The foundation of every RNCDR model is the Anderies pre-industrial subnetwork, which captures the natural carbon cycle among land ($$ A_1 $$), atmosphere ($$ A_2 $$), and ocean ($$ A_3 $$). This subnetwork is based on the carbon cycle model of Anderies et al.^[3] in The Topology of Non-linear Global Carbon Dynamics: From Tipping Points to Planetary Boundaries, which serves as the formal basis of the influential work of Steffen et al.^[33].

The model tracks carbon transfers among three pools, namely land ($$ A_1 $$), atmosphere ($$ A_2 $$), and ocean ($$ A_3 $$). Solid arrows represent direct carbon transfers between pools, while dashed arrows indicate biogeochemical feedbacks (fertilization and temperature) that modulate the transfer rates; this structure recurs, highlighted, within each technology-specific system introduced in Section "FOUR CDR STRATEGIES AND THEIR REACTION NETWORK REPRESENTATIONS". The total carbon stock is absent from the pre-industrial model since fossil fuel use is not considered.

The model is formulated as a generalized mass action (GMA) system^[34,35,36]. This is an ordinary differential equation (ODE) system in which each carbon transfer is individually approximated by a power-law term via Taylor linearization in logarithmic coordinates, with a positive sign for incoming transfers and a negative sign for outgoing transfers. The resulting CRN representation is dynamically equivalent to the original ODE system. The power-law approximation is summarized in Table 1, and the resulting ODE system, due to the carbon-cycle model of Anderies et al.^[3] in the power-law representation of Fortun et al.^[32], is:

(1) \begin{equation} \dot{A}_1 = k_1 A_1^{p_1} A_2^{q_1} - k_2 A_1^{p_2} A_2^{q_2}, \end{equation}

(2) \begin{equation} \dot{A}_2 = -k_1 A_1^{p_1} A_2^{q_1} + k_2 A_1^{p_2} A_2^{q_2} - a_m A_2 + a_m \beta A_3, \end{equation}

(3) \begin{equation} \dot{A}_3 = a_m A_2 - a_m \beta A_3, \end{equation}

where the kinetic orders $$ p_1, p_2, q_1, q_2 $$ encode the influence of each carbon transfer on the sizes of the land biota and the atmosphere pools, respectively.

The CRN representation of the subnetwork consists of four reactions:

\begin{align} R_1 &: A_1 + 2A_2 \xrightarrow{k_1} 2A_1 + A_2 \quad \text{(atmosphere-to-land)} \\ R_2 &: 2A_1 + A_2 \xrightarrow{k_2} A_1 + 2A_2 \quad \text{(land-to-atmosphere)} \\ R_3 &: A_2 \xrightarrow{a_m} A_3 \quad \text{(atmosphere-to-ocean)} \\ R_4 &: A_3 \xrightarrow{a_m \beta} A_2 \quad \text{(ocean-to-atmosphere)} \end{align}

Reactions $$ R_1 $$ and $$ R_2 $$ represent photosynthesis and respiration, respectively. The unusual-looking stoichiometry (e.g., $$ A_1 + 2A_2 \rightarrow 2A_1+A_2 $$ rather than simply $$ A_2 \rightarrow A_1 $$) arises from the power-law kinetic representation of the original nonlinear dynamics; it encodes how each carbon pool influences the carbon transfer. Reactions $$ R_3 $$ and $$ R_4 $$ are simpler: they represent the exchange of CO₂ between the atmosphere and the ocean.

The industrial carbon cycle introduces the total carbon stock $$ A_4 $$, representing the geological and inorganic carbon reserves. Fossil fuel combustion is modeled by:

(4) \begin{equation} R_5 : A_4 \xrightarrow{K_5} A_2, \end{equation}

which transfers carbon from the total carbon stock directly into the atmosphere through the combustion of its geological component. The rate function $$ K_5 $$ depends on the CDR technology in use, as described in Section "Connecting storage to the atmosphere through the emission reaction".

Each CDR technology introduces one additional storage pool $$ A_i $$ and at least two reactions:

\begin{align*} R_{i,6} &: A_i \to A_4 && \text{(CDR storage to total carbon stock)}, \\ R_{i,j} & && \text{(CDR-specific reactions)}, \end{align*}

where $$ i = 8, 9, \ldots, 17 $$ for both, and $$ j \geq 7 $$ for the CDR-specific reactions. The reaction $$ R_{i,6} $$ models the possibility of a "leak"^[37,38]: even after carbon is captured, some fraction may eventually return to the total carbon stock, from which it can be re-emitted. The notation for the CDR storage species is summarized in Table 2. Including this leak allows the framework to evaluate CDR performance even under imperfect storage conditions. The reactions $$ R_{i,j} $$ are CDR-specific and represent the additional interactions derived from each particular CDR technology.

Not all CDR technologies store carbon in the same way. Two parameters characterize the output of each technology:

● $$ \lambda_i \in [0,1] $$: the fraction of captured carbon that ultimately enters the geological reserve within the total carbon stock $$ A_4 $$, whether directly or through the leak reaction $$ R_{i,6} $$, and is therefore available for subsequent release by fossil fuel combustion. It is not a statement about where the carbon is held while it remains in the CDR pool $$ A_i $$, which is what the physical storage column of Table 3 records. Carbon in the geological reserve is not thereby permanent: it is precisely the carbon from which the emission reaction $$ R_5 : A_4 \to A_2 $$ draws. A value of $$ \lambda_i = 1 $$ means all captured carbon enters the geological reserve; $$ \lambda_i = 0 $$ means none of it does.

● $$ \mu_i \in [0,1] $$: the fraction of the long-term stored carbon that is inorganic. Inorganic carbon (e.g., carbonate minerals) behaves differently from organic carbon (e.g., biomass) and does not contribute to fossil fuel emissions.

Only organic carbon held in the geological reserve can be released by fossil fuel combustion. The emittable fraction of captured carbon is therefore $$ \lambda_i(1-\mu_i) $$, and the fraction withheld from emission is $$ 1 - \lambda_i(1-\mu_i) = (1-\lambda_i) + \mu_i\lambda_i $$, which is exactly the bracketed term in the emission rate function (5) below. The leak reaction $$ R_{i,6} : A_i \to A_4 $$ and that bracketed term represent distinct mechanisms: the former is a physical transfer of carbon from the CDR pool into the total carbon stock, whereas the latter weights the availability of stock carbon for combustion.

These parameters (see Table 3) are best understood as relative estimates rather than fixed empirical quantities; they allow the model to explore system behavior under different assumptions and can be recalibrated as real-world data become available.

Because these values are nominal rather than measured, it is worth being explicit about how far the results depend on them. The two parameters enter the analysis only through the single combination

$$ \begin{equation} \begin{aligned} c_i = (1-\lambda_i)+\mu_i\lambda_i, \end{aligned} \end{equation} $$

the fraction of CDR storage withheld from emission, which, together with the positive steady state about which (5) is linearized, determines the emission kinetic orders $$ e_i $$ and $$ f_i $$. The storage parameters alone do not fix $$ e_i $$ and $$ f_i $$; Supplementary Section 2 gives the relation and the operating-point dependence explicitly. Conditions in Section "CONDITIONS FOR MULTISTATIONARITY IN CDR REACTION NETWORKS" that require only $$ e_i \geq 1 $$ and $$ f_i \leq 0 $$ hold for any admissible pair $$ (\lambda_i,\mu_i) $$ and are unaffected by recalibration. Conditions that turn on a strict inequality, or on the exact value $$ c_i = 0 $$, are not: two of the results below depend on the particular values in Table 3, and we identify each where it arises. Supplementary Section 2 gives the derivation and a proposition-by-proposition account.

The emission reaction $$ R_5: A_4 \to A_2 $$, which transfers carbon from the total carbon stock into the atmosphere by combustion of its geological component, is a defining feature of the RNCDR framework^[29]. Its rate function $$ K_5 $$, introduced in that work, depends on both the total carbon stock $$ A_4 $$ and the CDR storage $$ A_i $$:

(5) \begin{equation} K_5(A_4, A_i) = k_5 \bigl(A_4 - \bigl[(1-\lambda_i) + \mu_i \lambda_i\bigr] A_i\bigr), \end{equation}

where the term $$ [(1-\lambda_i) + \mu_i \lambda_i] A_i $$ represents the portion of CDR storage that cannot contribute to fossil fuel emissions. In other words, the emission rate is proportional to the "available" geological carbon after subtracting what has been effectively locked away by CDR. Since (5) is not in power-law form, a power-law approximation around a positive steady state gives

(6) \begin{equation} K_5 \approx k_5 \, A_4^{e_i} \, A_i^{f_i}, \end{equation}

where the kinetic orders satisfy

(7) \begin{equation} e_i \geq 1, \quad f_i \leq 0, \quad e_i + f_i = 1. \end{equation}

These three conditions have clear physical interpretations: $$ e_i \geq 1 $$ means emission is at least as sensitive to the total carbon stock as in a basic linear model; $$ f_i \leq 0 $$ means more CDR storage reduces emission; and $$ e_i + f_i = 1 $$ reflects the complementarity of the two effects. When $$ \lambda_i = 1 $$ and $$ \mu_i = 0 $$ (as in BECCS^[29] and DAC^[30]), these reduce to $$ e_i = 1 $$, $$ f_i = 0 $$, and $$ K_5 = k_5 A_4 $$, recovering linear mass action.

The power-law kinetics of the entire RNCDR system is encoded in the kinetic order matrix F, whose entry $$ F_{ij} $$ gives the kinetic order of the j-th species in the i-th reaction. For a single-CDR system, the first six rows of F are common to all RNCDR models:

(8) $$ \begin{array}{rcl} & \begin{array}{ccccc} A_1 & A_2 & A_3 & A_4 & A_i \end{array} & \\[-2pt] F= & \left[ \begin{array}{ccccc} p_1 & q_1 & 0 & 0 & 0 \\ p_2 & q_2 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & e_i & f_i \\ 0 & 0 & 0 & 0 & 1 \\ & & \vdots & & \end{array} \right] & \begin{array}{l} R_1 \\ R_2 \\ R_3 \\ R_4 \\ R_5 \\ R_{i,6} \\ R_{i,j},\ j \geq 7 \end{array} \end{array} $$

The rows for $$ R_1 $$ and $$ R_2 $$ encode the kinetic influence of land biota ($$ p_1, p_2 $$) and atmosphere ($$ q_1, q_2 $$) on the natural carbon cycle. The row for $$ R_5 $$ encodes the coupled dependence of emission on both the total carbon stock and the CDR storage, with kinetic orders satisfying $$ e_i \geq 1 $$, $$ f_i \leq 0 $$, and $$ e_i + f_i = 1 $$ as established in (7). The row for $$ R_{i,6} $$ reflects the common storage-to-total carbon stock reaction, with mass-action kinetics of order 1 in $$ A_i $$. The remaining rows, indicated by the ellipsis, correspond to the CDR-specific reactions $$ R_{i,j} $$ for $$ j \geq 7 $$, whose number and kinetic orders vary by technology and are specified for each CDR system in Section "FOUR CDR STRATEGIES AND THEIR REACTION NETWORK REPRESENTATIONS".

The dynamic behavior of an RNCDR system is strongly shaped by the properties of its Anderies subnetwork. Following^[31], we classify Anderies systems – and by extension, RNCDR systems^[29]– according to the ratio

(9) \begin{equation} R = \frac{p_2 - p_1}{q_2 - q_1}, \quad q_2 - q_1 \neq 0, \end{equation}

which compares how much respiration and photosynthesis differ in their influence on land biota versus atmosphere. The kinetic orders $$ p_1, p_2 $$ encode the degree to which photosynthesis and respiration are influenced by the size of the land biota pool, while $$ q_1, q_2 $$ encode the degree to which they are influenced by the level of atmospheric CO₂. A value of $$ p_2 > p_1 $$ means that respiration responds more strongly to land biota than photosynthesis does; in other words, as forests grow, they release carbon faster than they absorb it. Similarly, $$ q_2 > q_1 $$ means respiration is more strongly driven by atmospheric CO₂ concentration than photosynthesis is. When both inequalities hold simultaneously, these asymmetries reinforce each other, creating the nonlinear feedback conditions under which multiple equilibria can arise. Four primary classes arise from the ratio R, each with distinct dynamic consequences:

● Positive class ($$ R > 0 $$): both $$ p_2 > p_1 $$ and $$ q_2 > q_1 $$ (or both reversed), so the same process dominates the feedback in both pools: respiration when $$ p_2 > p_1 $$ and $$ q_2 > q_1 $$, photosynthesis when both inequalities are reversed. What makes R positive is that the two inequalities point in the same direction, not the identity of the dominant process; the Anderies subnetwork may admit multiple steady states, meaning the Earth could settle into more than one equilibrium carbon distribution.

● Negative class ($$ R < 0 $$): photosynthesis dominates one feedback while respiration dominates the other, leading to more stabilizing dynamics; the Anderies subnetwork contains monostationary subsets (unique steady states under appropriate kinetic order conditions).

● P-null class ($$ R = 0 $$): the Anderies subnetwork is monostationary.

● Q-null class ($$ q_2 - q_1 = 0,\ p_2 - p_1 \neq 0 $$): defined for the case where the ratio R is undefined but $$ Q = (q_2 - q_1)/(p_2 - p_1) = 0 $$. This class shares the monostationarity of the P-null class in the Anderies subnetwork.

These classes characterize the dynamic behavior of the Anderies subnetwork in isolation. Once the Anderies subnetwork is embedded in an RNCDR system, the addition of the fossil fuel emission reaction and the CDR-specific reactions can change the steady-state behavior: a class that is monostationary at the Anderies level need not remain so for the full RNCDR system, and the propositions in Section "CONDITIONS FOR MULTISTATIONARITY IN CDR REACTION NETWORKS" make this explicit for each of the four CDR strategies analyzed.

Note the asymmetry in these definitions: the P-null class arises when the numerator of R vanishes (i.e., $$ p_2 = p_1 $$), while the Q-null class arises when the denominator vanishes (i.e., $$ q_2 = q_1 $$) with a nonzero numerator. The two classes are therefore structurally distinct, and as Propositions 3 and 4 demonstrate, they can exhibit different multistationarity behavior within the same CDR system.

Therefore, an RNCDR kinetic system for a single CDR technology consists of five species ($$ A_1 $$, $$ A_2 $$, $$ A_3 $$, $$ A_4 $$, and one CDR storage species $$ A_i $$) and at least seven reactions. The latter include the five reactions $$ R_1 $$–$$ R_5 $$ that are common to all RNCDR models, the common CDR storage reaction $$ R_{i,6}: A_i \to A_4 $$, and at least one CDR-specific reaction $$ R_{i,j} $$ for $$ j \geq 7 $$, whose number and form depend on the particular CDR technology deployed. The dynamics of the system follow power-law kinetics encoded by the matrix F in (8), with the emission kinetic orders $$ e_i $$ and $$ f_i $$ determined by the storage parameters $$ \lambda_i $$ and $$ \mu_i $$ via (5).

Figure 2 summarizes how the RNCDR framework is applied in practice. The analysis proceeds in five steps: (1) selecting a CDR strategy; (2) parametrizing its storage through $$ \lambda_i $$ and $$ \mu_i $$; (3) representing the system as a chemical reaction network together with its kinetic order matrix F; (4) classifying the underlying Anderies subnetwork through the ratio $$ R = (p_2 - p_1)/(q_2 - q_1) $$; and (5) testing for multistationarity or monostationarity through the Deficiency One Algorithm, the Multistationarity Algorithm, and the injectivity test (discussed in Section "CONDITIONS FOR MULTISTATIONARITY IN CDR REACTION NETWORKS"). The outcome of this workflow is a determination of whether a given CDR system can admit multiple steady states, and therefore whether it may be vulnerable to tipping-point behavior under varying conditions. Notably, this characterization is obtained from the graphical and kinetic structure of the network alone, without requiring precise knowledge of rate constants. The four CDR strategies analyzed in this study, presented in Section "FOUR CDR STRATEGIES AND THEIR REACTION NETWORK REPRESENTATIONS", all follow this same workflow, with the differences between them arising from the specific reactions and kinetic orders introduced at steps 2 and 3.

The key question the framework is designed to answer is: Under what conditions can a CDR system settle into more than one steady state, and therefore become vulnerable to a tipping point that would compromise the carbon removal it was designed to deliver? As explored in the subsequent sections, the answer depends critically on the class (positive, negative, or null) of the underlying Anderies subnetwork, and on the structural features of the CDR-specific reactions, including their kinetic orders and the resulting network deficiency. This provides a basis for screening CDR strategies for carbon-accounting risk before their projected mitigation potential is committed to in deployment plans or long-term climate policy.

Throughout this paper, multistationarity means the existence of more than one positive steady state, and monostationarity means that at most one exists. Relating these to tipping requires care, because tipping is not a single phenomenon. We take a tipping point to be an abrupt transition from one positive steady state to a different one, and on that definition multistationarity is necessary: the destination state must exist. The definition is narrower than the term as used in the wider literature. In the classification of Ashwin et al.^[46], now standard in this field^[47], a system may tip through a bifurcation of a quasi-static attractor as a control parameter drifts (B-tipping), through noise carrying it out of a basin of attraction (N-tipping), or through failing to track a moving attractor when that parameter changes too quickly (R-tipping). Each can end somewhere other than a second steady state, in a limit cycle or an excitable excursion, and R-tipping in particular can occur in systems whose steady state is unique and globally attracting^[46,48]. Those cases lie outside our definition, and outside what the framework addresses.

Within that definition, what the framework establishes can be stated exactly. A system admitting at most one positive steady state throughout a parameter range has no second state to move to, and so cannot undergo a transition of this kind however it is forced. That is a conclusion about the dynamics obtained from the structure of the network alone, and it holds whichever mechanism might otherwise have driven the transition. The best studied of these is the fold, or saddle-node, bifurcation, in which two steady states merge and annihilate as a slowly varying control passes a threshold; monostationarity excludes it directly, since the merging requires two states to begin with.

Nor does the argument require the system to be at rest. A system need not settle into a steady state for the number of steady states to matter, since the steady states and their stability determine which transitions are available to it. The results below should accordingly be read asymmetrically. A monostationarity result is an exclusion: under the stated kinetic-order conditions the model cannot support multiple positive steady states, and so cannot support a transition between them at all. A multistationarity result is a flag rather than a diagnosis: the necessary condition is met, and a dynamical study of the configuration is warranted before the strategy is relied upon. Establishing that a transition would actually occur requires stability, basin geometry, perturbation magnitude and timescale, none of which is determined here, and we retain the language of tipping-point risk only because it is the concern that makes the structural question worth asking.

Biochar sequestration, hereafter referred to as the BCS system, involves incorporating carbon-rich biochar into soils to enhance long-term carbon storage^[40]. Biochar is produced from biomass through pyrolysis, a process of burning organic material in an oxygen-limited environment, creating a stable, carbon-rich material that resists decomposition. In the RNCDR framework, biochar storage is represented by $$ A_{11} $$, with parameters $$ \lambda_{11} = 1 $$ and $$ \mu_{11} = 0.5 $$: all captured carbon enters long-term storage, half as inorganic biocharcoal in soil and half as organic biofuel in the geological stock.

The box model is shown in Figure 3. The BCS system involves five carbon pools: land biota ($$ A_1 $$), atmosphere ($$ A_2 $$), ocean ($$ A_3 $$), total carbon stock ($$ A_4 $$), and biochar storage ($$ A_{11} $$). Biomass is converted into biochar through pyrolysis ($$ A_1 \to A_{11} $$). When applied to soil, biochar stimulates plant growth and photosynthesis, facilitating further carbon capture ($$ A_{11} \to A_1 $$). Biochar may also undergo fossilization into the total carbon stock ($$ A_{11} \to A_4 $$), from which carbon can be re-emitted to the atmosphere through fossil fuel combustion ($$ A_4 \to A_2 $$). The CDR-specific reactions are:

\begin{align*} R_{11,6} &: A_{11} \to A_4 && \text{(biochar to total carbon stock)}, \\ R_{11,7} &: A_1 \to A_{11} && \text{(land-to-biochar capture)}, \\ R_{11,8} &: A_{11} \to A_1 && \text{(biochar-to-land transfer)}. \end{align*}

Together with the four Anderies reactions $$ R_1 $$–$$ R_4 $$ and the emission reaction $$ R_5 $$, this gives a BCS system with 5 species and 8 reactions. The dynamics of this system are encoded in the kinetic order matrix:

\begin{equation*} F_{BCS} = \begin{array}{c} A_1 \;\;\; A_2 \;\;\; A_3 \;\;\; A_4 \;\;\; A_{11} \\ \left[\begin{array}{[ccccc]} p_1 & q_1 & 0 & 0 & 0 \\ p_2 & q_2 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & e_{11} & f_{11} \\ 0 & 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ \end{array} \right] \end{array} \quad \begin{array}{l} \\ R_1 \\ R_2 \\ R_3 \\ R_4 \\ R_5 \\ R_{11,6} \\ R_{11,7} \\ R_{11,8} \end{array} \end{equation*}

The row for $$ R_5 $$ encodes the coupled dependence of emission on both the total carbon stock and biochar storage, with $$ e_{11} \geq 1 $$, $$ f_{11} \leq 0 $$, and $$ e_{11} + f_{11} = 1 $$. The rows for $$ R_{11,6} $$, $$ R_{11,7} $$, and $$ R_{11,8} $$ reflect mass-action kinetics for the biochar-specific transfers.

Ocean fertilization, hereafter referred to as the OF system, seeks to stimulate the growth of marine phytoplankton to enhance the oceanic uptake of CO₂^[41]. The process introduces nutrients such as iron to the ocean surface, inducing phytoplankton blooms that absorb atmospheric carbon into their biomass. When the phytoplankton die, their biomass sinks to the deep ocean where the carbon is stored for a long period of time. The efficacy of ocean fertilization is contested. Field experiments have generally found that only a small and highly variable fraction of the carbon fixed in a bloom reaches the deep ocean, and estimates of the cost per tonne removed span roughly two orders of magnitude, driven largely by uncertainty in that export efficiency^[41,51]. Deployment beyond legitimate scientific research has been restricted since 2008 by resolution of the parties to the London Convention and Protocol, an approach codified in a 2013 amendment to the London Protocol that has not yet entered into force^[41]. We retain it here as a modeling case rather than as an endorsement: it continues to appear in CDR portfolio inventories, and it is the analytically distinctive member of our set, being the only one of the four whose emission reaction reduces exactly to linear mass action. In the RNCDR framework, OF storage is represented by the separate pool $$ A_{12} $$, which holds carbon in the deep ocean and is distinct from the total carbon stock $$ A_4 $$. Carbon leaves $$ A_{12} $$ for $$ A_4 $$ only through the leak reaction $$ R_{12,6} $$, which represents fossilization on geological timescales. The parameters $$ \lambda_{12} = 1 $$ and $$ \mu_{12} = 0 $$ describe the disposition of that transferred carbon, namely that it is organic and therefore available for subsequent emission, rather than the physical location of the OF store itself. Since $$ \lambda_{12} = 1 $$ and $$ \mu_{12} = 0 $$, the emission kinetics of the OF system reduce to linear mass action: $$ K_5 = k_5 A_4 $$.

The box model is shown in Figure 4. The OF system involves five carbon pools: land biota ($$ A_1 $$), atmosphere ($$ A_2 $$), ocean ($$ A_3 $$), total carbon stock ($$ A_4 $$), and OF storage ($$ A_{12} $$). The OF process introduces an enhanced photosynthesis pathway from the atmosphere to the ocean, a transfer from the ocean to OF storage representing biomass sinking to the deep ocean ($$ A_3 \to A_{12} $$), and a reverse transfer from OF storage back to the ocean representing ocean water mixing ($$ A_{12} \to A_3 $$). The CDR-specific reactions are:

\begin{align*} R_{12,6} &: A_{12} \to A_4 && \text{(OF to total carbon stock)}, \\ R_{12,7} &: 2A_2 + A_3 \to A_2 + 2A_3 && \text{(atmosphere-to-ocean photosynthesis)}, \\ R_{12,8} &: 2A_3 \to A_3 + A_{12} && \text{(ocean-to-OF capture)}, \\ R_{12,9} &: A_3 + A_{12} \to 2A_3 && \text{(OF-to-ocean release)}. \end{align*}

Together with $$ R_1 $$–$$ R_4 $$ and $$ R_5 $$, this gives the OF system with 5 species and 9 reactions. The dynamics of the OF system are encoded in the kinetic order matrix:

\begin{equation*} F_{OF} = \begin{array}{c} A_1 \;\;\; A_2 \;\;\; A_3 \;\;\; A_4 \;\;\; A_{12} \\ \left[\begin{array}{[ccccc]} p_1 & q_1 & 0 & 0 & 0 \\ p_2 & q_2 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ 0 & p_3 & q_3 & 0 & 0 \\ 0 & 0 & p_4 & 0 & 0 \\ 0 & 0 & p_5 & 0 & q_5 \\ \end{array}\right] \end{array} \quad \begin{array}{l} \\ R_1 \\ R_2 \\ R_3 \\ R_4 \\ R_5 \\ R_{12,6} \\ R_{12,7} \\ R_{12,8} \\ R_{12,9} \end{array} \end{equation*}

The row for $$ R_5 $$ reflects linear mass action kinetics ($$ e_{12} = 1 $$, $$ f_{12} = 0 $$) since $$ \lambda_{12} = 1 $$ and $$ \mu_{12} = 0 $$. The rows for $$ R_{12,7} $$, $$R_{12,8} $$, and $$ R_{12,9} $$ introduce additional kinetic orders $$ p_3, q_3, p_4, p_5 $$, and $$ q_5 $$ that encode the influence of the atmosphere and ocean pools on the OF processes.

Soil carbon sequestration, hereafter referred to as the SCS system, aims to increase the organic carbon content of agricultural and natural soils through various land management practices^[42]. Carbon from the atmosphere is absorbed by plants through photosynthesis and stored in the soil as soil organic carbon. Practices such as cover cropping, crop rotation, and reduced tillage are used to increase the rate at which carbon is added to the soil while decreasing the rate at which it is lost through decomposition. In the RNCDR framework, SCS storage is represented by $$ A_{13} $$, with parameters $$ \lambda_{13} = 0.01 $$ and $$ \mu_{13} = 0 $$: only a very small fraction of sequestered carbon enters long-term geological storage, and this fraction is entirely organic.

The box model is shown in Figure 5. The SCS system involves five carbon pools: land biota ($$ A_1 $$), atmosphere ($$ A_2 $$), ocean ($$ A_3 $$), total carbon stock ($$ A_4 $$), and SCS storage ($$ A_{13} $$). Carbon decays from land biota into SCS storage ($$ A_1 \to A_{13} $$), from which it may enter the total carbon stock through fossilization ($$ A_{13} \to A_4 $$). From there, carbon can be re-emitted to the atmosphere through fossil fuel combustion ($$A_4 \to A_2 $$). The CDR-specific reactions are:

\begin{align*} R_{13,6} &: A_{13} \to A_4 && \text{(SCS to total carbon stock)}, \\ R_{13,7} &: A_1 \to A_{13} && \text{(land-to-SCS capture)}. \end{align*}

Together with $$ R_1 $$–$$ R_4 $$ and $$ R_5 $$, this gives the SCS system with 5 species and 7 reactions. The dynamics of the SCS system are encoded in the kinetic order matrix:

\begin{equation*} F_{SCS} = \begin{array}{c} A_1 \;\;\; A_2 \;\;\; A_3 \;\;\; A_4 \;\;\; A_{13} \\ \left[\begin{array}{[ccccc]} p_1 & q_1 & 0 & 0 & 0 \\ p_2 & q_2 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & e_{13} & f_{13} \\ 0 & 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 & 0 \\ \end{array}\right] \end{array} \quad \begin{array}{l} \\ R_1 \\ R_2 \\ R_3 \\ R_4 \\ R_5 \\ R_{13,6} \\ R_{13,7} \end{array} \end{equation*}

The row for $$ R_5 $$ encodes the coupled dependence of emission on both the total carbon stock and SCS storage, with $$ e_{13} \geq 1 $$, $$ f_{13} \leq 0 $$, and $$ e_{13} + f_{13} = 1 $$. The rows for $$ R_{13,6} $$ and $$ R_{13,7} $$ reflect mass-action kinetics for the SCS-specific transfers. As with the BCS system, the CDR storage species $$ A_{13} $$ does not appear in the rows corresponding to the natural carbon cycle reactions $$ R_1 $$–$$ R_4 $$, so the SCS-specific reactions extend the natural carbon cycle without directly modulating its rates.

Wetland restoration, hereafter referred to as the WR system, leverages the natural carbon sequestration capacity of wetland ecosystems^[43]. Wetlands are ecosystems where plants and soil are submerged in water; they can capture large amounts of CO₂ from the atmosphere and store it in soils for hundreds or even thousands of years. In the RNCDR framework, WR storage is represented by $$ A_{14} $$, with parameters $$ \lambda_{14} = 0.01 $$ and $$ \mu_{14} = 0.9 $$: only a very small fraction of sequestered carbon enters long-term geological storage, and most of this fraction is inorganic.

The box model is shown in Figure 6. The WR system involves five carbon pools: land biota ($$ A_1 $$), atmosphere ($$ A_2 $$), ocean ($$ A_3 $$), total carbon stock ($$ A_4 $$), and WR storage ($$ A_{14} $$). The WR system is the most complex of the four, with carbon transfers between the atmosphere and WR storage in both directions, mediated by respiration and photosynthesis processes. Carbon also transfers between land biota and WR storage in both directions ($$ A_1 \to A_{14} $$ and $$ A_{14} \to A_1 $$), reflecting the interaction between land biota and wetland ecosystems. The CDR-specific reactions are:

\begin{align*} R_{14,6} &: A_{14} \to A_4 && \text{(wetland to total carbon stock)}, \\ R_{14,7} &: A_2 + 2A_{14} \to 2A_2 + A_{14} && \text{(wetland respiration)}, \\ R_{14,8} &: 2A_2 + A_{14} \to A_2 + 2A_{14} && \text{(wetland photosynthesis)}, \\ R_{14,9} &: A_1 \to A_{14} && \text{(land-to-wetland capture)}, \\ R_{14,10} &: A_{14} \to A_1 && \text{(wetland-to-land transfer)}. \end{align*}

Together with $$ R_1 $$–$$ R_4 $$ and $$ R_5 $$, this gives the WR system with 5 species and 10 reactions. The dynamics of the WR system are encoded in the kinetic order matrix:

\begin{equation*} F_{WR} = \begin{array}{c} A_1 \;\;\; A_2 \;\;\; A_3 \;\;\; A_4 \;\;\; A_{14} \\ \left[\begin{array}{[ccccc]} p_1 & q_1 & 0 & 0 & 0 \\ p_2 & q_2 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & e_{14} & f_{14} \\ 0 & 0 & 0 & 0 & 1 \\ 0 & q_4 & 0 & 0 & p_4 \\ 0 & q_5 & 0 & 0 & p_5 \\ 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ \end{array}\right] \end{array} \quad \begin{array}{l} \\ R_1 \\ R_2 \\ R_3 \\ R_4 \\ R_5 \\ R_{14,6} \\ R_{14,7} \\ R_{14,8} \\ R_{14,9} \\ R_{14,10} \end{array} \end{equation*}

The WR system has the largest kinetic order matrix among the four systems, reflecting its greater complexity. The rows for $$ R_{14,7} $$ and $$ R_{14,8} $$ introduce additional kinetic orders $$ q_4, p_4, q_5 $$, and $$ p_5 $$ that encode the influence of both the atmosphere ($$ A_2 $$) and WR storage ($$ A_{14} $$) on the respiration and photosynthesis processes within the wetland ecosystem. This bidirectional coupling between $$ A_2 $$ and $$ A_{14} $$ is a structural feature unique to the WR system.

Table 4 summarizes the steady-state capacity of the four CDR systems by Anderies class. The results established in Section "CONDITIONS FOR MULTISTATIONARITY IN CDR REACTION NETWORKS" are sufficient conditions. They identify specific parameter combinations under which a system is guaranteed to contain multistationary or monostationary subsets. They do not preclude other behaviors under different parameter combinations. The Anderies class, defined by the signs of $$ p_1, p_2, q_1, q_2 $$, is a necessary first input for every entry in the table but not a complete one: each proposition additionally requires sign conditions on kinetic orders specific to the CDR technology itself, so no cell in Table 4 is determined by the Anderies class alone.

It bears emphasis that a system belonging to the positive class is not guaranteed to be multistationary for all parameter combinations; rather, there exist specific parameter regions within that class where multiple positive steady states can occur. Establishing whether a particular real-world deployment falls within such a region requires empirical estimation of the kinetic orders, a potential research direction discussed in Section "SUMMARY, CONCLUSION, AND FUTURE RESEARCH".

With this caveat in mind, the following observations can be made. The OF system stands alone in having multistationary subsets identified exclusively in the positive Anderies class. This pattern does not mean the Anderies class alone determines the outcome for OF: as for the other three systems, every case in Proposition 2 additionally requires sign conditions specific to the OF reactions themselves. The BCS and SCS systems share the same multistationarity profile: multistationary subsets have been identified in the positive class and also in both the P-null and Q-null classes, with monostationarity guaranteed only in the negative class. This shared profile follows from shared structural properties of the two networks. Both are deficiency-one networks in which the CDR-specific reactions extend the natural carbon cycle without directly modulating its rates. For both, the DOA identifies parameter conditions under which multistationarity arises in the positive, P-null, and Q-null classes. The WR system departs most significantly from the others: monostationary subsets have been identified in the positive class and multistationary subsets in the negative class, with the Q-null class containing both monostationary and multistationary subsets depending on the sign of $$ (q_4 - q_5)/(p_5 - p_4) $$.

These differences can be understood in terms of the structural features of each CDR system. Among the four, the OF system is the deficiency-two network in which the CDR-specific reactions do not directly modulate the rates of the natural carbon cycle reactions $$ R_1 $$–$$ R_4 $$, while adding a parallel atmosphere-to-ocean flux. This does not make the technology's own parameters irrelevant, however: whether the multistationarity flagged by the positive Anderies class is actually realized still depends on sign conditions specific to the OF reactions (Proposition 2). The BCS and SCS systems are both deficiency-one networks whose CDR-specific reactions also leave the natural carbon cycle reactions $$ R_1 $$–$$ R_4 $$ structurally unchanged. Applied to these networks, the DOA identifies parameter conditions under which multistationarity arises in the P-null and Q-null classes. Here the additional pathways to multistationarity come from network-level structural properties rather than from direct coupling between the CDR storage and the natural cycle. The WR system goes furthest in structural complexity. Bidirectional transfers between the atmosphere and WR storage introduce additional kinetic orders $$ q_4, p_4, q_5 $$ and $$ p_5 $$. These give rise to a new ratio $$ (q_4 - q_5)/(p_5 - p_4) $$, which governs the Q-null class behavior independently of the Anderies subnetwork class. This structural richness produces multistationarity behavior that is the most parameter-dependent among the four systems, as captured in Proposition 4.

The findings of this study have three important implications for CDR risk assessment and carbon neutrality planning.

First, the Anderies class of the natural carbon cycle should be treated as a prerequisite screening criterion in CDR risk assessment. Before evaluating any specific technology, policymakers and modelers could check whether the background carbon cycle is in a positive, negative, or null class. For the OF system, multistationary subsets were identified in the positive Anderies class alone. That determination is not by itself sufficient, however: Proposition 2 additionally requires sign conditions on the OF-specific kinetic orders $$ p_3, q_3, p_4, p_5, q_5 $$ and the condition $$ p_3\mu_{A_2} > M_3 $$, so the Anderies class is a necessary first input rather than a complete characterization. For the BCS, SCS, and WR systems, the Anderies class remains an important first input, but it does not by itself characterize tipping-point risk. BCS and SCS exhibit multistationary subsets across the positive, P-null, and Q-null classes, while WR exhibits a more nuanced pattern that includes a reversal between the positive and negative classes. Empirical estimates of $$ p_1, p_2, q_1, q_2 $$ from observed carbon flux data (even rough ones) therefore provide substantial risk-screening value at low computational cost, but should be paired with attention to CDR-specific structural features for the latter three systems.

Second, the four systems demonstrate that the network-level structure of a CDR system is itself risk-relevant, and that this structure can take qualitatively different forms. The OF system illustrates a deficiency-two case in which the CDR-specific reactions do not directly modulate the natural carbon cycle reactions; even so, the system's steady-state behavior is not organized by the Anderies class alone, since realizing the multistationarity it flags additionally requires sign conditions specific to the OF reactions (Proposition 2). The BCS and SCS systems illustrate the deficiency-one case in which the CDR-specific reactions extend the natural carbon cycle without modulating its rates, and the DOA identifies parameter conditions under which multistationarity arises in classes that would be monostationary at the Anderies level. The WR system illustrates the case in which bidirectional coupling between the storage pool and the atmosphere, encoded in the kinetic orders $$ q_4, p_4, q_5, p_5 $$, can actually suppress multistationarity in the positive class, a potentially counterintuitive protective effect. Taken together, these contrasts suggest that the structural design of a CDR technology, in particular its network deficiency and whether and how its storage pool feeds back into natural carbon cycle reactions, warrants attention as a risk-relevant design parameter in CDR evaluation frameworks.

Third, the parameter-minimal nature of the RNCDR framework means that the conclusions above hold without requiring precise knowledge of rate constants. The framework relies primarily on the graphical and kinetic structure of the network, rather than on specific parameter values, making it particularly well-suited to systems in which parameter uncertainty is high and exhaustive numerical simulation is impractical. As a practical example, the sign conditions on $$ p_1, p_2, q_1, q_2 $$ identified in Propositions 1–4 provide a concrete set of criteria that could guide the empirical estimation of kinetic orders from observed carbon flux data in future work. This approach complements existing numerical modeling studies by providing a systematic and tractable basis for assessing tipping-point risk across CDR strategies.

Table 5 places the RNCDR framework alongside the other families of methods used to assess CDR strategies. The framework is not an alternative to numerical simulation. It is a screen applied before simulation, and the asymmetry noted in Section "Multistationarity and tipping points: what this framework does and does not establish" defines its practical scope: a guaranteed monostationarity result excludes a configuration from further dynamical investigation, whereas a multistationarity result identifies where such investigation is warranted. Its computational cost is negligible by comparison, because the propositions follow from the structure of the network rather than from integrating the governing equations.

Several limits bear on how these results should be used. The conditions established in Section "CONDITIONS FOR MULTISTATIONARITY IN CDR REACTION NETWORKS" are sufficient rather than necessary. They identify parameter regions in which multistationarity or monostationarity is guaranteed, but they do not partition the parameter space, and behavior outside those regions is not determined by the present analysis. Nor does the existence of multiple positive steady states imply that more than one of them is an attractor. Establishing bistability would require a linearized stability analysis that we do not perform, which is the distinction drawn in Section "Multistationarity and tipping points: what this framework does and does not establish": multistationarity is a structural precondition for one tipping mechanism rather than a demonstration of tipping. The scope is also narrower than the word "tipping" suggests. We take a tipping point to be an abrupt transition from one positive steady state to another, and the results speak only to that. Abrupt behavior that ends somewhere else (e.g., in a limit cycle) lies outside the definition; so does rate-induced tipping in a system with a unique steady state. This is not a remote concern for the systems studied here. The compost-bomb instability, in which soil carbon is released explosively above a critical rate of warming, is a rate-induced phenomenon. It arises in a soil-carbon model whose steady state is unique and globally attracting^[48,56]. A monostationarity result for the SCS system would not exclude behavior of that kind.

The representation is also highly aggregated. It has five carbon pools, no spatial resolution and no explicit temperature state variable, so climate feedbacks enter only through the kinetic orders rather than as dynamical variables in their own right.

Two sets of parameters remain unconstrained by observation. The Anderies classification depends on $$ p_1, p_2, q_1 $$ and $$ q_2 $$, which have not been estimated from carbon flux data; the sign conditions in Propositions 1–4 therefore identify which qualitative behaviors are possible for this class of model rather than which occur in any particular deployment. The storage parameters in Table 3 are likewise nominal, chosen to span the parameter space rather than measured. Supplementary Section 2 sets out which propositions are sensitive to them.

Finally, the analysis covers single-technology deployments only; portfolios of simultaneously deployed strategies are not treated here, although the framework accommodates them. And a steady-state analysis is silent on time. It says nothing about how long a transition between states would take, which is precisely the quantity that determines whether monitoring, reporting and verification systems could detect one.