A Grammar of Power - Repertoires and the Reproduction of Asymmetry

ENGLISH

A Grammar of Power

Repertoires and the Reproduction of Asymmetry

Wanhong Huang · huangwanhong@serendip.ngo

---

Abstract

Asymmetries of capacity arise continually in social life, and most of them dissolve. A few do not. Accounts that measure power by the resources an actor commands cannot say which will do which, and are embarrassed by the cases in which a thin administration outlasts an overwhelming military advantage, or a body with no coercive capacity outlives every regime that tolerates it. This paper locates the difference in what the exercise of an asymmetry does to the conditions of its own exercise: some acts consume the preconditions that made them possible, and others manufacture them. Power, on the account offered here, is the asymmetry that survives both its own exercise and the continual production of difference against which it must hold. The paper first surveys the formal languages available for describing social dynamics and finds that each locates variability at one level while fixing the level above it, so that none makes the grammar of a system into a moving object; operator algebra, which comes nearest to the requirement, is assessed and set aside for reasons given in detail. A structure of typed operational repertoires and the transitions they induce upon themselves is then constructed, and the mechanism restated within it. Two consequences follow. The Nietzschean thesis that power consists in its own maintenance is reconstructed without the will that carried it: self-maintenance is treated as constitutive of the relation rather than as something any arrangement pursues, and the rarity of maintenance, which Nietzsche read as weakness, is explained instead by the continual production of the difference against which any arrangement must hold. And generativity is shown not to reduce to the number of possibilities a configuration admits: constitutional prohibition contracts the field of available action and remains generative, while the collapse of norms expands it and does not. What distinguishes the two is whose possibilities are enlarged, and whether what is generated returns to the relation that generated it. The argument for this is given by counterexample rather than by a general impossibility result, and the paper is explicit about which of its claims are established and which are proposed.

Keywords: power; asymmetry; relational generation; repertoire; self-reproduction; generativity; formal modelling of social dynamics.

§1 Introduction

Power has been understood relationally for some decades now. It is not a substance an actor holds in reserve, on this understanding, but something that exists only in the relation between parties and only in its exercise; it is dispersed rather than concentrated at a centre, productive of subjects and categories rather than merely repressive of them, and immanent to the practices through which social life is conducted rather than imposed upon them from outside. [The formulations here follow Foucault [8, 7] most closely. Arendt [1] arrives at a relational account by a different route, locating power in concerted action rather than in any party’s capacity to compel, and Emerson [6] makes the relational commitment operational through dependence, a line developed in the network literature surveyed by Wasserman and Faust [25]. The contrast with the older capacity-based formulations of Weber [26] and Dahl [4] is what the relational turn consists in; Lukes [12] maps the resulting debates.] The turn was salutary and is not in question here. What is in question is that the relational account has not been given a formal language adequate to its own ontology, at least in the treatments that have shaped the field. Where power is modelled quantitatively it is generally modelled as a quantity after all, whether a resource endowment, a network centrality, or a bargaining weight, which returns in the formalism precisely what the relational turn removed in the theory. The gap is not a failure of rigour on either side. It reflects the absence of a mathematical language in which a relation, rather than a magnitude, can be the object that moves.

The difficulty this paper takes up can be seen without any formalism at all, in cases where asymmetries of very different magnitude meet very different fates. Consider first a state that has achieved overwhelming military superiority over its neighbours and its own population, and a colonial administration that governs a territory of many millions with a few thousand officials and a garrison too small to hold any province against a determined rising. By any measure of resources commanded, the first asymmetry is the greater by a wide margin. Yet military supremacy of this kind has repeatedly proved among the most perishable of advantages, exhausting the fiscal base that sustains it and provoking the coalitions that end it, while thin colonial administrations have in several instances persisted for the better part of a century. Consider next a firm holding a dominant share of its market, and a professional body that certifies practitioners and can compel no one to do anything. The first commands capital, distribution, and the capacity to price rivals out; the second commands only the recognition of a credential. Dominant market shares have proved unstable on a timescale of decades. Certifying bodies have in several cases survived the regimes, constitutions, and economic orders under which they were founded.

An account that measures power by resources predicts the opposite in each pair. The natural repair is to say that resources are not everything and that legitimacy, institutionalisation, or embeddedness must be added to the account. This is true and insufficient. It names further quantities to be added to the ledger without saying why the ledger is the wrong instrument, and it leaves untouched the feature the cases actually share, which is not the magnitude of anything but a relation between an act and its own preconditions. Overwhelming military superiority is exercised by campaigning, and campaigning consumes the fiscal and demographic base from which superiority was built; the exercise of the advantage subtracts from the conditions of the advantage. A credential is exercised by being required, and each occasion on which it is required confirms that the distinction it marks is a real one; the exercise of the advantage adds to the conditions of the advantage. The two cases differ not in how much is held but in the sign of what the holding does to itself when used.

This suggests the problem the paper takes as its own. Asymmetries of capacity arise continually wherever relations are produced, and the overwhelming majority of them dissolve without leaving institutional trace. A small minority do not dissolve. If what distinguishes the minority is that the exercise of the asymmetry reproduces rather than consumes the conditions of its exercise, then power is not a magnitude to be measured but a property of the relation between an operation and the field of operations that remains available after it. Stating that property precisely requires a language in which the field of available operations is itself the thing that changes, and this is what the available formal languages do not supply. Differential and stochastic models fix a state space and move a point within it. Hybrid models add jumps but paste the jump rule on from outside. Agent-based models let a topology rewire while fixing the space of relations that may be instantiated. Game-theoretic models represent rule-following exactly and receive the rules as exogenous input. Rewriting systems generate genuinely, and generate syntax rather than relations among parties who bear differing capacities. Operator algebra comes nearest of all, taking composable operations as primitive and deriving the trajectory from them rather than the reverse, and it is fixed in its own way: its generators and relations are given in advance, as a differential model’s coordinates are given in advance. Each language locates variability at one level and holds fixed the level above. None makes the grammar itself the moving object.

The account developed here does. Its central construction is an operational repertoire, understood as a typed structure of partially composable operations, together with the transitions that operations induce upon the repertoire from which they were drawn. An asymmetry, on this construction, is a power relation when it is returned both by the transition its own exercise induces and by composition with the transitions the field supplies meanwhile. The first is closure and states the constitutive claim; the second is maintenance and states what closure must withstand. Power is maintenance, and the distinction between the two is what allows the account to record an arrangement that reproduces itself perfectly and is undone by what arises elsewhere.

Three consequences are drawn, and it is worth stating them here rather than reserving them. The first concerns a thesis that descends from Nietzsche and recurs, stripped of its original modality, in systems theory and elsewhere: that power consists in its own maintenance. The thesis is right and the modality is the difficulty. Nietzsche has power extending and discharging itself, with persistence following as a consequence; later formulations retain the self-maintenance and drop the will, but say little about why maintenance succeeds when it does. The reading offered here is that self-maintenance is constitutive rather than purposive. An asymmetry exists only in exercise, and exercise alters the conditions of exercise, so the relation between an asymmetry and its own continuation is definitional rather than causal, and power is the case in which that relation closes. What the present framework adds is an account of why the closure is rare. Because relation is continually produced, ordered difference arises constantly and cheaply, and the field does not stand still beneath any arrangement. Maintenance is therefore work performed against a productive background rather than persistence in a static one, and asymmetry is insufficient for power because asymmetry is abundant.

The second consequence is a negative result, and it is the one the paper would most want defended. It is tempting to identify generativity with the enlargement of possibility and degeneration with its restriction, so that a configuration is generative in proportion to the options it admits. The identification fails in both directions. A constitutional prohibition removes operations from the field and is among the most generative arrangements known; the collapse of shared norms enlarges the field taken as a whole, since restraints that formerly held cease to hold, and is degenerative. What the second case conceals, and §8 sets out, is that the enlargement accrues to those already positioned to act while the capacities of everyone else contract. Generativity therefore reduces neither to the number of available options, nor to entropy, nor to freedom in any sense that counts. What separates the two is whose possibilities are enlarged, and whether what the arrangement generates returns to the relation that generated it or is diverted to an end the relation did not require.

The third is a schema. Durable asymmetries, the paper argues, are assembled from four operations working across distinct registers: a differentiation that produces variation, a selection that ranks the variation so that one side’s difference confers what the other’s does not, a normalisation that renders the ranking unremarkable, and a reproduction that transmits it. The registers are distinct in kind, being severally a distribution, an ordering, a meaning, and an inheritance, and the composite crosses between them rather than remaining within any one. The schema is offered as general rather than illustrative, and the argument for the necessity of each register is given case by case. Its most useful consequence concerns intervention: normalisation is the only register with no material substrate, and an asymmetry deprived of it does not cease but must be continually re-imposed, which is a different and more expensive thing than being reproduced.

The paper proceeds as follows. §2 surveys the available formal languages along a common axis, that of what each takes as primitive and what it thereby holds fixed, and derives from the survey the requirements any adequate language must meet. §3 assesses operator algebra against those requirements, finds that it meets them and cannot serve nonetheless, and gives the four obstructions in detail. §4 states the mechanism philosophically and without formal apparatus. §5 constructs the formal object, and §6 restates the mechanism within it. §7 applies three distinctions borrowed from operator algebra to the constructed object, which is where they can do work they could not have done earlier. §8 sets out the refutation of the counting criterion. §9 generalises to the schema of durable asymmetry. §10 states what the framework forgoes and what it leaves open.

§2 Survey of Existing Formal Languages

2.1 Criteria of Comparison

The languages surveyed here were built for different purposes and are not in competition. Comparing them requires an axis on which each can be placed without distortion. The axis used throughout is the pair consisting of the language’s primitive object and its standing commitment: what the formalism takes as given, and what it thereby holds fixed. Every formalism must hold something still in order to let something else move, so the presence of a commitment is not itself a defect. What matters is where the commitment falls relative to the phenomenon at hand.

A useful diagnostic runs through the whole survey. In each canonical form below, some symbols carry a time index and others do not. Those that carry one are what the language permits to move; those that do not are the commitment. Reading the subscripts is sufficient to locate each language on the axis.

2.2 Differential and Stochastic Languages

Ordinary and partial differential equations, together with their stochastic and delayed variants, take a located point as primitive and evolve it under a law of motion. The canonical form is

$$\dot{x}(t) = f\bigl(x(t)\bigr), \qquad x(t) \in M,$$

with the state space $M$ and the vector field $f$ both given in advance. Only $x$ carries the index. Within that commitment these languages are powerful: they express continuous quantitative change, yield equilibria and their stability, and supply rates, which no other language on this list provides with comparable ease.

Composition lies outside what the equation above can express. A trajectory records the positions a system has occupied, not the operations performed upon it or the order of their performance, and the two are not interchangeable. Let $\Phi_a$ and $\Phi_b$ denote the time-one flow maps of two distinct vector fields, representing two social operations. The inequality

$$\Phi_a \circ \Phi_b \neq \Phi_b \circ \Phi_a$$

does hold in general, so order-dependence is present. The inequality holds, however, as a consequence of vector fields the modeller supplied, not as a structural feature the language represents. Nothing in the equation distinguishes an operation from any other displacement, and admissibility is not a structural feature of the language. One may of course restrict the domain of a flow map and thereby forbid a composition, and models routinely do so; the restriction is a stipulation added to the equation rather than something the form expresses, and it does not change as the system evolves. The problem of §1 is stated in exactly these terms. An act that consumes its own preconditions is an act after which certain further acts are unavailable, which is a claim about admissibility rather than about position.

A natural response brings the rules inside the state space. If an institutional arrangement governs which acts are available, one may write $M’ = M \times R$, with $R$ a space of arrangements, and evolve the pair. This is always possible and it does not help. The enlarged space $M’$ has no natural coordinates on the $R$ factor, an arrangement not being a quantity; the dimension of $M’$ is not fixed, since arrangements bring distinctions into existence and retire others; and the enlargement discards the compositional structure that motivated it, leaving a point moving through a space whose axes cannot be named. The manoeuvre secures a formal representation at the cost of the explanation the representation was to supply.

2.3 Hybrid and Piecewise Languages

Hybrid systems extend the differential languages by admitting discontinuous transitions [24]. Leaky integrate-and-fire models, hybrid automata, impact mechanics and piecewise-smooth systems all take the form of a tuple

$$H = \bigl(M,, f,, G,, R\bigr),$$

in which the state flows according to $\dot{x} = f(x)$ while $x \notin G$, and upon reaching the guard set $G$ is assigned a new value by the reset map $R : G \to M$. The extension is genuine and handles a large class of phenomena that pure differential models handle badly.

Its commitment is easily missed. Of the four components of the tuple, none carries a time index. The reset map $R$ is stipulated alongside the flow rather than generated by it, so a hybrid model contains a part the mathematics governs and a part the modeller supplies, and the supplied part does not itself change. The state space is likewise fixed across the jump: $R$ maps into $M$, so the trajectory arrives somewhere else within the same space rather than somewhere the space did not previously contain. It is worth contrasting the alternative treatment in which no discontinuity appears at all, the steep but smooth conductance dynamics of Hodgkin and Huxley [10] being the canonical instance; there the jump is an artefact of timescale rather than a feature of the model. Hybrid languages capture interruption without capturing structural change. For the present problem this is the wrong extension. The difficulty is not that social change is discontinuous, which hybrid models would accommodate, but that the exercise of an asymmetry alters which exercises remain possible, which is a change in $R$ rather than a jump under it.

2.4 Agent-Based and Network Languages

Agent-based and network models take the agent and the edge as primitive and permit the topology to change [25]. Writing $V$ for the set of agents and $T$ for the set of relation types the model can represent, a configuration at time $t$ is a graph

$$G_t = \bigl(V,, E_t\bigr), \qquad E_t \subseteq V \times V \times T,$$

evolved by a rewiring rule $\rho : G_t \mapsto G_{t+1}$. This is the first language on the list in which something structural moves, and it earns its place for that reason. Models of this kind capture emergent macro-patterns arising from local interaction with a fidelity the aggregate languages cannot match.

The subscript pattern, shown in Figure 1, locates the commitment precisely. The edge set $E_t$ carries an index and the type set $T$ does not, and neither does $\rho$. A network model may therefore create an edge $(u, v, \tau)$ that did not previously exist, but $\tau$ must already belong to $T$: the new edge instantiates a relation the model already knew how to represent. A relation of a kind not in $T$ cannot come into existence.

Figure 1 (The standing commitment of network languages). The rewiring rule $\rho$ may add an edge of a type already in $T$, shown at centre. A relation $\sigma$ of a kind not in $T$, shown at right, cannot be brought into existence, because $T$ carries no time index.

This distinction is not pedantic in the present context. When a certifying body is founded, the change is not that new links appear among existing nodes. A relation which previously had no representation, being certified by and being disqualified under, becomes available, and with it facts that were formerly not false but unformulable. The requirement is that $T$ carry the index rather than $E$, and no network formalism supplies that.

2.5 Game-Theoretic and Mechanism-Design Languages

Games take the strategy set and the payoff function as primitive [20]:

$$\Gamma = \bigl(N,, {S_i}{i \in N},, {u_i}{i \in N}\bigr).$$

These are the languages closest in spirit to the present problem, since they alone represent parties who anticipate one another’s responses when choosing what to do. Rule-governed transformation is available elsewhere on this list; strategic anticipation is not. Extensive-form games represent sequence, so a limited compositional structure is available, and mechanism design goes further by treating the rules as objects of choice rather than as fixed background.

The commitment is the specification $\Gamma$ itself, which enters as exogenous input. Within the intended scope this is not a defect, a mechanism designer being external to the mechanism by construction, but it locates rule-change outside the system rather than within it. The standard extension models rule-change as a meta-game

$$\Gamma’ = \bigl(N,, {\Sigma_i}{i \in N},, {u_i’}{i \in N}\bigr), \qquad \Sigma_i \subseteq {\Gamma_1, \Gamma_2, \ldots},$$

whose players choose among institutional arrangements.

Figure 2 (The meta-game regress). Each level treats the specification of the level below as an object of choice while fixing its own, so the commitment is relocated at every step and discharged at none.

The manoeuvre relocates the commitment without discharging it, as Figure 2 records: the specification is itself fixed, and iterating produces $\Gamma’’$ with the same property. No level of the regress is itself endogenous. The present problem requires that the alteration of the available operations be performed by those same operations, which the separation between player and designer forbids.

2.6 Rewriting Systems and Process Calculi

Rewriting systems take the production rule as primitive and are the nearest relatives of what this paper proposes. Formal grammars, term and graph rewriting [22], Petri nets [21] and process calculi are all specified by a set of productions

$$\mathcal{R} = \bigl{, \ell_k \rightarrow r_k ,\bigr}_{k \in K},$$

under which a configuration $c$ transforms to $c’$ whenever some $\ell_k$ matches a subterm of $c$. These languages generate rather than evolve: a configuration is a structured object rather than a point, and a rule transforms it into another. They carry a native notion of well-formedness, so not every sequence of transformations is admissible, and composition is therefore partial rather than total. All three of the requirements derived below are satisfied by rewriting systems, and any account of the present kind owes an explanation of why it does not simply use them.

The explanation lies in the anonymity of the production set. A grammar rewrites strings, a graph rewriting system rewrites graphs, a process calculus rewrites process terms, and in each case the object transformed is syntactic while the rules apply wherever a pattern matches. Take the phenomenon of a constitutional prohibition, which removes an operation, say the suspension of an election, from what may be done. A graph rewriting system represents this as the deletion of a production from $\mathcal{R}$, and the representation is faithful as far as it goes. It leaves unstated the fact that makes the prohibition interesting: the operation was removed from one party while the possibilities of every other party were thereby enlarged. The rule set contracted, but a rule set belongs to no one, and the question of whose capacity was reduced and whose extended has no formulation in a language whose rules are anonymous.

Figure 3 (Deletion of a production under the two presentations). On the left the rule set contracts and the question of whose capacity was reduced has no formulation. On the right the same deletion is attributed, and the enlargement of every other party’s relative position becomes stateable.

Further encoding does not relieve this. One may attribute rules to parties by annotation, replacing the anonymous set with

$$\mathcal{R}^{\ast} = \bigl{, \ell_k \xrightarrow{;p_k;} r_k ,\bigr}_{k \in K}, \qquad p_k \in P,$$

where $P$ is a set of parties and $p_k$ names the party permitted to fire the production. The annotation is available and it is exactly the point. What this adds is a structure in which operations are indexed by those who bear them, and that structure, rather than the rewriting, is where the content of the account lies. A system so annotated is on the way to becoming the object constructed in §5, and the difference between them is which structure is treated as primary. The account here treats party-indexed availability as the primary object and rewriting as what it undergoes. Nothing prevents a rewriting-theoretic presentation of the same content; the claim is that the content does not reside in the rewriting.

2.7 Operator-Algebraic Languages

Operator algebras take the composable operation as primitive [3]. A $C^*$-algebra may be specified by generators and relations,

$$\mathcal{A} = \bigl\langle, G \mid R ,\bigr\rangle,$$

and a dynamics upon it is a one-parameter family of automorphisms

$$\alpha_t \in \operatorname{Aut}(\mathcal{A}), \qquad \alpha_s \circ \alpha_t = \alpha_{s+t}.$$

A trajectory, where one is wanted, is derived from the algebra rather than the reverse. The formalism is coordinate-free in the sense that it does not require the modeller to name variables, and non-commutativity is native, so the order-dependence that differential languages receive as input is here a structural feature of the object. These languages further distinguish, with a precision no other formalism on this list matches, the symmetries of a dynamics from those of the observables and from those of a state.

The commitment is visible in the two displays taken together. The index sits on $\alpha$, not on $G$ or $R$. Passing from one representation to another, as the standard construction does by associating a representation to each state, leaves $\langle G \mid R \rangle$ untouched: it exhibits a single algebra realised in different ways, which is variability of realisation rather than of the algebra. Operator algebra fixes generators and relations in the way a differential model fixes coordinates, and both are dynamics upon a fixed structure. Because operator algebra comes nearest of any established language to what is required here, and because it is the language most likely to be reached for by anyone approaching this problem, the reasons it cannot serve are set out separately and at length in §3.

2.8 Derivation of the Requirements

The table below records the survey and Figure 4 displays its common shape. The pattern is consistent across the six entries: each language admits variability at one level and holds fixed the level immediately above. Differential models move a point and fix the space; hybrid models move a point discontinuously and fix the reset map; network models move an edge set and fix the type set; games move strategies and fix the specification; rewriting systems move configurations and fix the rule set, whose rules bear no parties; operator algebras move representations and fix the presentation. In no case does the grammar of the system, meaning what may be done, by whom, and in what combinations, become the object that moves.

Figure 4 (The common pattern of the survey). In every language the commitment sits one level above whatever is permitted to vary, so the grammar of the system is at no point the object that moves.

Language	Primitive	Held fixed	Unavailable for the present problem
Differential, stochastic	Point $x(t) \in M$	$M$ and $f$	Composition; admissibility cannot be posed
Hybrid, piecewise	Point with reset $R$	$M$, $G$, $R$	Reset stipulated, not generated
Agent-based, network	Edge in $V \times V \times T$	Type set $T$; rule $\rho$	New instances only; $T$ carries no index
Game-theoretic, mechanism design	Strategy set, payoff	Specification $\Gamma$	Endogenous rule change; the meta-game regress
Rewriting, process calculi	Production $\ell \rightarrow r$	Rule set $\mathcal{R}$; syntactic objects	Anonymous rules; whose capacity changed is unstateable
Operator-algebraic	Composable operation	Generators $G$, relations $R$	A fixed presentation is a fixed grammar

Three requirements follow. The first is that the primitive be a composable operation rather than a located point, the phenomenon being stated in terms of what may follow what. The second is that composition be order-sensitive natively rather than by stipulation, since the sequence in which operations are performed determines what remains available. The third is that composition be partial, since not every operation may follow every other, and the boundary between the admissible and the inadmissible is itself subject to change.

A fourth requirement is implied by the survey and met by no entry in it, and it is the one that motivates the construction of §5. Operations must be borne by parties. An operation available to one party and not to another is the elementary form of the asymmetry under study, and a language whose operations belong to no one cannot state the difference between an arrangement enlarging what one party may do and one enlarging what another may do. The contrast between the anonymous and the party-indexed production sets is the contrast at issue, and this fourth requirement is what separates the present account from rewriting systems, which satisfy the first three.

§3 Assessment of the Operator-Algebraic Candidate

3.1 Satisfaction of the Requirements

Of the languages surveyed, operator algebra alone satisfies the first three requirements without amendment. Its primitive is a composable operation. Composition is order-sensitive natively, non-commutativity being the structural feature from which the subject takes its interest rather than a property added to it. And a $C^*$-algebra generated by a specified set of relations admits partiality in the sense that matters here: which products are non-zero, which operations annihilate one another, and which compositions are constrained is determined by the relations rather than stipulated case by case.

The temptation to proceed on this basis is considerable, and the reasons are worth stating plainly rather than dismissing. The precedent is the algebraic reformulation of quantum field theory due to Haag and Kastler [9], in which the algebra of observables rather than any particular Hilbert space is taken as the primary object, and inequivalent representations are read as distinct phases rather than as competing descriptions. Operator algebra carries a developed theory of representation, so that one structure may be realised in many ways; a theory of inequivalence, so that realisations may be distinguished as belonging to genuinely separate sectors; a theory of subalgebras and the maps that project onto them; and a precise separation of the symmetries belonging to a dynamics, to the observables, and to a state. Each of these has an evident bearing on the questions of this paper, and §7 returns to three of them and puts them to use. What follows here is not an argument that operator algebra has nothing to offer the study of power. It is an argument that its axioms cannot carry the object this paper requires, and that the difference between borrowing a question and adopting a formalism is one the account must observe rather than blur.

Four obstructions are given. They are independent, and any one of them would suffice.

3.2 Obstruction from the Involution

A $C^*$-algebra is equipped with an involution, a map $a \mapsto a^*$ satisfying $(a^*)^* = a$ and reversing the order of products, $(ab)^* = b^a^$. The involution is not a decoration attached to an algebra that would otherwise be complete without it. It is what makes the notions of self-adjointness, positivity, and state available, and these in turn are what connect the algebra to anything that could be called an observable or a probability.

The involution admits no plausible social reading. Let $a$ denote the suppression of a publication. The involution is not the inverse: suppression cannot be undone, and the algebra in any case does not require $a$ to be invertible. Nor is it the adjoint in the sense of some inner product on social states, since no such product has been defined and defining one would import precisely the quantitative structure the relational account was meant to avoid. Nor is it a reversal of the parties, exchanging who suppresses and who is suppressed, since that operation does not satisfy the required identities and in most cases does not exist at all. The candidates are exhausted without a plausible reading having been found.

The point generalises beyond a single unfortunate example. A great many social operations are irreversible in a strong sense: they alter what is available afterward in ways that no subsequent operation restores. An operation of this kind has no natural partner under an order-reversing involution because the structure that would pair it with another has been dissolved by its own performance. One may of course impose an involution formally, declaring $a^* := a$ for every generator and verifying that the axioms hold. This yields a well-defined algebra and no interpretation: the resulting self-adjointness of every element would assert, if it asserted anything, that every social operation is an observable, which is false in an obvious way and useless in a subtle one.

3.3 Obstruction from the $C^*$-Identity

Suppose the involution problem were somehow discharged. There remains the identity from which the subject takes its name, $|a^a| = |a|^2$, and it is not one axiom among several. Its consequence is that the norm on a $C^$-algebra is determined by the algebraic structure [3, Ch. 2]. An algebra admits at most one norm satisfying it, so the metric and the algebraic content are not independent pieces of data but two aspects of a single object. Everything rigid in the theory descends from this. The uniqueness of the norm gives the automatic continuity of homomorphisms; that gives the isometry of injective homomorphisms; and the structure theorems that make the subject powerful follow in turn.

For a social repertoire of operations no such norm is available. The quantity $|a|$ would have to measure the magnitude of a suppression or the size of an act of certification, and no candidate suggests itself that is not either arbitrary or a smuggled-in resource quantity. A resource quantity is what the account set out to displace. Nor would an arbitrary choice help, since the identity constrains the norm rather than merely accompanying it: a norm not satisfying it yields a Banach $$-algebra, and the structure theory does not apply to Banach $$-algebras.

The consequence deserves to be stated without softening. An account that calls itself $C^*$-algebraic while supplying neither a norm nor the identity has borrowed the name without the structure the name denotes. It possesses an associative algebra with a formal involution, which is a considerably weaker object than the one whose reputation it has invoked, and the results that depend on the norm, which is to say the greater part of the structure theory, are unavailable to it.

3.4 Obstruction from Typing

An algebra is a structure with one object: every element may be multiplied by every other, and the product is again an element of the same algebra. Partial vanishing is expressible, but the ambient totality is not optional, composition being defined throughout whatever its value.

The social composites this paper is concerned with are not of that form. The sequence by which a durable asymmetry is assembled, treated at length in §9, consists of a differentiation that produces variation among parties, a selection that ranks the variation, a normalisation that renders the ranking unremarkable, and a reproduction that transmits it. The first takes a distribution and yields a distribution with greater variance. The second takes variance and yields an ordering. The third takes an ordering and yields a meaning, namely the ordering as it is understood, which is an object of a different kind from the ordering itself. The fourth takes a meaning together with an ordering and yields their transmission across a generation. These operations have distinct domains and distinct codomains, and their composite is well-formed only in one direction. The reverse composites are not merely uninteresting; most of them are undefined, there being no sense in which a normalisation may be applied to an undifferentiated population.

A structure whose operations carry domains and codomains of different kinds, in which composition is defined exactly when the codomain of one matches the domain of the next, is a category and is not an algebra [16]. An algebra is the degenerate case in which every domain and codomain coincide. Forcing the social composite into that case requires one of two moves. The first discards the typing, and with it the distinction between a distribution and a meaning that made the composite interesting. The second reintroduces the typing through auxiliary projections, which reconstructs the categorical structure inside the algebra while retaining the advantages of neither.

3.5 Obstruction from Reversibility

The three obstructions above are technical. The fourth is not, and it would remain even if the others were somehow answered.

The symmetries of an operator algebra are its automorphisms, and a dynamics is a group of them. The group structure is essential rather than conventional: a time evolution in this setting is a family of maps each of which has an inverse, and the framework was built for physical systems where this is appropriate. Irreversibility enters the theory, when it enters at all, as a derived phenomenon: a semigroup of completely positive maps describing an open system, or a modular flow associated with a state. Both are constructed atop a reversible foundation rather than replacing it.

The primitive of the present account is generation, and generation does not invert. A distinction brought into existence is not removed by any subsequent operation, though it may be superseded; an institution once founded leaves a residue in what remains formulable even after its dissolution; and the whole interest of the mechanism described in §4 lies in the asymmetry between an exercise that consumes its conditions and one that manufactures them, which is a statement about a direction that cannot be run backward. To build such an account on a foundation whose basic maps are invertible is to place the phenomenon in the derived layer and the idealisation in the primitive layer, which inverts the order the subject matter requires.

3.6 Grounds for Constructing a New Object

The requirements derived in §2 survive. The candidate does not. Operator algebra fixes generators and relations in the same way, and for the same kind of reason, that a differential model fixes coordinates: both are languages for dynamics upon a fixed structure, and the object of this paper is the dynamics of the structure.

What is needed, accordingly, is an object meeting the three requirements of §2 together with the fourth, that operations be borne by parties, and admitting irreversible transformation of the operational field as its basic move rather than as a derived special case. §5 builds it. Before that, and deliberately before any formal apparatus is introduced, §4 states in ordinary language the mechanism the construction is meant to capture. The order matters. A reader who declines the construction should still be in a position to accept or reject the mechanism, and a mechanism stated only in the vocabulary of its own formalism cannot be assessed independently of it.

§4 The Relational Mechanism of Power

This section states the mechanism the remainder of the paper formalises. It does so without formal apparatus and without the vocabulary introduced in §5, and the omission is deliberate. A mechanism stated only in the terms of its own formalism cannot be assessed independently of that formalism, and a reader who finds the construction of the following section unpersuasive should still be in a position to accept or reject what is claimed here. The argument proceeds in seven steps, of which the second and the sixth are where most accounts of power move too quickly.

4.1 From Change to Difference

The account begins from a commitment about what social reality consists in. Relations are not a static web upon which events occur; they are continually produced, and their production is what social life is. On this view change is not something that befalls an otherwise stable arrangement but the ordinary condition of the arrangement, which persists only insofar as it is reproduced.

A consequence follows immediately. Where relations are continually produced, they are not produced identically. Production under varying circumstance yields variation, so difference arises wherever relation is generated, and it arises constantly rather than exceptionally. Two parties to a relation come to differ in what they have done, in what has been done to them, in what they are positioned to do next. This much requires no argument beyond the commitment itself, and it is the starting point rather than a result.

4.2 From Difference to Asymmetry

The step from difference to asymmetry is where accounts of power most often move too quickly, and the distinction matters for what follows.

Two parties differing is not yet an asymmetry between them. A difference is symmetric in the relevant sense: each party has something the other lacks, and nothing in the bare fact of difference determines which lack is consequential. For an asymmetry to obtain, the difference must be ordered, so that what one party has confers a capacity that what the other has does not. The ordering is a further fact and not a restatement of the first.

Nothing in the production of difference supplies the ordering. Change generates variation; it does not rank the variation it generates. The ranking is imposed by whatever selects among differences: a market that prices one skill and not another, a court that recognises one claim and not another, a professional body that certifies one training and not another. This selection is an operation in its own right, distinct from the differentiation that preceded it, and it is treated as such in the schema of §9.

The consequence is that an account deriving power directly from change is incomplete. It has passed over the step at which difference becomes consequential, and that step is where a good deal of what is interesting about power resides. Where an asymmetry is contested, the contest is very often not over the difference but over the ordering: not whether two parties differ, which is rarely in doubt, but whether the difference should confer what it presently confers.

4.3 Exercise and Its Own Conditions

An ordered difference makes certain things possible for one party that are not possible for the other. This is what it is for an asymmetry to exist rather than merely to be recorded.

The exercise of such a possibility has two effects rather than one. The first is the effect intended: something is done, and the situation of the parties is altered accordingly. The second effect is the one this paper is concerned with. The alteration includes what is thereafter available, so the exercise of an asymmetry is also an intervention in the conditions of its own exercise. Every act performed in virtue of an asymmetry adjusts the standing of the asymmetry that permitted it.

The adjustment carries no fixed sign. An act may leave the conditions of its own repetition intact, or diminished, or enhanced, and which of these obtains is a fact about the act and its setting rather than about the magnitude of the advantage that licensed it. It is this variability of sign, rather than any variation in size, that the account takes as the locus of the phenomenon.

4.4 The Ordinary Case of Self-Consumption

Most asymmetries do not survive their own exercise. This is worth stating as the default rather than as an interesting exception, because the reverse assumption is what makes power appear more puzzling than it is.

A military advantage is exercised by campaigning, and campaigning consumes the fiscal capacity, the demographic base, and the willingness that underwrote the advantage. A seizure of authority is exercised by suspending the procedures under which authority was previously conferred, and the suspension destroys the source from which the seizure drew whatever recognition it had. A monopoly is exercised by pricing, and pricing at monopoly levels invites the entry that ends the monopoly. In each case the act is effective and the advantage is real; what fails is the reproduction of the conditions that made the act available.

Asymmetries of this kind arise constantly and leave little trace. They are not failed power, or weak power, or power in an early stage. They are asymmetries that were exercised and thereby dissolved, which is the ordinary fate of an ordered difference in a system where relations are continually produced.

4.5 Persistence as the Mark of Power

A minority of asymmetries behave otherwise. Their exercise reproduces the conditions of their exercise, so that the advantage stands after use as it stood before, and often more securely.

A credential is exercised by being required. Each occasion on which it is required confirms that the distinction it marks is a real one, and the confirmation is what a credential consists in. A bureaucratic procedure is exercised by being followed, and each following establishes the procedure more firmly as what is done. A norm is exercised by being invoked, and invocation is how a norm acquires the standing that makes invocation effective. In these cases the act does not draw down a stock; it deposits.

Power, on the account offered here, is the asymmetry that survives its own exercise. Survival is the mark rather than the substance: what makes an arrangement a power relation is the structural condition described in the following subsection, and persistence is how that condition shows itself. The formulation is deliberately narrow. It does not measure how much a party can compel, or how large the advantage is, or how many others are subject to it. It asks a single question about the relation between an act and what remains available afterward, and it counts as power only what answers that question in one particular way.

Two things follow that recommend the definition. It classifies the cases of §1 correctly, assigning the thin administration and the certifying body to power and the overwhelming military advantage and the dominant market share to the ordinary case, which resource accounts cannot do. And it does not require an arrangement to be intended, designed, or understood by anyone in order to count. What reproduces itself reproduces itself whether or not anybody planned that it should.

4.6 Reconstruction of the Self-Maintenance Thesis

A formulation recurs across otherwise unrelated accounts: that power consists in its own maintenance, that what power is about, beneath whatever else it is about, is its going on being power. The reading given to it here differs from the readings usually offered, and the differences are worth setting out, since the thesis is older and better defended than the account of this paper.

The formulation originates with Nietzsche [19], and it is commonly attributed to him in a form he explicitly rejected. Nietzsche does not hold that life or power aims at self-preservation. He holds the reverse: that a living thing seeks above all to discharge its strength, that self-preservation is among the indirect and frequent consequences of this rather than its object, and that treating preservation as the cardinal drive is to indulge a superfluous teleological principle, a charge he lays against the conatus of Spinoza. The formulation of the thesis is therefore not preservation but discharge and extension, with persistence following as a by-product. The account developed here departs from this and the departure should be recorded rather than concealed.

Two later positions retain the self-maintenance without the will. Luhmann [14] treats power as a symbolically generalised medium of communication rather than as a capacity held by anyone, and Luhmann [15] develops the autopoietic account on which social systems reproduce themselves from their own elements, so that a legal or political order sustains itself through recursive operation rather than through any external warrant. Foucault [8, 7] makes power immanent to the relations it obtains in and existent only in exercise, and adds a requirement of some importance: a power relation requires that the party subject to it be maintained throughout as one who acts, since a party reduced to no capacity at all is no longer a party to a relation but an object of force.

The reading offered here is that self-maintenance is constitutive rather than purposive. An asymmetry is not a state that happens to endure; it exists only in exercise, and exercise is what alters the conditions of exercise, so the relation between an asymmetry and its own continuation is not causal but definitional. What is called power is the case in which that relation closes. This preserves what Nietzsche saw, that the continuation of power is not incidental to power, while declining the modality in which he saw it. No aim is attributed to any arrangement, and nothing is said to seek anything.

A structural precedent for a self-maintaining process without an aim exists outside the study of power. In Lacan [13], desire is constituted by a lack that demand cannot meet, so that desire sustains itself precisely by remaining unsatisfied and its apparent object functions as a vehicle for its continuation rather than as a terminus. The parallel holds in three respects. Both are constituted by a process rather than by a substrate; both would end in their own satisfaction, which is why Foucault’s requirement that the subordinate party remain one who acts has a counterpart in the analytic setting; and in both, an appearance of purposiveness arises from the structure with no aim present. That the same form appears in a domain unconnected with social asymmetry is some evidence that it belongs to self-constituting processes generally rather than to power in particular.

The disanalogy is where the present account owes work that the analytic one does not. Desire’s self-maintenance is structurally guaranteed, demand being unmeetable, so no explanation is required of why desire persists. Power’s self-maintenance is neither guaranteed nor common: the overwhelming majority of ordered differences dissolve, and an account on which power maintains itself owes an explanation of why maintenance is so rarely achieved. Nietzsche has an answer available here and it is one this account rejects. On his view the failure to persist is weakness, an inadequate discharge of strength; the strong have no need to concern themselves with preservation and it is the fearful who pursue it. The cases of §1 do not support that reading. An advantage may be exercised vigorously, effectively, and without any deficiency of will, and consume its own conditions precisely through the vigour of its exercise. Self-consumption is not a defect of the party but a property of the relation between an act and what it leaves available.

The explanation offered instead is the generative commitment with which §4 began, and it is the contribution of the present framework to a thesis several traditions have reached independently. Because relation is continually produced, difference is continually produced with it, and ordered difference arises constantly and cheaply. The field does not stand still beneath an asymmetry: new difference is generated, new orderings become available, and the arrangement that was consequential yesterday is unsettled by what has been produced since. Nietzsche saw the same background and read it as conquest, holding that whatever exists is repeatedly reinterpreted to new ends and redirected by some power superior to it, with each subduing obscuring the meaning that went before [18]. The present account keeps the restlessness and drops the hierarchy: what unsettles an arrangement need not be a superior power, and is more often the ordinary productivity of relation itself. Maintenance is therefore work performed against a productive background rather than persistence in a static one, and its rarity follows from the abundance of what it must be maintained against. Neither the systems-theoretic nor the analytic account supplies this, the first because autopoietic reproduction is the definition rather than the achievement, the second because desire faces no competition from what it does not desire.

Two consequences follow for what may be claimed. The first is that asymmetry is not sufficient for power, and the insufficiency is not an independent stipulation but a consequence of generativity: were asymmetry scarce it might have sufficed, and it is scarce nowhere. The second is that the observed population of power relations is the maintained population, so that any description of an arrangement is a description of what has so far been maintained. This is a real epistemic constraint and it is not the whole of the account. The constitutive claim concerns what power is; the observation concerns what is available to be studied.

4.7 Generation and Degeneration

A last distinction is needed, and it is stated here qualitatively.

That an asymmetry reproduces itself says nothing about how the reproduction is achieved. Two arrangements may both survive their own exercise and differ entirely in what the survival costs the parties to them. A certifying body persists by making a distinction that others find worth having, and the persistence is purchased by enlarging what those others can do: certified practitioners can undertake what they could not undertake before, and those who deal with them can rely on what they could not previously rely on. A system of exclusion may equally persist, and its persistence is purchased by contracting what the excluded can do, each exercise removing a further possibility from those it acts upon.

Both are power under the account given above, and the account is not thereby defective. What separates them is not whether the asymmetry survives but at whose expense. Where the survival of an asymmetry is sustained by enlarging the possibilities of those it acts upon, the arrangement is generative; where it is purchased by contracting them, the arrangement is degenerative. The two are values of a single condition rather than two conditions, which is why the definition of power does not require amendment to accommodate them.

The distinction is easy to state and easy to state wrongly. It is tempting to make it a matter of how many possibilities the arrangement admits in total, so that generative arrangements are the permissive ones and degenerative arrangements the restrictive ones. §8 shows that this fails in both directions, and that the criterion cannot be recovered by counting anything.

§5 Construction of the Formal Object

The object constructed here is intended to satisfy the four requirements of §2 and to state the mechanism of §4 precisely. It is assembled in stages: first the collection of available operations, then the typing that makes composition partial, then the transitions that operations induce upon the collection, and finally the presentation that holds the two kinds of movement together.

Two things should be said at the outset. The construction is not a new branch of mathematics and does not claim to be. Its ingredients are standard, and the contribution lies in which structure is treated as primary rather than in any technical novelty. Nor does the construction yield a predictive apparatus; what it yields is a vocabulary in which the distinctions of §4 can be stated without ambiguity, together with the conditions under which those distinctions are well defined.

5.1 Repertoires

Let $P$ be a set of parties. A repertoire is a collection of operations, each of which is borne by a party and each of which acts upon the relational field the parties inhabit. Writing $\mathcal{A}$ for a repertoire and $a \in \mathcal{A}$ for an operation, every operation carries an attribution

$$\pi : \mathcal{A} \longrightarrow P,$$

so that $\pi(a)$ names the party for whom $a$ is available. The attribution map is what discharges the fourth requirement, and it is the feature that distinguishes a repertoire from a rule set. The fibre

$$\mathcal{A}_p ;=; \pi^{-1}(p) ;\subseteq; \mathcal{A}$$

collects what is available to the party $p$, and it is at the level of these fibres, rather than at the level of $\mathcal{A}$ as a whole, that the distinctions of §4 are stated.

An asymmetry between two parties is now expressible directly. Where $\mathcal{A}_p$ and $\mathcal{A}_q$ differ, the parties differ in what is available to them, which is the difference of §4. The ordering that converts difference into asymmetry is not supplied by the fibre and must be added; this is taken up in §6, and its absence here is deliberate, the point of §4 having been that the ordering is a separate fact.

5.2 Typed Partial Composition

Operations act upon relational facts of different kinds. A differentiation acts upon a distribution and yields a distribution; a selection acts upon a distribution and yields an ordering; a normalisation acts upon an ordering and yields a meaning. To record this, let $\mathcal{O}$ be a collection of registers, each register being a kind of relational fact, and equip every operation with a domain and a codomain,

$$a : X \longrightarrow Y, \qquad X, Y \in \mathcal{O}.$$

Composition is then defined exactly when the types agree:

$$b \circ a \ \text{ is defined} \iff \operatorname{cod}(a) = \operatorname{dom}(b).$$

This discharges the third requirement without stipulation. Partiality is not a constraint imposed upon a total operation but a consequence of the operations having types at all, and the inadmissible compositions are inadmissible because they are not well formed rather than because a rule forbids them. Order-sensitivity follows in the same way: where both $b \circ a$ and $a \circ b$ are defined, they are in general distinct, and where only one is defined the asymmetry of order is absolute rather than a matter of degree.

A repertoire equipped with the typing and the partial composition is a category whose objects are registers and whose arrows are operations, together with the attribution. It will be convenient to write $\mathcal{A}$ for this whole structure and to call it a repertoire without further qualification.

5.3 Induced Transitions

The mechanism of §4 turns on the second effect of an exercise: that performing an operation alters what is thereafter available. This is recorded by letting each operation induce a transition of the repertoire itself.

Write $\mathfrak{R}$ for the collection of repertoires over the party set $P$ and register collection $\mathcal{O}$. Each operation $a \in \mathcal{A}$ determines a map

$$\Theta_a : \mathcal{A} \longrightarrow \mathcal{A}’,$$

where $\mathcal{A}’ \in \mathfrak{R}$ is the repertoire in force after $a$ has been performed. The map $\Theta_a$ may add operations, remove them, or reattribute them from one party to another, and these are the three ways in which the exercise of an asymmetry alters the conditions of its own exercise.

Not every transition is induced by an operation an asymmetry licenses. Because relation is continually produced, the repertoire is also carried forward by what happens independently of any given arrangement: parties enter and leave, new difference arises among them, and new orderings become available by which difference may be made consequential. Write

$$\Delta : \mathcal{A} \longrightarrow \mathcal{A}^{\Delta}$$

for a transition of this second kind, and call it a background transition. The distinction between the induced and the background transition is not one of mechanism, both being transitions of the same structure, but of provenance: the first is induced by the exercise of an asymmetry and the second is not. Background transitions may enlarge the register collection, admit operations no party previously held, or supply orderings under which differences that were inconsequential become consequential. They are the formal counterpart of the productive background described in §4, and §6 shows that the distinction between an asymmetry that merely closes upon itself and one that is maintained turns on them.

Three further features of the induced transition deserve emphasis, since they are what the languages of §2 could not supply.

The transition is induced by the operation rather than stipulated alongside it. There is no external rule playing the part of the reset map in the hybrid tuple; $\Theta$ is determined by the operation, and an operation is not fully specified until its induced transition is given.

The target $\mathcal{A}’$ need not have the same registers as $\mathcal{A}$. A transition may enlarge $\mathcal{O}$, which is how a relational fact that was previously not merely false but unformulable comes into existence, and it is what the fixed type set $T$ of the network graph forbids.

The attribution is what changes. Since $\Theta_a$ acts on a structure carrying the attribution map, its effect is stated fibrewise: $\Theta_a$ may contract $\mathcal{A}_p$ while enlarging $\mathcal{A}_q$, and this is a well-formed description of what a prohibition does. The corresponding description is unavailable for anonymous rules, as Figure 3 recorded.

5.4 The Double-Category Presentation

Two kinds of movement have now been introduced. Operations compose with one another within a repertoire, and operations induce transitions between repertoires. These are movements of different kinds and they are not interchangeable, but neither are they independent: an operation performed inside a repertoire is the very thing that carries the repertoire to its successor.

The structure holding both is a double category, in the sense introduced by Ehresmann [5]. Its objects are registers. Its horizontal arrows are operations, composing according to the type condition. Its vertical arrows are repertoire transitions, composing by performance in sequence. Its cells are squares

$$\begin{array}{ccc} X & \stackrel{a}{\longrightarrow} & Y \[2pt] \big\downarrow{\scriptstyle,\theta} & & \big\downarrow{\scriptstyle,\theta’} \[2pt] X’ & \stackrel{a’}{\longrightarrow} & Y’ \end{array}$$

recording that the operation $a$, performed under the transition $\theta$, appears as $a’$ in the successor repertoire.

Figure 5 (Cells of the double category). On the left an operation whose induced transition leaves it available, which is the formal counterpart of persistence. At centre an operation absent from the successor repertoire, which is self-consumption. On the right the fibrewise effect of a transition, which the attribution map makes stateable and an anonymous rule set does not.

The commuting square is the point of the presentation rather than a technical convenience. An operation that rewrites the conditions of its own availability is a single object in this setting, not two coupled stories about the same event, because the cell states the horizontal fact and the vertical fact as one. The failure modes of §4 become properties of such cells. Where $a’$ does not exist, the operation has removed itself from what may be done, which is self-consumption. Where $a’$ exists and is again $a$, the operation has left its own availability intact.

5.5 Irreversibility of the Vertical Arrows

A final stipulation completes the construction, and it is the one that separates this object from the operator-algebraic candidate assessed in §3.

The vertical arrows are not required to be invertible, and in the intended reading they are typically not. A transition $\theta : \mathcal{A} \to \mathcal{A}’$ has no assumed partner $\theta^{-1} : \mathcal{A}’ \to \mathcal{A}$. This is not an omission to be repaired in a fuller treatment. Generation does not invert: a register brought into existence is not removed by any subsequent transition, though it may cease to be inhabited; an attribution once contracted is not restored by running anything backward; and the whole content of §4 lies in a directional difference between exercises that deposit and exercises that draw down.

The vertical arrows accordingly form a category and not a groupoid, and the dynamics of the object is not a group action. Where the automorphism family placed reversibility in the primitive layer and irreversibility in the derived one, the present construction places irreversibility in the primitive layer. Reversible transitions are the special case, recovered when $\theta$ happens to be invertible, and nothing in the framework privileges them.

Two properties follow that are worth recording, since they are used later. A transition may fail to be surjective on operations, which is how possibilities are lost, and it may fail to be injective on registers, which is how distinctions collapse. §7 takes up the first of these under the heading of institutional filtering, and the second is the formal counterpart of the collapse of distinction familiar from accounts of relational degradation.

§6 Formalization of the Mechanism

The mechanism of §4 is now stated in the vocabulary of §5. The order of the two sections was chosen so that this one adds precision rather than content, and the reader should be able to check that nothing new is claimed here beyond what was claimed there.

6.1 Ordering as Added Structure

§5 supplies difference between parties without supplying asymmetry. Where the fibres $\mathcal{A}_p$ and $\mathcal{A}_q$ fail to coincide, the parties differ in what is available to them, and nothing in the construction determines which of the two is thereby advantaged. This corresponds exactly to the step of §4 at which difference is not yet asymmetry, and the correspondence is deliberate: an object that delivered asymmetry directly from difference would have built in the step the account holds to be separate.

The ordering is added as follows. Let $\preceq$ be a partial order on the fibres of a repertoire, and write

$$\alpha ;=; \bigl(p, q, \preceq\bigr), \qquad \mathcal{A}_q \prec \mathcal{A}_p,$$

for the assertion that $p$ stands above $q$ under it. The relation $\preceq$ is not derived from set inclusion, and this matters. That $\mathcal{A}_q \subsetneq \mathcal{A}_p$ neither implies nor is implied by $\mathcal{A}_q \prec \mathcal{A}_p$: a party may have fewer operations available and stand higher, as a sovereign bound by few procedures and empowered by them illustrates, and a party may have more operations available and stand lower, as anyone whose many permitted actions are all inconsequential illustrates. Counting operations does not order fibres, and the independence of $\preceq$ from cardinality recorded here is what §8 generalises.

What supplies $\preceq$ is the selection of §4: a market, a court, a certifying body, some operation that ranks differences and thereby makes them consequential. In the schema of §9 this is the second register, and its status as a distinct operation rather than a derived quantity is the formal counterpart of the philosophical point that change generates variation without ranking it.

6.2 Closure Under Self-Induced Transition

An asymmetry $\alpha$ in a repertoire $\mathcal{A}$ licenses a set of operations, namely those available to $p$ and not to $q$ under the ordering:

$$L(\alpha) ;=; \bigl{, a \in \mathcal{A}_p ;:; a \notin \mathcal{A}_q ,\bigr}.$$

Each such operation induces a transition, and the composite of the licensed operations performed in sequence induces

$$\Theta_{L(\alpha)} : \mathcal{A} \longrightarrow \mathcal{A}’,$$

carrying the repertoire to its successor. The asymmetry $\alpha$ has an image $\alpha’$ in the target whenever the parties, the fibres, and the ordering all survive the transition.

Definition. An asymmetry $\alpha$ is closed when the transition induced by the operations it licenses returns it: $\Theta_{L(\alpha)}(\alpha) = \alpha$.

Closure is the formal counterpart of the constitutive claim of §4: the exercise of the asymmetry is also an intervention in the conditions of that exercise, and in the closed case the intervention returns what it acted upon. Three points about the form of the condition are worth making explicit, since each answers an objection that would otherwise stand.

The condition is on the asymmetry and not on the repertoire. A repertoire may be reproduced entire without any asymmetry being reproduced, since a repertoire admits fixed points in abundance whether or not any asymmetry obtains within it. The stable egalitarian arrangement is therefore excluded by the condition rather than by a clause appended to it. Two grounds of exclusion should be distinguished. Where all fibres coincide, no difference obtains and $\alpha$ has no subject. Where fibres differ but $\preceq$ ranks none of them above another, difference obtains and $\alpha$ is undefined for want of an ordering. Both cases fall outside the condition, and the second is the one that matters, since egalitarian arrangements are rarely uniform and are characterised by the absence of consequential ranking rather than by the absence of difference.

The condition quantifies over the licensed set rather than over all operations. An arrangement is not disqualified because some unrelated operation would disturb it; what is asked is whether the exercise of the advantage sustains the advantage. This is what makes the condition a statement about self-reference rather than about stability in general.

The condition is silent on magnitude. No cardinality, measure, or weight appears in it. Two asymmetries may differ by any amount in what they permit and stand alike under it, which is the intended consequence: the account holds that the durability of an asymmetry is not a function of its size, and a condition mentioning size would have contradicted that at the outset.

6.3 Maintenance Under Background Transition

Closure is not yet sufficient for power, and the reason is the one §4 gave. An asymmetry that returns itself under its own exercise has been tested against one thing only, namely itself. The field meanwhile does not stand still. Because relation is continually produced, background transitions carry the repertoire forward independently: parties enter, difference accumulates, and orderings become available under which what was inconsequential becomes consequential. An asymmetry closed under its own transition may nonetheless fail to survive what the field has done in the interval.

Definition. An asymmetry $\alpha$ is maintained when it is returned by its own induced transition composed with the background transitions the field supplies, that is, when $\bigl(\Theta_{L(\alpha)} \circ \Delta\bigr)(\alpha) = \alpha$ for the background transitions $\Delta$ admissible in the setting under consideration. Power is maintenance rather than closure.

The strengthening is what the generative commitment requires, and three consequences follow from it.

The distinction between closure and maintenance separates two cases that the earlier condition could not tell apart. An asymmetry may be closed and unmaintained, which is the position of an arrangement that reproduces itself perfectly under its own exercise and is undone by what arises elsewhere. The military advantage of §1 is of this kind: campaigning may reproduce the command structure that campaigning requires, and the coalition that forms in response was produced by no operation the advantage licensed. Nothing internal to the arrangement failed. What defeated it was the field’s own productivity, and an account with only the closure condition would have to record the case as inexplicable.

Maintenance is a modal condition where closure is a factual one. Closure asks what one transition does; maintenance asks what would happen under the perturbations the setting admits. This is the formal difference between an arrangement that has persisted and one that is disposed to persist, and it is why the account can speak of a tendency without attributing an aim: a disposition is not a purpose, and a modal claim about behaviour under perturbation carries no teleology whatever.

The quantifier over $\Delta$ is where empirical content enters, and its scope must be specified rather than left universal. Nothing is maintained against every conceivable perturbation, and an asymmetry required to survive arbitrary background transition would be an asymmetry no arrangement satisfies. What is at issue is the class of background transitions a given setting actually supplies, and this is a matter of fact about the setting rather than something the framework determines. §10 records the consequence: the account states the condition and does not decide, for any actual arrangement, which perturbations it must withstand.

6.4 The Two Modes of Failure

The complementary cases are the ones §4 identified as ordinary, and the strengthened condition distinguishes two of them where the closure condition alone distinguished one.

An asymmetry is self-consuming when its own licensed operations induce a transition under which it fails to return, either because those operations are no longer available,

$$L(\alpha) \not\subseteq \mathcal{A}’,$$

or because the ordering does not survive, the parties having ceased to stand in the relation that made the operations consequential. Closure fails, and it fails from within. The centre panel of Figure 5 depicts this case. The seizure of authority that suspends the procedures conferring authority is of this kind, and so is the advantage whose exercise exhausts the base that produced it.

An asymmetry is unmaintained when it is closed but not maintained: its own exercise returns it, and it does not survive composition with what the field supplies meanwhile. Nothing internal to the arrangement has gone wrong. What has happened is that new difference has arisen, or a new ordering has become available, or parties have entered who stand outside the relation, and the arrangement that reproduced itself perfectly a moment ago no longer finds the conditions it reproduced.

The second mode is the one the account most wants to record, and it bears directly on the reading of §4. On a view that treats failure to persist as inadequacy in the party holding the advantage, the unmaintained case must be assimilated to the self-consuming one and read as a defect of exercise. The distinction drawn here denies that assimilation. An arrangement may be exercised with complete competence, return itself under every operation it licenses, and be undone by a productivity that no operation of its own occasioned. Weakness is not the explanation, and vigour would not have been the remedy.

A further observation follows, and it is epistemic rather than constitutive. Under repeated exercise and repeated background transition, asymmetries failing either condition are absent from the repertoires that succeed, while those meeting both remain. Any description of an arrangement made at a later stage is therefore a description of a repertoire in which the failed asymmetries no longer figure, and the population available for study is the maintained population. This explains something about the evidence rather than about the phenomenon: what power is, is stated by the condition of §6; what we are in a position to observe, is stated here. Conflating the two would make the account trivial, since it would reduce the claim that power maintains itself to the observation that what remains has remained.

The account does not predict which asymmetries in a given repertoire will prove to be maintained, and §10 takes up the reasons. What it supplies is a condition under which the question is well posed, together with the finding that the answer is not read off the size of the advantage.

6.5 The Generative and Degenerative Parameter

The definition given above is satisfied by arrangements that differ in what their persistence costs the parties subject to them, and §4 distinguished these qualitatively. The distinction is now stated fibrewise.

Let $\alpha = (p, q, \preceq)$ be a power relation, and let $\Theta = \Theta_{L(\alpha)}$ be the transition it induces. The relevant comparison is between the fibre of the subordinate party before and after:

$$\mathcal{A}_q \quad \text{against} \quad \mathcal{A}’_q ;=; \Theta(\mathcal{A})_q .$$

Definition. A power relation $\alpha$ is degenerative when its maintenance is purchased by contraction of the subordinate fibre, and generative when maintenance is sustained while that fibre is enlarged.

The right-hand panel of Figure 5 depicts the fibrewise comparison, and the attribution map is what makes this a well-formed expression. In a formalism whose operations belong to no one, the comparison could not be written down, and this is the point at which the fourth requirement of §2 earns its place.

Two features of the definition deserve comment.

Generation and degeneration are values of one condition rather than two conditions. Both presuppose maintenance, and they differ in the fibrewise sign of the transition that sustains it. This is why the definition of a power relation required no amendment to accommodate them, and why an arrangement cannot be generative without being a power relation in the first place. A transient enlargement of the subordinate fibre, not sustained by any self-reproducing asymmetry, is not generative on this account; it is simply an event.

The comparison is fibrewise and not aggregate. What is compared is $\mathcal{A}_q$ with $\mathcal{A}’_q$, not $\mathcal{A}$ with $\mathcal{A}’$, and the two comparisons can run in opposite directions. A transition may contract the repertoire as a whole while enlarging the subordinate fibre, and the definition attends to the second and disregards the first. §8 shows that this is not a technicality but the substance of the matter, since the aggregate comparison yields the wrong verdict in both directions.

§7 Application of Operator-Algebraic Distinctions

§3 set aside operator algebra as a foundation. It does not follow that the subject has nothing to contribute, and this section takes up three of its distinctions and applies them to the object constructed in §5. The placement is deliberate. Had these appeared alongside the assessment, they would have been analogies with nothing underneath, since the object they are applied to did not then exist. Applied to repertoires and their transitions, they yield distinctions that the fixed-point definition alone does not supply.

The borrowings are of questions rather than of axioms, and the difference should be kept in view. Nothing below asserts that a repertoire is an algebra, that a transition is an automorphism, or that any theorem of operator algebra transfers. What is claimed is that three questions the subject has learned to ask are worth asking here, and that asking them produces results.

7.1 Individuation of Regimes

Operator algebra distinguishes representations of a single algebra that cannot be connected by any transformation preserving the observables [9]. Such representations belong to separate superselection sectors, and the distinction is structural: no continuous path, and no automorphism, carries one to the other. The physical reading is that the two describe genuinely different phases of the same system rather than different states within one phase.

The corresponding question for repertoires is when two arrangements should count as different regimes rather than as one regime in different conditions. The maintenance condition of §6 does not answer this. It tells us which asymmetries persist within an arrangement and says nothing about the individuation of arrangements themselves, so a large change and a change of kind are on its account indistinguishable.

The borrowed distinction supplies a criterion. Call two repertoires connected when some sequence of transitions carries one to the other while preserving the register structure, and separated when no such sequence exists. Separation is not a matter of distance. Two arrangements may differ in almost every attribution and remain connected, a sequence of transitions carrying the one to the other; and two arrangements may differ in a single register and be separated, no transition preserving the structure being available to bridge them.

This yields a reading of regime change that the magnitude languages of §2 cannot state. A constitutional order and its successor are not related as two points along a continuum of arrangements, differing by more than usual. Where the successor lies in a separate sector, the transition between them was not the accumulation of ordinary transitions and cannot be described as such, whatever the sequence of events that in fact produced it. The criterion also disposes of a familiar ambiguity: two states sharing a constitutional text may be separated, if no transition preserving the register structure connects them, and the shared text is then a fact about documents rather than about arrangements.

The criterion is offered as a distinction rather than as a test. Determining whether a bridging sequence exists is not something the framework makes tractable, and §10 returns to this.

7.2 Institutional Filtering

A conditional expectation in operator algebra is a projection from an algebra onto a subalgebra, compatible with the multiplication in the sense that elements of the subalgebra pass through it unchanged [23]. Its interpretation is that of a coarse description: what the larger algebra records is retained only insofar as the smaller one can express it. Where the construction is available, an index measures the size of the discrepancy [11], quantifying how much of the larger structure the smaller one fails to see.

The corresponding structure for repertoires arises whenever an institution acts upon a relational field through a vocabulary of its own. A court recognises claims of certain kinds; a bureaucracy processes cases under categories it maintains; a certifying body registers distinctions its own procedures can express. In each case the institution operates not upon the field as it stands but upon its own reading of it, and the reading is a subrepertoire.

Let $\mathcal{B} \subseteq \mathcal{A}$ be such a subrepertoire, and write

$$E : \mathcal{A} \longrightarrow \mathcal{B}$$

for the assignment carrying each operation to whatever the institution recognises it as. The requirement corresponding to compatibility is that operations already belonging to $\mathcal{B}$ pass unchanged under $E$, so an institution reads its own categories correctly and reads everything else through them.

Two consequences follow, and the second is the one worth having.

An institution is thereby blind in a specific and describable way. What $E$ discards is not noise but relational fact: distinctions present in $\mathcal{A}$ that $\mathcal{B}$ cannot express are, from the institution’s standpoint, absent. This is the non-surjectivity noted at the end of §5, and it gives a formal shape to a familiar observation about administration without recourse to metaphor.

The blindness admits comparison. Where an index analogue is available, arrangements may be ordered by how much of the field their institutions fail to register, and the ordering is not the same as the ordering by how much they do. An institution with a large vocabulary may discard more than one with a small vocabulary, if what it discards is what mattered. The account does not supply such an index in general, and constructing one for a class of cases would be a contribution this paper does not make.

The bearing on what follows is direct. Normalisation, the third register of §9, is precisely the operation by which a contingent ordering is rendered as a category an institution can recognise. In the present vocabulary, normalisation moves an ordering from $\mathcal{A}$ into the subrepertoire $\mathcal{B}$ through which the institution acts, after which the ordering is no longer something the institution imposes but something it merely reads.

7.3 Levels of Symmetry

The third borrowing is the most useful and the least technical. Operator algebra distinguishes the symmetries of a dynamics, the symmetries of the observables, and the symmetries of a state, and holds them apart as three questions rather than one. The corresponding levels here are the transition, the repertoire, and the fibrewise distribution.

Consider an arrangement in which every party has the same operations available. This is symmetry at the level of the repertoire: for parties $p$ and $q$,

$$\mathcal{A}_p ;=; \mathcal{A}_q .$$

Consider next an arrangement in which the ordering places no party above another, so that no asymmetry obtains. This is symmetry at the level of the distribution, and it is a different condition. And consider finally an arrangement whose transitions treat parties alike, so that $\Theta$ commutes with the exchange of $p$ and $q$. This is symmetry at the level of the transition, and it is a third condition again.

Figure 6 (The three levels held apart). Formal equality asserts the first panel and is compatible with the failure of the second, which is the shape of the distinction between formal and substantive equality. The third panel is a further condition that discussions recognising only the first two have no place for.

The three are independent, as Figure 6 sets out, and the independence is the result. Equality before the law asserts $\mathcal{A}_p = \mathcal{A}_q$: the same operations are available to all, and this is what a formal guarantee of equal treatment guarantees. Substantive inequality is the failure of symmetry at the level of the distribution, and it is entirely compatible with $\mathcal{A}_p = \mathcal{A}_q$ holding exactly. An arrangement may therefore be symmetric in its repertoire and asymmetric in its distribution without contradiction, which is the formal shape of the long-standing observation that formal and substantive equality are distinct. The gap between the two is what the second and third dimensions of power in Lukes [12] are addressed to.

What the distinction adds to that observation is a third term usually omitted. Reform that alters what may be done, as against reform that alters who stands where, is a change at the level of the transition, and it is neither of the first two. An arrangement may be reformed in this third sense without either the repertoire or the distribution changing at the moment of reform, the alteration consisting in what subsequent exercises will do. Discussions that recognise only formal and substantive equality have no place for this, and are obliged to classify such reforms as one or the other, generally as merely formal.

The table below records the three borrowings and what each contributes.

Distinction	Operator-algebraic content	Applied to repertoires
Superselection	Representations unconnectable by observable-preserving transformation	Regimes separated when no transition preserving register structure bridges them; regime change is not accumulated ordinary change
Conditional expectation	Projection onto a subalgebra, with an index measuring the discrepancy	An institution acts through its own subrepertoire; what it cannot express is absent from its standpoint
Levels of symmetry	Symmetries of dynamics, observables, and state held apart	Symmetry of the transition, the repertoire, and the distribution; formal equality is $\mathcal{A}_p = \mathcal{A}_q$, substantive equality is not

§8 Refutation of the Counting Criterion

The distinction between generative and degenerative power relations was given in §6 as a fibrewise condition. A simpler criterion suggests itself, and it is held widely enough to deserve refutation rather than dismissal. This section states it, shows that it fails in both directions, and identifies what the failure reveals about the criterion that replaces it.

8.1 The Counting Intuition

The intuition is that generativity consists in the enlargement of possibility. An arrangement that admits more is more generative than one that admits less, so that generativity may be measured by the size of what is available and degeneration by its diminution. Written in the vocabulary of §5, the proposal is that a transition $\Theta$ is generative when

$$\lvert \Theta(\mathcal{A}) \rvert ;>; \lvert \mathcal{A} \rvert$$

and degenerative when the inequality runs the other way.

The proposal has evident attractions. It is simple, it requires no attribution map, it agrees with ordinary usage in a wide range of cases, and it connects generativity to notions with established formal treatments, whether the cardinality of an option set, the entropy of a distribution over available actions, or the size of a feasible region. Nothing in what follows disputes that these are well-defined quantities. What is disputed is that any of them tracks the distinction the account requires.

Variants weighting operations by importance, or counting only operations that are in some sense significant, do not evade what follows. The counterexamples turn on which party holds the operations rather than on how the operations are counted, and a weighted count remains a count over the repertoire as a whole.

8.2 Counterexamples in Both Directions

Contraction that generates. A constitutional prohibition removes operations from a repertoire. Where a constitution forbids the suspension of elections, the detention of persons without process, or the expropriation of property without compensation, the operations so forbidden are no longer available, and the repertoire is smaller after the prohibition than before. By the counting criterion the transition is degenerative.

The verdict is wrong, and it is wrong in a way that indicates where the criterion has gone astray. What the prohibition removes is drawn almost entirely from one fibre. The operations forbidden are those available to the party in a position to suspend, detain, or expropriate, and their removal is what permits the other parties to act at all in ways that were previously subject to arbitrary reversal. Contracts become worth entering, since they cannot be voided at discretion; associations become worth forming, since they cannot be dissolved at will; and each of these is an enlargement of $\mathcal{A}_q$ effected by a contraction of $\mathcal{A}_p$. The aggregate may fall while the subordinate fibre rises, and the second is what the account attends to.

Expansion that degrades. Consider the dissolution of shared norms, the condition Merton [17] analyses under the name of anomie, in which what was formerly not done becomes possible. Operations previously excluded, whether by convention, by expectation, or by the anticipated response of others, are now available. The repertoire is larger after the dissolution than before, and by the counting criterion the transition is generative.

This verdict is wrong in the mirror image of the first case. The operations that become available are available in principle to all and in practice to those positioned to use them, so the enlargement accrues to the fibre of whoever can act with least constraint. For the party whose position depended on the dissolved expectations, what was available has contracted sharply, since operations whose effectiveness rested on others’ compliance with a norm are no longer effective operations at all. The aggregate rises while the subordinate fibre falls.

Figure 7 sets the two cases side by side. The left panel shows what the counting criterion registers, which is the aggregate; the remaining panels show the fibrewise decomposition that the criterion discards. The aggregate trajectories run in opposite directions from the fibrewise ones in both cases.

Figure 7 (The counting criterion against the fibrewise one). The left panel plots the aggregate size of the repertoire under the two transitions, which is the whole of what the counting criterion reads. The centre and right panels decompose the same transitions by fibre. Prohibition contracts the aggregate while enlarging the subordinate fibre; the dissolution of norms enlarges the aggregate while contracting it. The illustrated trajectories are constructed rather than measured, and no claim is made that any actual arrangement follows them.

The two counterexamples run in opposite directions, which forecloses the natural repair. Were the criterion wrong only about prohibition, one might reverse it and hold that contraction generates. The second case rules this out. No monotone function of $\lvert \mathcal{A} \rvert$ can agree with the account on both cases, since the account assigns opposite verdicts to two transitions whose aggregate behaviour is itself opposite, and a monotone function of the aggregate must order them as the aggregate orders them. The failure is not of calibration but of the quantity chosen.

8.3 Reformulation of the Criterion

What the counterexamples share is that the aggregate discards the attribution. The quantity $\lvert \mathcal{A} \rvert$ is computed over the repertoire as a whole and is therefore insensitive to which fibre gained and which lost, and it is precisely that information the distinction requires. This is why the criterion of §6 was stated fibrewise, and why the attribution map was built into the object rather than added to it.

The reformulation may now be stated as the moral of the two cases. Generativity is not a property of how much an arrangement permits. It is a property of the relation between what an arrangement permits its dominant party and what it thereby permits the others. A quantity insensitive to the attribution cannot express that relation, since the two counterexamples agree in aggregate behaviour where the account separates them and differ in fibrewise behaviour where the account joins them. The consequence for the notions mentioned above is direct: option count, entropy over available actions, and the size of a feasible region are all aggregate quantities and all fail for the same reason.

A related consequence bears on freedom as ordinarily discussed. If freedom is taken to consist in the number of available options, then the second counterexample is a gain in freedom and the first a loss, which reverses the assessment that almost anyone would make of the two cases. The account does not offer a theory of freedom, and nothing here settles what freedom consists in. It does show that whatever freedom is, it is not the aggregate quantity in the counting criterion, since that quantity delivers the wrong verdict on the two cases considered here.

8.4 Relation to the Value Loop

A further characterisation follows, and it connects the present result to the account of generation given in §4.

Where a power relation is degenerative, what the arrangement generates returns to the party that generated it. The subordinate fibre is contracted, the dominant fibre is thereby secured, and the value produced by the exercise accrues to the position from which the exercise was made. The loop closes upon itself, and the arrangement is sustained by what it takes from those subject to it.

Where a power relation is generative, what the arrangement generates does not return in this way. The certifying body’s persistence is purchased by the enlargement of what certified practitioners may do, and the value generated accrues to the relation rather than to either party alone. The loop remains open, in the sense that what is produced is not recovered by the producer.

The two characterisations agree on the cases considered here, and no argument is offered that they agree in general; the loop formulation is a gloss on the fibrewise criterion rather than an independent test, and establishing an equivalence would require a treatment of value that this paper does not attempt. Its usefulness is that it connects this result to the treatment of appropriation elsewhere in the programme to which this paper belongs, where the question of whether generated value is left with the relation that generated it recurs in settings having nothing to do with power.

§9 Generalization to a Schema of Durable Asymmetry

The account has so far given a condition under which an asymmetry persists without saying how such asymmetries are assembled. This section proposes that they are assembled in a common way, and offers the proposal as a general schema rather than as an illustration of the preceding sections.

9.1 The Four Registers

A durable asymmetry, on the proposal, is the composite of four operations.

Differentiation takes a population and yields variation within it. Parties come to differ in training, in holdings, in position, in what they have done. This is the step of §4 at which relation is produced and, being produced under varying circumstance, produces difference.

Selection takes variation and yields an ordering. Some differences are made consequential and others are not, and the operation that ranks them is distinct from the operation that produced them. §6 recorded this as the addition of $\preceq$, which the construction does not supply on its own.

Normalisation takes an ordering and yields a meaning: the ordering as it is understood, which is to say the ordering rendered unremarkable. What was a contingent ranking becomes the way things are, and the parties to it cease to experience it as something imposed. In the vocabulary of §7, normalisation is what moves an ordering into the subrepertoire through which an institution reads the field, after which the institution does not impose the ordering but merely recognises it.

Reproduction takes a normalised ordering and yields its transmission. What holds among the present parties comes to hold among their successors, whether through inheritance, through training, through the persistence of the institutions that recognise it, or through the simple continuation of the practices that constitute it.

Writing the four as arrows,

$$X ;\xrightarrow{\ D\ }; V ;\xrightarrow{\ S\ }; O ;\xrightarrow{\ N\ }; M ;\xrightarrow{\ R\ }; X,$$

the composite $R \circ N \circ S \circ D$ carries a population to a population in which the same ordering obtains, which is the closure condition of §6 exhibited as a cycle. What the composite must further withstand is the background productivity of the field, and §9 argues that one register in particular is what makes it able to.

9.2 Typing Across Registers

The four operations do not compose within a single register, and this is why the cycle was written with distinct objects. A distribution, an ordering, a meaning, and an inheritance are relational facts of different kinds, and the arrows between them have correspondingly different domains and codomains.

The typing is not decoration. It is the reason the composite could not be written as a product in an algebra, as §3 recorded, and it is what makes the reverse composites undefined rather than merely uninteresting: a normalisation cannot be applied to an undifferentiated population, there being no ordering for it to act upon. The partiality of composition is therefore doing work here rather than sitting idle.

It follows that a durable asymmetry is not a single kind of thing occurring four times. It is a passage across four kinds of relational fact, and an account that treats power as one substance moving through a system will not have the resources to state it.

9.3 Argument from Necessity

The proposal is that all four registers are required, and the argument is given register by register. In each case the question is what becomes of an asymmetry when the register is absent.

Without differentiation there is nothing to order. This case is degenerate and is recorded for completeness.

Without selection there is difference and no asymmetry, which is the step of §4 at which most accounts move too quickly. Parties differ, and no difference confers anything. Such arrangements are common and unremarkable, which is precisely the point: variation is ubiquitous and only some of it is made to matter.

Without normalisation an asymmetry may still be held in place, but only by continual re-imposition. The ordering does not stand on its own, so each occasion of its operation requires that it be asserted afresh against parties who experience it as imposed. Such arrangements exist and are recognisable, and what characterises them is expense: surveillance, enforcement, the maintenance of a coercive apparatus, and the periodic demonstration that the ordering still holds. The asymmetry does not cease, but it ceases to reproduce itself and must instead be reproduced by someone.

Without reproduction the asymmetry does not survive the parties to it. An ordering that is differentiated, selected, and normalised, but not transmitted, holds for a generation and lapses. This is the case of the charismatic arrangement that does not outlive its founder, the routinisation problem Weber [26] identifies, and its familiarity is evidence for the necessity of the fourth register rather than against it.

9.4 Limits of the Argument

Sufficiency is not claimed, and the reasons should be stated rather than left to inference.

The four registers may be present and the asymmetry fail to persist for causes lying outside the composite: exogenous shock, the arrival of parties not subject to the ordering, the exhaustion of a material base on which the whole arrangement depended. The schema states what a durable asymmetry is assembled from, not what guarantees that an assembly of those parts will hold.

Nor does the argument from necessity establish that the four registers are the right decomposition rather than one decomposition among several. A different partition of the same passage, into three registers or five, might do the same work. What recommends this one is that each register corresponds to a distinct kind of relational fact and that the necessity argument runs separately for each, but neither consideration excludes alternatives.

9.5 Instantiation Across Domains

The schema is offered as general, and the case for generality rests on instances.

In credentialism, analysed at length by Bourdieu and Passeron [2], differentiation is the variation in training and aptitude; selection is the examination that ranks it; normalisation is the treatment of the credential as evidence of what it certifies rather than as a convention about who may practise; reproduction is the transmission through institutions that require the credential and schools that supply it.

In colonial administration, differentiation is the difference in armament, organisation, and disease exposure; selection is the conversion of that difference into a claim of authority; normalisation is the elaboration of a doctrine under which the authority is proper rather than merely effective; reproduction is the administrative apparatus, the schooling of a local intermediary class, and the legal forms that outlast the personnel.

In scientific prestige, differentiation is the variation in what is discovered; selection is peer judgement; normalisation is the treatment of eminence as tracking merit rather than as constituting it; reproduction is the training of students by the eminent and the transmission of position through appointment.

In market position, differentiation is the variation in efficiency and holdings; selection is competition; normalisation is the treatment of the resulting distribution as an outcome of desert or of natural process; reproduction is the reinvestment of returns and the inheritance of capital.

The instances differ in almost every respect that matters descriptively, and in the four registers they agree. That agreement is what the generality of the schema amounts to.

9.6 Identification of the Load-Bearing Register

The registers are not equally exposed, and the asymmetry among them has a consequence for intervention.

Differentiation cannot be removed. Parties differ, and an arrangement that prevented them from differing would have to prevent relation from being produced at all. Selection can be altered but not abolished, since any arrangement in which some differences are consequential involves a selection, and the question is which one rather than whether. Reproduction can be interrupted, and much redistributive policy consists in interrupting it, but interruption must be repeated in each generation and the interruption is itself expensive.

Normalisation is different in kind, and the distinction of §6 says why. Closure is achieved by the composite as a whole; maintenance is what the composite must further withstand, and normalisation is the register through which it withstands it. An ordering taken as unremarkable is not defended against each new difference the field produces, because new difference is read through the ordering rather than tested against it. This is the same operation §7 described as institutional filtering: once an ordering has passed into the subrepertoire through which an institution acts, background transitions arrive already interpreted. Alone among the four registers, normalisation has no material substrate. Differentiation is grounded in the actual variation among parties, selection in the institutions that rank them, reproduction in the mechanisms of transmission. Each of these three can be pointed to independently of what anyone believes about it. Normalisation cannot: it consists in the understanding under which the ordering is taken as unremarkable, and an understanding is sustained by nothing except its continuing to be held. The asymmetry among the registers claimed here is this one, and it is offered as an observation about the instances considered above rather than as a theorem about registers in general.

The consequence is illustrated in Figure 8. Where normalisation is removed, closure may still be achieved but maintenance is not: the composite returns to its source under its own operation and no longer withstands what the field supplies, so each new difference must be met as a difference rather than absorbed as a confirmation. The ordering may still be maintained, and frequently is, but its maintenance now requires that it be asserted rather than assumed, and the assertion must be repeated indefinitely. What was reproduced becomes something that must be continually re-imposed, and the difference between these is the difference between an arrangement that costs nothing to hold and one that costs something every cycle.

Figure 8 (The effect of removing normalisation, under a constructed model). The left panel shows that the ordering may be held at the required level in either case. The right panel shows what holding it costs: with the composite intact the required re-imposition falls toward nothing, since the ordering stands of itself; with normalisation absent the ordering decays between cycles and the cost of restoring it does not fall. The model is illustrative and its parameters are stipulated; no claim is made that any actual arrangement follows these trajectories.

This identifies the register at which intervention is cheapest, and it does so without recommending any intervention. The finding is structural: an asymmetry deprived of its normalisation is not thereby ended, but it is converted from something that reproduces itself into something that must be reproduced by an identifiable party at a recurring cost, and arrangements of the second kind are considerably easier to end than arrangements of the first. This is also why contests over durable asymmetries are so often contests over description rather than over distribution, a fact that accounts treating power as a quantity have difficulty explaining and that follows directly here.

§10 Scope and Directions

An account of this kind is best judged by what it declines to claim as much as by what it asserts, and several sections above deferred a limit to this one. They are collected here, together with the work the account leaves open.

10.1 Limits of Recoverability

It is tempting to read the framework as offering an inverse problem: given an observed history of a social arrangement, recover the repertoire and the transitions that generated it. The temptation should be resisted, and the reason is not that the problem is difficult.

The problem is underdetermined in a way that no quantity of data relieves. Any observed sequence of arrangements is compatible with many generating repertoires, since the operations available to a party are not observed directly but only through the occasions on which they are exercised, and an operation that is available and never exercised leaves at best indirect evidence of its availability. Two arrangements identical in every observed respect may differ in what was possible but not done, and the difference is precisely what the account holds to be decisive. The situation is worse than the corresponding one for differential models, where at least the state is observed even if the vector field is not.

What follows is that positing a repertoire is a theoretical commitment rather than an inference. The account does not claim otherwise, and the claim it does make is a different one: a posited generating set is more falsifiable than a posited quantity, because relations among the generators forbid certain sequences outright, and a single observed occurrence of a forbidden sequence refutes the posit. A quantitative model accommodates an anomalous observation by adjusting a parameter. A generative posit that forbids what is observed to occur has been refuted. This is a claim about the form of the two kinds of theory rather than a result, and it is taken up below as a direction rather than a finding.

10.2 Capacities Relinquished by the Account

Three things available in the languages of §2 are not available here, and the account is worse than they are in each respect.

There are no quantitative dynamics. The framework supports no differential equation, yields no trajectory, and permits no calculation of how fast anything happens. Where a question concerns rates, the differential languages are the appropriate instrument and this one is not.

There is no spectral theory and nothing corresponding to it. The structure theorems that make operator algebra powerful were purchased by the axioms §3 found unavailable, and declining the axioms means declining the theorems. Nothing here decomposes an arrangement into components, identifies invariants, or classifies arrangements up to any equivalence that a theorem certifies.

There is no prediction. The account states a condition under which an asymmetry persists and does not determine which asymmetries in a given arrangement satisfy it. It is a framework for describing and distinguishing rather than for forecasting, and its usefulness stands or falls on whether the distinctions it draws are the right ones.

A further limit arises from the strengthened condition of §6. Maintenance is defined against the background transitions a setting admits, and the framework does not determine that class. Which perturbations an actual arrangement must withstand is a question about the setting, to be settled by whoever knows the setting, and an account that fixed the class in advance would be legislating rather than describing. The condition is therefore well posed only relative to a specification the framework does not supply.

To these should be added the two limits recorded earlier. The criterion of §7 for separating regimes is a distinction and not a test, since determining whether a bridging sequence of transitions exists is not made tractable by anything in the construction. And the index analogue that would measure institutional blindness was described but not constructed, so the comparison it would license is at present unavailable.

10.3 Proposal of the Generating-Set Inversion

The direction most worth pursuing is the one identified above and not there developed.

Explanation in the differential languages runs from a law of motion to a trajectory: the vector field is posited and the observed history is what it produces. The inversion proposed here runs from a generating repertoire to the admissible compositions and thence to the possible histories, so that what is posited is not how the arrangement moves but what may be done within it and in what combinations.

The two are not equivalent in explanatory content. A vector field is compatible with any history it produces and forbids nothing that it does not produce; a generating set with relations among its generators forbids compositions outright, and the forbidden compositions are commitments the theory can be held to. The proposal is that this makes the second form of theory more falsifiable in a specific sense, and the proposal has not been tested. What would test it is a case in which relations among posited generators yield a prohibition that survives confrontation with the record, and no such case is offered here. Until one is, the inversion is a direction rather than a result, and the paper marks it as such.

10.4 The Unresolved Coupling of Fast and Slow Change

The framework’s principal unresolved difficulty concerns the relation between two kinds of change it holds apart.

Within a repertoire, ordinary social dynamics proceed: parties act, distributions shift, the events that occupy most social description occur. Between repertoires, the operational field is itself transformed. The double category of §5 presents these as horizontal and vertical movement respectively, and the commuting square records that an operation performed within a repertoire is what carries the repertoire to its successor. That is a statement of compatibility rather than a theory of the coupling.

What is missing is an account of the timescales. Ordinary dynamics are fast and structural transition is slow, and the two are not merely different in rate but different in kind, so the standard techniques for separating fast and slow variables do not apply. A transition is not the accumulation of many small horizontal movements; it is a change in what horizontal movement consists in. How a great many exercises within a repertoire combine into a transition of it, and under what conditions they fail to, is not answered by anything in this paper.

The question matters because the cases of §1 turn on it. An asymmetry that consumes its own preconditions does so through the accumulated effect of ordinary exercise, and the account states the outcome without describing the accumulation. Supplying that description would convert several of the paper’s distinctions from classifications into mechanisms, and it is the work this paper most wants done.

10.5 Standing of the Account

The account is offered as a conceptual framework with a formal presentation, and the two should be assessed separately.

The mechanism of §4 stands or falls on its own. It claims that asymmetry arises from ordered difference rather than from difference alone, that exercise alters the conditions of exercise, that most asymmetries consume themselves, and that what remains is what reproduced its own conditions. A reader may accept all of this and reject the construction that follows.

The construction of §5 is offered as a way of stating that mechanism without ambiguity, and its claim is modest. It is assembled from standard ingredients and contributes no technical novelty; what it contributes is a decision about which structure is primary, namely the party-indexed availability of operations, with rewriting and composition as what that structure undergoes. A reader may accept the construction and dispute the schema of §9, which is a further and more exposed proposal.

The result the account would most want defended is the negative one of §8. Generativity is not the enlargement of possibility, and a quantity computed over an arrangement as a whole does not distinguish the cases the account must distinguish. That result requires none of the formal apparatus, survives the rejection of everything else in the paper, and is stated as an invitation to counterexample.

---

A note on method. The formal apparatus of this paper is deliberately thin, and the thinness is a judgement rather than an omission. Two of the paper’s results, the reconstruction of the self-maintenance thesis as a selection effect and the refutation of the counting criterion, require no apparatus at all and are stated in §4 and §8 in terms a reader may assess without accepting any construction. The construction of §5 is introduced to state the remaining distinctions unambiguously, and it goes no further than that use requires. A more elaborate presentation was available and was declined: the categorical machinery admits considerable development, and developing it in advance of results that would need it would have borrowed a rigour the argument has not earned. Where the account has reached for an established formalism and found it unsuitable, as with operator algebra in §3, the reasons have been given at length rather than passed over, on the view that a declined borrowing should be visible in the record.

References

[1] Arendt, Hannah (1970). On Violence. New York: Harcourt, Brace & World.

[2] Bourdieu, Pierre, and Jean-Claude Passeron (1990). Reproduction in Education, Society and Culture. 2nd ed. Trans. Richard Nice. London: Sage. [French orig. 1970.]

[3] Bratteli, Ola, and Derek W. Robinson (1987). Operator Algebras and Quantum Statistical Mechanics 1: $C^$- and $W^*$-Algebras, Symmetry Groups, Decomposition of States*. 2nd ed. New York: Springer.

[4] Dahl, Robert A. (1957). The Concept of Power. Behavioral Science, 2(3), 201–215.

[5] Ehresmann, Charles (1963). Catégories doubles et catégories structurées. Comptes Rendus de l’Académie des Sciences de Paris, 256, 1198–1201.

[6] Emerson, Richard M. (1962). Power-Dependence Relations. American Sociological Review, 27(1), 31–41.

[7] Foucault, Michel (1977). Discipline and Punish: The Birth of the Prison. Trans. Alan Sheridan. New York: Pantheon. [French orig. 1975.]

[8] Foucault, Michel (1978). The History of Sexuality, Volume 1: An Introduction. Trans. Robert Hurley. New York: Pantheon. [French orig. 1976.]

[9] Haag, Rudolf, and Daniel Kastler (1964). An Algebraic Approach to Quantum Field Theory. Journal of Mathematical Physics, 5(7), 848–861.

[10] Hodgkin, Alan L., and Andrew F. Huxley (1952). A Quantitative Description of Membrane Current and Its Application to Conduction and Excitation in Nerve. Journal of Physiology, 117(4), 500–544.

[11] Jones, Vaughan F. R. (1983). Index for Subfactors. Inventiones Mathematicae, 72(1), 1–25.

[12] Lukes, Steven (2005). Power: A Radical View. 2nd ed. Basingstoke: Palgrave Macmillan. [1st ed. 1974.]

[13] Lacan, Jacques (2006). Écrits: The First Complete Edition in English. Trans. Bruce Fink. New York: Norton. [French orig. 1966.]

[14] Luhmann, Niklas (1979). Trust and Power. Trans. Howard Davis, John Raffan, and Kathryn Rooney. Chichester: Wiley. [German orig. 1975.]

[15] Luhmann, Niklas (1995). Social Systems. Trans. John Bednarz Jr. with Dirk Baecker. Stanford: Stanford University Press. [German orig. 1984.]

[16] Mac Lane, Saunders (1998). Categories for the Working Mathematician. 2nd ed. New York: Springer.

[17] Merton, Robert K. (1938). Social Structure and Anomie. American Sociological Review, 3(5), 672–682.

[18] Nietzsche, Friedrich (1998). On the Genealogy of Morality. Trans. Maudemarie Clark and Alan J. Swensen. Indianapolis: Hackett. [German orig. 1887.]

[19] Nietzsche, Friedrich (2002). Beyond Good and Evil: Prelude to a Philosophy of the Future. Ed. Rolf-Peter Horstmann and Judith Norman, trans. Judith Norman. Cambridge: Cambridge University Press. [German orig. 1886.]

[20] Osborne, Martin J., and Ariel Rubinstein (1994). A Course in Game Theory. Cambridge, MA: MIT Press.

[21] Reisig, Wolfgang (2013). Understanding Petri Nets: Modeling Techniques, Analysis Methods, Case Studies. Berlin: Springer.

[22] Rozenberg, Grzegorz, ed. (1997). Handbook of Graph Grammars and Computing by Graph Transformation, Volume 1: Foundations. Singapore: World Scientific.

[23] Takesaki, Masamichi (1972). Conditional Expectations in von Neumann Algebras. Journal of Functional Analysis, 9(3), 306–321.

[24] van der Schaft, Arjan, and Hans Schumacher (2000). An Introduction to Hybrid Dynamical Systems. London: Springer.

[25] Wasserman, Stanley, and Katherine Faust (1994). Social Network Analysis: Methods and Applications. Cambridge: Cambridge University Press.

[26] Weber, Max (1978). Economy and Society: An Outline of Interpretive Sociology. Ed. Guenther Roth and Claus Wittich. Berkeley: University of California Press. [German orig. 1922.]

---

中文

权力的一种语法

剧目与不对称的再生产

黄万宏 · huangwanhong@serendip.ngo

---

摘要

能力上的诸不对称在社会生活中持续不断地生起，而它们中的大多数消散。少数不消散。以一个行动者所支配的资源来度量权力的诸说明，无法说出哪些将会如何，并且被这样的案例所困窘：一个单薄的行政当局比一个压倒性的军事优势更持久，或一个没有任何强制能力的团体比每一个容忍它的政权都更长寿。本文把这一差异定位在一个不对称之行使对它自身行使之诸条件所做之事上：有些行动消耗使它们成为可能的那些前提条件，而另一些则制造它们。权力，依此处所提出的说明，是那个既在它自身的行使中、又在它必须据以维持的差异之持续生产中存活下来的不对称。本文首先概览可用于描述社会动力学的诸形式语言，并发现每一个都把可变性定位在一个层级、而固定它上面的那个层级，以致没有一个把一个系统的语法弄成一个运动中的对象；算子代数最接近这一要求，被评估并因详加给出的诸理由而被搁置。随后，一个由带类型的操作剧目、以及它们在自身上所诱导的诸转变所构成的结构被建构出来，而那个机制在其中被重述。有两个后果随之而来。尼采式的论题，即权力在于它自身的维持，被重构而去掉了承载它的那个意志：自我维持被当作那个关系的构成性的东西、而不是当作任何安排所追求的东西，而维持的稀有，尼采读作软弱，在此则由那个任何安排都必须据以维持的差异之持续生产所解释。而生成性被表明并不化约为一个格局所容许的诸可能性之数目：宪法性的禁止收缩可用行动之场域、却仍然是生成性的，而诸规范的崩塌扩张它、却不是。把这两者区别开来的，是谁的诸可能性被扩大，以及所生成之物是否返回生成它的那个关系。为此所作的论证是通过反例、而不是通过一个一般的不可能性结果给出的，而本文明确地说出它的诸主张中哪些是被确立的、哪些是被提议的。

关键词： 权力；不对称；关系性生成；剧目；自我再生产；生成性；社会动力学的形式建模。

§1 引论

权力已被关系性地理解了数十年之久。依此理解，它不是一个行动者持留作储备的一种实体，而是某种只在诸方之间的关系之中、并且只在它的行使之中存在的东西；它是弥散的、而不是集中于一个中心的，是生产诸主体与诸范畴的、而不仅仅是压制它们的，并且是内在于社会生活据以被展开的诸实践的、而不是自外部被强加于它们的。[此处的诸表述最贴近福柯 [8, 7]。阿伦特 [1] 经由一条不同的路线抵达一个关系性的说明，把权力定位于协同的行动、而不是任何一方强迫的能力之中，而爱默生 [6] 通过依赖使那项关系性承诺可运作，这是一条在瓦瑟曼与福斯特 [25] 所概览的网络文献中被发展的线索。与韦伯 [26] 和达尔 [4] 那些较老的、基于能力的表述的对比，正是那个关系性转向之所在；卢克斯 [12] 绘制了由此而来的诸争论。] 那个转向是有益的，并且在此不受质疑。受质疑的是：那个关系性的说明尚未被给予一种适足于它自身本体论的形式语言，至少在那些塑造了该领域的诸处理中是如此。凡权力被定量地建模之处，它一般而言毕竟被建模为一个量，无论是一份资源禀赋、一个网络中心性，还是一个讨价还价的权重，而这在形式体系中恰恰归还了那个关系性转向在理论中所移除之物。这个缺口不是任何一方在严格性上的失败。它反映一种数学语言的缺席，在其中一个关系、而不是一个量值，可以是那个运动中的对象。

本文所接过的困难，无需任何形式体系就可以被看见，在那些量级极为不同的诸不对称遭遇极为不同之命运的案例中。首先考虑一个已对它的诸邻邦和它自己的人口达成压倒性军事优势的国家，以及一个以几千名官员和一支小到无法据任何一省以抗一场决意之起义的驻军来治理一片数百万人之领土的殖民行政当局。以任何所支配资源的度量衡量，第一个不对称都以宽阔的差距为更大者。然而这一类军事至上已一再被证明属于最易消逝的诸优势之列，耗尽维持它的那个财政基础，并激起终结它的那些联盟，而单薄的殖民行政当局在数个实例中已持续了将近一个世纪。其次考虑一个持有其市场之支配份额的公司，以及一个证明从业者、却不能强迫任何人做任何事的专业团体。第一个支配资本、分销以及把对手挤出定价的能力；第二个只支配一份资格证书的认可。支配性的市场份额已被证明在数十年的时间尺度上是不稳定的。证明团体在数个案例中已比它们据以被创立的那些政权、宪法与经济秩序更长寿。

一个以资源来度量权力的说明，在每一对中都预测相反者。那个自然的修补是说，资源不是一切，而合法性、制度化或嵌入性必须被添加到该说明中。这是真的、而不充分的。它点出更多有待被添加到那个账本上的量，却不说出为何那个账本是错误的工具，而它把这些案例实际所共有的那个特征留如不动，那个特征不是任何东西的量级，而是一个行动与它自身诸前提条件之间的一个关系。压倒性的军事优势是通过征战而被行使的，而征战消耗至上由之被建立起来的那个财政与人口基础；那个优势的行使从那个优势的诸条件中减去。一份资格证书是通过被要求而被行使的，而它被要求的每一个场合都确认它所标记的那个区分是一个真实的区分；那个优势的行使向那个优势的诸条件添加。这两个案例的区别不在于持有多少，而在于当被使用时那个持有对它自身所做之事的正负号。

这提示了本文取为它自己的那个问题。能力上的诸不对称在凡关系被生产之处持续不断地生起，而它们中的绝大多数消散而不留下制度痕迹。一小部分少数不消散。如果把那个少数区别开来的，是那个不对称的行使再生产、而不是消耗它行使的诸条件，那么权力就不是一个有待被度量的量值，而是一个操作与在它之后仍然可用的那个操作之场域之间的关系的一个属性。精确地陈述那个属性，要求一种语言，在其中可用操作之场域本身就是那个改变着的东西，而这正是可用的诸形式语言所不提供的。微分与随机模型固定一个状态空间、并在其内移动一个点。混合模型添加诸跳跃、却把那个跳跃规则自外部贴上。基于主体的模型让一个拓扑重新布线、同时固定那个可被实例化的诸关系之空间。博弈论模型精确地表示规则遵循、并把那些规则作为外生输入接收。改写系统真正地生成，而生成句法、而不是负有不同能力的诸方之间的诸关系。算子代数最接近于一切，把可复合的操作取为原始的、并从它们导出那个轨迹、而不是相反，而它以它自己的方式被固定：它的诸生成元与诸关系是预先给定的，正如一个微分模型的诸坐标是预先给定的。每一种语言都把可变性定位在一个层级、而固定它上面的那个层级。没有一个把那个语法本身弄成运动中的对象。

此处所发展的说明做到了。它的核心建构是一个操作剧目，被理解为一个由部分可复合的操作所构成的、带类型的结构，连同诸操作在它们由之被抽取的那个剧目上所诱导的诸转变。依此建构，一个不对称是一个权力关系，当它既被它自身行使所诱导的那个转变、又被与那个场域此间所提供的诸转变的复合所归还之时。第一个是闭合，并陈述那个构成性的主张；第二个是维持，并陈述闭合必须承受什么。权力是维持，而这两者之间的区别正是那个允许该说明去记录一个完美地再生产它自身、却被别处所生起之物所解除的安排的东西。

三个后果被推出，而值得在此陈述它们、而不是保留它们。第一个关乎一个下承于尼采、并在系统理论及别处以剥去了它原初模态的形式复现的论题：即权力在于它自身的维持。那个论题是对的，而那个模态是那个困难。尼采让权力延伸并释放它自身，而持续作为一个后果随之而来；较后的诸表述保留那个自我维持而去掉那个意志，却对”维持何时成功、为何成功”说得很少。此处所提出的读法是：自我维持是构成性的、而不是有目的的。一个不对称只在行使中存在，而行使改变行使的诸条件，因此一个不对称与它自身的延续之间的关系是定义性的、而不是因果性的，而权力是那个关系闭合的那个情形。当前框架所添加的，是一个关于为何那个闭合是稀有的说明。因为关系被持续地生产，有序的差异持续而廉价地生起，而那个场域并不在任何安排之下静立不动。因此，维持是针对一个富有产出的背景所执行的工作、而不是在一个静态背景中的持续，而不对称对于权力是不充分的，因为不对称是丰盈的。

第二个后果是一个否定结果，而它是本文最想要被辩护的那个。把生成性等同于可能性的扩大、把退化等同于它的限制，是诱人的，以致一个格局在它所容许的诸选项的比例上是生成性的。这个等同在两个方向上都失败。一个宪法性的禁止从那个场域中移除操作、并且属于已知最具生成性的诸安排之列；共有诸规范的崩塌扩大那个作为整体的场域，因为先前所持的诸约束停止所持，而它是退化性的。第二个案例所隐瞒的、并且 §8 所列出的，是那个扩大累积给那些已然被置于可行动之地位者，而其他每个人的诸能力收缩。因此，生成性既不化约为可用选项之数目、也不化约为熵、也不化约为任何算数的意义上的自由。把这两者区别开来的，是谁的诸可能性被扩大，以及那个安排所生成之物是返回生成它的那个关系、还是被转向那个关系并不要求的一个目的。

第三个是一个图式。本文论证，持久的诸不对称是由跨越彼此有别的诸层域而工作的四个操作所装配而成的：一个产生变异的分化，一个对那个变异排名以致一方的差异授予另一方的差异所不授予之物的选择，一个把那个排名弄得不足为奇的规范化，以及一个传递它的再生产。这些层域在种类上是彼此有别的，分别是一个分布、一个排序、一个意义与一个继承，而那个复合在它们之间穿越、而不是停留在任何一个之内。这个图式是作为一般的、而不是作为例示的被提出的，而每一个层域之必要性的论证是逐案给出的。它最有用的后果关乎干预：规范化是唯一一个没有物质基底的层域，而一个被剥夺了它的不对称并不停止、而是必须被持续地重新强加，而这是一件不同于、并且比被再生产更昂贵的事。

本文如下推进。§2 沿一条共同的轴线概览可用的诸形式语言，即每一个把什么取为原始的、以及它因此固定什么的那条轴线，并从那个概览导出任何适足的语言必须满足的诸要求。§3 对照那些要求评估算子代数，发现它满足它们、却仍然无法充任，并详加给出那四个障碍。§4 在哲学上、并且不带形式装置地陈述那个机制。§5 建构那个形式对象，而 §6 在它之内重述那个机制。§7 把三个从算子代数借来的区分应用于那个被建构的对象，而那正是它们能够做它们早先所不能做的工作之处。§8 列出对计数判准的反驳。§9 一般化到持久不对称的那个图式。§10 陈述那个框架所放弃之物、以及它所留下之物。

§2 现有诸形式语言的概览

2.1 比较的诸判准

此处所概览的诸语言是为不同的目的而被建造的，并且不处于竞争之中。比较它们要求一条轴线，在其上每一个都能够被安放而不失真。全篇所使用的那条轴线是由该语言的原始对象及其常驻承诺所构成的那一对：即那个形式体系取什么为给定的，以及它因此固定什么。每一个形式体系都必须持某种东西不动、以便让别的东西运动，因此一项承诺的在场本身不是一项缺陷。要紧的是那项承诺相对于手边的现象落在何处。

一条有用的诊断贯穿整个概览。在以下每一个规范形式中，某些符号携带一个时间指标而另一些不携带。那些携带一个的，是该语言所允许运动的东西；那些不携带的，是那项承诺。读那些下标就足以把每一种语言定位在那条轴线上。

2.2 微分与随机诸语言

常微分方程与偏微分方程，连同它们的随机与延迟变体，把一个被定位的点取为原始的，并在一条运动定律之下演化它。那个规范形式是

$$\dot{x}(t) = f\bigl(x(t)\bigr), \qquad x(t) \in M,$$

其中状态空间 $M$ 与向量场 $f$ 二者都是预先给定的。只有 $x$ 携带那个指标。在那项承诺之内，这些语言是有力的：它们表达连续的定量变化，产出诸平衡及其稳定性，并提供诸速率，而这没有任何这个清单上的别的语言以可比的轻易提供。

复合躺在上面那个方程所能表达之外。一个轨迹记录一个系统所占据过的诸位置，而不是被施行于它之上的诸操作、或它们施行的次序，而这两者不可互换。设 $\Phi_a$ 与 $\Phi_b$ 记两个彼此有别的向量场的单位时间流映射，表示两个社会操作。不等式

$$\Phi_a \circ \Phi_b \neq \Phi_b \circ \Phi_a$$

一般而言确实成立，因此次序依赖性是在场的。然而那个不等式之成立，是作为那个建模者所提供的诸向量场的一个后果、而不是作为那个语言所表示的一个结构性特征。那个方程中没有任何东西把一个操作与任何别的位移区别开来，而可容许性不是该语言的一个结构性特征。人们当然可以限制一个流映射的定义域并因此禁止一个复合，而诸模型例行地这样做；那个限制是被添加到那个方程上的一个规定、而不是那个形式所表达的某种东西，而它不随系统的演化而改变。§1 的那个问题恰恰以这些术语被陈述。一个消耗它自身诸前提条件的行动，是一个在它之后某些进一步的行动不可用的行动，而这是一个关于可容许性、而不是关于位置的主张。

一个自然的回应把那些规则带入那个状态空间之内。如果一个制度性安排治理哪些行动是可用的，人们可以写 $M’ = M \times R$，其中 $R$ 是一个诸安排的空间，并演化那一对。这总是可能的，而它无济于事。那个被扩大的空间 $M’$ 在那个 $R$ 因子上没有自然的坐标，一个安排并不是一个量；$M’$ 的维数不是固定的，因为诸安排把诸区分带入存在、并使别的诸区分退役；而那个扩大丢弃了那个促动它的复合结构，留下一个点移动通过一个其诸轴无法被命名的空间。这个花招以那个表示本应提供的解释为代价，保得了一个形式表示。

2.3 混合与分段诸语言

混合系统通过容许不连续的诸转变而扩展那些微分语言 [24]。漏积分-发放模型、混合自动机、碰撞力学与分段光滑系统，全都取一个元组的形式

$$H = \bigl(M,, f,, G,, R\bigr),$$

在其中那个状态依 $\dot{x} = f(x)$ 而流动，只要 $x \notin G$，而在抵达那个卫集 $G$ 时被那个重置映射 $R : G \to M$ 赋予一个新的值。这个扩展是真正的，并处理一大类纯微分模型处理得很差的现象。

它的承诺容易被错失。那个元组的四个组成部分中，没有一个携带一个时间指标。那个重置映射 $R$ 是与那个流并列地被规定的、而不是由它所生成的，因此一个混合模型含有一个那个数学所治理的部分和一个那个建模者所提供的部分，而那个被提供的部分本身不改变。那个状态空间同样在那个跳跃之间被固定：$R$ 映到 $M$ 之内，因此那个轨迹抵达那同一个空间之内的别处、而不是那个空间先前所不含的某处。值得对比其中根本没有任何不连续出现的那个替代处理，霍奇金与赫胥黎 [10] 那个陡峭但光滑的电导动力学是那个规范实例；在那里那个跳跃是时间尺度的一个赝象、而不是那个模型的一个特征。混合语言捕捉打断、而不捕捉结构性变化。对于当前的问题，这是错误的扩展。那个困难不是社会变化是不连续的，那是混合模型会容纳的，而是一个不对称的行使改变哪些行使仍然可能，而这是 $R$ 中的一个变化、而不是它之下的一个跳跃。

2.4 基于主体与网络诸语言

基于主体与网络的模型把主体与边取为原始的，并允许那个拓扑改变 [25]。写 $V$ 为诸主体之集、$T$ 为该模型所能表示的诸关系类型之集，在时刻 $t$ 的一个格局是一个图

$$G_t = \bigl(V,, E_t\bigr), \qquad E_t \subseteq V \times V \times T,$$

由一个重连规则 $\rho : G_t \mapsto G_{t+1}$ 所演化。这是这个清单上第一个其中某种结构性的东西运动的语言，而它因那个理由赢得它的位置。这一类模型以那些聚合语言无法匹敌的忠实度，捕捉从局部互动中生起的涌现宏观模式。

那个下标模式，如图 1 所示，精确地定位那项承诺。那个边集 $E_t$ 携带一个指标，而那个类型集 $T$ 不携带，$\rho$ 也不携带。因此，一个网络模型可以创造一条先前不存在的边 $(u, v, \tau)$，但 $\tau$ 必须已经属于 $T$：那条新的边实例化一个该模型已经知道如何表示的关系。一个不在 $T$ 之中的种类的关系无法进入存在。

图 1（网络诸语言的常驻承诺）。 重连规则 $\rho$ 可以添加一条其类型已经在 $T$ 之中的边，示于中央。一个不在 $T$ 之中的种类的关系 $\sigma$，示于右侧，无法被带入存在，因为 $T$ 不携带时间指标。

这个区分在当前的语境中不是迂腐的。当一个证明团体被创立时，那个变化不是新的诸链接在现有的诸节点之间出现。一个先前没有任何表示的关系，即被…所证明、以及在…之下被取消资格，变得可用，而随之诸事实，它们先前不是假的、而是不可表述的。那个要求是 $T$ 携带那个指标、而不是 $E$，而没有任何网络形式体系提供那一点。

2.5 博弈论与机制设计诸语言

诸博弈把策略集与支付函数取为原始的 [20]：

$$\Gamma = \bigl(N,, {S_i}{i \in N},, {u_i}{i \in N}\bigr).$$

这些是在精神上最接近当前问题的诸语言，因为它们独独表示那些在选择做什么时预期彼此之回应的诸方。规则治理的变换在这个清单上的别处是可用的；策略性的预期不是。扩展式博弈表示序列，因此一个有限的复合结构是可用的，而机制设计更进一步，把那些规则当作选择的对象、而不是当作固定的背景。

那项承诺是那个规范 $\Gamma$ 本身，它作为外生输入而进入。在那个意图的范围之内，这不是一项缺陷，一个机制设计者按构造是外在于那个机制的，但它把规则变化定位在系统之外、而不是它之内。那个标准的扩展把规则变化建模为一个元博弈

$$\Gamma’ = \bigl(N,, {\Sigma_i}{i \in N},, {u_i’}{i \in N}\bigr), \qquad \Sigma_i \subseteq {\Gamma_1, \Gamma_2, \ldots},$$

它的诸参与者在诸制度性安排之间选择。

图 2（元博弈的无穷回退）。 每一个层级都把它下面那个层级的规范当作选择的对象、而固定它自己的，因此那项承诺在每一步被重新安置、而在没有一步被卸除。

这个花招重新安置那项承诺而不卸除它，如图 2 所记录的：那个规范本身是固定的，而迭代产生具有同样属性的 $\Gamma’’$。那个回退的没有一个层级本身是内生的。当前的问题要求可用操作的更改由那些同样的操作所执行，而这是参与者与设计者之间的那个分离所禁止的。

2.6 改写系统与进程演算

改写系统把生产规则取为原始的，并且是本文所提议之物的最近亲属。形式文法、项改写与图改写 [22]、Petri 网 [21] 与进程演算，全都由一个生产之集所规定

$$\mathcal{R} = \bigl{, \ell_k \rightarrow r_k ,\bigr}_{k \in K},$$

在其之下，一个格局 $c$ 变换为 $c’$，只要某个 $\ell_k$ 匹配 $c$ 的一个子项。这些语言生成、而不是演化：一个格局是一个被结构化的对象、而不是一个点，而一个规则把它变换为另一个。它们携带一个原生的良构性概念，因此并非每一个变换的序列都是可容许的，而复合因此是部分的、而不是全部的。以下所导出的三个要求全都被改写系统所满足，而任何当前这一类的说明都欠一个关于它为何不直接使用它们的解释。

那个解释在于那个生产之集的匿名。一个文法改写字符串，一个图改写系统改写图，一个进程演算改写进程项，而在每一种情形中，那个被变换的对象是句法性的，而那些规则应用于凡一个模式匹配之处。取一个宪法性禁止的现象，它从可被做之物中移除一个操作，比方说一次选举的中止。一个图改写系统把这表示为从 $\mathcal{R}$ 中删除一个生产，而那个表示就其所及是忠实的。它把那个使那个禁止有趣的事实留作未陈述：那个操作从一方被移除，而每一个其他方的诸可能性因此被扩大。那个规则集收缩了，但一个规则集不属于任何人，而”谁的能力被减少、谁的被扩展”这一问题，在一种其规则是匿名的语言中没有任何表述。

图 3（在那两种呈现之下一个生产的删除）。 在左边那个规则集收缩，而”谁的能力被减少”这一问题没有任何表述。在右边那同一个删除被归属，而每一个其他方之相对位置的扩大变得可陈述。

进一步的编码不缓解这一点。人们可以通过标注把诸规则归属于诸方，把那个匿名之集替换为

$$\mathcal{R}^{\ast} = \bigl{, \ell_k \xrightarrow{;p_k;} r_k ,\bigr}_{k \in K}, \qquad p_k \in P,$$

其中 $P$ 是一个诸方之集，而 $p_k$ 命名被允许发放那个生产的那一方。那个标注是可用的，而它恰恰是那个要点。这所添加的，是一个其中诸操作被那些负有它们者所指标化的结构，而那个结构、而不是那个改写，正是该说明的内容之所在。一个如此被标注的系统是在成为 §5 中所建构的那个对象的途中，而它们之间的区别是哪一个结构被当作首要的。此处的说明把方指标化的可用性当作首要的对象、而把改写当作它所经历之物。没有任何东西阻止对同样内容的一个改写论式的呈现；那个主张是那个内容不寓于那个改写之中。

2.7 算子代数诸语言

算子代数把可复合的操作取为原始的 [3]。一个 $C^*$-代数可以由诸生成元与诸关系所规定，

$$\mathcal{A} = \bigl\langle, G \mid R ,\bigr\rangle,$$

而它之上的一个动力学是一个单参数的自同构族

$$\alpha_t \in \operatorname{Aut}(\mathcal{A}), \qquad \alpha_s \circ \alpha_t = \alpha_{s+t}.$$

一个轨迹，凡想要一个之处，是从那个代数导出的、而不是相反。那个形式体系是无坐标的，在它不要求那个建模者命名诸变量的意义上，而非交换性是原生的，因此那个微分语言作为输入接收的次序依赖性，在此是那个对象的一个结构性特征。这些语言进一步以一种没有这个清单上的别的形式体系匹敌的精确性，把一个动力学的诸对称性与诸可观测量的诸对称性、以及一个态的诸对称性区别开来。

那项承诺在那两个展示式合起来看时是可见的。那个指标坐落在 $\alpha$ 上、而不在 $G$ 或 $R$ 上。从一个表示过渡到另一个，正如那个标准的建构通过把一个表示关联到每一个态所做的，把 $\langle G \mid R \rangle$ 留如不动：它展示单一一个以不同方式被实现的代数，而这是实现的可变性、而不是那个代数的可变性。算子代数固定诸生成元与诸关系，以一个微分模型固定诸坐标的那种方式，而二者都是一个固定结构之上的动力学。因为算子代数在任何已确立的语言之中最接近此处所要求之物，也因为它是任何趋近这个问题者最可能伸手去取的语言，它无法充任的那些理由被分开地、并且详加地列于 §3。

2.8 诸要求的导出

下面的表格记录那个概览，而图 4 展示它的共同形状。那个模式在这六个条目之间是一致的：每一种语言都在一个层级容许可变性、而固定紧接其上的那个层级。微分模型移动一个点而固定那个空间；混合模型不连续地移动一个点而固定那个重置映射；网络模型移动一个边集而固定那个类型集；诸博弈移动诸策略而固定那个规范；改写系统移动诸格局而固定那个规则集，其诸规则不负任何方；算子代数移动诸表示而固定那个呈示。在没有任何一种情形中，那个系统的语法，意即什么可被做、由谁、以及以什么组合，成为那个运动中的对象。

图 4（那个概览的共同模式）。 在每一种语言中，那项承诺坐落在无论什么被允许变化之物上面一个层级，因此那个系统的语法在没有任何一点上是那个运动中的对象。

语言	原始	被固定	对当前问题不可用
微分、随机	点 $x(t) \in M$	$M$ 与 $f$	复合；可容许性无法被提出
混合、分段	带重置 $R$ 的点	$M$、$G$、$R$	重置被规定、而非被生成
基于主体、网络	$V \times V \times T$ 中的边	类型集 $T$；规则 $\rho$	只有新的实例；$T$ 不携带指标
博弈论、机制设计	策略集、支付	规范 $\Gamma$	内生的规则变化；元博弈的无穷回退
改写、进程演算	生产 $\ell \rightarrow r$	规则集 $\mathcal{R}$；句法对象	匿名规则；谁的能力改变了是不可陈述的
算子代数	可复合的操作	生成元 $G$、关系 $R$	一个固定的呈示是一个固定的语法

三个要求随之而来。第一个是那个原始的是一个可复合的操作、而不是一个被定位的点，那个现象以什么可跟随什么的术语被陈述。第二个是复合是原生地对次序敏感的、而不是靠规定，因为诸操作被施行的序列决定什么仍然可用。第三个是复合是部分的，因为并非每一个操作都可以跟随每一个别的操作，而可容许者与不可容许者之间的界限本身受制于改变。

一个第四要求被那个概览所蕴涵、而被它之中没有任何条目所满足，而它是那个促动 §5 之建构的那个。诸操作必须被诸方所负。一个对一方可用、而对另一方不可用的操作，是所研究的那个不对称的初等形式，而一种其诸操作不属于任何人的语言，无法陈述一个扩大一方可做之物的安排与一个扩大另一方可做之物的安排之间的区别。匿名的与方指标化的诸生产之集之间的对比，是所论的那个对比，而这个第四要求正是那个把当前说明与改写系统区别开来的东西，后者满足前三个。

§3 对算子代数候选者的评估

3.1 诸要求的满足

在所概览的诸语言之中，唯有算子代数无需修正就满足前三个要求。它的原始是一个可复合的操作。复合是原生地对次序敏感的，非交换性是那个主题由之取得它的旨趣的那个结构性特征、而不是一个被添加到它上面的属性。而一个由一组被规定的关系所生成的 $C^*$-代数容许此处所要紧的意义上的部分性：哪些乘积是非零的、哪些操作彼此湮灭、以及哪些复合被约束，是由那些关系所决定的、而不是逐案被规定的。

据此推进的诱惑是相当大的，而那些理由值得被明白地陈述、而不是被打发。那个先例是哈格与卡斯特勒 [9] 所作的量子场论的代数重述，在其中诸可观测量的代数、而不是任何特定的希尔伯特空间被取为首要的对象，而不等价的诸表示被读作彼此有别的诸相、而不是相互竞争的诸描述。算子代数携带一个发达的表示理论，以致一个结构可以以许多方式被实现；一个不等价性理论，以致诸实现可以被区别为属于真正分开的诸扇区；一个子代数以及投影到它们之上的诸映射的理论；以及属于一个动力学、属于诸可观测量、属于一个态的诸对称性的一个精确的分离。这些中的每一个都对本文的诸问题有一个明显的关涉，而 §7 返回到它们中的三个并把它们投入使用。此处所随之而来的，不是一个”算子代数对权力的研究没有任何可贡献”的论证。它是一个”它的诸公理无法承载本文所要求的那个对象、并且借用一个问题与采纳一个形式体系之间的区别是那个说明必须遵守、而不是模糊的一个区别”的论证。

四个障碍被给出。它们是独立的，而它们中的任何一个都会是充分的。

3.2 来自对合的障碍

一个 $C^*$-代数配备一个对合，一个满足 $(a^*)^* = a$ 并逆转诸乘积之次序 $(ab)^* = b^a^$ 的映射 $a \mapsto a^*$。那个对合不是一个附着于一个否则会无它而完备的代数之上的装饰。它是那个使自伴性、正性与态这些概念可用的东西，而这些转而是那个把那个代数连接到任何可被称为一个可观测量或一个概率之物的东西。

那个对合不容许任何可信的社会读法。设 $a$ 记一份出版物的压制。那个对合不是那个逆：压制无法被撤销，而那个代数无论如何也不要求 $a$ 是可逆的。它也不是某个社会诸态上的内积意义上的伴随，因为没有任何这样的积被定义，而定义一个会恰恰引入那个关系性说明本欲避免的那个定量结构。它也不是诸方的一个逆转，交换谁压制与谁被压制，因为那个操作不满足所要求的诸恒等式，并且在大多数情形中根本不存在。那些候选者被穷尽而没有一个可信的读法被找到。

那一点一般化到超出单一一个不幸的例子。极多的社会操作在一个强的意义上是不可逆的：它们以没有任何后续操作所恢复的方式改变此后可用之物。这一类的一个操作在一个逆转次序的对合之下没有任何自然的伙伴，因为那个本会把它与另一个配对的结构已被它自身的施行所消解。人们当然可以形式地强加一个对合，为每一个生成元声明 $a^* := a$ 并验证那些公理成立。这产出一个良定义的代数和没有任何解释：每一个元素由此而来的自伴性，如果它断言任何东西，会断言每一个社会操作是一个可观测量，而这以一种明显的方式为假、并以一种微妙的方式无用。

3.3 来自 $C^*$-恒等式的障碍

假设那个对合问题以某种方式被卸除了。仍然存在那个该主题由之取得它的名字的恒等式，$|a^a| = |a|^2$，而它不是若干公理之中的一个。它的后果是一个 $C^$-代数上的范数由那个代数结构所决定 [3, 第 2 章]。一个代数容许至多一个满足它的范数，因此那个度量与那个代数内容不是两块独立的数据、而是单一一个对象的两个方面。那个理论中一切僵硬的东西都下承于此。那个范数的唯一性给出诸同态的自动连续性；那个给出诸单射同态的等距性；而那些使那个主题有力的结构定理转而随之而来。

对于一个操作的社会剧目，没有任何这样的范数是可用的。量 $|a|$ 会不得不度量一次压制的量级、或一次证明之行动的大小，而没有任何候选者自荐，它不是要么任意的、要么一个被夹带进来的资源量。一个资源量正是那个说明着手去取代的东西。一个任意的选择也无济于事，因为那个恒等式约束那个范数、而不仅仅伴随它：一个不满足它的范数产出一个巴拿赫 $$-代数，而那个结构理论不适用于巴拿赫 $$-代数。

那个后果值得不加软化地被陈述。一个自称 $C^*$-代数的、却既不提供一个范数也不提供那个恒等式的说明，已借用那个名字而没有那个名字所指称的结构。它拥有一个带一个形式对合的结合代数，而这是一个比它所援引其声誉的那个对象相当弱的对象，而那些依赖于那个范数的结果，也就是说那个结构理论的较大部分，对它不可用。

3.4 来自类型化的障碍

一个代数是一个带一个对象的结构：每一个元素都可以被每一个别的元素相乘，而那个乘积又是那同一个代数的一个元素。部分的消没是可表达的，但那个环绕的全部性不是可选的，复合是遍处被定义的、无论它的值为何。

本文所关切的诸社会复合不是那个形式的。一个持久不对称由之被装配的那个序列，在 §9 中被详加处理，由一个产生诸方之间变异的分化、一个对那个变异排名的选择、一个把那个排名弄得不足为奇的规范化，以及一个传递它的再生产所构成。第一个取一个分布并产出一个具有更大方差的分布。第二个取方差并产出一个排序。第三个取一个排序并产出一个意义，即如它所被理解的那个排序，而这是一个不同于那个排序本身的种类的对象。第四个取一个意义连同一个排序，并产出它们跨越一代的传递。这些操作有彼此有别的定义域和彼此有别的陪定义域，而它们的复合只在一个方向上是良构的。那些逆向的复合不仅仅是无趣的；它们中的大多数是无定义的，没有任何意义可言，在其中一个规范化可以被应用于一个未分化的人口。

一个其诸操作携带不同种类的诸定义域与诸陪定义域、并且其中复合恰当一个的陪定义域匹配下一个的定义域时被定义的结构，是一个范畴、而不是一个代数 [16]。一个代数是每一个定义域与陪定义域都重合的那个退化情形。把那个社会复合强行纳入那个情形要求两个举动之一。第一个丢弃那个类型化，并随之丢弃那个使那个复合有趣的、一个分布与一个意义之间的区别。第二个通过辅助的诸投影重新引入那个类型化，这在那个代数之内重构那个范畴结构、而保留二者的优势中的哪一个都不。

3.5 来自可逆性的障碍

上面那三个障碍是技术性的。第四个不是，而它会保留、即便别的以某种方式被回答。

一个算子代数的诸对称性是它的诸自同构，而一个动力学是它们的一个群。那个群结构是本质性的、而不是约定性的：这个设置中的一个时间演化是一个诸映射的族，它们中的每一个都有一个逆，而那个框架是为其中这适当的物理系统所建造的。不可逆性进入那个理论，当它进入之时，作为一个导出的现象：一个描述一个开放系统的完全正映射之半群，或一个关联于一个态的模流。二者都是在一个可逆的基础之上被建构的、而不是替换它。

当前说明的那个原始是生成，而生成不逆转。一个被带入存在的区分不被任何后续的操作所移除，尽管它可能被取代；一个一旦被创立的制度即便在它的消解之后也在仍然可表述之物中留下一道残余；而 §4 中所描述的那个机制的整个旨趣，在于一个消耗它诸条件的行使与一个制造它们的行使之间的那个不对称，而这是一个关于一个不能被倒着运行的方向的陈述。在一个其基本诸映射是可逆的基础之上建造这样一个说明，是把那个现象置于那个导出的层、而把那个理想化置于那个原始的层，而这颠倒了那个主题所要求的次序。

3.6 建构一个新对象的诸根据

§2 中所导出的那些要求存活。那个候选者不存活。算子代数固定诸生成元与诸关系，以同样的方式、并因同样种类的理由，而一个微分模型固定诸坐标：二者都是一个固定结构之上的动力学的诸语言，而本文的对象是那个结构的动力学。

因此，所需要的，是一个满足 §2 的三个要求连同那第四个、即诸操作被诸方所负、并容许那个操作场域的不可逆变换作为它的基本举动、而不是作为一个导出的特例的对象。§5 建造它。在那之前，并且刻意地在任何形式装置被引入之前，§4 以日常语言陈述那个建构本欲捕捉的那个机制。那个次序要紧。一个谢绝那个建构的读者仍然应当处于一个能够接受或拒绝那个机制的地位，而一个只以它自身形式体系的词汇被陈述的机制无法独立于它而被评估。

§4 权力的关系性机制

本节陈述本文其余部分所形式化的那个机制。它不带形式装置、并且不带 §5 中所引入的词汇地这样做，而那个省略是刻意的。一个只以它自身形式体系的术语被陈述的机制无法独立于那个形式体系而被评估，而一个发现下一节的建构不能说服人的读者，仍然应当处于一个能够接受或拒绝此处所主张之物的地位。那个论证以七个步骤推进，其中第二个与第六个是大多数权力说明移动得太快之处。

4.1 从变化到差异

那个说明从一个关于社会实在由什么构成的承诺开始。诸关系不是一张诸事件在其上发生的静态之网；它们被持续地生产，而它们的生产正是社会生活之所是。依此看法，变化不是某种降临于一个否则稳定的安排之物，而是那个安排的寻常状况，它只在被再生产的限度内持续。

一个后果立刻随之而来。凡诸关系被持续地生产之处，它们不是被同一地生产的。在变化着的境况之下的生产产出变异，因此差异在凡关系被生成之处生起，而它持续地、而不是例外地生起。一个关系的两方逐渐在他们所做过之事上、在被对他们所做过之事上、在他们被置于接下来去做之事上有别。这么多无需超出那个承诺本身的任何论证，而它是那个起点、而不是一个结果。

4.2 从差异到不对称

从差异到不对称的那个步骤，是权力的诸说明最经常移动得太快之处，而那个区分对随后所述要紧。

两方有别尚不是它们之间的一个不对称。一个差异在那个相关的意义上是对称的：每一方都有另一方所缺之物，而那个赤裸的差异之事实中没有任何东西决定哪一个缺是有后果的。为使一个不对称成立，那个差异必须被排序，以致一方所有之物授予一个另一方所有之物所不授予的能力。那个排序是一个进一步的事实、而不是第一个的一个重述。

差异的生产中没有任何东西提供那个排序。变化生成变异；它不对它所生成的变异排名。那个排名是由无论什么在诸差异之间选择者所强加的：一个对一种技能定价、而对另一种不定价的市场，一个承认一项诉求、而对另一项不承认的法庭，一个证明一种训练、而对另一种不证明的专业团体。这个选择是一个它自己权利上的操作，与先于它的那个分化彼此有别，而它在 §9 的那个图式中作为如此被处理。

那个后果是，一个直接从变化导出权力的说明是不完备的。它已越过那个差异成为有后果之物的步骤，而那个步骤正是关于权力的许多有趣之物所寓之处。凡一个不对称被争之处，那个争非常经常不是关于那个差异、而是关于那个排序：不是两方是否有别，那鲜少有疑，而是那个差异是否应当授予它目前所授予之物。

4.3 行使与它自身的诸条件

一个有序的差异使某些事对一方成为可能、而对另一方不可能。这正是一个不对称之存在、而不仅仅之被记录的所在。

这样一个可能性的行使有两个效果、而不是一个。第一个是那个被意图的效果：某种东西被做了，而诸方的处境相应地被改变。第二个效果是本文所关切的那个。那个改变包括此后可用之物，因此一个不对称的行使也是对它自身行使之诸条件的一个干预。每一个凭借一个不对称而被施行的行动，都调整那个允许它的不对称的地位。

那个调整不携带任何固定的正负号。一个行动可能把它自身重复的诸条件留如完好、或减损、或增强，而这些之中哪一个成立，是一个关于那个行动及其设置的事实、而不是关于那个许可它的优势之量级的事实。正是这个正负号的可变性、而不是任何大小的变化，是那个说明取为那个现象之所在的东西。

4.4 自我消耗的寻常情形

大多数不对称不在它们自身的行使中存活。这值得作为那个默认、而不是作为一个有趣的例外被陈述，因为那个相反的假定正是那个使权力显得比它所是更令人费解的东西。

一个军事优势是通过征战而被行使的，而征战消耗承保那个优势的那个财政能力、那个人口基础、以及那个意愿。一次权威的攫取是通过中止权威先前据以被授予的那些程序而被行使的，而那个中止摧毁那次攫取据以取得它所有的无论什么认可的那个源头。一个垄断是通过定价而被行使的，而在垄断水平上定价招致终结那个垄断的进入。在每一种情形中，那个行动是有效的，而那个优势是真实的；所失败的是那个使那个行动可用的诸条件的再生产。

这一类的诸不对称持续不断地生起、并留下很少痕迹。它们不是失败的权力、或软弱的权力、或处于一个早期阶段的权力。它们是被行使、并因此被消解的诸不对称，而这是一个有序差异在一个其中诸关系被持续地生产的系统中的寻常命运。

4.5 持续作为权力的标记

一小部分少数的不对称表现得不然。它们的行使再生产它们行使的诸条件，以致那个优势在使用之后一如它在之前所立而立，并且经常更牢固。

一份资格证书是通过被要求而被行使的。它被要求的每一个场合都确认它所标记的那个区分是一个真实的区分，而那个确认正是一份资格证书之所由构成。一个官僚程序是通过被遵循而被行使的，而每一次遵循都把那个程序更牢固地确立为那个被做之事。一个规范是通过被援引而被行使的，而援引正是一个规范据以取得使援引有效的那个地位的方式。在这些情形中，那个行动不提取一个储量；它存入。

权力，依此处所提出的说明，是那个在它自身的行使中存活下来的不对称。存活是那个标记、而不是那个实质：那个使一个安排成为一个权力关系的东西，是下一小节中所描述的那个结构性条件，而持续是那个条件显示它自身的方式。那个表述是刻意地狭窄的。它不度量一方能够强迫多少、或那个优势有多大、或有多少他人受制于它。它就一个行动与在它之后仍然可用之物之间的关系提出一个单一的问题，而它只把以一种特定方式回答那个问题之物算作权力。

有两件事随之而来，它们推荐那个定义。它正确地分类 §1 的诸案例，把那个单薄的行政当局与那个证明团体指派给权力、把那个压倒性的军事优势与那个支配性的市场份额指派给那个寻常情形，而这是资源诸说明所不能做的。而它不要求一个安排为了算数而被任何人所意图、所设计、或所理解。凡再生产它自身之物就再生产它自身，无论是否有任何人计划它应当如此。

4.6 自我维持论题的重构

一个表述跨越别样不相关的诸说明复现：即权力在于它自身的维持，即权力所关乎的、在它所关乎的无论什么别的之下的，是它继续是权力。此处给予它的读法有别于通常所提供的诸读法，而那些区别值得被列出，因为那个论题比本文的说明更古老、更被良好地辩护。

那个表述源于尼采 [19]，而它通常以一个他明确地拒绝的形式被归于他。尼采并不主张生命或权力以自我保存为目标。他主张相反者：即一个活的东西首先寻求释放它的力量，即自我保存属于这的诸间接而频繁的后果之列、而不是它的对象，而把保存当作那个基本的驱力，是纵容一个多余的目的论原则，一个他对斯宾诺莎的 conatus 所提的指控。因此那个论题的表述不是保存、而是释放与延伸，而持续作为一个副产品随之而来。此处所发展的说明背离于此，而那个背离应当被记录、而不是被隐瞒。

两个较后的立场保留那个自我维持而无那个意志。卢曼 [14] 把权力当作一个被象征地一般化的传播媒介、而不是当作任何人所持有的一种能力，而卢曼 [15] 发展那个自创生的说明，据它社会系统从它们自己的诸元素中再生产它们自身，以致一个法律或政治秩序通过递归的运作、而不是通过任何外部的凭据而维持它自身。福柯 [8, 7] 使权力内在于它所成立于其中的诸关系、并只在行使中存在，并添加一个有相当重要性的要求：一个权力关系要求受制于它的那一方自始至终被维持为一个行动者，因为一个被减少到根本没有任何能力的一方不再是一个关系的一方、而是力的一个对象。

此处所提出的读法是：自我维持是构成性的、而不是有目的的。一个不对称不是一个碰巧持续的状态；它只在行使中存在，而行使正是那个改变行使之诸条件的东西，因此一个不对称与它自身的延续之间的关系不是因果性的、而是定义性的。所谓的权力是那个关系闭合的那个情形。这保存尼采所看见的东西，即权力的延续对权力不是偶然的，而谢绝他据以看见它的那个模态。没有任何目标被归于任何安排，而没有任何东西被说成寻求任何东西。

一个无一个目标的自我维持过程的结构性先例，在权力的研究之外存在。在拉康 [13] 中，欲望由一个需求所不能满足的缺失所构成，以致欲望恰恰通过保持不被满足而维持它自身，而它表面上的对象充任它延续的一个载体、而不是一个终点。那个平行在三个方面成立。二者都由一个过程、而不是由一个基底所构成；二者都会在它们自身的满足中终结，这正是为何福柯那个”那个下位的一方保持为一个行动者”的要求在那个分析的设置中有一个对应；而在二者中，一个有目的的表象从那个结构中生起而无一个目标在场。那同一个形式在一个与社会不对称无关的领域中出现，这是某种证据，表明它一般地属于自我构成的诸过程、而不是特别地属于权力。

那个不平行是当前说明欠一个那个分析的说明所不欠之工作之处。欲望的自我维持是结构上被保证的，需求是不可满足的，因此不要求任何关于欲望为何持续的解释。权力的自我维持既不被保证、也不常见：有序诸差异的绝大多数消散，而一个据它权力维持它自身的说明，欠一个关于为何维持如此鲜少被达成的解释。尼采在此处有一个可用的答案，而它是一个这个说明所拒绝的。依他的看法，未能持续是软弱，是力量的一个不充分的释放；强者无需去关心保存，而追求它的是那些恐惧者。§1 的诸案例不支持那个读法。一个优势可能被有力地、有效地、并且不带任何意志之缺陷地行使，而恰恰通过它行使的那个有力去消耗它自身的诸条件。自我消耗不是那一方的一项缺陷、而是一个行动与它所留下之物之间的关系的一个属性。

所提供的解释反而是 §4 由之开始的那个生成性承诺，而它是当前框架对一个若干传统已独立抵达的论题的贡献。因为关系被持续地生产，差异随它被持续地生产，而有序的差异持续而廉价地生起。那个场域不在一个不对称之下静立不动：新的差异被生成，新的诸排序变得可用，而昨天有后果的那个安排被此后所生产之物所动摇。尼采看见那同一个背景并把它读作征服，主张无论什么存在之物都被反复地重新解释到新的目的、并被某个高于它的权力所重新导向，而每一次征服都遮蔽先前之意义 [18]。当前的说明保留那个不安、而去掉那个等级：动摇一个安排之物无需是一个高于它的权力，而更经常是关系本身的寻常产出性。因此，维持是针对一个富有产出的背景所执行的工作、而不是在一个静态背景中的持续，而它的稀有从它必须据以被维持之物的丰盈随之而来。系统理论的说明与那个分析的说明都不提供这一点，第一个因为自创生的再生产是那个定义、而不是那个成就，第二个因为欲望不面对来自它所不欲之物的任何竞争。

有两个后果随之而来，就什么可被主张而言。第一个是不对称对于权力是不充分的，而那个不充分不是一个独立的规定、而是生成性的一个后果：设若不对称是稀缺的，它可能会足够了，而它无处稀缺。第二个是被观察到的权力关系的人口是那个被维持的人口，以致对一个安排的任何描述都是对迄今所被维持之物的一个描述。这是一个真实的认识论约束，而它不是那个说明的全部。那个构成性的主张关乎权力是什么；那个观察关乎什么可供被研究。

4.7 生成与退化

一个最后的区分是需要的，而它在此被定性地陈述。

一个不对称再生产它自身，这对那个再生产是如何被达成的什么也没说。两个安排可能都在它们自身的行使中存活，而在那个存活对它们的诸方所耗费之物上全然有别。一个证明团体通过作出一个别人发现值得拥有的区分而持续，而那个持续是通过扩大那些别人所能做之物而被购得的：被证明的从业者能够着手他们此前所不能着手之物，而那些与他们打交道的人能够倚赖他们此前所不能倚赖之物。一个排斥的系统同样可能持续，而它的持续是通过收缩那些被排斥者所能做之物而被购得的，每一次行使都从它所作用于其上者那里移除一个进一步的可能性。

二者依上面所给出的说明都是权力，而那个说明不因此就是有缺陷的。把它们区别开来的，不是那个不对称是否存活、而是以谁的代价。凡一个不对称的存活通过扩大它所作用于其上者的诸可能性而被维持之处，那个安排是生成性的；凡它通过收缩它们而被购得之处，那个安排是退化性的。这两者是单一一个条件的诸值、而不是两个条件，这正是为何权力的那个定义无需修正就能容纳它们。

那个区分容易被陈述、也容易被错误地陈述。把它弄成一件关于那个安排在总体上容许多少可能性的事，是诱人的，以致生成性的诸安排是那些宽松的、而退化性的诸安排是那些限制性的。§8 表明这在两个方向上都失败，而那个判准无法通过计数任何东西而被复得。

§5 那个形式对象的建构

此处所建构的对象意在满足 §2 的四个要求、并精确地陈述 §4 的那个机制。它分阶段被装配：首先是可用操作的那个集合，然后是那个使复合成为部分的类型化，然后是诸操作在那个集合上所诱导的诸转变，最后是那个把这两种运动持在一起的呈示。

在开端应当说两件事。那个建构不是数学的一个新分支，并且不声称是。它的诸成分是标准的，而那个贡献在于哪一个结构被当作首要的、而不在于任何技术上的新颖。那个建构也不产出一个预测性的装置；它所产出的，是一个词汇，在其中 §4 的诸区分能够被无歧义地陈述，连同那些区分在其下是良定义的诸条件。

5.1 剧目

设 $P$ 为一个诸方之集。一个剧目是一个操作的集合，它们中的每一个都被一方所负、并且每一个都作用于那些方所栖居的那个关系性场域之上。写 $\mathcal{A}$ 为一个剧目、$a \in \mathcal{A}$ 为一个操作，每一个操作都携带一个归属

$$\pi : \mathcal{A} \longrightarrow P,$$

以致 $\pi(a)$ 命名 $a$ 对之可用的那一方。那个归属映射正是那个卸除那第四个要求的东西，而它是那个把一个剧目与一个规则集区别开来的特征。那个纤维

$$\mathcal{A}_p ;=; \pi^{-1}(p) ;\subseteq; \mathcal{A}$$

汇集对方 $p$ 可用之物，而正是在这些纤维的层级上、而不是在那个作为整体的 $\mathcal{A}$ 的层级上，§4 的诸区分被陈述。

两方之间的一个不对称现在是可直接表达的。凡 $\mathcal{A}_p$ 与 $\mathcal{A}_q$ 有别之处，那些方在对他们可用之物上有别，而这是 §4 的那个差异。那个把差异转换为不对称的排序不是由那个纤维所提供的、而必须被添加；这在 §6 中被接过，而它在此处的缺席是刻意的，§4 的那个要点曾是那个排序是一个分开的事实。

5.2 带类型的部分复合

诸操作作用于不同种类的诸关系性事实。一个分化作用于一个分布并产出一个分布；一个选择作用于一个分布并产出一个排序；一个规范化作用于一个排序并产出一个意义。为记录这，设 $\mathcal{O}$ 为一个层域的集合，每一个层域是一个种类的关系性事实，并为每一个操作配备一个定义域与一个陪定义域，

$$a : X \longrightarrow Y, \qquad X, Y \in \mathcal{O}.$$

复合于是恰当那些类型一致时被定义：

$$b \circ a \ \text{ 被定义 } \iff \operatorname{cod}(a) = \operatorname{dom}(b).$$

这无需规定就卸除那第三个要求。部分性不是一个被强加于一个全部操作之上的约束、而是那些操作根本带有类型的一个后果，而那些不可容许的复合之为不可容许，是因为它们不是良构的、而不是因为一个规则禁止它们。对次序的敏感以同样的方式随之而来：凡 $b \circ a$ 与 $a \circ b$ 二者都被定义之处，它们一般而言是彼此有别的，而凡只有一个被定义之处，次序的那个不对称是绝对的、而不是一件程度的事。

一个配备那个类型化与那个部分复合的剧目，是一个其诸对象是层域、其诸箭头是操作的范畴，连同那个归属。写 $\mathcal{A}$ 为这整个结构、并无进一步限定地称它为一个剧目，将是方便的。

5.3 被诱导的诸转变

§4 的那个机制转在一次行使的第二个效果上：即施行一个操作改变此后可用之物。这通过让每一个操作诱导那个剧目本身的一个转变而被记录。

写 $\mathfrak{R}$ 为在那个方集 $P$ 与层域集 $\mathcal{O}$ 上的诸剧目的集合。每一个操作 $a \in \mathcal{A}$ 决定一个映射

$$\Theta_a : \mathcal{A} \longrightarrow \mathcal{A}’,$$

其中 $\mathcal{A}’ \in \mathfrak{R}$ 是在 $a$ 已被施行之后有效的那个剧目。那个映射 $\Theta_a$ 可以添加操作、移除它们、或把它们从一方重新归属给另一方，而这些是一个不对称的行使据以改变它自身行使之诸条件的那三种方式。

并非每一个转变都是由一个不对称所许可的操作所诱导的。因为关系被持续地生产，那个剧目也被独立于任何给定安排所发生之物携带向前：诸方进入并离开，新的差异在他们之间生起，而新的诸排序变得可用，差异凭它们可被弄得有后果。写

$$\Delta : \mathcal{A} \longrightarrow \mathcal{A}^{\Delta}$$

为这第二种的一个转变，并称它为一个背景转变。那个被诱导的转变与那个背景转变之间的区别不是一件机制之事，二者都是那同一个结构的诸转变，而是一件出处之事：第一个是由一个不对称的行使所诱导的，而第二个不是。背景转变可以扩大那个层域集、容许没有任何方此前所持的操作、或提供诸排序，在它们之下先前无后果的诸差异变得有后果。它们是 §4 中所描述的那个富有产出的背景的形式对应，而 §6 表明一个仅仅闭合于它自身之上的不对称与一个被维持的不对称之间的区别转在它们上。

那个被诱导的转变的三个进一步的特征值得强调，因为它们正是 §2 的诸语言所不能提供的。

那个转变是由那个操作所诱导的、而不是与它并列地被规定的。没有任何外部规则扮演那个混合元组中的重置映射的角色；$\Theta$ 由那个操作所决定，而一个操作在它的被诱导转变被给出之前不是被完全规定的。

那个目标 $\mathcal{A}’$ 无需有与 $\mathcal{A}$ 同样的层域。一个转变可以扩大 $\mathcal{O}$，而这正是一个先前不仅仅是假的、而是不可表述的关系性事实进入存在的方式，而它正是那个网络图的固定类型集 $T$ 所禁止的。

那个归属正是那个改变着的东西。因为 $\Theta_a$ 作用于一个携带那个归属映射的结构，它的效果是逐纤维地被陈述的：$\Theta_a$ 可以收缩 $\mathcal{A}_p$ 而扩大 $\mathcal{A}_q$，而这是一个关于一个禁止所做之事的良构描述。对匿名规则而言那个相应的描述是不可用的，如图 3 所记录的。

5.4 双重范畴的呈示

现在已引入两种运动。诸操作在一个剧目之内彼此复合，而诸操作在诸剧目之间诱导诸转变。这些是不同种类的运动，而它们不可互换，但它们也不是独立的：一个在一个剧目之内被施行的操作，正是那个把那个剧目携带到它后继者之物。

那个持有二者的结构是一个双重范畴，在埃雷斯曼 [5] 所引入的意义上。它的诸对象是层域。它的水平箭头是操作，依那个类型条件而复合。它的垂直箭头是剧目转变，通过依序的施行而复合。它的诸胞元是诸方块

$$\begin{array}{ccc} X & \stackrel{a}{\longrightarrow} & Y \[2pt] \big\downarrow{\scriptstyle,\theta} & & \big\downarrow{\scriptstyle,\theta’} \[2pt] X’ & \stackrel{a’}{\longrightarrow} & Y’ \end{array}$$

记录那个操作 $a$，在那个转变 $\theta$ 之下被施行，在那个后继剧目中作为 $a’$ 出现。

图 5（那个双重范畴的诸胞元）。 在左边一个其被诱导转变把它留如可用的操作，而这是持续的形式对应。在中央一个自那个后继剧目缺席的操作，而这是自我消耗。在右边一个转变的逐纤维效果，而这是那个归属映射所使之可陈述、而一个匿名规则集所不使之可陈述的。

那个交换的方块是那个呈示的要点、而不是一个技术上的便利。一个改写它自身可用性之诸条件的操作，在这个设置中是单一一个对象、而不是关于那同一个事件的两个被耦合的故事，因为那个胞元把那个水平的事实与那个垂直的事实作为一个陈述。§4 的诸失败模式成为这样的诸胞元的诸属性。凡 $a’$ 不存在之处，那个操作已把它自身从可被做之物中移除，而这是自我消耗。凡 $a’$ 存在并又是 $a$ 之处，那个操作已把它自身的可用性留如完好。

5.5 那些垂直箭头的不可逆性

一个最后的规定完成那个建构，而它是那个把这个对象与 §3 中所评估的那个算子代数候选者区别开来的。

那些垂直箭头不被要求是可逆的，而在那个意图的读法中它们典型地不是。一个转变 $\theta : \mathcal{A} \to \mathcal{A}’$ 没有任何被假定的伙伴 $\theta^{-1} : \mathcal{A}’ \to \mathcal{A}$。这不是一个有待在一个更充分的处理中被修补的省略。生成不逆转：一个被带入存在的层域不被任何后续的转变所移除，尽管它可能停止被栖居；一个一旦被收缩的归属不通过倒着运行任何东西而被恢复；而 §4 的整个内容在于一个存入的行使与一个提取的行使之间的一个方向性的区别。

因此，那些垂直箭头构成一个范畴、而不是一个广群，而那个对象的动力学不是一个群作用。凡那个自同构族把可逆性置于那个原始的层、而把不可逆性置于那个导出的层之处，当前的建构把不可逆性置于那个原始的层。可逆的诸转变是那个特例，在 $\theta$ 碰巧是可逆的时被复得，而那个框架中没有任何东西优待它们。

有两个属性随之而来，值得记录，因为它们在后面被使用。一个转变可能在诸操作上不是满射的，而这是诸可能性被丧失的方式，而它可能在诸层域上不是单射的，而这是诸区分崩塌的方式。§7 在制度性过滤的标题下接过这些中的第一个，而第二个是那个从关系性退化的诸说明中熟悉的区分之崩塌的形式对应。

§6 那个机制的形式化

§4 的那个机制现在以 §5 的词汇被陈述。这两节的次序被如此选择，以致这一节添加精确性、而不是内容，而读者应当能够查验此处所主张的没有任何东西超出那里所主张之物。

6.1 排序作为被添加的结构

§5 提供诸方之间的差异而不提供不对称。凡那些纤维 $\mathcal{A}_p$ 与 $\mathcal{A}_q$ 未能重合之处，那些方在对他们可用之物上有别，而那个建构中没有任何东西决定这两者中哪一个因此被给予优势。这恰恰对应于 §4 的那个差异尚不是不对称的步骤，而那个对应是刻意的：一个直接从差异交付不对称的对象，会已内建那个说明持为分开的那个步骤。

那个排序如下被添加。设 $\preceq$ 为一个剧目的诸纤维上的一个偏序，并写

$$\alpha ;=; \bigl(p, q, \preceq\bigr), \qquad \mathcal{A}_q \prec \mathcal{A}_p,$$

为 $p$ 在它之下立于 $q$ 之上的那个断言。那个关系 $\preceq$ 不是从集合包含导出的，而这要紧。$\mathcal{A}_q \subsetneq \mathcal{A}_p$ 既不蕴涵、也不被蕴涵于 $\mathcal{A}_q \prec \mathcal{A}_p$：一方可能有更少的操作可用而立得更高，正如一个被少数程序所约束、并被它们所赋权的主权者所例示的，而一方可能有更多的操作可用而立得更低，正如任何一个其许多被允许的行动全都无后果者所例示的。计数操作不对诸纤维排序，而此处所记录的 $\preceq$ 与基数性的独立正是 §8 所一般化的。

提供 $\preceq$ 之物，是 §4 的那个选择：一个市场、一个法庭、一个证明团体，某个对诸差异排名并因此把它们弄得有后果的操作。在 §9 的那个图式中这是那第二个层域，而它作为一个彼此有别的操作、而不是一个导出的量的地位，是那个”变化生成变异而不对它排名”的哲学要点的形式对应。

6.2 在自我诱导转变之下的闭合

一个剧目 $\mathcal{A}$ 中的一个不对称 $\alpha$ 许可一组操作，即那些在那个排序之下对 $p$ 可用、而对 $q$ 不可用者：

$$L(\alpha) ;=; \bigl{, a \in \mathcal{A}_p ;:; a \notin \mathcal{A}_q ,\bigr}.$$

每一个这样的操作诱导一个转变，而那些被许可的操作依序施行的复合诱导

$$\Theta_{L(\alpha)} : \mathcal{A} \longrightarrow \mathcal{A}’,$$

把那个剧目携带到它的后继者。那个不对称 $\alpha$ 在那个目标中有一个像 $\alpha’$，只要那些方、那些纤维与那个排序全都在那个转变中存活。

定义。 一个不对称 $\alpha$ 是闭合的，当它所许可的诸操作所诱导的那个转变归还它：$\Theta_{L(\alpha)}(\alpha) = \alpha$。

闭合是 §4 那个构成性主张的形式对应：那个不对称的行使也是对那个行使之诸条件的一个干预，而在那个闭合的情形中那个干预归还它所作用于其上之物。关于那个条件的形式的三点值得被显明地说出，因为每一点都回答一个否则会成立的反对。

那个条件是关于那个不对称、而不是关于那个剧目。一个剧目可以整个被再生产而没有任何不对称被再生产，因为一个剧目丰盈地容许诸不动点，无论是否有任何不对称在它之内成立。因此那个稳定的平等主义安排被那个条件所排除、而不是被一个附加于它的从句所排除。两个排除的根据应当被区别开来。凡所有纤维重合之处，没有差异成立，而 $\alpha$ 没有主体。凡诸纤维有别、但 $\preceq$ 对它们中的没有一个排名于另一个之上之处，差异成立，而 $\alpha$ 因缺一个排序而无定义。二者都落在那个条件之外，而第二个是那个要紧的，因为平等主义的诸安排鲜少是一律的，而是以有后果之排名的缺席、而不是以差异的缺席为特征的。

那个条件量化于那个被许可之集、而不是于所有操作之上。一个安排不因为某个无关的操作会扰动它而被取消资格；所问的是那个优势的行使是否维持那个优势。这正是那个使那个条件成为一个关于自我指涉、而不是关于一般稳定性的陈述的东西。

那个条件对量级沉默。没有任何基数性、测度或权重在它之中出现。两个不对称可能在它们所允许之物上相差任何数量而在它之下立得一样，而这是那个被意图的后果：那个说明主张一个不对称的持久不是它的大小的一个函数，而一个提及大小的条件会已在开端就与那个相矛盾。

6.3 在背景转变之下的维持

闭合尚不是权力的充分条件，而那个理由是 §4 所给出的那个。一个在它自身的行使之下归还它自身的不对称，只被对照一件东西被检验过，即它自身。那个场域此间并不静立不动。因为关系被持续地生产，背景转变独立地把那个剧目携带向前：诸方进入，差异累积，而诸排序变得可用，在它们之下无后果之物变得有后果。一个在它自身的转变之下闭合的不对称，可能仍然未能在那个场域在那个间隔中所做之事中存活。

定义。 一个不对称 $\alpha$ 是被维持的，当它被它自身的被诱导转变与那个场域所提供的诸背景转变的复合所归还，也就是说，当 $\bigl(\Theta_{L(\alpha)} \circ \Delta\bigr)(\alpha) = \alpha$，对于在所考虑的设置中可容许的诸背景转变 $\Delta$。权力是维持、而不是闭合。

那个强化正是那个生成性承诺所要求的，而三个后果从它随之而来。

闭合与维持之间的区别把那个较早的条件所不能分辨的两个情形区别开来。一个不对称可能是闭合的且未被维持的，而这是一个在它自身的行使之下完美地再生产它自身、而被别处所生起之物所解除的安排的处境。§1 的那个军事优势是这一类的：征战可能再生产征战所要求的那个指挥结构，而作为回应而形成的那个联盟不是由那个优势所许可的任何操作所产生的。那个安排内部没有任何东西失败。击败它之物是那个场域自己的产出性，而一个只有那个闭合条件的说明会不得不把那个情形记录为不可解释的。

维持是一个模态的条件、而闭合是一个事实的条件。闭合问一个转变做什么；维持问在那个设置所容许的诸扰动之下会发生什么。这是一个已持续的安排与一个被安置以持续的安排之间的形式区别，而这正是为何那个说明能够谈论一个倾向而不归属一个目标：一个安置不是一个目的，而一个关于在扰动之下之行为的模态主张不携带任何目的论。

那个对 $\Delta$ 的量词是那个经验内容进入之处，而它的范围必须被规定、而不是被留作普遍的。没有任何东西被对照每一个可设想的扰动而维持，而一个被要求在任意的背景转变中存活的不对称，会是一个没有任何安排满足的不对称。所论的是一个给定的设置实际所提供的那类背景转变，而这是一件关于那个设置的事实、而不是那个框架所决定之物。§10 记录那个后果：那个说明陈述那个条件，而对任何实际的安排都不裁定它必须承受哪些扰动。

6.4 那两种失败的模式

那些互补的情形是 §4 所指认为寻常的那些，而那个被强化的条件把它们中的两个区别开来，而那个闭合条件独独只区别一个。

一个不对称是自我消耗的，当它自身所许可的诸操作诱导一个在其之下它未能归还的转变，或者因为那些操作不再可用，

$$L(\alpha) \not\subseteq \mathcal{A}’,$$

或者因为那个排序不存活，那些方已停止立于那个使那些操作有后果的关系之中。闭合失败，而它从内部失败。图 5 的中央面板描绘这个情形。那个中止授予权威的诸程序的权威攫取是这一类的，而那个其行使耗尽产生它的那个基础的优势也是。

一个不对称是未被维持的，当它是闭合的、但未被维持的：它自身的行使归还它，而它不在那个场域此间所提供之物的复合中存活。那个安排内部没有任何东西出错。所发生的是新的差异已生起、或一个新的排序已变得可用、或诸方已进入他们立于那个关系之外，而一刻之前完美地再生产它自身的那个安排不再找到它所再生产的诸条件。

第二个模式是那个说明最想要记录的那个，而它直接关涉 §4 的那个读法。依一个把未能持续当作持有那个优势的一方之不充分的看法，那个未被维持的情形必须被同化于那个自我消耗的情形、并被读作行使的一项缺陷。此处所划的那个区分否认那个同化。一个安排可能被以完全的胜任行使、在它所许可的每一个操作之下归还它自身、而被一个它自己的任何操作都未曾促成的产出性所解除。软弱不是那个解释，而有力不会曾是那个补救。

一个进一步的观察随之而来，而它是认识论的、而不是构成性的。在反复的行使与反复的背景转变之下，未能满足任一条件的诸不对称自那些后继的剧目缺席，而那些满足二者的保留。因此，在一个较晚的阶段所作的对一个安排的任何描述，是对一个其中那些失败的不对称不再出现的剧目的描述，而可供研究的那个人口是那个被维持的人口。这解释某种关于那个证据、而不是关于那个现象之物：权力是什么，是由 §6 的那个条件所陈述的；我们所处于一个能够观察的地位之物，是在此被陈述的。把这两者混同会使那个说明成为平凡的，因为它会把”权力维持它自身”这一主张化约为”所保留之物已保留”这一观察。

那个说明不预测一个给定剧目中哪些不对称将被证明是被维持的，而 §10 接过那些理由。它所提供的，是一个在其下那个问题是良好提出的条件，连同”那个答案不是从那个优势的大小读出”这一发现。

6.5 那个生成与退化参数

上面所给出的那个定义被那些在它们的持续对受制于它们的诸方所耗费之物上有别的安排所满足，而 §4 定性地区别了这些。那个区分现在被逐纤维地陈述。

设 $\alpha = (p, q, \preceq)$ 为一个权力关系，而设 $\Theta = \Theta_{L(\alpha)}$ 为它所诱导的那个转变。那个相关的比较是在那个下位方之前与之后的纤维之间：

$$\mathcal{A}_q \quad \text{对比} \quad \mathcal{A}’_q ;=; \Theta(\mathcal{A})_q .$$

定义。 一个权力关系 $\alpha$ 是退化性的，当它的维持通过那个下位纤维的收缩而被购得；而是生成性的，当维持被维持、同时那个纤维被扩大。

图 5 的右手面板描绘那个逐纤维的比较，而那个归属映射正是那个使这成为一个良构表达式的东西。在一个其诸操作不属于任何人的形式体系中，那个比较无法被写下，而这正是 §2 的那第四个要求赢得它位置之处。

那个定义的两个特征值得评注。

生成与退化是一个条件的诸值、而不是两个条件。二者都预设维持，而它们在那个维持它的转变的逐纤维正负号上有别。这正是为何一个权力关系的那个定义无需修正就能容纳它们，也是为何一个安排不能在首先不是一个权力关系的情况下是生成性的。那个下位纤维的一个暂态的扩大，不被任何自我再生产的不对称所维持，依这个说明不是生成性的；它只是一个事件。

那个比较是逐纤维的、而不是聚合的。所比较的是 $\mathcal{A}_q$ 与 $\mathcal{A}’_q$，而不是 $\mathcal{A}$ 与 $\mathcal{A}’$，而这两个比较可以在相反的方向上运行。一个转变可能收缩那个作为整体的剧目、同时扩大那个下位纤维，而那个定义关注第二个而无视第一个。§8 表明这不是一个技术细节、而是那件事的实质，因为那个聚合的比较在两个方向上都产出错误的裁决。

§7 算子代数诸区分的应用

§3 把算子代数作为一个基础搁置了。这并不推出那个主题没有任何可贡献，而这一节接过它的三个区分并把它们应用于 §5 中所建构的那个对象。这个安置是刻意的。倘若这些与那个评估并列地出现，它们会曾是下面什么也没有的诸类比，因为它们所被应用于其上的那个对象当时并不存在。被应用于诸剧目及其诸转变，它们产出那个不动点定义独独所不提供的诸区分。

那些借用是问题的、而不是公理的借用，而那个区别应当被保持在视野中。下面没有任何东西断言一个剧目是一个代数、一个转变是一个自同构、或算子代数的任何定理转移。所主张的，是那个主题已学会去问的三个问题在此值得被问，而问它们产出结果。

7.1 诸政体的个别化

算子代数区别一个单一代数的、无法被任何保持诸可观测量的变换所连接的诸表示 [9]。这样的诸表示属于分开的诸超选择扇区，而那个区分是结构性的：没有任何连续的道路、也没有任何自同构，把一个携带到另一个。那个物理的读法是，这两个描述那同一个系统的真正不同的诸相、而不是一个相之内的不同的诸态。

对诸剧目的那个相应的问题，是两个安排何时应当算作不同的诸政体、而不是不同的诸条件中的一个政体。§6 的那个维持条件不回答这。它告诉我们哪些不对称在一个安排之内持续，而对诸安排本身的个别化什么也不说，因此一个大的变化与一个种类的变化在它的说明上是无从分辨的。

那个被借用的区分提供一个判准。称两个剧目为连通的，当某个转变的序列把一个携带到另一个、同时保持那个层域结构；而分离的，当没有任何这样的序列存在。分离不是一件距离之事。两个安排可能在几乎每一个归属上都有别而保持连通，一个转变的序列把一个携带到另一个；而两个安排可能在一个单一的层域上有别而分离，没有任何保持那个结构的转变可用以桥接它们。

这产出一个 §2 的那些量级语言所不能陈述的、对政体变化的读法。一个宪法秩序与它的后继者不是作为一个诸安排之连续统上的两个点、相差多于寻常那样被关联的。凡那个后继者躺在一个分开的扇区之中之处，它们之间的那个转变不是寻常诸转变的累积、并且不能被如此描述，无论事实上产生它的那个事件的序列为何。那个判准也解决一个熟悉的歧义：两个共享一个宪法文本的国家可能是分离的，如果没有任何保持那个层域结构的转变连接它们，而那个被共享的文本于是是一个关于文档、而不是关于安排的事实。

那个判准是作为一个区分、而不是作为一个检验被提出的。确定一个桥接的序列是否存在，不是那个框架所使之可处理之物，而 §10 返回到这。

7.2 制度性过滤

算子代数中的一个条件期望是从一个代数到一个子代数的一个投影，与那个乘法相容，在那个子代数的诸元素不变地通过它的意义上 [23]。它的解释是一个粗略描述的解释：那个较大的代数所记录之物只在那个较小的能够表达它的限度内被保留。凡那个建构可用之处，一个指数度量那个差异的大小 [11]，量化那个较小的未能看见那个较大的结构之多少。

对诸剧目的那个相应的结构，在凡一个制度通过一个它自己的词汇作用于一个关系性场域之处生起。一个法庭承认某些种类的诸诉求；一个官僚机构在它所维持的诸范畴之下处理诸案例；一个证明团体登记它自己的诸程序所能表达的诸区分。在每一种情形中，那个制度不作用于如其所立的那个场域、而作用于它自己对它的读法，而那个读法是一个子剧目。

设 $\mathcal{B} \subseteq \mathcal{A}$ 为这样一个子剧目，并写

$$E : \mathcal{A} \longrightarrow \mathcal{B}$$

为那个把每一个操作携带到那个制度所认作它之物的赋值。那个对应于相容性的要求是，已经属于 $\mathcal{B}$ 的诸操作在 $E$ 之下不变地通过，因此一个制度正确地读它自己的诸范畴、并通过它们读每一个别的东西。

有两个后果随之而来，而第二个是那个值得拥有的。

一个制度因此以一种具体而可描述的方式是盲的。$E$ 所丢弃之物不是噪声、而是关系性事实：$\mathcal{A}$ 中在场的、$\mathcal{B}$ 所不能表达的诸区分，从那个制度的立场看，是缺席的。这是 §5 结尾处所指出的那个非满射性，而它给一个关于行政的熟悉观察一个形式的形状而不诉诸隐喻。

那个盲容许比较。凡一个指数的类似物可用之处，诸安排可以被以它们的诸制度未能登记那个场域之多少排序，而那个排序不同于以它们所登记之多少的那个排序。一个有一个大词汇的制度可能丢弃多于一个有一个小词汇的，如果它所丢弃之物是那个曾要紧之物。那个说明一般而言不提供这样一个指数，而为一类案例建构一个会是一个本文所不作的贡献。

对随后所述的关涉是直接的。规范化，§9 的那第三个层域，恰恰是那个把一个偶然的排序弄成一个制度所能承认的范畴的操作。在当前的词汇中，规范化把一个排序从 $\mathcal{A}$ 移入那个制度据以作用的那个子剧目 $\mathcal{B}$，此后那个排序不再是那个制度所强加之物、而是它仅仅所读之物。

7.3 诸对称性的诸层级

那第三个借用是最有用的、并且最不技术的。算子代数区别一个动力学的诸对称性、诸可观测量的诸对称性、以及一个态的诸对称性，并把它们作为三个问题、而不是一个持在分开。此处的诸相应的层级是那个转变、那个剧目、以及那个逐纤维的分布。

考虑一个其中每一方都有同样的操作可用的安排。这是那个剧目层级上的对称：对方 $p$ 与 $q$，

$$\mathcal{A}_p ;=; \mathcal{A}_q .$$

其次考虑一个其中那个排序把没有任何方排名于另一个之上、以致没有不对称成立的安排。这是那个分布层级上的对称，而它是一个不同的条件。而最后考虑一个其诸转变一样地对待诸方、以致 $\Theta$ 与 $p$ 和 $q$ 的交换交换的安排。这是那个转变层级上的对称，而它又是一个第三条件。

图 6（那三个层级被持在分开）。 形式的平等断言那第一个面板、并与那第二个的失败相容，而这是形式平等与实质平等之间的区分的形状。那第三个面板是一个进一步的条件，而只承认前两个的诸讨论对它没有位置。

那三个是独立的，如图 6 所列出的，而那个独立是那个结果。法律面前的平等断言 $\mathcal{A}_p = \mathcal{A}_q$：同样的操作对所有人可用，而这是一个对平等对待的形式保证所保证之物。实质的不平等是那个分布层级上的对称的失败，而它与 $\mathcal{A}_p = \mathcal{A}_q$ 精确地成立完全相容。因此一个安排可以在它的剧目上对称、而在它的分布上不对称而无矛盾，而这是”形式平等与实质平等是彼此有别的”这一由来已久的观察的形式形状。这两者之间的那个缺口正是卢克斯 [12] 中权力的那第二个与第三个维度所针对之物。

那个区分对那个观察所添加之物，是一个通常被略去的第三项。改变什么可被做的改革，相对于改变谁立于何处的改革，是那个转变层级上的一个变化，而它不是前两个中的任何一个。一个安排可以在这第三个意义上被改革而剧目与分布二者在那个改革的时刻都不改变，那个更改在于随后的诸行使将做什么。只承认形式平等与实质平等的诸讨论对这没有位置，而被迫把这样的诸改革分类为二者之一，一般而言为仅仅形式的。

下面的表格记录那三个借用以及每一个所贡献之物。

区分	算子代数的内容	被应用于诸剧目
超选择	无法被保持可观测量的变换所连接的诸表示	诸政体分离，当没有保持层域结构的转变桥接它们；政体变化不是被累积的寻常变化
条件期望	到一个子代数的投影，带一个度量那个差异的指数	一个制度通过它自己的子剧目作用；它所不能表达之物从它的立场缺席
诸对称性的诸层级	动力学、诸可观测量、态的诸对称性被持在分开	那个转变、那个剧目、那个分布的对称；形式平等是 $\mathcal{A}_p = \mathcal{A}_q$，实质平等不是

§8 对计数判准的反驳

生成性权力关系与退化性权力关系之间的区分在 §6 中被作为一个逐纤维的条件给出。一个更简单的判准自荐，而它被持得足够广泛以值得反驳、而不是打发。本节陈述它，表明它在两个方向上都失败，并指认那个失败关于那个取代它的判准所揭示之物。

8.1 那个计数直觉

那个直觉是生成性在于可能性的扩大。一个容许更多的安排比一个容许更少的更具生成性，以致生成性可以被以可用之物的大小度量、而退化被以它的减损度量。以 §5 的词汇写出，那个提议是一个转变 $\Theta$ 是生成性的，当

$$\lvert \Theta(\mathcal{A}) \rvert ;>; \lvert \mathcal{A} \rvert$$

而是退化性的，当那个不等式向另一个方向运行。

那个提议有明显的吸引力。它是简单的，它不要求任何归属映射，它在一大范围的案例中与日常用法一致，而它把生成性连接到有已确立的形式处理的诸概念，无论是一个选项集的基数性、一个在可用行动上的分布的熵、还是一个可行域的大小。以下所述中没有任何东西争议这些是良定义的量。所争议的是它们中的任何一个追踪那个说明所要求的那个区分。

那些以重要性对操作加权、或只计数在某种意义上重要的操作的变体，不规避随后所述。那些反例转在哪一方持有那些操作、而不在那些操作如何被计数上，而一个加权的计数仍然是一个在那个作为整体的剧目上的计数。

8.2 两个方向上的诸反例

生成的收缩。 一个宪法性的禁止从一个剧目中移除操作。凡一个宪法禁止诸选举的中止、诸人未经程序的拘留、或诸财产未经补偿的征收之处，如此被禁止的那些操作不再可用，而那个剧目在那个禁止之后比之前更小。依那个计数判准，那个转变是退化性的。

那个裁决是错的，而它以一种指出那个判准在何处已走岔的方式为错。那个禁止所移除之物几乎全然从一个纤维中被抽取。那些被禁止的操作是那些对那个被置于一个能够中止、拘留或征收之地位的一方可用者，而它们的移除正是那个允许那些其他方竟能以先前受制于任意逆转的方式行动的东西。诸合同变得值得订立，因为它们不能被任意作废；诸社团变得值得组成，因为它们不能被随意解散；而这些中的每一个都是一个由 $\mathcal{A}_p$ 的一个收缩所施行的 $\mathcal{A}_q$ 的扩大。那个聚合可能下降、而那个下位纤维上升，而第二个是那个说明所关注之物。

退化的扩张。 考虑共有诸规范的消解，即默顿 [17] 在失范的名称之下所分析的那个状况，在其中先前不被做之物变得可能。先前被排除的诸操作，无论是被约定、被期望、还是被他人的被预期的回应所排除，现在是可用的。那个剧目在那个消解之后比之前更大，而依那个计数判准，那个转变是生成性的。

这个裁决在那第一个案例的镜像中为错。那些变得可用的操作在原则上对所有人可用、而在实践中对那些被置于使用它们之地位者可用，因此那个扩大累积给无论谁能够以最少约束行动者的纤维。对那个其位置依赖于那些被消解的诸期望的一方，可用之物已尖锐地收缩，因为其有效性搁在他人对一个规范的遵从之上的诸操作不再是有效的操作。那个聚合上升、而那个下位纤维下降。

图 7 把这两个案例并排安放。左面板展示那个计数判准所登记之物，即那个聚合；其余的诸面板展示那个判准所丢弃的那个逐纤维的分解。那些聚合的轨迹在两个案例中都从那些逐纤维的轨迹相反的方向运行。

图 7（那个计数判准对照那个逐纤维的判准）。 左面板绘出那个剧目在那两个转变之下的聚合大小，而这是那个计数判准所读的全部。中央与右面板把那两个同样的转变逐纤维地分解。禁止收缩那个聚合、同时扩大那个下位纤维；诸规范的消解扩大那个聚合、同时收缩它。那些被展示的轨迹是被构造的、而不是被度量的，而没有主张说任何实际的安排遵循它们。

那两个反例在相反的方向上运行，而这排除了那个自然的修补。倘若那个判准只关于禁止是错的，人们可能会逆转它并主张收缩生成。第二个案例排除这。没有任何 $\lvert \mathcal{A} \rvert$ 的单调函数能够在两个案例上与那个说明一致，因为那个说明对两个其聚合行为本身相反的转变指派相反的裁决，而一个聚合的单调函数必须一如那个聚合对它们排序那样对它们排序。那个失败不是校准的、而是那个被选择的量的失败。

8.3 那个判准的重新表述

那些反例所共有的，是那个聚合丢弃那个归属。量 $\lvert \mathcal{A} \rvert$ 是在那个作为整体的剧目上被计算的，因此对哪一个纤维得利、哪一个失利是不敏感的，而恰恰是那个信息是那个区分所要求的。这正是为何 §6 的那个判准被逐纤维地陈述，也是为何那个归属映射被内建进那个对象、而不是被添加到它。

那个重新表述现在可以被陈述为那两个案例的寓意。生成性不是一个关于一个安排容许多少的属性。它是一个关于一个安排所允许它的支配方之物与它因此所允许他人之物之间的关系的属性。一个对那个归属不敏感的量无法表达那个关系，因为那两个反例在那个说明把它们分开之处在聚合行为上一致、而在那个说明把它们连接之处在逐纤维行为上有别。对上面所提及的诸概念的后果是直接的：选项计数、在可用行动上的熵、以及一个可行域的大小，全都是聚合的量，并且全都因同样的理由失败。

一个相关的后果关涉通常所讨论的自由。如果自由被取为在于可用选项之数目，那么第二个反例是一个自由的得利、而第一个是一个失利，而这颠倒了几乎任何人会对这两个案例所作的评估。那个说明不提供一个自由的理论，而此处没有任何东西裁定自由在于什么。它确实表明，无论自由是什么，它不是那个计数判准中的那个聚合的量，因为那个量在此处所考虑的两个案例上交付错误的裁决。

8.4 与那个价值回路的关系

一个进一步的刻画随之而来，而它把当前的结果连接到 §4 中所给出的那个生成的说明。

凡一个权力关系是退化性的之处，那个安排所生成之物返回生成它的那一方。那个下位纤维被收缩，那个支配纤维因此被保得，而那个行使所产生的价值累积给那个行使由之被作出的那个位置。那个回路闭合于它自身之上，而那个安排被它从受制于它者那里所取之物所维持。

凡一个权力关系是生成性的之处，那个安排所生成之物不以这种方式返回。那个证明团体的持续是通过被证明的从业者所可做之物的扩大而被购得的，而那个被生成的价值累积给那个关系、而不是给任一方独自。那个回路保持敞开，在所生产之物不被那个生产者所收回的意义上。

这两个刻画在此处所考虑的诸案例上一致，而没有论证被提供说它们一般地一致；那个回路的表述是对那个逐纤维判准的一个注解、而不是一个独立的检验，而确立一个等价会要求一个本文所不尝试的对价值的处理。它的用处是它把这个结果连接到本文所属的那个纲领中别处对挪用的处理，在那里”被生成的价值是否被留给生成它的那个关系”这一问题在与权力毫无关系的诸设置中复现。

§9 一般化到持久不对称的一个图式

那个说明迄今给出了一个在其下一个不对称持续的条件，而没有说出这样的诸不对称如何被装配。本节提议它们以一个共同的方式被装配，并把那个提议作为一个一般的图式、而不是作为前面诸节的一个例示提出。

9.1 那四个层域

依那个提议，一个持久的不对称是四个操作的复合。

分化取一个人口并产出它之内的变异。诸方逐渐在训练、在持有、在位置、在他们所做过之事上有别。这是 §4 的那个关系被生产、并且被在变化着的境况之下生产、而产生差异的步骤。

选择取变异并产出一个排序。某些差异被弄得有后果而另一些不，而那个对它们排名的操作与那个产生它们的操作彼此有别。§6 把这记录为 $\preceq$ 的添加，而那个建构不自行提供它。

规范化取一个排序并产出一个意义：如它所被理解的那个排序，也就是说被弄得不足为奇的那个排序。曾是一个偶然的排名之物成为事情所是的样子，而它的诸方停止把它经验为某种被强加之物。在 §7 的词汇中，规范化是那个把一个排序移入一个制度据以读那个场域的那个子剧目的东西，此后那个制度不强加那个排序、而仅仅承认它。

再生产取一个被规范化的排序并产出它的传递。在当前诸方之间所持之物逐渐在他们的诸后继者之间所持，无论是通过继承、通过训练、通过承认它的那些制度的持续、还是通过构成它的那些实践的简单延续。

把这四个写作诸箭头，

$$X ;\xrightarrow{\ D\ }; V ;\xrightarrow{\ S\ }; O ;\xrightarrow{\ N\ }; M ;\xrightarrow{\ R\ }; X,$$

那个复合 $R \circ N \circ S \circ D$ 把一个人口携带到一个其中同样的排序成立的人口，而这是 §6 的那个闭合条件作为一个循环被展示。那个复合必须进一步承受之物，是那个场域的背景产出性，而 §9 论证一个层域尤其是那个使它能够承受之物。

9.2 跨越诸层域的类型化

那四个操作不在一个单一的层域之内复合，而这正是为何那个循环被以彼此有别的诸对象写出。一个分布、一个排序、一个意义与一个继承是不同种类的诸关系性事实，而它们之间的诸箭头有相应地不同的诸定义域与诸陪定义域。

那个类型化不是装饰。它是那个复合不能被写作一个代数中的一个乘积的理由，如 §3 所记录的，而它正是那个使那些逆向的复合无定义、而不仅仅无趣的东西：一个规范化不能被应用于一个未分化的人口，没有任何排序可供它作用于其上。因此，复合的那个部分性在此处做工作、而不是闲置。

由此可得，一个持久的不对称不是一个单一种类的东西发生四次。它是一个跨越四个种类的关系性事实的通行，而一个把权力当作一个移动通过一个系统的单一实体的说明，将没有资源去陈述它。

9.3 来自必要性的论证

那个提议是全部四个层域都是被要求的，而那个论证是逐层域被给出的。在每一种情形中那个问题是当那个层域缺席时一个不对称变成什么。

没有分化就没有任何东西可供排序。这个情形是退化的，并为完备起见被记录。

没有选择就有差异而没有不对称，而这是 §4 的那个大多数说明移动得太快的步骤。诸方有别，而没有差异授予任何东西。这样的诸安排是常见而不足为奇的，而这恰恰是那个要点：变异是无处不在的，而只有它的一些被弄得要紧。

没有规范化一个不对称仍然可能被持在原地，但只通过持续的重新强加。那个排序不自行站立，因此它运作的每一个场合都要求它被针对那些把它经验为被强加的诸方重新断言。这样的诸安排存在并且是可辨认的，而刻画它们的是花费：监视、执行、一个强制装置的维护，以及那个排序仍然所持的周期性证明。那个不对称不停止，但它停止再生产它自身、而必须反而被某人所再生产。

没有再生产那个不对称不在它的诸方中存活。一个被分化、被选择、被规范化、而不被传递的排序，持一代而失效。这是那个不比它的创立者更长寿的有魅力的安排的情形，即韦伯 [26] 所指认的那个例行化问题，而它的熟悉是那第四个层域之必要性的证据、而不是反对它的证据。

9.4 那个论证的诸界限

充分性不被主张，而那些理由应当被陈述、而不是被留待推断。

那四个层域可能在场、而那个不对称因躺在那个复合之外的诸原因而未能持续：外生的冲击、不受制于那个排序的诸方的到来、整个安排所依赖的一个物质基础的耗尽。那个图式陈述一个持久的不对称由什么被装配，而不是什么保证那些部分的一个装配将会持。

来自必要性的那个论证也不确立那四个层域是那个正确的分解、而不是若干分解中的一个。同一个通行的一个不同的划分，成三个层域或五个，可能做同样的工作。推荐这一个之物，是每一个层域对应于一个彼此有别的种类的关系性事实、并且那个必要性论证对每一个分开地运行，但这两个考虑中的哪一个都不排除诸替代方案。

9.5 跨越诸领域的实例化

那个图式是作为一般的被提出的，而那个一般性的理由搁在诸实例上。

在资格证书主义中，被布迪厄与帕斯隆 [2] 详加分析，分化是训练与才能上的变异；选择是对它排名的考试；规范化是把那份资格证书当作它所证明之物的证据、而不是当作一个关于谁可以从业的约定的对待；再生产是通过那些要求那份资格证书的制度与那些供应它的学校的传递。

在殖民行政中，分化是军备、组织与疾病暴露上的区别；选择是那个区别向一个权威之诉求的转换；规范化是一个学说的精心阐发，在它之下那个权威是正当的、而不仅仅是有效的；再生产是那个行政装置、一个本地中间阶级的教育，以及比人员更长寿的那些法律形式。

在科学声望中，分化是所发现之物上的变异；选择是同行判断；规范化是把卓越当作追踪功绩、而不是当作构成它的对待；再生产是学生被卓越者的训练以及位置通过任命的传递。

在市场位置中，分化是效率与持有上的变异；选择是竞争；规范化是把那个所得的分布当作应得或自然过程的一个结果的对待；再生产是诸回报的再投资与资本的继承。

这些实例在几乎每一个描述上要紧的方面都有别，而在那四个层域上它们一致。那个一致正是那个图式的一般性所相当于之物。

9.6 那个承重层域的指认

那些层域不是同等地暴露的，而它们之间的那个不对称对干预有一个后果。

分化不能被移除。诸方有别，而一个阻止他们有别的安排会不得不阻止关系竟被生产。选择能够被更改而不能被废除，因为任何一个其中某些差异是有后果的安排都涉及一个选择，而那个问题是哪一个、而不是是否。再生产能够被打断，而许多再分配的政策在于打断它，但打断必须在每一代被重复、而那个打断本身是昂贵的。

规范化在种类上是不同的，而 §6 的那个区分说出为何。闭合是由那个作为整体的复合所达成的；维持是那个复合必须进一步承受之物，而规范化是那个它据以承受它的层域。一个被当作不足为奇的排序不被针对那个场域所产生的每一个新差异所防御，因为新的差异是通过那个排序被读、而不是被对照它检验的。这是 §7 所描述为制度性过滤的那同一个操作：一旦一个排序已通行进一个制度据以作用的那个子剧目，背景转变便已被解释地到来。在那四个层域之中独独地，规范化没有任何物质基底。分化被扎根在诸方之间的实际变异中，选择在对它们排名的诸制度中，再生产在传递的诸机制中。这三个中的每一个都能够独立于任何人关于它所相信之物而被指出。规范化不能：它在于那个在其下那个排序被当作不足为奇的理解，而一个理解不被除了它继续被持之外的任何东西所维持。此处所主张的诸层域之间的那个不对称是这一个，而它是作为一个关于上面所考虑的诸实例的观察、而不是作为一个关于一般诸层域的定理被提出的。

那个后果被例示于图 8。凡规范化被移除之处，闭合仍然可能被达成、而维持不：那个复合在它自己的运作之下返回它的源头、而不再承受那个场域所提供之物，因此每一个新的差异都必须被作为一个差异所迎接、而不是被作为一个确认所吸收。那个排序仍然可能被维持，并且经常是，但它的维持现在要求它被断言、而不是被假定，而那个断言必须被无限地重复。曾被再生产之物成为某种必须被持续地重新强加之物，而这两者之间的区别是一个持起来不耗费任何东西的安排与一个每一个循环都耗费某种东西的安排之间的区别。

图 8（移除规范化的效果，在一个被构造的模型之下）。 左面板展示那个排序在任一种情形中都可能被持在所要求的水平。右面板展示持它耗费什么：以那个复合完好，那个所要求的重新强加朝向无物下降，因为那个排序自行站立；以规范化缺席，那个排序在诸循环之间衰减，而恢复它的成本不下降。那个模型是例示性的，而它的参数是被规定的；没有主张说任何实际的安排遵循这些轨迹。

这指认那个干预最便宜的层域，而它不推荐任何干预地这样做。那个发现是结构性的：一个被剥夺了它的规范化的不对称并不因此被终结，但它被从某种再生产它自身之物转换为某种必须被一个可指认的方以一个反复的成本所再生产之物，而第二种的诸安排比第一种的诸安排相当地更容易被终结。这也是为何对持久诸不对称的诸争夺如此经常是对描述、而不是对分配的争夺，而这是一个把权力当作一个量的诸说明有困难去解释、而在此处直接随之而来的事实。

§10 范围与方向

这一类的一个说明最好被以它所拒绝主张之物、一如以它所断言之物来评判，而上面若干节把一个界限推迟到这一节。它们在此被汇集，连同那个说明所留下的工作。

10.1 可复原性的诸界限

把那个框架读作提供一个逆问题，是诱人的：给定一个社会安排的一个被观察到的历史，复原那个生成它的剧目与诸转变。那个诱惑应当被抵制，而那个理由不是那个问题是困难的。

那个问题以一种没有任何数量的数据缓解的方式是欠决定的。任何被观察到的诸安排的序列都与许多个生成的剧目相容，因为对一方可用的诸操作不是被直接观察到的、而只通过它们被行使的那些场合，而一个可用而从不被行使的操作至多留下它可用性的间接证据。两个在每一个被观察到的方面都相同的安排，可能在什么曾可能而不被做上有别，而那个区别恰恰是那个说明所持为决定性之物。那个情形比对微分模型的那个相应情形更糟，在那里至少那个状态被观察到、即便那个向量场不。

由此可得，设定一个剧目是一个理论承诺、而不是一个推断。那个说明不声称相反，而它确实作出的那个主张是一个不同的：一个被设定的生成集比一个被设定的量更可证伪，因为诸生成元之间的诸关系彻底禁止某些序列，而一个被禁止序列的单一一个被观察到的发生反驳那个设定。一个定量模型通过调整一个参数容纳一个反常的观察。一个禁止被观察到发生之物的生成性设定已被反驳。这是一个关于那两种理论的形式的主张、而不是一个结果，而它在下面作为一个方向、而不是作为一个发现被接过。

10.2 那个说明所放弃的诸能力

三样在 §2 的诸语言中可用之物在此不可用，而那个说明在每一方面都比它们更糟。

没有定量动力学。那个框架支持不了任何微分方程、产出不了任何轨迹、并允许不了任何关于任何东西发生得多快的计算。凡一个问题关乎诸速率之处，那些微分语言是那个适当的工具、而这一个不是。

没有谱理论、也没有任何对应于它之物。那些使算子代数有力的结构定理是被 §3 发现不可用的那些公理所购得的，而谢绝那些公理意味着谢绝那些定理。此处没有任何东西把一个安排分解为诸组成部分、指认诸不变量、或把诸安排分类到任何一个定理所证明的等价。

没有预测。那个说明陈述一个在其下一个不对称持续的条件，而不决定一个给定安排中哪些不对称满足它。它是一个用于描述与区别、而不是用于预报的框架，而它的用处系于它所划的诸区分是否是那些正确的。

一个进一步的界限从 §6 的那个被强化的条件生起。维持是被对照一个设置所容许的诸背景转变所定义的，而那个框架不决定那个类。一个实际的安排必须承受哪些扰动，是一个关于那个设置的问题，有待被无论谁知道那个设置者所裁定，而一个预先固定那个类的说明会是在立法、而不是在描述。因此，那个条件只相对于一个那个框架所不提供的规范才是良好提出的。

对这些应当添加那两个较早所记录的界限。§7 的那个分离诸政体的判准是一个区分、而不是一个检验，因为确定一个桥接的转变序列是否存在，不被那个建构中的任何东西所使之可处理。而那个会度量制度性盲的指数类似物被描述、而不被建构，因此它会许可的那个比较目前是不可用的。

10.3 那个生成集反演的提议

最值得追求的那个方向是上面所指认的、而在那里不被发展的那个。

那些微分语言中的解释从一条运动定律运行到一个轨迹：那个向量场被设定，而那个被观察到的历史是它所产生之物。此处所提议的那个反演从一个生成的剧目运行到那些可容许的复合、并从那里到那些可能的历史，以致所设定的不是那个安排如何运动、而是什么可被做在它之内、以及以什么组合。

这两者在解释内容上不是等价的。一个向量场与它所产生的任何历史相容、并且不禁止它所不产生的任何东西；一个带它诸生成元之间诸关系的生成集彻底禁止复合，而那些被禁止的复合是那个理论可被追究的诸承诺。那个提议是这使第二种形式的理论在一个具体的意义上更可证伪，而那个提议尚未被检验。会检验它之物，是一个其中被设定诸生成元之间的诸关系产出一个在与那个记录的对质中存活的禁止的案例，而没有任何这样的案例在此被提供。直到一个被提供，那个反演是一个方向、而不是一个结果，而本文把它作为如此而标出。

10.4 快变化与慢变化的未解决的耦合

那个框架的主要未解决的困难关乎它持在分开的两种变化之间的关系。

在一个剧目之内，寻常的社会动力学进行：诸方行动，诸分布移动，占据大多数社会描述的那些事件发生。在诸剧目之间，那个操作场域本身被变换。§5 的那个双重范畴分别把这些呈现为水平的与垂直的运动，而那个交换的方块记录一个在一个剧目之内被施行的操作正是那个把那个剧目携带到它后继者之物。那是一个相容性的陈述、而不是那个耦合的一个理论。

所缺的是一个关于那些时间尺度的说明。寻常的动力学是快的而结构性的转变是慢的，而这两者不仅仅在速率上不同、而是在种类上不同，因此那些分离快变量与慢变量的标准技术不适用。一个转变不是许多小的水平运动的累积；它是一个水平运动在于什么的变化。一个剧目之内的极多的行使如何组合成它的一个转变、以及在什么条件下它们未能，不被本文中的任何东西所回答。

那个问题要紧，因为 §1 的诸案例转在它上。一个消耗它自身诸前提条件的不对称通过寻常行使的累积效果而如此做，而那个说明陈述那个结局而不描述那个累积。提供那个描述会把本文的若干区分从诸分类转换为诸机制，而它是本文最想要被做的工作。

10.5 那个说明的地位

那个说明是作为一个带一个形式呈示的概念框架被提出的，而这两者应当被分开地评估。

§4 的那个机制自行成败。它主张不对称从有序的差异、而不是从差异独自生起，行使改变行使的诸条件，大多数不对称消耗它们自身，而所保留之物是那个再生产了它自己诸条件之物。一个读者可以接受这一切并拒绝随之而来的那个建构。

§5 的那个建构是作为一个无歧义地陈述那个机制的方式被提出的，而它的主张是谦抑的。它是由标准的诸成分所装配的，并贡献不了任何技术上的新颖；它所贡献的，是一个关于哪一个结构是首要的决定，即诸操作的方指标化的可用性，而以改写与复合作为那个结构所经历之物。一个读者可以接受那个建构并争议 §9 的那个图式，而它是一个进一步的、更暴露的提议。

那个说明最想要被辩护的结果是 §8 的那个否定的。生成性不是可能性的扩大，而一个在一个作为整体的安排上被计算的量不区别那个说明必须区别的诸案例。那个结果不要求任何那个形式装置，在本文中其他一切的拒绝中存活，并作为一个对反例的邀请被陈述。

---

关于方法的一则说明。本文的形式装置是刻意地单薄的，而那个单薄是一个判断、而不是一个省略。本文的两个结果，即自我维持论题作为一个选择效应的重构以及对计数判准的反驳，根本不要求任何装置，并在 §4 与 §8 中以读者可以在不接受任何建构的情况下评估的术语被陈述。§5 的那个建构被引入以无歧义地陈述其余的诸区分，而它不比那个使用所要求的更进。一个更精致的呈示是可用的、而被谢绝了：那个范畴机制容许相当的发展，而在会需要它的诸结果之前发展它，会已借用一个那个论证尚未挣得的严格。凡那个说明已伸手去取一个已确立的形式体系并发现它不合适之处，如同 §3 中的算子代数，那些理由已被详加给出、而不是被越过，本于”一个被谢绝的借用应当在那个记录中可见”这一看法。

参考文献

[1] Arendt, Hannah (1970). On Violence. New York: Harcourt, Brace & World.

[2] Bourdieu, Pierre, and Jean-Claude Passeron (1990). Reproduction in Education, Society and Culture. 2nd ed. Trans. Richard Nice. London: Sage. [French orig. 1970.]

[3] Bratteli, Ola, and Derek W. Robinson (1987). Operator Algebras and Quantum Statistical Mechanics 1: $C^$- and $W^*$-Algebras, Symmetry Groups, Decomposition of States*. 2nd ed. New York: Springer.

[4] Dahl, Robert A. (1957). The Concept of Power. Behavioral Science, 2(3), 201–215.

[5] Ehresmann, Charles (1963). Catégories doubles et catégories structurées. Comptes Rendus de l’Académie des Sciences de Paris, 256, 1198–1201.

[6] Emerson, Richard M. (1962). Power-Dependence Relations. American Sociological Review, 27(1), 31–41.

[7] Foucault, Michel (1977). Discipline and Punish: The Birth of the Prison. Trans. Alan Sheridan. New York: Pantheon. [French orig. 1975.]

[8] Foucault, Michel (1978). The History of Sexuality, Volume 1: An Introduction. Trans. Robert Hurley. New York: Pantheon. [French orig. 1976.]

[9] Haag, Rudolf, and Daniel Kastler (1964). An Algebraic Approach to Quantum Field Theory. Journal of Mathematical Physics, 5(7), 848–861.

[10] Hodgkin, Alan L., and Andrew F. Huxley (1952). A Quantitative Description of Membrane Current and Its Application to Conduction and Excitation in Nerve. Journal of Physiology, 117(4), 500–544.

[11] Jones, Vaughan F. R. (1983). Index for Subfactors. Inventiones Mathematicae, 72(1), 1–25.

[12] Lukes, Steven (2005). Power: A Radical View. 2nd ed. Basingstoke: Palgrave Macmillan. [1st ed. 1974.]

[13] Lacan, Jacques (2006). Écrits: The First Complete Edition in English. Trans. Bruce Fink. New York: Norton. [French orig. 1966.]

[14] Luhmann, Niklas (1979). Trust and Power. Trans. Howard Davis, John Raffan, and Kathryn Rooney. Chichester: Wiley. [German orig. 1975.]

[15] Luhmann, Niklas (1995). Social Systems. Trans. John Bednarz Jr. with Dirk Baecker. Stanford: Stanford University Press. [German orig. 1984.]

[16] Mac Lane, Saunders (1998). Categories for the Working Mathematician. 2nd ed. New York: Springer.

[17] Merton, Robert K. (1938). Social Structure and Anomie. American Sociological Review, 3(5), 672–682.

[18] Nietzsche, Friedrich (1998). On the Genealogy of Morality. Trans. Maudemarie Clark and Alan J. Swensen. Indianapolis: Hackett. [German orig. 1887.]

[19] Nietzsche, Friedrich (2002). Beyond Good and Evil: Prelude to a Philosophy of the Future. Ed. Rolf-Peter Horstmann and Judith Norman, trans. Judith Norman. Cambridge: Cambridge University Press. [German orig. 1886.]

[20] Osborne, Martin J., and Ariel Rubinstein (1994). A Course in Game Theory. Cambridge, MA: MIT Press.

[21] Reisig, Wolfgang (2013). Understanding Petri Nets: Modeling Techniques, Analysis Methods, Case Studies. Berlin: Springer.

[22] Rozenberg, Grzegorz, ed. (1997). Handbook of Graph Grammars and Computing by Graph Transformation, Volume 1: Foundations. Singapore: World Scientific.

[23] Takesaki, Masamichi (1972). Conditional Expectations in von Neumann Algebras. Journal of Functional Analysis, 9(3), 306–321.

[24] van der Schaft, Arjan, and Hans Schumacher (2000). An Introduction to Hybrid Dynamical Systems. London: Springer.

[25] Wasserman, Stanley, and Katherine Faust (1994). Social Network Analysis: Methods and Applications. Cambridge: Cambridge University Press.

[26] Weber, Max (1978). Economy and Society: An Outline of Interpretive Sociology. Ed. Guenther Roth and Claus Wittich. Berkeley: University of California Press. [German orig. 1922.]