explanative framework to address the thesis of self-regulation at the planetary scale. Homeostasis is framed mathematically in terms of the first-order cybernetics (Wiener, 1948), self-organization (Ashby, 1947;1962a;Von Foerster, 1960) and ultrastability (Ashby, 1956a(Ashby, , 1952. The possibility of self-regulation at the planetary scale is explored with heuristic models such as Daisyworld (Watson and Lovelock, 1983), 'complex adaptive' systems (Lenton and van Oijen, 2002;Levin, 2005Levin, , 1998Schwartzman, 2002) and sequential selection (ultrastability) . When Gaia is seen as a system with many degrees of freedom, the possibility that it may optimize certain quantities finds some justification in the theory of irreversible far-from thermodynamic equilibrium (Nicolis and Prigogine, 1977) along with information theory (Shannon, 1948). Life would enhance and maximize entropy production (Karnani and Annila, 2009;Kleidon, 2009Kleidon, , 2004Kleidon and Lorenz, 2004;Toniazzo et al., 2005) and reduce planetary scale gradients (Sagan and Whiteside, 2004).
In conclusion, the present-day Gaia theory is neo-Darwinized (it has been made compatible with the adaptationist programme) and physicalized (Gaia can be simulated by dynamical systems using thermodynamics and cybernetics) to fit the notion of Life as a planetary regulator (Fig. 1). They are important contributions because they outline possible mechanisms of planetary self-regulation across scales. However, we argue that they obliterate Schrödinger's question as the fundamental motivation of Gaia hypothesis. Indeed, certain abiotic systems display a form of self-regulation. For example, the thermostat is an evolving ultrastable machine able to regulate the temperature by feedback mechanisms (Ashby, 1952(Ashby, , 1956b. Mechanisms of regulation by feedback loops have been described on lifeless planets such as Mars (Chassefière, 1991;Ng and Zuber, 2006). Abiotic geochemical cycles, such as that of nitrogen, are plausible on a planetary scale (Laneuville et al., 2018). The concept of negative entropy also appears to fall short for characterizing life. Irreversible, far-from equilibrium states, symmetry breaking, and self-organization take place in abiotic systems such as Bénard convection cells or dissipative systems (Fodor et al., 2016;Sanchez et al., 2012). Chemical disequilibrium and far-fromthermodynamic equilibrium of exoplanets atmospheres have been identified in inhabitable planets (Krissansen-Totton et al., 2016). Thus, self-regulation, self-organization, chemical disequilibrium, geochemical cycles and even adaptive behaviour can and do appear in inanimate systems. Furthermore, from a strict thermodynamic perspective, the entropy of living systems does not appear to be particularly low (Martyushev, 2013). With these considerations at hand, we see that the Gaia hypothesis as the Schrödinger's question support the claim that the Gaia phenomenon is the manifestation of something more than self-regulation and the postulations behind the present-day Gaia theory. It involves more than organisms-environment coupled co-evolution, and more than the Earth system viewed as a planetary system far from thermodynamic states with feedbacks and geochemical cycles due to geochemistry and geophysics alone .
A parallel may be drawn with the cellular phenomenon. To fabricate a living cell, we need more than the concatenation of biosynthetic cycles (Calvin cycle, tricarboxylic cycle, etc), gene regulation by feedback circuits (such as operon Lac or ribosomal operon) and active matter (Fodor et al., 2016;Sanchez et al., 2012).
Likewise, bringing an inert planet far from thermodynamic equilibrium states with feedbacks and a selection of geochemical cycles would arguably not yield the Gaia phenomenon. Therefore, the Schrödinger's question remains the key to how best pose the Gaia hypothesis (Lovelock, 1986;Margulis and Sagan, 1995). The answer to this question is the scope of the next section.  (Thomas, 1974, p. 5). More recently, Harold Morowitz wrote "the metabolic character of life is a planetary phenomenon, no less than the atmosphere, hydrosphere, or geosphere" (Morowitz et al., 2008, p. 8). These claims appear today compatible with the Gaia hypothesis, if not defining it.

Towards the theoretical biology of the Gaia phenomenon
Since the beginning of physiology, the realization of the living of a biological system is characterized by the maintenance of its internal physiological bounds, despite forcing fluctuations and dissipation. Claude Bernard noted that "All the vital processes, varied as they are, have only one object, to maintain the uniformity of the conditions of life in the internal milieu" (Bernard, 1878, p. 84). This asymmetry between external and internal conditions is thus broadly recognized as important characterization of biological systems. Following the notion of conservation of the internal milieu,  and Margulis (1990Margulis ( , 1997 considered that the Gaia phenomenon can be characterized as a biological system: "Different from a physicist, biochemist and neo-

Darwinist's view…[Gaia is]… a bounded system that is open to a flux of energy and matter, and that is able to
keep its internal conditions constant, despite changing external conditions" ( Lovelock, 1991, p. 29 brackets are ours). Or: "Cells and Gaia display a general property...: as their surroundings change unpredictably, they maintain their structural integrity and internal organization" (Margulis, 1997, p. 267). Furthermore, they recognized that instead of planetary homeostasis, "Gaia's history [ontogeny] is characterized by homeorhesis with periods of constancy punctuated by shifts to new, different states of constancy" (Lovelock, 1991, p. 141 brackets are ours), or "Gaia…like the physiology of an embrio, is more homeorhetic, than homeostatic" (Margulis, 1990, p. 866). Homeorhesis indicates system dynamics with potential transitions to multistability and stability of dynamics rather than stability of states. Waddington (1968, p. 12) indeed suggested that what "is being held constant is not a single parameter but is a time-extended course of change, that is to say, a trajectory". In summary, a biological system responds to a perturbation (forcing fluctuations and dissipation), thus maintains its internal physiological bounds, by returning to a homeorhetic trajectory, not to a homeostatic state.
Since the development of cybernetic systems, the response to perturbances appealed to the notion of self-regulation by feedback mechanisms. There is, however, an alternative explanation why biological systems maintain their homeorhetic trajectories within physiological bounds. It relies on self-fabrication by closure to efficient causation in the (M,R)-system (M=metabolism, R=repair) (Rosen, 1991a(Rosen, , 1972(Rosen, , 1958 and selfproduction by operational closure in autopoiesis (auto=self, poiesis=production) (Maturana and Varela, 1980) characterization (Fig. 2). These explanations have become important, because for a process to be regulated, it must first be produced. That is to say, the material production precedes the regulation (Rosen, 1991a, Maturana andVarela, 1980). Therefore, it is logical to pay attention first to how the systems are constituted materially, to later study how this material production may or may not be regulated. This reflects 'what life does' as to best represent 'what life is', thus a way to answer Schrödinger's question (Cornish-Bowden and Cárdenas, 2017;Friston, 2013;Letelier et al., 2011;Maturana, 1980;Piaget, 1967;Ramstead et al., 2018;Rosen 1985Rosen , 1991avon Foerster, 1974).
The (M,R)-system outlines the inferential aspects of a formal system, hence the theoretical characterization of biological systems in mathematical language ( Fig. 2A). Based on the (M, R) formal system, Rosen states: "a material system is an organism if, and only if, it is closed to efficient causation" (Rosen, 1991a, p. 244). Rosen (1972Rosen ( , 1985aRosen ( , 1994b identified an efficient cause with the enzymatic 'active site' which is the operator that induces or constrains the material flow of unlikely chemical reactions, and thus, it is the generator of forcing (Jee et al., 2018). The active site is only found in the tertiary structure and three-dimensional form of an enzyme. It determines the catalytic activity of the enzyme, and is configured in the protein folding (phenotype at very basic molecular level). Protein folding is not predetermined in the DNA/RNA sequence, hence the 'active site' is not coded in the genomic sequence. It is non-isolable and non-fractionable from the folded enzyme which is produced, specified and determined only in the realization of the living of biological systems (Rosen, 1991a).
Self-fabrication by closure to efficient causation take place when the (M,R)-system models a biological system selecting a set of elements A (matter and energy) of its environment and effectively transforms them in B by its catalytic function F. The function F represents the efficient cause of this transformation. This reads F: A→B ( Fig. 2A). However, the efficient cause F is subject to dissipation, and therefore needs to be repaired, fabricated, produced and replaced. A new efficient cause is introduced, the repair function φ which takes B as material cause to produce F. This reads φ: B→F. But the efficient cause φ is also subject to dissipation, and therefore needs to be fabricated and produced as well. It requires another efficient cause, and then yet another efficient cause for the production of this efficient cause, and so on infinitely. The (M,R)-system could, of course, be simply augmented with another mapping that would do it, but this would lead to an incipient infinite regress 2 . Instead of invoking efficient causes ad infinitum, the nontrivial key point in the (M,R)-system formal representation of biological systems is that the efficient cause (repair function) φ is fabricated from inside, rather than from the outside the system. This is done by proving, 'under stringent but not prohibitively strong conditions', the existence of a map β that replicates φ. The condition is related to the existence of a function that assigns a map β to an least one b∈β such that, for those b∈β, ! satisfies the relation ! : F → φ.
The satisfaction of this condition generates closure to efficient causation, thus a clear-cut formal characterization of a biological system. In this case, each efficient cause in the system is materially produced within the system itself by metabolism (F), repair (φ) and replication (β). They are mutually dependent on each other, or, in the terms of formal logic, they are mutually 'entailed' (Rosen, 1972). The fabrication of every efficient cause is caused (internally entailed) by another efficient cause produced within the biological system. The organization of the fabrication of all catalytic elements is embedded in a metabolically closed loop -a circularity of efficient causes -implying that biological systems do not require any catalyst from their environment (Cornish-Bowden, 2015). When the product of a self-fabricated catalyst ( itself" (Rosen, 1999, p. 265). As the (M,R)-system refers to itself in its operations: "...an (M,R)-system in which the operator φ is itself entailed from within the system is an (M,R)-system" (Rosen, 1989, p. 28), the (M,R)system theory is also a theory of self-referential systems (also known as impredicative systems). This suggests, on formal grounds, the existence and physical realization of self-referential causal systems (Rosen, 1985a).
Autopoiesis rather outlines and characterize the physical realization of the living of biological systems, thus the causal aspects of natural systems. That is, autopoiesis "describes the molecular processes…taking place in the realization of the living of living systems… not a philosophical proposition of a formalization of the phenomenon of life...the molecular autopoiesis of a cell is its living in the continuous realization of their selfproduction without the participation of any organizing principle" (Maturana, 2011, p. 144). An autopoietic unity is characterized as "a network of processes of production (transformation and destruction) of components which: (i) through their interactions [openness to flux of matter and energy] and transformations continuously regenerate and realize the network of processes (relations) that produced them; and (ii) constitute it (the system) as a concrete unity in space in which they (the components) exist by specifying the topological domain of its realization as such a network" (Maturana and Varela, 1980, p. 78, brackets are ours) (Fig. 2B). There is no separation between the product and the producer: Their operational closure results in the same continuous selfreferential process from which it is produced: "a closed domain of operational relations specified only with respect to the system organization that these relations constitute, and thus it defines a space whose dimensions are the relations of production of the components that realize it as a concrete biological unity" (Maturana and Varela, 1980, p. 97).
Because the closure of both systems is explicitly related (Nomura, 2001(Nomura, , 1997Zaretzky and Letelier, 2002;Letelier et al., 2003) in the notion of metabolic closure (Letelier et al., , 2006Soto-Andrade et al., 2011), Autopoesis and the (M,R)-system satisfy the modelling relation, in the way Rosen (1985, p.15) defined it: "a process of bringing into relation two entailment structures, a material one governed by causality, and a mathematical governed by inferential rules". A biological system is then characterized by the autopoiesis as the physical realization of an (M,R)-system, and the (M,R)-system as the theoretical model of autopoiesis (Fig. 2C).
Following this, here, we shall distinguish a biological system as an (M,R)-autopoietic unity, (Fig. 2D), thereby bringing about an equivalence between closure to efficient causation and operational closure.
The operational closure to efficient causation can be contrasted with a chemical reaction, where the catalysts (the 'efficient cause') that transform the reactant into products are not a product of the reaction, nor the material of the reactants: the catalysts are external to the chemical reaction. Therefore, the chemical reaction is 'open to efficient cause' which makes its causal entailments unrelated, and thus fundamentally different from biological systems. Operational closure to efficient causation can also be contrasted with cybernetic systems (Maturana, 2011;Rosen, 1985b) and artificial intelligence (AI) in general (Maturana 1980;Rosen, 1985a). The latter have an organization determined (designed, fabricated, programmed and controlled) from outside by input / output control feedback mechanisms, while the (M,R)-autopoietic unity have an organization defined from within, which is not always amenable to input / output control parameters (Casti 1988). However, the latter is distinct from self-organized systems (Rosen, 1959(Rosen, , 1994aMaturana, 1987Maturana, , 2002Maturana and Varela, 1980;McMullin, 2001, far-from thermodynamic equilibrium systems (Maturana 1980;Rosen 1978a) or the chemoton (Cornish-Bowden, 2015). That is, the chemoton (Gánti, 2003), dissipative structures (Nicolis and Prigogine, 1977), hypercycles (Eigen et al., 1977) and autocatalytic sets (Kauffman, 1986), either at molecular (Hordijk et al., 2019;Jain and Krishna, 2001) (Letelier et al., 2006;Louie, 2007;Luz Cárdenas et al., 2010;Maturana, 1980;2011;Rosen, 1985aRosen, , 1988bRosen, , 1991a)(see section 6). In summary, the (M,R)-Autopoiesis explains how, despite forcing fluctuations and dissipation, biological systems maintain their homeorhetic trajectories within physiological bounds and why they are distinct from chemical, self-organized, thermodynamic and cybernetic systems. We conclude, therefore, that the (M,R)-Autopoiesis represents a necessary and sufficient rigorous and formal clear-cut criterion to answer the Schrodinger' question.
Although (M,R)-Autopoiesis' scope is for cellular and multicellular phenomena rather than for planetary systems, it is nonetheless necessary to formally express the Gaia hypothesis, and it would make sense, at first glance, to consider it as the most generic and less restrictive definition to characterize whether the Gaia phenomenon is (or not) a biological phenomenon. If so (M,R)-Autopoiesis may be common to large classes of systems, if not universal throughout a range between different discrete scales, from planets to cells and vice versa, as suggested before (Lovelock 1987;Margulis 1997). Mikulecky (2000) previously suggested that the categorical expression of the (M,R)-system may provide the theoretical biology for approaching formally the Gaia hypothesis. To this end, our main question here is whether operational closure to efficient causation takes place on a planetary scale, i.e. whether the Earth operation implements causal and inferential relations of processes of production and fabrication that are embedded in metabolic closure (Fig. 3A). To our knowledge, this has never been seriously examined. In the next section we materialize this proposal.

Mathematical biology of Geophysiology: A formal expression of the Gaia hypothesis
We begin by denoting all the Gaia's fluxes of matter and energy and its efficient causes of transformation ( Fig. 3B)(cf. Volk, 1998) as a single mapping in which a planetary abiotic (Cornish-Bowden and Cárdenas, 2017;Goldford and Segrè, 2018;Hordijk and Steel, 2018; or inorganic catalytic network I reacts and transforms matter and energy (M) into ongoing geochemical cycles (C) (Fig. 3B): that is, for any m ∈ M, we have: The efficient cause I (hence, a mapping or function) is thus identified with a multi-scale pre-metabolism (nonenzymatic autocatalytic network) which, in a given environment, transforms M into C. M is the collection of all sets of energy and matter m (m ∈ M) of the atmosphere, lithosphere and hydrosphere which can be transformed.
C includes everything that is being produced by I: the numerous geophysical and geochemical processes that generate the collection of all sets of geochemical cycles c ∈ C (Fig. 3B). As I, the abiotic pre-metabolic-like chemical network, is a physical entity subject to decline, dissipation and disruption, it needs to be effectively repaired and/or replaced. Likewise in an (M,R)-system every efficient cause F must have an associate φ that maintains the concentration of F. On the Earth system, the metabolic enzymatic activity of the biosphere O (from organic) mainly linked to origin and activity of microbes (Cavicchioli et al., 2019;Magnabosco et al., 2018;Margulis and Sagan, 1986;McGenity, 2018;Falkowski et al., 2008;Raoult, 2010;Stolz, 2016) may correspond to φ. The efficient cause O acts as a selector of C, which becomes the material reserve to produce I.
O is therefore a second efficient cause (mapping) which, in a given environment or boundary condition, metabolizes selectively, catalyses and transforms C into I through multi-scale enzymatic metabolism (Caetano-Anollés et al., 2007;Medini et al., 2005;Sun et al., 2017) allowing continuous operation of I from existing C (Fig. 3C). As the metabolic transformation of C into I by O removes, geologically transforms and relocalizes compounds (Atekwana and Slater, 2009;Falkowski et al., 2008;McGenity, 2018;Tornos et al., 2018), it effectively constrains Earth's material dynamics, thermodynamics, electromagnetism and geomorphological structural changes such as the configuration of the continents, tectonic and volcanic activity, ocean salinity, pH, redox potential, ocean circulation, distribution of ice sheets, etc (Hinkle, 1996;Kleidon, 2002;Kump, 2004;Lowman and Lowman, 2002). This can be written as: where H (M, C) is a set of mappings from M to C, and O is the efficient cause that acts as a metabolic selector of existing C that converts its products into new I (Fig. 3C). However, O is subject to ageing, senescence, decline and thus potential disruption. It therefore requires another efficient cause to be produced and remain operating.
Like in the (M,R)-system we can avoid efficient causes ad infinitum if, in the formal expression of the Gaia hypothesis, we can identify an efficient cause (a replication function) using the content of the set C, in which case will be entailed inside the Earth system and act on I to ensure the continuous operation of the efficient cause O, the metabolic enzymatic activity of the biosphere. This systemic and relational replication may generate closure to efficient causation at the planetary scale (Fig. 3C).
Rosen's approach (Rosen 1972) (see Mossio et al., 2009 for an computable attempt using Lambda calculus), to formalize it, and thus to prove the existence of effective self-referential functions in general, and the existence of an efficient cause within the set C in our formal definition of Gaia hypothesis in particular, is by having an inverse evaluation map as follows. Let X and Y be sets, and H (X, Y) is a set of mappings from X to Y. In general, we always regard the elements ∈ as operators on H (X, Y), by defining ( ) = ( ) is in this case called the evaluation map associated with → . Then, : , → .
One such evaluation map has an inverse !! then we can define this inverse as

The inverse exists if and only if
that is, once x is known, ( ) contains enough information to determine within the set , So, to formally identify we can identify: where M and C are the sets A and B in the (M, R)-system respectively (see Fig. 2A). This shows that if an evaluation map in this case is invertible, it is precisely a map into the set , , to which belongs.
The map can be parameterized as a function of ∈ , where c is a particular product produced by the system for which the inversion condition is satisfied: such that ! = That is, once the inversion condition is satisfied for a given , it is possible to define ! which will ensure the replacement of using information provided by the product of ( Fig. 3C).
Under these terms, the Gaia hypothesis is expressed as the existence of the closure to efficient causation: → ! → → . The existence of an inverse evaluation map would therefore imply that the operation of the Earth system is encoded in the entire system, but not in a specific component of it. Indeed, ! is an efficient cause of O, thus, an efficient cause of the efficient cause of C, which contains c, and which determines ! . This closed entailment loop implies that a subset of C (c, in this case) needs to contain information about C itself. C is, however, multifaceted. It involves multiple time scales, and contains the numerous geophysical and geochemical structures generate by Earth's metabolism, including orography, topography, heat and chemical gradients. Similar to enzymes in a cell, some of these structures play the role of an efficient cause necessary for producing the metabolism. The subset (c) represents this efficient cause. It is on this basis that one can say, formally, that the process of fabrication of the Earth system produces O based on what the Earth system recognizes of itself, like a biological unity. Finally, equation (5) formalizes a one-to-one map between the Gaia phenomenon and the (M,R)-system (Fig. 3A, D). The Earth processes of production would satisfy the (M,R)-system formal model, in which the efficient causes are uniquely determined by their products, requiring no external efficient cause (Fig. 3D). The efficient causes of Earth's operation need not be added from the outside. They exist inside the Earth system. The (M,R)-system is the formal expression of the Gaia hypothesis and essentially the theoretical model of Gaia phenomenon. The latter is the realization of the former. Having suggested this, in the following section, we will provide a plausible, empirically supported account of autopoiesis as the realization of the living of Gaia phenomenon. Vernadsky (1945) suggested that Life is not a form of energy and is not merely a geological force, rather it is the geological force. So, the natural question he asked was: 'How on Earth can it (Life) change material processes?'. Given the limitations of characterizing life phenomenon as a mere regulator, autopoiesis was suggested as the realization of the living of Gaia phenomenon (von Foerster, 1975): the relations of production on Earth system may correspond to an autopoietic system. Several authors consider that the production of the atmosphere (mainly troposphere and stratosphere), in the same domain in which the planetaryscale metabolism continuously, generates the very same relations of production offers evidence for autopoiesis in the planetary scale (Capra, 1996;Capra and Luisi, 2014;Clarke, 2012;Jantsch, 1980;Kazansky, 2004;Levchenko et al., 2012;Margulis, 1990Margulis, , 1997Sagan, 1995, 1986;Onori and Visconti, 2012;Sahtouris, 1996). Margulis wrote: 'Whereas the smallest recognizable autopoietic entity… is a tiny bacterial cell the largest is Gaia' (Margulis, 1990, p. 861). Or that 'planetary physiology…is the autopoiesis of the cell write large' (Margulis and Sagan, 1995, p. 54).

The Metabolism and Repair of Gaian Autopoiesis implies conservation of organization under structural change
Is it, however, possible to identify on the Earth system an organization of processes of self-production?
"Since it is a defining feature of an autopoietic system that it should specify its own boundaries, a proper recognition on an autopoietic system as a unity requires a proper recognition […] that defines the limits of the system in the same domain in which it specifies them through […] relations of production of components that generate these relations and define it as a unity in a given space' (Maturana and Varela, 1980, pp. 108-109).
Accordingly, a proper recognition of Gaian Autopoiesis should offer evidence of the continuous processes of self-production by closure to efficient causation on the planetary scale. We consider here that to recognize the autopoesis as the realization of the living of Gaia phenomenon, both natural systems, Gaia and an autopoietic system should satisfy, through empirical evidence, the modelling relation as indicated in the figure (4A).
Today, it is generally accepted that the productions of troposphere: aerosols (Dani and Loreto, 2017;Fröhlich-Nowoisky et al., 2016;Gantt and Meskhidze, 2013;Lana et al., 2011), methane greenhouse gas (Conrad, 2009;Izon et al., 2017), some clouds (Charlson et al., 1987;Hughes et al., 2014), and even stratospheric ozone (Falkowski, 2006) are immediately related to multi scale planetary metabolic activity (Goldford and Segrè, 2018). The Earth's metabolism takes place in the subsurface and surface of the lithosphere and ocean, the atmosphere, the rhizosphere, the gut microbiomes, in the dark, rare and deep biosphere (Cavicchioli et al., 2019;Magnabosco et al., 2018). It is carried out mainly by prokaryotes (Margulis and Sagan, 1986;McGenity, 2018;Falkowski et al., 2008), the so-called 'Gaia microbiome' (Stolz, 2016) and its pangenome (Medini et al., 2005), which comprises autotrophic, heterotrophic, anoxic and oxide activity. It has been suggested that the Earth's prokaryotes constitute along with eukaryotes an interconnected single rhizome of life (Raoult, 2010). Morowitz states that "all organisms interact (are connected) through the gas-phase components that they take up from and give off to the atmosphere...life…persists over long geological periods only because of integrated processes of the atmosphere, the hydrosphere, the lithosphere, and the biosphere" (Morowitz, 1993, pp. 5-6). That is, the atmosphere is not only the product of the planetary-scale metabolism, but allows its continuous operation through the hydrosphere and lithosphere suggesting a plausible scenario operational closure at planetary-scale. Indeed, one of the main motivations of proposing the Gaia hypothesis was that the planetary-scale metabolism does not merely affect the chemical composition of the atmosphere and vice versa; they self-produce in a single mutual continuous process (Margulis and Lovelock, 1975).
Recent additional empirical evidence somehow may support this possibility. The different values of the water deuterium-to-hydrogen ratio (D/H) observed recently on various bodies in the Solar system, suggest a non-asteroid origin of Earth's cycled water (Altwegg et al., 2014)(Deep mantle Earth's water is not found, so far, to be cycled water (Hallis et al., 2015)). In biological systems, a significant amount of water is produced metabolically (endogenously) (Li et al., 2016). In addition, the conservation of the core signatures of the early Earth metabolism in contemporary cellular biogeochemistry (Morowitz, 1993) suggests that most prokaryotes notably, methanogens (CO 2 + 4H 2 → CH 4 + 2H 2 O) (Bardgett et al., 2008), sulphate-reducing bacteria (CO 2 + The evidence accumulated so far offers thus a plausible account of continuous processes of molecular self-production of the terrestrial environment. Furthermore, the fact that Earth's metabolism defines molecularly the boundary (hydrosphere, troposphere and stratosphere) in the same domain in which it is specified indicates molecular metabolic closure in the planetary scale, and henceforth, an "Autopoietic Gaia" (Margulis, 1990) ( Fig. 4C). In fact, the explanatory scope of autopoiesis covers the phenomenological basis of broader domains of biological realization than the cellular and multicellular scales. Maturana refers to such larger-scale domain of biological realization as 'higher order autopoietic unity'. Crucially, he indicates that such realization must be molecular: "There are autopoietic systems of higher order, integrated by (populated by) lower order autopoietic unities that may not be the components realizing them as autopoietic systems... there are higher order autopoietic systems whose components are molecular entities produced through the autopoiesis of lower autopoietic unities" (Maturana, 1980, p. 53, parhentesis and subline are ours). This implies that Gaia phenomenon is the molecular constitution of planetary scale metabolic closure.
Some proposals such as the 'molecular biology of Gaia' (Williams, 1996), a 'wasteworld of byproducts' (Volk, 2004) and the biogeochemical cycles as the unit of natural 'selection' bypassing taxonomic individual species (Doolittle, 2017(Doolittle, , 2014) support somehow that Gaia phenomenon is molecular. However, operational closure to efficient causation at the planetary scale is more than these proposals. For example, biogeochemical cycles, like cellular biochemical cycles, are loops of material causes, but not loops of efficient causes (Louie and Poli, 2011). Loops of material causes take place because there are efficient causes that make the material elements be cycled. In other words, the biogeochemical cycles or by-products ( ∈ in the (M,R)system) are necessary for enabling Gaian autopoeisis, but they are only partially constitutive of it.
The metabolism and repair of Gaian autopoiesis implies that the Earth is an organized system as a biological system, i.e. the present terrestrial environment is itself the result of its own fabrication processes. That is, at geological scales, there is no separation between product and producer, between biotic and abiotic elements. As such, Gaia represents a sui generis scale of (M,R)-autopoietic organization in the planetary scale, and thus a proper biological domain of realization of the living (Fig. 4C, 3D). With domain we refer to a biological unity, such as the cellular or multicellular organism, which is distinct from self-organized emergent systems, i.e., Gaia is different to a colony-like stigmergic assemblages of agents that can form a 'global' superorganism that emerges out of their interactions. This description fails to capture molecular operational closure to efficient causation. The (M,R) of Gaian Autopoiesis suggests that the continuous process of terrestrial self-production is a biological realization of its own at the planetary domain.
In the formulation of autopoiesis, biological systems are distinguished by their organization and structure (Maturana and Varela, 1980;Nomura, 2006). While the former is preserved, i.e. self-production by the closure to efficient causation must persist, the latter may change through the flows, cycling and continuous transformation of materials (Letelier et al., 2006;Maturana and Varela, 1980). This explains why life phenomenon persists despite structural changes during ontogeny and phylogeny (Maturana, 1980;Maturana and Mpodozis, 2000). For example, despite large structural changes, abrupt catastrophic shifts (e.g. loss of almost 90% of the biosphere) and tipping points  from Pangea to the current continental configuration (Lenton and Watson, 2011) Gaia phenomenon has persisted. Indeed, whereas self-production by closure to efficient causation persists, multiple interdependent structural changes can take place and when one structural dimension is changed, the complete structure of the system may undergo correlative changes (Maturana, 1980;Maturana and Mpodozis, 2000). In this interpretation, bifurcations, critical transitions, tipping points and tipping cascades (Scheffer, 2009;Ashwin et al., 2012;Lenton and Williams, 2013;Steffen et al., 2018), may be associated with structural changes constrained by the conservation of Earth's self-production by closure to efficient causation. The structural changes in the Precambrian Vendian shows, for example, that the phenotypic transformation of multicellular organisms took place along with the transformation of the lithosphere, atmosphere and the complete biosphere through the conservation of the Gaia phenomenon (Levchenko et al., 2012). Even when the Earth has been impacted by planetesimals (Abramov and Mojzsis 2009), the Gaia phenomenon has persisted.
However, the explanatory scope of the theory and realization of (M,R)-autopoietic systems foresees that some structural perturbations (e.g. the loss of crucial information of the replication map in the M,Rsystem) can send Gaia into an 'autopoietic oscillator dead' (Friston 2013) or more broadly into 'system degeneracy' (see Rosen (1988aRosen ( , 1978b), such that the conservation of system's organization, therefore its living, is lost (Letelier et al., 2006;Maturana, 1980;Maturana and Varela, 1980).
What we have discussed so far provides a plausible, empirically supported account of (M,R)autopoiesis behind the Gaia phenomenon that goes beyond the somehow superficial 'aquarium poetic view' ascribed to it by Doolittle (2017). It turns out that the (M,R)-autopoiesis surrogates the Gaia phenomenon to a self-referential system (Soto-Andrade et al., 2011). Next, we discuss the implication of it on simulable approaches of the Gaia phenomenon.

Daisyworld is a mechanism, the Gaia phenomenon is not.
According to Rosen's Life Itself, in a mechanism "there can be no closed path of efficient causation" (Rosen, 1991a, p. 241). Moreover, a natural system is said to be a "mechanism if every model of it can be simulated on a mathematical Turing machine" (Rosen, 1991a)(the standard form of computing today). In other words, mechanisms are simple systems which obey dynamics that can be encoded in the form of algorithms, i.e.
the system is open to efficient causation, hence its model is simulable. Mechanisms can have very complicated simulations, and the complicated connections between chemical and physical processes are captured with complicated codes resolving atmospheric and oceanic fluid motion such as general circulation models and modern Earth system simulations.
With this point of view, the Gaia phenomenon can be captured by mechanisms with even simpler simulation programmes such as the Daisyworld(s) (Watson and Lovelock, 1983). The chain of feedbacks described by Daisyworld constitutes a mechanism, in the sense that it can be captured by a recursive algorithm.
It is standard to attempt capturing biological systems with algorithms. Von Neumann's self-reproducing automata (Von Neumann, 1966), neural networks, agent-based models, machine learning, deep learning and AI in general follow this principle. Under this rubric, the 'degree' of complexity would be related to connectivity, nonlinearity and size: "there exists a critical size below which the process of synthesis is degenerative, but above which the phenomenon of synthesis, if properly arranged, can become explosive" (Von Neumann, 1966, p. 80). Following this view, the Gaia phenomenon can be identified with 'self-organized criticality' (Bak, 1993), 'adaptive' systems (Lenton and van Oijen, 2002;Levin, 1998) or more complicated phenomena representing learning (ultrastability) at the planetary scale showing an 'emergent' and nonlinear and adaptive behaviour ). Such a system can be large, but which is still formally equivalent to a dynamical system that can be encoded as a (large) numerical algorithm.
However, the Gaia phenomenon by means of the closure to efficient causation suggests on formal grounds, that there is a fundamental gap between the Gaia as a biological phenomenon and any algorithmic representation (Letelier et al., 2006;Louie, 2007;Luz Cárdenas et al., 2010;Maturana, 1980;Rosen, 1988b).
That is, the realization of the living cannot be implemented in a Turing machine and "accordingly must have a nonsimulable model" (Rosen, 1999, p. 292). No matter the resolution, in principle it will never be filled in with computing power or be completely surrogated to algorithmic procedures. The discrepancy of efficient causation between the Gaia as a biological phenomenon and algorithms is so severe that the former cannot even be mapped onto a dynamical system (Rosen, 1991b(Rosen, , 1973, self-reproducing automata (Rosen 1959(Rosen , 1975Maturana, 1980) or a collection of sequential recursive algorithms (Ashby's ultrastability) (Maturana, 2011;Rosen, 1985b). In the Daisyworld(s), sequential selection or even in any other Earth system simulation likewise as in chemical reactions described above, the recursive functions of the algorithms correspond to the repetition of a set of rules that are explicitly specified to the system from the outside and are implementable in a Turing machine.
This suggest that the distinction between mechanisms of cybernetic systems and the living of biological systems is not a matter of degree, but of character. Such as in a cellular or any other biological system, Gaia  (Rosen, 1999, p. 44, parentheses are ours).
Moreover, the Gaia phenomenon by means of the closure to efficient causation entails the existence of an self-referential (impredicative) "effective" process ("effective" because it is physical) (Rosen, 1991a;Soto-Andrade et al., 2011) that rises paradoxes to Von Neumann's self-reproducing automata and AI in general (Ashby, 1962b;Rosen, 1959;Ben-David et al., 2019, Reyzin, 2019. Therefore, Gaia as a biological phenomenon may be of a non-simulable, non-algorithmic, and hence non-computable character (Letelier et al., 2006;Louie, 2007;Luz Cárdenas et al., 2010;Maturana, 1980;Rosen, 1988bRosen, , 1989. That is, the Gaia phenomenon, in principle, may have at least one model that cannot be simulated by finite-state machines (e.g., Turing machines) (Louie, 2007;Luz Cárdenas et al., 2010). Therefore, in a fundamental way, the Gaia phenomenon, although physical, is neither a mechanism nor a machine (Maturana 1980;Rosen, 1991aRosen, , 1985a. The relation between the biological character of the Gaia phenomenon and the mechanical simulable (algorithmic) models of it (Daisyworld(s)) may be linked to the situation faced by early cartographers, who were attempting to map the surface of a sphere while armed only with pieces of (tangent) planes. As long as we only map local regions, the planar approximations (mechanisms) suffice, but as we try to map larger and larger regions, the discrepancy between the map and the surface grows. Thus, if we want to make accurate maps of large regions of the sphere, we have to keep shifting our tangent planes. The surface of the sphere is in some sense a limit of its planar approximations, but to specify it in this way requires a new concept (the topology of the sphere) that cannot be inferred from local planar maps alone (Rosen 1985). Similarly, the (M,R)-system model of Gaian autopoiesis implies that we need to widen our concept of what the Gaia phenomenon is, or should be.

Concluding Remarks
In this paper, we have examined the Gaia hypothesis from a fundamental biological standpoint offered by the (M,R)-system as a formal theory and autopoiesis as the realization of the living. We have considered that the Gaia phenomenon was identified by observing the qualitative difference among Earth, Mars and Venus, i.e. from differential observables in planetary systems. Such differences between these planets, and the unique quality on Earth, has given rise to the Gaia hypothesis, which raises Schrodinger's question, 'What is life?', a central question of biology, at the planetary scale. We have pointed out that the lack of use of a clear-cut formal criterion and characterization of biological systems has led to use the adaptationist programme and the regulator thesis, with so-called strong and weak interpretations of the Gaia hypothesis. This has configured the presentday Gaia theory, but obliterated the Gaia hypothesis and thus Schrodinger's question.
We have further argued that (M,R)-Autopoiesis offers a necessary and sufficient answer to self-produce by closure to efficient causation. Then, based on the modelling relation, a set of interacting causal processes operating on a wide range of spatial time scales through the atmosphere, lithosphere, hydrosphere, and biosphere of the Earth system has been categorized within the relational and systemic entailments of (M,R)-Autopoiesis. Consequently, we suggested a one-to-one realization map between (M,R)-Autopoiesis and the Gaia phenomenon. In other words, the Gaia phenomenon realizes the inferential and causal entailments of (M,R)-Autopoiesis, such that it generates metabolic molecular closure to efficient causation on the planetary domain.
This suggests that the Earth is an organized system, not of cells, multicellular or ecosystems, but of their molecular products that together with the atmosphere, the hydrosphere and the lithosphere self-produce by metabolic closure. The Gaia phenomenon is, therefore, a sui generis biological system and the embodiment of Life itself in the planetary domain. Moreover, if something has to be regulated, it has to be produced in the first place. Hence, self-production by closure to efficient causation is more fundamental than self-regulation by feedback mechanisms.
Biologizing Gaia theory as such provides a rigorous basis to the claim that planetary biology elsewhere in the universe must involve and have a formal equivalence to self-referential physical processes, which has at least one model that cannot be implemented by a Turing machine and, therefore, is of a non-simulable, nonalgorithmic, and non-computable character.

Footnotes
1 The Nobel-laureate physicist Erwin Schrödinger suggested that biological phenomenon, contrary to the second law of thermodynamics, decreases or maintains its entropy by feeding on negative entropy. Schrödinger remarks on his usage of the term negative entropy: 'Let me say first, that if I had been catering for them [physicists] alone I should have let the discussion turn on free energy instead. It is the more familiar notion in this context. But this highly technical term seemed linguistically too near to energy for making the average reader alive to the contrast between the two things' (Schrödinger, 1945, p. 74).  hence the living realization (implementation) of a (M,R)-system. Autopoiesis take place by a form of operational closure involving the molecular network (f, Φ, β in the M,R-system) and system's boundary. The self-production process must occur in structural coupling (openness)(arrows in both directions) with the ambience (curved grey line). C) Modelling relation depicting the Autopoiesis realization as a natural causal system and the (M,R)-system as a formal inferential system such that self-production by operational closure and self-fabrication by closure to efficient causation are equivalent (D) such that an (M,R)-autopoietic unity is distinguished.  238). C) The autopoietic organization of the Earth as a system of causal processes of self-production through operational closure to efficient causation (black circular arrow) in structural coupling (black arrows in both directions) with its ambience (curve grey line).