Abstract

Field-Structure Theory proceeds along the sequence of “cognition—formation—change” to examine how reality enters finite cognition, how structures form within fields, and how objects persist through change. A probability cloud represents the open possibilities present in finite cognition; the law field, state field, and agent field respectively describe transformation relations, the current configuration, and the agent component. Nodes and edges constitute structures, while cross-scale compatibility tests whether a higher-level description can omit detail yet preserve the relevant changes. An agent’s cognition, purposes, and feedback participate in the formation of paths through actual mechanisms. This paper further proposes a variational unification hypothesis constrained by the dynamical class, the form of the action, and boundary conditions, and uses a two-node model to demonstrate exact compatibility, perturbation error bounds, and the construction of a feedback action. The model’s domain of applicability is determined through predictive, intervention, and perturbation tests.

Keywords: Field-Structure Theory; probability cloud; three fields; structural formation; cross-scale compatibility; object identity; stationary action

Introduction: From Objects to Formation

Understanding the world often begins with identifying objects. Field-Structure Theory shifts attention toward the conditions under which objects become established:

How does an object form, and by virtue of what does it remain itself through change?

The paper’s basic position is that the field precedes the object in the order of explanation. To understand an object, one must explain the configuration, relations, boundaries, and constraints that support it, and thereby account for its formation, persistence, transformation, and dissolution.

Here, “precedes” refers to the starting point of explanation; it does not presuppose a temporal origin. Lower-level structures can support higher-level fields, and a whole at one scale can become a part at another. Fields, structures, and objects are organizational roles assumed by reality at different scales.

The paper moves from cognition to formation and then from formation to change: it first distinguishes descriptions of laws, states, and agents; it then examines the conditions under which structures and objects become established; and finally, it discusses actual paths and their variational representation.

I. Probability Clouds and the Three Fields

1. Probability Clouds and Fields

Human beings know reality through perception, measurement, language, and reasoning. With respect to states, relations, and explanations that have not yet been determined, cognition retains multiple possibilities. This paper calls such an open, local, and revisable representation of possibilities a probability cloud:

Here, $\mathfrak C_t$ is the probability cloud at time $t$; $A_t$ is the index set of questions and contexts that can currently be expressed; $\alpha$ labels one such context; $\mathcal H_{\alpha,t}$ is the set of possible states, relations, or explanations within that context; and $w_{\alpha,t}$ is a weighting rule on $\mathcal H_{\alpha,t}$.

When the probability conditions are satisfied, the weights constitute a probability distribution. When information is insufficient, the representation instead retains probability intervals, evidential weights, or possibility rankings, each updated according to its own rules. Different contexts need not share a common additive scale, but they must still be cross-checked against one another with respect to shared observational content.

Physical objects, relations, languages, and theories can all be represented within a probability cloud. Their modes of representation may be shared, while their modes of existence need not be the same. Change in reality, randomness in a process, and uncertainty in cognition are distinct issues and cannot substitute for one another.

A candidate explanation should state which observations it predicts and which results would weaken it. Evidence changes the weights assigned to possibilities, while new observations, concepts, and methods can expand $A_t$ and reorganize the space of possibilities. The probability cloud characterizes current cognition and its open boundary; it does not exhaustively represent a total distribution of reality.

This paper uses a field to describe the configuration, couplings, and conditions of transformation within a specified scope. The configuration and couplings at scale $\ell$ and time $t$ are written as a field state:

Here, $\Omega_\ell$ is the domain of the relevant degrees of freedom; $\phi_{\ell,t}$ is the configuration within that domain; and $\Lambda_{\ell,t}$ represents the couplings, dependencies, and constraints among the degrees of freedom. The transformations permitted between field states are described by the corresponding laws.

The domain may be a spatial, state-space, or relational range. A configuration may be represented degree by degree, or it may retain the necessary joint information. A whole is not presupposed to be an assemblage of independent local parts, and couplings are not restricted to pairwise relations.

In what follows, $\chi$ denotes the analytical context as a whole, including the scale, research scope, time interval, admissible perturbations, and observational metrics. Variables, intervention protocols, and error criteria are fixed before testing. Once the context is fixed, the corresponding subscripts may be omitted. A hat over a field symbol indicates a model or estimate; an unhatted symbol refers to the real conditions and structures being described.

Within a fixed context, each candidate explanation $\theta\in\mathcal H_{\alpha,t}$ supplies a transformation model $k_\theta$ and a compatible state history $r_\theta(\tau)$, where $\tau$ denotes time within the history. When models use different variables, correspondences between those variables and common observational content must first be established before the following decompositions are carried out.

2. The Law Field: Laws and Constraints of Transformation

The law field $\mathcal K$ describes the transformations permitted between configurations and the constraints on those transformations.

The law partition retains the transformation model from each candidate explanation:

Here, $\pi_{\mathcal K}$ is the projection that extracts the transformation model; the braced expression collects the candidate transformation models; and $\widehat{\mathcal K}_t$ is the law model selected for use. When the evidence is insufficient, multiple candidates and their weights may be retained.

This partition retains “how transformation occurs.” “Horizontal” here means moving across states and histories to identify transformation relations that hold under the relevant conditions. The projection specifies what is retained, whereas model selection depends on mechanisms and evidence. A revision of cognition does not directly imply that the laws themselves have changed.

Current connections, internal rules, and resource configurations belong to the state; how they operate and how they change are specified by the description of transformations. Rules that an agent can modify should be treated as variable states, and the conditions of applicability must be stated for any higher-level effective laws that depend on those rules.

3. The State Field: Current Configuration and Its Couplings

The state field $\mathcal R_t$ describes the overall configuration and its couplings within the research scope at time $t$. It records local states, current relations, and the necessary joint information.

The state partition fixes the same time and extracts the corresponding cross-section from each candidate history:

Here, $\pi_{\mathcal R,t}$ is the projection that extracts the state at time $t$; $r_\theta(t)$ is the state model supplied by the candidate explanation, including the domain, configuration, and couplings in equation (2); and $\widehat{\mathcal R}_t$ is the selected model of the overall state.

This partition retains “what state the system is in at this moment.” “Vertical” here means extracting a time slice from a history. Taking a cross-section at one time does not mean excluding motion: rates of change and relevant memory may still belong to the state.

Laws and states are drawn from the same candidate space. After they are partitioned, their pairings and uncertainties must be preserved so that mutually incompatible models are not spliced together.

The state field contains the agents and environment within the research scope, as well as their relations. Only influences that cross the boundary of the research scope count as external inputs to the overall description; what lies outside an agent does not necessarily lie outside that scope.

4. The Agent Field: A Partition Along the Agent Boundary

The agent field $\mathcal P_t$ is the agent component obtained by partitioning the state field $\mathcal R_t$ along the agent boundary. In the model, it is written as:

Here, $\widehat{\mathcal P}_t$ is the agent-field model; $B_t$ is the agent boundary adopted by the model; and $\operatorname{Cut}_{B_t}$ denotes the extraction of the agent’s states and internal relations along that boundary. The partition delimits a scope; it does not sever the connection between agent and environment.

The agent boundary must be supported by an actual organization that carries perception, evaluation, action, and feedback. Arbitrarily drawing a boundary around some region is not sufficient to establish agency.

The agent field contains the agent’s actual states, capacities, resources, and internal relations. Three aspects are especially relevant to the orientation of action.

Cognition of the self. This includes the agent’s judgments about its own states, capacities, resources, and limitations, as well as its expectations regarding its own changes. Such cognition helps determine which actions appear feasible and which costs appear bearable, but it may diverge from actual conditions.

Cognition of the environment. This includes the agent’s judgments about external conditions, other agents, resources, and constraints, together with its expectations regarding environmental change and feedback from action. Cognition of the environment belongs to the agent field; the environment itself remains outside the agent boundary.

The agent’s purposes. These are the states that the agent seeks to maintain, realize, change, or avoid, together with the priority relations among these orientations. Purposes may appear as explicit goals, or as persistent tendencies and preferences.

Cognition of self and environment jointly contributes to anticipation, while purposes contribute to selection. Actual action is also constrained by capacities, resources, habits, and the environment. Cognition need not be accurate or verbally articulable, and purposes need not always be objects of explicit reflection.

The agent field belongs to the state field; the three fields are not mutually independent domains at the same level.

II. Fields and Structural Formation

1. Stable Differences and Effective Relations

Within a given temporal and perturbational range, a persistently identifiable difference can serve as a node. A node must be distinguishable from its surroundings and must retain characteristics sufficient to track its state and role; its internal composition and local state may nevertheless change.

An edge represents a persistently identifiable interaction, dependency, or constraint between nodes. It may have direction, type, strength, and temporal scope, and it may also connect multiple nodes. Statistical correlation provides a clue for identification; assigning causal significance to an edge additionally requires support from a mechanism, a controlled comparison, or testable consequences of intervention.

Nodes and edges together constitute a structure:

Here, $\mathcal N_\ell$ is the structure at scale $\ell$, $V_\ell$ is the set of nodes, and $E_\ell$ is the set of relations. Node states, relational attributes, and any necessary joint constraints are retained with them; connectivity alone is generally insufficient to determine evolution.

A node’s boundary is supported by differences between internal and external relations, while the identification of relations in turn presupposes that a pattern can be tracked. Lower-level couplings connect a field’s degrees of freedom, whereas structural relations connect patterns identified within the field; the two belong to different levels. A transient interaction can contribute to an explanation as a state change or input without first becoming a stable node.

2. Structural Formation and Cross-Scale Compatibility

A generative explanation asks how a structure forms and persists; cross-scale compatibility tests whether a higher-level description can omit details while preserving the relevant changes.

In a deterministic model, lower-level evolution is written as:

Here, $x_t$ is the relevant state at scale $\ell$; $\Delta t$ is the time interval; $T_\ell^{\Delta t}$ is the evolution map; and $\xi_t$ is the input that crosses the boundary of the research scope during that interval, including external influences and perturbations, rather than random noise alone. Explicit time dependence must be incorporated into the state, the input, or a declared transition rule.

Given an initial state, couplings, boundaries, and inputs, a generative explanation must determine which differences persist and which are amplified, eliminated, or reorganized. A pattern may persist through internal updating and exchange across a boundary, or it may disintegrate; whether a new stable structure forms depends on the specific mechanism.

A coarse-graining map $\pi_\ell:\mathcal X_\ell\to\mathcal X_{\ell+1}$ is obtained by selecting macroscopically relevant features from the lower-level configuration and the identified structure. Here, $\mathcal X_\ell$ and $\mathcal X_{\ell+1}$ are the lower- and higher-level state spaces, respectively. The map must specify which states, relations, and global differences are retained and which information is omitted. It connects descriptions of the same process at different scales and is distinct from the partition of candidate explanations in Part I.

Whether a higher-level description is valid is determined by comparing two routes: first evolve the lower level and then map it to the higher level; or first map it to the higher level and then evolve it according to the higher-level law.

Let $T_{\ell+1}^{\mathrm{eff},\Delta t}$ be a candidate higher-level evolution and $D_\chi$ the predeclared range of lower-level states and inputs. Cross-scale compatibility requires:

Here, $(x,\xi)$ is a lower-level state–input pair; $\bar\xi$ is the higher-level input obtained according to a predetermined rule; $d_{\ell+1}$ measures differences between higher-level states; $\varepsilon\geq0$ is the admissible error; and the superscript $\mathrm{eff}$ denotes an effective description. A higher-level input must not make use of undeclared lower-level details or information that is unavailable at the time of prediction.

An effective coarse-graining must preserve the differences required by the observational task while genuinely omitting lower-level information. Mapping every state to the same label cannot perform the required distinctions; retaining the complete lower-level state and merely renaming it does not establish a new level of description either.

When $d_{\ell+1}$ is a metric, equation (8) also yields a necessary condition: different lower-level realizations corresponding to the same higher-level state and input must, after evolution, map to higher-level states no more than $2\varepsilon$ apart. Otherwise, a deterministic higher-level prediction cannot satisfy the error requirement for all of them simultaneously.

If omitted differences still push predictions beyond the admissible range, the partition must be adjusted, variables added, or memory retained. When a statistical description is used, the conditional distribution of lower-level realizations must also be specified, and the resulting distributions of successor states must be compared. If those distributions depend on history, the relevant memory must be retained as well. Coarse-grained molecular dynamics with state-dependent memory provides one class of examples (Lyu & Lei, 2023).

When a structure is identifiable as a whole, supports the foregoing relation of evolutionary compatibility, and imposes testable organizational constraints on local change, it can support a higher-level field:

Here, $\mathcal F_\ell$ denotes the lower-level field process over the interval under consideration, and $\mathcal F_{\ell+1}$ is the corresponding higher-level field. The first arrow denotes the actual formation and persistence of a pattern; through macroscopic variables, information reduction, and evolutionary compatibility, the second arrow establishes a higher-level description. The higher-level field belongs to the same process; the mapping establishes a description and does not create the process described.

Organizational constraints are identified through realizable controlled comparisons: when relevant external conditions and local initial conditions are comparable, one changes the organization of global connections, boundaries, or resources and tests whether the range of locally realizable changes is altered accordingly. Through its lower-level realization, a higher-level structure constrains, coordinates, or amplifies local changes.

The same field can support multiple effective partitions, and a higher-level field can continue to form structures. Each cross-level description must still satisfy the corresponding criteria. The appendix presents an exact coarse-graining and shows how its error is controlled when the conditions supporting it are perturbed.

3. Structural Criticality and Robustness

The way in which a component affects the whole can be examined through the response of the whole after that component is altered.

Let $C$ be a component of the structure $\mathcal N$, such as a node, an edge, or a local substructure; let $\Phi_\chi(\mathcal N)$ be a macroscopic characterization obtained under a fixed evolution and observation protocol, recording the overall state, function, and operational response. Define structural criticality as:

Here, $\kappa_\chi(C;\mathcal N)$ is the criticality of component $C$ to the whole; $\mathfrak I_\chi(C)$ is the set of permitted interventions that act directly on $C$; $\iota$ is a particular intervention; $\iota(\mathcal N)$ is the structure after intervention; $d_\chi$ measures the difference between macroscopic characterizations; and $\sup$ denotes the supremum over the permitted interventions.

Intervention strength, external conditions, observation time, and recovery rules must be fixed, and the comparison range must include both loss of function and organizational disintegration. Direct intervention is restricted to $C$, but indirect consequences may propagate along relations. Criticality measures an upper bound on impact, not the probability of occurrence. It depends on the component’s properties, relational position, and available substitutes, and combined effects cannot generally be obtained by simply adding separate effects.

Robustness is the capacity of a structure, within a specified perturbational range and observation period, either to preserve specified organizational conditions or to keep the relevant macroscopic changes within an admissible range. It must always answer three questions: Against what changes, which features are preserved, and for how long?

Structural robustness concerns whether the structure persists; model robustness concerns whether predictions remain accurate under measurement error, parameter variation, boundary perturbation, and reasonable changes in partition. Accurately predicting structural instability can itself be evidence of a model’s effectiveness.

One-step compatibility must still be tested over time. Let $e_n$ be the higher-level prediction error at step $n$, with an exact initial higher-level state so that $e_0=0$, and suppose the higher-level transition amplifies state differences by at most a factor $c$ at each step. Provided that the distance obeys the triangle inequality, both trajectories remain within the validated range, and their inputs are the same, equation (8) gives:

Here, $c\geq0$ is a uniform error-amplification factor and $j$ is the summation index. If $c<1$, the error is bounded above by $\varepsilon/(1-c)$; if $c=1$, the upper bound is $n\varepsilon$; and if $c>1$, the upper bound may grow rapidly, so a small one-step error is insufficient to guarantee long-term accuracy.

A model must determine the range within which these constants are valid and must test its predictions using states, inputs, and perturbations that were not used to build it. When a criterion is exceeded, the domain of failure should be stated explicitly. A model updated or reconstructed according to a predetermined rule must also be tested anew. Revisability is not a substitute for robustness.

III. Objects and the Direction of Change

1. Objects and Persistent Identity

At a given scale and under given criteria of identification, a structure in a field process that possesses a relative boundary and identifiable organizational relations can be identified as an object. Its boundary, organizational relations, and characteristic interactions jointly provide the basis for identification.

Persistent identity means that structures examined at different times remain the same real instance. It requires an actual evolutionary connection and specified conditions of organizational preservation. Composition, local relations, and modes of interaction may change; a change in any single function does not by itself determine that identity has ended.

Persistence permits change, but it also requires limits. Identification criteria should state which features may change and which changes imply the termination of identity. Features that must be strictly preserved are called structural invariants; features allowed to vary within limits must be assigned tolerances. The same structure can occur in different objects; similarity, common origin, or causal connection cannot, by itself, determine identity.

For a segment of history identified as belonging to the same object, let its path be $\gamma_O:I_O\to\mathcal X_O$, with $\gamma_O(t)=x_O(t)$. Here, $O$ labels the object, $I_O$ is the temporal domain over which its persistence is examined, $\mathcal X_O$ is the effective state space, and $x_O(t)$ is the object’s state at time $t$. The path records the history of states; its continuity and differentiability are specified by the particular model.

An interruption in observation does not amount to an interruption of the object. When the evidence is insufficient, uncertainty in the judgment should be retained. Branching, merging, and disintegration also require a distinction between object persistence and lineage inheritance: the continuation of a related process does not guarantee the persistence of the original object.

2. Object Motion and Agent Action

This paper understands motion as change in an object’s state over time, rather than restricting it to change in spatial position. Studying change at the level of the object does not imply that the object has only one degree of freedom or that it is isolated from the environment.

The transition form in equation (7) applies to objects as well. Let $T_O^{\Delta t}$ denote the object transition and $\eta_{O,t}$ the input acting on the object boundary during the corresponding time interval. The distinction between $\eta_{O,t}$ and $\xi_t$ lies in the boundary: the former is defined relative to the object, while the latter is defined relative to the entire research scope. If the object changes the environment’s subsequent response, the corresponding coupling laws or necessary environmental variables must be retained.

When the object is an agent, the relevant states and mechanisms of the agent field $\mathcal P_t$ are included in $x_O(t)$. Let $\widehat\gamma$ denote a candidate history formed by the agent on the basis of its cognition of itself and its environment. When evaluation can be scalarized, write it as $J_t[\widehat\gamma]$; $J_t$ is the path-evaluation functional at time $t$, that is, a rule assigning an evaluative value to each candidate history. When evaluation cannot be scalarized, the corresponding rules of evaluation and selection are retained.

Anticipation is not equivalent to feasibility: an agent may omit genuinely feasible paths or include infeasible ones among its options. Choice produces action under the current representation and evaluation; reappraisal changes the purpose, model, or mode of evaluation; inertia perpetuates existing states and mechanisms, including habits, commitments, and switching costs. Limited search, satisficing solutions, and established rules can all contribute to action; an evaluative extremum need not be reached on every occasion (Simon, 1955).

In a deterministic effective description, the agent mechanism is incorporated into the object dynamics:

Here, $\beta$ is the decision/control mechanism, and $a_t$ is the resulting control command or action policy for the interval. The superscript $\mathrm{ctrl}$ indicates that the action channel is retained explicitly, whereas $\mathrm{cl}$ indicates the closed-loop transition obtained after substituting the agent mechanism. The object state and transition rule must retain the information required to determine that mechanism and its updating. A closed-loop system still receives environmental inputs; it is not thereby an isolated system.

Purposes influence action through current goal representations, evaluative orientations, and actual mechanisms; future events themselves do not act directly on the present. Feedback uses the results of action to revise cognition, adjust purposes, or alter subsequent action. Evaluation, execution, and environmental response must be modeled separately before their relations are established. Obtaining an optimized trajectory likewise does not guarantee its stable realization under feedback (Tedrake, n.d.).

Agent action and physical change belong to the same actual history. An agent participates in forming its own path and can, through learning and feedback, alter the conditions under which subsequent paths are formed. Here, the direction of change means the order of transformations and the actual history that arise from a given state and set of conditions; it does not presuppose that all change obeys a single monotonic measure.

3. Stationary Action and Variational Unification

Dynamics explains how states transform; a variational representation examines the same process in terms of an entire path.

For models that admit a real-valued scalar formulation, introduce an action functional $\mathscr A_O:\Gamma_O\to\mathbb R$. Here, $\Gamma_O$ is the set of candidate paths for the object and any necessary extended variables, $\mathbb R$ is the set of real numbers, and $\mathscr A_O[\gamma]$ assigns an action value to the candidate path $\gamma$. A real-valued output neither limits the dimensionality of the state nor gives the action any subjective meaning of good or bad.

Candidate paths satisfy specified kinematic, boundary, and constraint conditions; they are not screened in advance by the complete dynamics that is to be derived. Let $\gamma_s$ be a family of neighboring paths satisfying the constraints, with $\gamma_0=\gamma$. For a differentiable functional, a path is stationary when its first variation vanishes in every admissible two-sided direction $h$:

Here, $s$ is a variational parameter that can take positive and negative values near zero; $h$ is the admissible tangent direction of the path family at $s=0$; and $\delta\mathscr A_O[\gamma;h]$ is the corresponding first variation of the action. The parameter $s$ indexes neighboring histories, whereas time $t$ indexes states along any one history.

Stationarity does not mean zero velocity, nor does it guarantee global optimality, dynamical stability, or a decrease of the action over time. Cases involving inequality constraints, discrete choices, and related situations require conditions suited to the feasible variations.

The classical Lagrangian form is:

where $t_0$ and $t_1$ are the initial and final times, $q(t)$ is the configuration coordinate, $\dot q(t)$ is its rate of change over time, $L$ is the Lagrangian, and $\gamma$ denotes the configuration history. Under appropriate smoothness and fixed-endpoint conditions, the stationarity condition yields the Euler–Lagrange equations of motion (Tong, n.d.).

When candidate paths are compared, the environmental setup and response laws must remain the same. If the environment responds to the object’s path, it must vary together with the candidate path or be represented by joint variables; it cannot be held fixed as a feedback record copied from the actual history.

This paper proposes a restricted variational unification hypothesis: for a predeclared class of effective dynamics, an action of a specified form can be constructed from the laws, structure, and boundary conditions according to a common rule, such that its stationary description reproduces the corresponding dynamics. In a deterministic model, the requirement is:

Here, $T_O$ is the object dynamics, including the specified inputs and environmental response rules; $\operatorname{Sol}_O(T_O)$ is the set of solution paths satisfying the specified conditions; $\operatorname{Stat}_{\Gamma_O}(\mathscr A_O)$ is the set of stationary paths within the candidate set; and $\rho_O$ maps paths of variational variables to paths of object states.

When the same variables are used, $\rho_O$ is the identity map. When configuration, auxiliary, or environmental variables are introduced, state reconstruction, projection, or the physical limit must be specified. The equality requires both that the dynamical solutions be recovered and that no additional paths violating the original dynamics be produced. The time, initial, boundary, and constraint conditions on the two sides must correspond.

For an agent, this relation should be established for the complete effective closed loop in equation (12). Cognition, purposes, inertia, execution, and feedback enter the dynamics through mechanisms, and the corresponding action must reproduce the path formed jointly by those mechanisms. The subjective evaluation $J_t$ and the action $\mathscr A_O$ are not presumed to be equal.

The construction must specify the dynamical class, admissible variables, form of the action, and mapping rule, and it must use the same rule to handle new initial states and admissible perturbations. Fitting each realized history separately, or merely embedding the equations of motion as constraints, cannot by itself provide a unified explanation. Its explanatory value must additionally take the form of a shared structure, information reduction, or a testable connection.

The conditions under which a conventional Lagrangian representation exists are studied by the corresponding inverse problem of the calculus of variations and must be interpreted together with the specified restrictions on variables and functional form (Bucataru & Constantinescu, 2010). Nonconservative processes can use extended variational forms, but concrete correspondences among variables, boundary conditions, and dynamics must still be established (Galley, 2013).

Stochastic models must additionally match the probability law over paths, or transition rules sufficient to determine that law; merely having the same set of possible paths is not enough. A quantum description must specify the relation among states, complex amplitudes, and observation probabilities; the outcome of a single observation cannot be identified directly with a classical stationary trajectory (Feynman, 1948).

The appendix constructs an action for a class of two-node feedback models. Any broader unification claim must be tested class by class: if dynamics within the declared class cannot satisfy the specified correspondence, the claim fails within that range. Even when a correspondence has been established, a particular model must still undergo predictive tests, perturbation tests, and cross-scale error tests.

Appendix: Compatibility, Perturbation, and Variational Representation in a Two-Node Model

This example begins with two already identified nodes and examines the relation between a higher-level description and a dynamical representation. It does not explain the spontaneous formation of structure.

A.1 Exact Cross-Scale Compatibility

Let the configurations of two nodes be $q_1(t)$ and $q_2(t)$. Taking the inertial coefficients to be unity, their evolution is:

Here, one dot and two dots denote the first and second derivatives with respect to time, respectively; $\omega>0$ is the natural frequency; $b\geq0$ is the coupling coefficient under unit inertia; and $u(t)$ is a continuous input acting on both nodes.

Let the overall mean be $y=(q_1+q_2)/2$ and the relative configuration be $z=(q_1-q_2)/2$. Adding and subtracting the two equations gives:

Taking $(q_1,q_2,\dot q_1,\dot q_2)$ as the lower-level state and $(y,\dot y)$ as the higher-level state allows $(z,\dot z)$ to be omitted while leaving the evolution of the mean fully determined. For these equations and the same input, equation (8) can be satisfied with $\varepsilon=0$.

If the higher-level description retains only $y$, states with the same mean but different rates of change of the mean may produce different successor states. Such a description therefore cannot, in general, preserve the same deterministic closure.

A.2 Parameter Perturbation and Finite-Time Error

Change the intrinsic coefficients of the two nodes to $\omega^2+\delta$ and $\omega^2-\delta$, respectively, where $\delta$ is a coefficient deviation with the same dimensions as $\omega^2$ and $|\delta|<\omega^2$; leave all other conditions unchanged. In the $y,z$ variables, the equations become:

When $\delta\neq0$, the mean again depends on the omitted relative configuration, and the original exact closure generally no longer holds.

Let $y_\delta(t)$ be the actual mean under the perturbation and $\widehat y(t)$ the prediction of the original model in equation (A2), and define $e_y(t)=y_\delta(t)-\widehat y(t)$. The two use the same preset input, initial mean, and initial rate of change of the mean, with $t=0$ as the initial time. Subtraction gives $\ddot e_y+\omega^2e_y=-\delta z$, with $e_y(0)=\dot e_y(0)=0$. Therefore:

Suppose that, within the declared range of initial values, inputs, and perturbations, one can guarantee $|z(t)|\leq Z$ throughout $0\leq t\leq H$, where $H$ is the observation horizon and $Z$ is a uniform upper bound on amplitude. Equation (A4) and its time derivative then give, over the same interval:

Although exact closure has been broken, the prediction errors in the mean and its rate of change can still be controlled over a finite period. The value of $Z$ must be supported by predeclared initial values, inputs, and model conditions; it cannot be selected solely from the observed trajectory after the fact. These error bounds assume the same preset input. If a different closed-loop environmental response is introduced, the error bounds must be rederived.

A.3 An Action Construction for Configuration Feedback

Return to the unperturbed system (A1) and let the input be generated by configuration feedback: $u(t)=-g(y(t)-y_\star)$. Here, $y_\star$ is a fixed target and $g\geq0$ is the feedback coefficient. Observation and execution are instantaneous and exact, with no time delay, dissipation, or additional internal state.

A corresponding action can be chosen as:

The subscript $\mathrm{ex}$ marks this example. The kinetic terms arise from the unit inertial coefficients of the two nodes. The quadratic $y$- and $z$-terms encode the intrinsic restoring dynamics, with $2bz^2$ supplying the coupling contribution; the final term encodes configuration feedback.

For admissible variations with fixed configuration endpoints, the Euler–Lagrange equations for $y$ and $z$ are, respectively, $\ddot y=-(\omega^2+g)y+gy_\star$ and $\ddot z=-(\omega^2+2b)z$. These exactly reproduce equation (A2) after the feedback is substituted. The original state path can be recovered through $q_1=y+z$, $q_2=y-z$, and their time derivatives.

When both sides use the same endpoint boundary conditions, the correspondence in equation (14) follows. When treating an initial-value problem, the equations of motion are first obtained variationally and the particular solution is then determined using the corresponding initial values; the problem is not replaced by an arbitrarily specified two-endpoint boundary-value problem.

The terms involving $z$ are separate from those involving $y$ in the action. Retaining only the $y$-terms therefore yields an effective action for the mean, consistent with the exact coarse-graining in equation (8). Changing the initial state does not require reconstructing the functional; changing the feedback mechanism does require its form to be checked again.

The equilibrium position of the closed-loop mean is $y_{\mathrm{eq}}=g y_\star/(\omega^2+g)$, where $y_{\mathrm{eq}}$ denotes the mean at static equilibrium. It is generally unequal to a nonzero target $y_\star$, and an undamped model will generally continue to oscillate. This example establishes a correspondence between feedback dynamics and a stationary-action representation; it does not guarantee target attainment or asymptotic stability, nor does it incorporate mechanisms for updating an agent’s cognition and purposes.

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Abstract

Conatus Theory asks why states of the world matter to a subject, and how their significance acquires the power to move it to act. This paper understands conatus as the subject’s immanent striving, through its effective powers, to maintain continuity and actualize itself. Through preservation and change, the subject continually becomes itself.

Conatus unfolds along two dimensions: continuity and actualization. Continuity enables the subject to remain itself through change; actualization brings its capacities, values, and possibilities into reality.

The paper proposes that every motive capable of stably organizing purposive action is rooted in the actual effects of relevant states on the subject’s continuity or actualization, or in the subject’s understanding of those effects. This claim requires independent support from specific accounts of how motives form. Motives can acquire relative autonomy, conflict with one another, and even harm the subject. Conatus grounds purposes; the success and justification of actions remain separate questions.

Keywords: Conatus Theory; conatus; subjectivity; continuity; actualization; purposive action

Introduction: Where Do Purposes Come From?

Every subject tends, in some way, toward something.

A subject may pursue survival, knowledge, others, or creation. These purposes differ, yet a more fundamental question remains: How can something become a subject's purpose?

A state can acquire the power to move a subject to act only if it is not neutral for that subject. Integrity differs from damage, possibility from closure, continuation from ending. Because these differences bear on the subject itself, the world ceases to be merely a collection of facts and begins to reveal what is at stake and where to turn.

The subject thus approaches, avoids, chooses, persists, and changes.

1. Conatus

Spinoza uses conatus to name a thing’s striving to persist through its own powers. Following this line of thought, the present paper offers a constructive account of subjects, motives, and purposes.

Effective powers are the capacities a subject can actually exercise to sustain itself, perceive differences, form and revise courses of action, build relationships, and respond to change. Striving names the enduring direction within these activities toward its own continuation and unfolding.

The subject’s identity is formed and sustained through these activities. Preservation accommodates change; development also requires continuity. Learning, repair, and adjustment can all be ways for the subject to continue becoming itself.

2. The Subject: The Bearer of Conatus

A subject is a relatively unified entity with an identifiable boundary, able to retain its identity through time and regulate its activity in relation to its own states.

A boundary distinguishes the subject from its environment. Continuity through time links states at different moments as states of the same subject. Endogenous regulation allows differences between states to leave effects and alter subsequent activity. These are necessary conditions adopted in this paper; they do not exhaust the conditions of subjectivity.

A subject may depend on external support. Endogenous regulation concerns how differences enter its own activity and give action a direction that can be sustained and revised.

For subjects capable of feeling, the formation of action can be represented by a typical pathway:

Feeling → Desire → Intention → Action

Feeling discloses a situation; desire takes a direction; intention arranges a path; action answers the world. These stages interact, and the consequences of action inform subsequent adjustments.

3. The Two Dimensions of Conatus

Conatus is an ongoing activity. To distinguish its two dimensions, the profile of a subject $\mathit{s}$ at time $\mathit{t}$ can be written as:

Here, $\mathit{C}_{\mathit{s}}(\mathit{t})$ describes the subject’s continuity up to time $\mathit{t}$, and $\mathit{A}_{\mathit{s}}(\mathit{t})$ describes its actualization at that time. $\mathit{P}_{\mathit{s}}(\mathit{t})$ presents the two dimensions together, without assuming a common measure or an aggregate score.

3.1 Continuity

Continuity is the temporal relation through which a subject retains its identity amid change.

This relation is sustained by the carrying forward of the key states and regulatory relations that maintain the subject. Its material composition, beliefs, capacities, and relationships may change; continuity requires that these changes remain changes of the same subject. A merely causal connection is insufficient: a work may carry forward its author’s influence without becoming the author.

Continuity must also be distinguished from the capacity to maintain it. When a subject takes a risk, what is threatened is the set of conditions under which it can continue as itself. Identity itself need not be represented as a gradually diminishing quantity.

In seeking life, I had to create death.

To explain how a subject continues is also to explain where that continuation ends. Within this framework, death means the irreversible loss of the conditions that allow the same subject to persist; its continuity ends there. Thoughts and works may continue to exert influence. What remains is the outcome of what the subject once actualized.

3.2 Actualization

Actualization is the extent to which a subject’s capacities are exercised and its values and commitments become real through action.

It can be examined through three overlapping aspects: capacity, value, and externalization. The actualization of capacities involves exercising and developing perception, learning, judgment, creativity, and the ability to form relationships. The actualization of values brings beliefs, commitments, and an understanding of one’s identity into practice. Externalization gives these activities forms beyond the subject’s boundary: works, ideas, and lasting effects.

Writing can exercise a capacity, express a value, and leave a work behind. A subject who writes to be remembered may also seek a form of externalization: leaving thoughts, values, or an expression of identity to those who come later. Sharing ideas, working in the service of a faith, and honoring commitments to others can all be forms of actualization.

The capacities, values, and commitments invoked in an explanation should have support independent of the action being explained. They may also be formed and revised through learning and practice. Simply naming the pursuit of a desire “actualization” does not explain the motive’s origin.

Actualization has several aspects. Exercising one capacity may restrict another; realizing one value may incur other costs. Actualization in a particular respect therefore need not amount to an increase in the subject’s actualization as a whole.

3.3 Tensions Between Continuity and Actualization

Continuity allows the subject to persist; actualization allows its capacities and values to unfold in action. The two support each other, yet may also come into tension.

An action may protect the conditions of continued existence while limiting expression and creation. It may also risk continued existence for something important to be realized. The subject’s judgments of both dimensions are shaped by circumstances, experience, and the time horizon under consideration.

Such tensions give conatus different directions in action. A subject may preserve its present state or secure a future through change. What it seeks to actualize must still be distinguished from what its action brings about.

4. From Conatus to Stakes, Motives, and Purposes

The effects of states on a subject’s continuity and actualization constitute what is at stake for it. These differences can acquire motivational force as they enter perception, learning, and judgment. The subject responds both to actual effects and to its understanding of possible effects. The two may diverge.

A motive is a disposition that impels action and organizes its direction. A purpose is the state toward which a motive is directed. A stable motive, as understood here, is one that persistently, or recurrently in relevant situations, organizes purposive action.

Conatus Theory makes the further claim:

This claim offers a unified explanation of the origins of motives. It goes beyond the conditional claim that a subject must exist in order to act. Its validity depends on independent accounts of the specific pathways through which motives form.

Such an account must explain how relevant differences acquire stable motivational force through perception, learning, relationships, and history. It must explain both why a particular state is pursued and why similar states do not acquire the same force.

The origin of a motive, its object, and the outcome of an action answer different questions. The origin explains how the motive forms; the object identifies what the subject pursues; the outcome identifies what actually happens. A subject can genuinely care about others, truth, or works of creation without treating them all as means of preserving itself.

Once formed, motives can acquire relatively autonomous objects and standards. Dependence in origin does not determine subsequent function. Motives may support one another, conflict, or even turn against the subject.

5. Conatus in Relationships and Among Subjects

A subject takes shape within relationships and an environment. Its continuation and actualization often depend on others, resources, language, and knowledge. Support may extend beyond its boundary, while the one that persists and unfolds remains this subject.

When others enter a subject’s identity, judgments, and future through enduring relationships, their circumstances can become matters of genuine concern. Boundaries remain; concern can cross them.

The directions of different subjects’ conatus may support one another or conflict over resources, relationships, and scope for action. These dependencies and constraints constitute the real circumstances of conatus.

6. How Is Self-Termination Possible?

Conatus can give rise to different motives without ensuring their lasting coordination. Self-termination thus presents a limiting case: how can a motive rooted in conatus come to be directed toward ending the subject’s continuity?

Self-harm, suicidal ideation, suicidal action, and self-sacrifice have different intentional structures. The emergence of suicidal ideation is distinct from acting on it. The following discussion concerns possible motivational pathways and does not constitute a clinical explanation.

The first pathway is misjudgment. A subject seeks to relieve suffering, regain control, or protect a value, while misjudging how the action will affect the conditions of continued existence and the scope for future actualization.

The second pathway is a narrowing of the perceived future. When the subject understands its circumstances as unbearable and unchangeable, visible alternatives may narrow. The motive to “end the present state” may then outweigh the motive to preserve other possible futures.

The third pathway is conflict between dimensions. When a subject judges that preserving continuity and realizing an important value are difficult to reconcile, it may sacrifice continuity for actualization. This describes a motivational orientation toward a particular realization at the cost of continued existence. Whether the value can be realized, whether the cost is necessary, and whether the action is justified remain separate questions.

These pathways may overlap and do not exhaust all possibilities. Recognizing a cost does not ensure an accurate understanding of every consequence or alternative. Conatus Theory here explains how motives form; that explanation does not confer moral justification on an action.

7. Scope and Testing

This paper focuses on motives that stably organize purposive action. Accidents, reflexes, and activity without a purposive structure lie outside its central scope. Continuity and actualization provide a conceptual distinction; specific mechanisms require support from research in the relevant fields.

An adequate explanation should first specify how the subject and the two dimensions are to be identified and assessed. It should then explain their relation to motive formation and yield expectations that distinguish it from other explanations. These grounds should remain meaningful before the actual outcome is known. Renaming an action “conatus” after the event provides no independent support.

If conditions of continuity and actualization specified in advance repeatedly fail to explain how motives form, the theory must be revised. Not having found a pathway does not establish that none exists; equally, that reservation cannot justify indefinitely adding assumptions. If a competing account explains more with fewer assumptions, the explanatory necessity of Conatus Theory is also weakened.

Conclusion

The subject carries itself forward through change and unfolds itself in action. Its capacities can grow, its values can be revised, and its concern can cross boundaries. Through these activities, its identity takes shape and endures.

Conatus is the verb of the subject’s existence.

References

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Declarations

Funding and data. This study received no specific funding and generated or analyzed no datasets.

Competing interests. The author has disclosed an affiliation with StructureIntelligence Inc. and declares no other competing interests relevant to this study.

Author contributions. Kinyu Li: conceptualization, methodology, and writing and revision of the manuscript.

Use of generative artificial intelligence. Generative AI systems assisted with translation and revisions to language, argument structure, formulas, and layout. The author is responsible for scholarly judgments and final approval of the manuscript.

Copyright and license. Copyright © 2026 Kinyu Li. This work is licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0).