Beyond Unity and Fragmentation - Relational Crystallization and the Emergence of Generative Plurality 【（Preliminary）Draft】

ENGLISH

Abstract

A polity holding several coherent traditions faces two opposite dangers. It may dissolve them into a single order, and it may allow them to separate until nothing passes between them. Governance is ordinarily theorized as the avoidance of the second, with the first taken as the ideal it approaches. This paper examines a picture in which both are failures of the same kind. Polycrystalline materials hold their form through many domains of differing orientation, joined at boundaries that are structures in their own right, and such materials are neither single crystals nor powders. The paper asks what transfers from that picture to the governance of plural relational domains, states carefully what fails to transfer, and builds a formal model on the part that survives. The model is deliberately non-optimizing: it states two positivity conditions, one requiring that each domain retain the capacity to reproduce its own meaning system and one requiring that each boundary permit exchange without imposition, and it defines injustice as the loss of either. Four failure modes follow as a theorem rather than a list, of which unity and fragmentation are two. A three-domain case study is worked through with the dynamics displayed at each step. The survey concedes that the productivity of interfaces is established in the study of scientific subcultures, that plural centres without an apex are established in the study of polycentric governance, and that the rejection of maximization is settled in political philosophy. What appears to remain open is the boundary itself, treated as a structure with properties of its own, and the nucleation of novelty at boundaries, which is the point at which the crystallographic source supplies something its rivals do not. The account’s propositions hold of the model that generates them, and the conditions under which they would bear on a polity are set out with the questions that would have to be settled first. Those questions are held to be the more valuable half of what is offered.

Keywords: relational plurality; grain boundaries; viability conditions; co-creation; generativity.

A note on the standing of this paper. This is a discussion paper and the fourth of a series. It identifies a structure, surveys what existing scholarship has made of it, marks where each account appears to run out, and proposes a direction for work. It does not complete a theory. Two features require notice at the outset. The paper borrows from a natural science, and §4 states what licenses the borrowing and where it fails, since an undisciplined transfer of physical vocabulary into social theory is a known and deserved target of criticism. And the paper offers a formal model whose propositions are consequences of stipulated dynamics; they are true of the model, and their bearing on any polity is a further question the paper does not settle. Where a claim entered here as open is in fact occupied, or where a source has been read wrongly, correction is welcome at huangwanhong@serendip.ngo.

§1 Introduction

A polity containing several coherent traditions of practice may fail in two opposite ways. Its traditions may be dissolved into a single order, whether by assimilation, by the imposition of one tradition’s categories on the rest, or by the ordinary administrative preference for uniform procedure. Or its traditions may separate, until each is internally intact and nothing of consequence passes between them. The first failure leaves a polity that decides easily and has lost what it might have decided among. The second leaves a polity whose parts survive and which cannot act as one.

Political theory has treated these asymmetrically. Fragmentation is the recognized danger, and the literatures on cohesion, integration, and shared identity are directed against it. Unity, by contrast, tends to appear as the limiting ideal, approached where circumstances allow. A polity that achieved perfect agreement would be counted fortunate.

The picture examined here treats both as failures, and it takes its structure from a class of materials in which neither occurs. Most solids of practical importance are polycrystalline. They consist of many domains, each internally ordered, whose orientations differ, joined along boundaries where the two orders meet. A polycrystal is not a single crystal, since no orientation extends through the whole. It is not a powder, since the domains are joined and the material holds together and bears load. Its properties depend as much on the boundaries as on the domains, and in several respects the boundaries are where the interesting behaviour occurs.

The suggestion pursued below is that this structure supplies a picture of plural governance that is available in neither of the received alternatives. A polity may be ordered locally without being ordered globally. Difference of orientation may be a condition of its properties. The boundary between two domains may be a structure with characteristics of its own, and the site at which something arises that neither domain contained.

Three difficulties motivate the enquiry. The first concerns the form of the normative question. A theory of plural governance that proceeds by maximizing a measure of the good imports commitments that the plurality was introduced to resist: that there is a measurable good, that arrangements are rankable, that losses in one domain are compensable by gains in another. The second concerns the treatment of boundaries. Where boundaries between communities are theorized at all, they tend to appear as obstacles to be reduced or as defects to be managed, and the possibility that a boundary is a productive structure is seldom pursued formally. The third concerns the conditions of novelty. Accounts of plural governance that stop at coexistence describe an arrangement in which nothing new arises from the plurality, and an arrangement of that kind is difficult to distinguish from an elaborate separation.

Five bodies of work bear on these difficulties and each is treated below. The study of scientific subcultures has established that groups without a shared global framework coordinate at interfaces where new intermediate structures develop (Galison 1997; Star and Griesemer 1989). Postcolonial and ethnographic work has argued that encounter at zones of difference produces forms reducible to neither party (Bhabha 1994; Pratt 1992; Tsing 2005). The study of polycentric governance has established that order may be sustained by many centres of decision with overlapping jurisdictions and no apex (Polanyi 1951; Ostrom, Tiebout, and Warren 1961; Ostrom 2010). Political philosophy has, since the rejection of aggregative utilitarianism, developed conceptions of justice as thresholds, capabilities, and relational conditions in place of maxima (Rawls 1971; Frankfurt 1987; Anderson 1999; Sen 1999; Pettit 1997; Berlin 1969). Viability theory has, finally, developed a mathematics of systems constrained to remain within a set, with no optimization performed (Aubin 1991; Aubin, Bayen, and Saint-Pierre 2011). These literatures settle a considerable proportion of the subject, and one purpose of this paper is to record how much; the survey below accordingly proceeds by conceding first and locating what remains afterwards.

The account is developed within a framework the author has set out elsewhere, and which may be stated in a sentence:

Generative Relational Being. A theory of how subject, meaning, value, creation, and normativity co-emerge through generative relational processes.

Three features of the framework bear here. Values are treated as generated within relations, so that a plurality of relational structures yields values differing in kind. Relations are treated as generative, in that a relation in working order produces further determinations of meaning, of value, and of what the relation itself requires. The framework carries, third, a methodological commitment governing how its claims are advanced, its keynote being historical dialectics, so that propositions are entered as revisable within a historical frame, the finitude of knowledge and of practice is acknowledged as a standing condition, and philosophical argument is joined to operable formulation where the subject admits of it. A paper written under that commitment states its claims so that they may be given up.

Four contributions are offered, claimed with differing degrees of confidence. The first is a disciplined statement of the analogy, given in §4, which enumerates what transfers from crystallography and what fails to. The second is the pair of conditions stated in §9, which express plural governance as constraint satisfaction with no aggregation across domains. The third is the formal model of §10, whose propositions include a derivation of four failure modes from a single requirement, and a demonstration that the conditions leave the arrangement underdetermined. The fourth is the treatment of the boundary developed in §13, where the crystallographic source supplies a structure that the surveyed rivals leave thin. The questions set down in §16 are held to be the more valuable half of what is offered.

§2 Scope and Method

2.1 The Containment Device

Three questions organize the discussion, and material earns its place by discharging one of them:

(i) The transfer. What may be carried from the behaviour of polycrystalline materials to the governance of plural relational domains, and on what licence.

(ii) The conditions. What must hold of a plural arrangement for it to be defensible, given that maximization is unavailable.

(iii) The boundary. What a boundary between domains is, and under what conditions it produces something that neither domain contained.

2.2 The Conceding Register

A theory whose components are individually unoriginal may still contribute through the manner in which it joins them, and the joining becomes visible only where the components are correctly attributed. The argument here is architectural, and an architectural argument is weakened by any claim to have quarried its own stone. Three sections consist largely of concessions and are placed before the constructive material for that reason.

2.3 The Boundaries of the Undertaking

Three matters lie outside the enquiry, and marking them fixes what the conditions of §9 are conditions upon.

The object is a condition, with arrangements left aside. A condition specifies what an arrangement must leave possible; which arrangements would secure it in a given polity is a question of institutional design, and §16 enters it as such.

The object is a model, and the polity such a model might describe is a further matter. The case study of §11 works through the model’s behaviour under varied parameters, and its labels are chosen for legibility.

The individuation of domains is taken as given. Which relational structures a polity comprises, and by what procedure they are identified, is the presupposition on which every condition below rests, and §16 treats it as the account’s first outstanding question.

2.4 A Note on Sources

The survey draws on materials science, science and technology studies, political theory, philosophy of science, and applied mathematics. The transfer of concepts across such a range is the paper’s principal methodological risk, and §4 is written to contain it.

§3 Crystallization as a Source Phenomenon

The source is set out before anything is drawn from it, since a transfer from a misdescribed source is worthless. The account is given in the terms the physical literature uses, with the governing relations stated, because a qualitative paraphrase would conceal exactly the conditions under which each relation holds, and those conditions bear on what may be transferred.

3.1 Nucleation and the Existence of a Barrier

A liquid cooled below its freezing point does not order itself at once. For a spherical region of the ordered phase of radius $r$, the free-energy change relative to the disordered parent is a competition between a volume term that favours the new phase and a surface term that opposes it:

$$ \Delta G(r) ;=; -\tfrac{4}{3}\pi r^{3},\Delta g_{v} ;+; 4\pi r^{2}\gamma , \tag{1} $$

where $\Delta g_{v}>0$ is the magnitude of the bulk free-energy reduction per unit volume of the new phase and $\gamma>0$ is the interfacial free energy per unit area. The surface term dominates at small $r$ and the volume term at large $r$, so $\Delta G$ rises, attains a maximum, and falls. Small regions therefore lower their free energy by disappearing, and only a region carried past the maximum by fluctuation can lower its free energy by growing. Setting $\mathrm{d}\Delta G/\mathrm{d}r=0$,

$$ r^{}=\frac{2\gamma}{\Delta g_{v}}, \qquad \Delta G^{}=\frac{16\pi\gamma^{3}}{3,\Delta g_{v}^{2}} . \tag{2} $$

Panel (a) of Figure 1 plots equation (1) with its two components; the marked maximum agrees with equation (2) to the resolution of the plot. Two features matter below. Order begins at scattered points, leaving the rest of the field unchanged, and a threshold must be crossed before a new arrangement can persist at all.

Figure 1. The three relations of §3 that carry weight later, computed from the expressions given in the text. In (a) the free energy of a forming region, equation (1), with the volume and surface contributions dotted and dashed and the critical point marked. In (b) the factor $f(\theta)$ of equation (3), by which a pre-existing surface reduces the barrier. In (c) boundary energy against misorientation: the Read–Shockley form (5) below the low-angle limit, an approximate plateau above it, and a cusp at a special misorientation. Panel (c) is schematic in its vertical scale and is drawn to display the shape, in particular the non-monotonicity on which §4 turns.

3.2 Why New Order Appears at Boundaries

A region forming on a pre-existing surface need not create its whole interface anew. For a spherical cap meeting a flat substrate at contact angle $\theta$,

$$ \Delta G^{}_{\text{het}} ;=; f(\theta),\Delta G^{}_{\text{hom}}, \qquad f(\theta)=\frac{2-3\cos\theta+\cos^{3}\theta}{4}=\tfrac{1}{4}(1-\cos\theta)^{2}(2+\cos\theta), \tag{3} $$

with $f(180^{\circ})=1$, $f(90^{\circ})=\tfrac{1}{2}$, and $f\to0$ as $\theta\to0$. Panel (b) of Figure 1 plots it. One point of precision should be recorded, because it is often mistaken: $f(\theta)$ reduces the barrier and leaves the critical radius $r^{*}$ of equation (2) unchanged.

The case that matters here is formation on a boundary between two existing domains, treated by Clemm and Fisher (1955) and Cahn (1956). A region forming there destroys a patch of the boundary, and the energy of the destroyed boundary is a credit against the interface that must be created. With $k=\gamma_{\alpha\alpha}/2\gamma_{\alpha\beta}$, the ratio of the energy of the existing boundary to twice the energy of the new interface, the barrier falls as the credit rises, and it falls further at junctions where more boundary area is destroyed per unit volume formed. The ordering established there is

$$ \Delta G^{}_{\text{hom}} ;>; \Delta G^{}{\text{face}} ;>; \Delta G^{*}{\text{edge}} ;>; \Delta G^{*}_{\text{corner}}, \tag{4} $$

with formation on a given class of site occurring only below a critical $k$ for that class. New order therefore appears preferentially where existing domains meet, and most readily where several meet at once. This is the single relation from which §13 draws most, and it is stated here in the form its authors gave it.

3.3 Boundary Structure and Misorientation

A boundary between domains whose orientations differ by a small angle $\theta$ may be described as an array of dislocations of spacing $b/\theta$. Summing their elastic energies gives the Read–Shockley relation

$$ \gamma(\theta) ;=; \gamma_{0},\theta,(A-\ln\theta), \qquad \gamma_{0}=\frac{Gb}{4\pi(1-\nu)}, \tag{5} $$

with $G$ the shear modulus, $b$ the Burgers vector magnitude, $\nu$ Poisson’s ratio, and $A$ a constant absorbing the dislocation core. The relation holds for low-angle boundaries only, conventionally $\theta\lesssim15^{\circ}$; above that the description fails, and boundary energy approximately saturates.

Two facts about $\gamma$ as a function of misorientation are decisive for this paper and are recorded here in the body of the text. Boundary energy is not monotone in misorientation: it rises steeply at small angles, saturates at large ones, and possesses deep local minima at particular misorientations at which the two lattices share a high proportion of sites. The coherent twin is the standard instance, with an energy in copper roughly an order of magnitude below that of a general boundary. A boundary is specified, second, by five macroscopic parameters, three for the relative orientation and two for the plane on which the boundary lies, so that misorientation underdetermines boundary properties. Panel (c) of Figure 1 displays the shape. §4 states what this forbids.

3.4 Boundaries as Obstacle and as Weakness

Boundaries impede the propagation of deformation, and the yield stress of a polycrystal rises as domain size $d$ falls:

$$ \sigma_{y} ;=; \sigma_{0} + k,d^{-1/2}. \tag{6} $$

A material of many small domains is stronger than one of few large domains, which is the physical sense in which plurality of domains is not a defect. The relation reverses below a crossover size, where deformation passes to boundary-mediated mechanisms and further refinement weakens the material; simulations of copper place the maximum near ten to fifteen nanometres.

Boundaries are at the same time paths of enhanced transport, with diffusion along them exceeding diffusion through domain interiors by several orders of magnitude, and they are the sites at which impurities collect and at which fracture propagates when they do. Boundaries are accordingly neither defects nor advantages as such: what they do depends on the property in question and on what has collected at them.

3.5 Coarsening and the Status of Plurality

Boundary migration is driven by curvature. With mobility $M$ and driving pressure $P$,

$$ v = M P, \qquad P = \gamma\kappa = \frac{2\gamma}{r}, \tag{7} $$

so a boundary moves toward its centre of curvature and boundary area falls. In two dimensions the von Neumann–Mullins relation makes the consequence exact for an isotropic domain of $n$ sides:

$$ \frac{\mathrm{d}A}{\mathrm{d}t} ;=; \frac{\pi M\gamma}{3},(n-6), \tag{8} $$

so that domains with more than six neighbours grow, those with fewer shrink, and six is neutral. Domains are lost, the survivors enlarge, and the mean size advances as

$$ \bar D^{,n} - \bar D_{0}^{,n} = \kappa t , \tag{9} $$

with $n=2$ in the ideal isotropic case and values between two and four usual in real materials.

Figure 2. Coarsening in a simulated field of 900 domains under a local curvature-reducing rule. In (a) the number of surviving domains falls monotonically; in (b) the mean size of the survivors rises; in (c) the boundary network that remains at the end of the run. The simulated rule coarsens somewhat faster than curvature flow, giving a best-fit exponent near $1.5$ in equation (9) where the ideal value is $2$; the exponent carries no weight below. The simulation is used for the monotone and unforced loss of domains, which is robust to the choice of local rule.

Figure 2 shows the process in simulation. Two consequences follow, and the second bears directly on any normative reading of the source.

Coarsening requires no agent. No domain intends to consume its neighbours; each boundary responds to its own curvature, and the loss of plurality follows from local rules operating everywhere.

Plurality is, second, not the favoured state. Every boundary carries positive excess free energy, so the single crystal is the thermodynamic ground state and a polycrystal is metastable, held in being by kinetic obstruction: low mobility, and obstacles that pin boundaries. A dispersion of second-phase particles of radius $r$ and volume fraction $f$ exerts a pinning pressure that arrests growth at a limiting size of order

$$ R_{\text{lim}} ;\sim; \frac{r}{f}, \tag{10} $$

with the standard estimate $R_{\text{lim}}=4r/3f$ and prefactors varying with convention. Where such obstruction is absent or is removed, coarsening resumes; and where normal coarsening is pinned but a few boundaries escape, a small number of domains may grow to consume the rest, a condition known as abnormal growth, which is the route by which a polycrystal is deliberately converted into something approaching a single crystal.

One qualification belongs with this, since it will matter in §15. The metastability of plurality is not absolute. If a segregating species accumulates at boundaries it lowers their energy, in the simplest treatment

$$ \gamma ;=; \gamma_{0} - \Gamma_{s}!\left(\Delta H_{\text{seg}} + k_{B}T\ln X\right), \tag{11} $$

and where $\gamma$ is driven toward zero the driving force for coarsening vanishes and a plural state of finite domain size becomes a genuine equilibrium in place of a delay. This route is predicted and simulated; it is not experimentally settled, and it is reported here as a possibility whose confirmation is outstanding.

3.6 Findings and Residue

Order arises at scattered points and only past a threshold, extends by local rule with no global calculation and nothing optimized, and produces boundaries where independently oriented regions meet. New order forms preferentially at those boundaries, and most readily where several meet, because forming there destroys existing boundary and recovers its energy. Boundary energy rises with misorientation at small angles, saturates at large ones, and dips at special misorientations, so that difference and boundary property are not monotonically related. Boundaries both obstruct deformation and provide the paths along which transport and failure travel. Plurality of domains is metastable and decays by coarsening unless something obstructs it, and may in principle be stabilized by a species that accumulates at the boundaries themselves.

§4 The Discipline of the Analogy

Transfers of physical vocabulary into social theory have a poor record, and the criticism they have attracted is largely deserved (Sokal and Bricmont 1998). The apparatus for conducting such a transfer responsibly is available, and this section applies it.

4.1 The Status Claimed

Hesse (1966) distinguishes, within an analogy, the properties known to be shared, the properties known to differ, and the properties whose status is undetermined. Inference proceeds from the first, is blocked by the second, and is licensed as conjecture in the third. Black (1962) treats a model as an instrument that organizes a subject by importing a structure, and holds that its value lies in the questions it makes askable.

The status of the analogy. Crystallography is used here as a source of structure and as a generator of hypotheses. Its role is exhausted by those two functions: the claims advanced below concern relational domains and their boundaries, and they are answerable to grounds internal to political theory and to the model of §10. The formal model of §10 is a stipulated system whose propositions are true of it, and whose relation to any polity is conjectural.

4.2 The Positive Analogy

Four structural features are carried forward.

Local rules producing extended order. Coherent arrangements may extend across a population through interactions each of which is local, with no participant possessing the structure of the whole.

Independent nucleation producing differing orientations. Where order begins at several points independently, the resulting domains will differ in orientation as a matter of course, and their difference requires no explanation beyond the independence of their origins.

Impingement producing boundaries. Where two coherent domains meet, the surface of meeting is a structure determined by both and identical with neither.

Boundaries as sites of formation. The conditions at a boundary differ from those within either domain, and formation of something new occurs preferentially there.

4.3 The Negative Analogy

Eight disanalogies are entered. Each marks a feature of the source that is absent from the target, and the arguments below are constructed so as to turn on features common to both. Two of them correct a transfer the source will not license and are given first.

Units of a material have no interests, purposes, or interpretations. An atom does not understand its neighbours, and nothing in the physical case corresponds to a party’s reading of what another party is doing. Every argument below concerning imposition and rendering has no physical counterpart.

Grain boundaries are not evaluable. A physical boundary is neither just nor unjust. The normative content of every condition below is supplied by argument in political theory and is imported into the model, never derived from the physical source.

There is no social free energy. The physical account has thermodynamic potentials that make certain evolutions determinate. No quantity of that kind is claimed here, and the model below is stipulated, with no derivation from any such potential.

Orientation is not measurable. A crystallographic orientation is a physical quantity. The orientation of a relational domain, used below, is a placeholder for a mode of articulation and admits of no measurement procedure at present.

Domains are not sharply bounded. A grain has a definite extent. A relational domain does not, its membership is contested, and parties belong to several at once. The model treats domains as discrete because a tractable model requires it, and §15 records the cost.

Coarsening is not deliberate; homogenization often is. The physical process has no agent. The corresponding social process frequently has one, and the difference matters at every point where responsibility is at issue.

Difference does not vary monotonically with the property of the boundary it produces. This is the disanalogy most likely to be assumed away, and §3 recorded the physical fact that forbids it. Boundary energy rises with misorientation only at small angles, saturates thereafter, and falls sharply at special misorientations where the two orders happen to share structure. Two consequences follow for the model of §10. Where co-creation is made to increase with misorientation, that is a stipulation of the model and is licensed by the physical source only within the low-angle regime. The physical source positively suggests, second, something the model does not represent: that particular pairs of widely differing domains may stand in an unusually accommodating relation, cheap to maintain and correspondingly unproductive of anything new, which would be a special boundary in the sense of equation (5) and its exceptions. Whether relational analogues of such boundaries exist is entered among the open questions.

Boundary properties are underdetermined by misorientation. A physical boundary requires five parameters and misorientation supplies three. The model below characterizes a boundary by a single difference and two further quantities, and does so for tractability; nothing in the physical source suggests that so coarse a description is adequate.

4.4 Findings and Residue

Four structural features transfer and six do not. The transferred features concern the emergence of plural coherent domains, the structure of what lies between them, and the preferential formation of novelty there. The normative content of the account is imported from political theory, and the physical source supplies structure and vocabulary alone.

§5 Interfaces as Productive Sites

The largest concession of this paper is owed to work that reached its central observation some decades ago.

5.1 Trading Zones

Galison (1997) examines the coordination of scientific subcultures that hold incompatible commitments and share no global framework. Theorists, experimenters, and instrument builders coordinate, on his account, within a trading zone, where a local interlanguage develops for the purposes of exchange. The interlanguage begins as a restricted jargon adequate to a particular transaction, thickens into a pidgin as the exchange becomes routine, and may stabilize into a creole rich enough to support a practice of its own. His summary formulation, that what is achieved is local coordination without global meaning, states the position this paper’s picture arrives at from another direction.

The consequence for the present account should be stated without qualification. The proposition that coherent domains lacking a common framework generate, at their interface, a structure belonging to neither and capable of becoming a domain in its own right is not new. It is the central finding of the trading zone literature, and a paper presenting it as a discovery would be mistaken.

5.2 Boundary Objects

Star and Griesemer (1989) identify objects that inhabit several communities at once, plastic enough to be adapted to local requirements and robust enough to preserve a common identity across sites. Their treatment establishes that an interface may be inhabited by stable structures with characteristics of their own, and it develops a typology of such structures.

5.3 Contact, Hybridity, Friction

Three further treatments occupy the same terrain from the humanities. Pratt (1992) names the contact zone, in which parties previously separated encounter one another under conditions of asymmetry. Bhabha (1994) argues that cultural statements are constructed in a third space of enunciation, and that what emerges from encounter is irreducible to either source. Tsing (2005) treats friction at zones of difference as productive of new arrangements.

5.4 Coexistence and Co-Creation

A parallel debate has been conducted in policy terms. The critique of multiculturalism developed by Cantle (2005, 2012) holds that arrangements securing the standing of communities may produce parallel lives, in which groups occupy the same territory with little contact and develop separately. Interculturalism, developed by Bouchard (2012) and others, proposes that diversity be treated as a resource generating something new, and Parekh (2000) argues that cultures are transformed through dialogue.

The move from coexistence to co-creation is accordingly the constitutive move of an existing position, and this paper inherits it.

5.5 The Residue

Two things appear to remain.

The literature theorizes interlanguages, boundary objects, and third spaces without supplying a typology of boundaries by structure. Whether a boundary strengthens or weakens what it joins, and what distinguishes the two cases, is not the question these accounts were built to answer.

The literature is, second, largely descriptive. It establishes that interfaces are productive and examines the conditions under which coordination succeeds. A criterion distinguishing a defensible interface from an indefensible one is not among its products.

5.6 Findings and Residue

The productivity of interfaces is established, the move from coexistence to co-creation is the constitutive move of interculturalism, and the vocabulary of interlanguage, boundary object, contact zone, and third space is available. The residue lies in the structural typology of boundaries and in a normative criterion for them.

§6 Plural Centres without an Apex

The polycrystalline picture has a close relative in governance theory, and the relation must be stated plainly.

6.1 Polycentricity

Polanyi (1951) introduced polycentric order for arrangements in which many participants adjust to one another under general rules with no central direction. Ostrom, Tiebout, and Warren (1961) carried the term into political science, describing metropolitan governance as many formally independent centres of decision with overlapping jurisdictions and no single apex. Ostrom (2010) developed polycentric governance for common-pool resources, establishing that arrangements of this kind sustain themselves and explain outcomes that a dichotomy of market and state does not.

6.2 The Concession

The structural claim of the polycrystalline picture, that a stable order requires local coherence within domains together with workable relations among them and requires no single order extending through the whole, is the claim of polycentricity. The two words describe the same structure, and a reader familiar with the polycentric literature will find the picture familiar.

6.3 The Residue

One asymmetry between the two literatures leaves room, and it is the asymmetry this paper works in.

Polycentricity theorizes centres. Its unit is the decision centre, its questions concern the autonomy of centres, the overlap of jurisdictions, and the rules under which centres adjust to one another. What lies between centres is treated as relation, adjustment, or contract, with no structure of its own.

The crystallographic source theorizes boundaries as intensively as domains. A boundary has a structure determined by the orientations meeting at it, that structure varies, and the variation accounts for much of the material’s behaviour. §13 develops this asymmetry into the paper’s principal claim.

6.4 Findings and Residue

Plural centres without an apex are established ground, and the polycrystalline picture adds nothing to that claim. The residue lies in the treatment of what stands between centres, which polycentricity leaves thin and which the crystallographic source treats as a structure.

§7 Justice without Maximization

The refusal to write justice as an optimization problem is the paper’s normative starting point and is a settled position in political philosophy.

7.1 The Established Refusal

Rawls (1971) rejects aggregative utilitarianism on the ground that it fails to take seriously the distinction between persons, since a maximizing criterion permits a loss to one to be compensated by a gain to another as though the two occurred within a single life. The rejection is foundational to the subsequent literature.

Four developments carry it further. Frankfurt (1987) argues that what matters is whether each has enough, so that the criterion is a threshold and no maximum is required. Sen (1999) and Nussbaum (2011) frame justice in terms of capabilities secured, which is a condition on what a person is able to do and be. Anderson (1999) grounds equality in the character of the relations in which persons stand, so that the object of concern is relational in character. Pettit (1997) treats freedom as non-domination, which is a property of a relationship and admits of no natural maximand. Berlin (1969) argues, finally, that fundamental values are plural and incommensurable, so that no single scale is available on which arrangements could be ranked.

7.2 The Concession

A paper claiming credit for rejecting maximization would be claiming credit for the settled consensus of a discipline. The refusal is inherited here, and the sources above are its authors.

7.3 The Residue

What is comparatively less developed is the formal expression of the refusal. A criterion stated as a threshold or as a relational condition is ordinarily left in prose, and the mathematics ordinarily reached for in modelling work is optimization, which reintroduces what the prose refused. §10 states the conditions in a form that permits a model without reintroducing a maximand, and §14 records that the mathematics required already exists.

7.4 Findings and Residue

The rejection of maximization is settled and is inherited here. The residue lies in expressing plural conditions formally without smuggling an objective function back in.

§8 The Relational Field

The apparatus is stated here. The definitions are stipulative and are chosen for tractability.

Definition 1 (Relational domain). A relational domain $D_i$ is a set of parties together with the relations among them, such that the relations sustain a common mode of articulation: a way of stating what is at issue, what counts as a reason, and what would settle a question.

Definition 2 (Coherence). The coherence $c_i \in [0,1]$ of a domain is its capacity to reproduce its own mode of articulation across occasions and across changes of membership. A domain with $c_i = 0$ has lost that capacity: its mode of articulation is no longer transmitted, and questions arising within it are settled in the terms of some other domain.

Definition 3 (Orientation and misorientation). The orientation $\theta_i$ of a domain is its mode of articulation, represented as a point in a space $\Theta$. The misorientation $\Delta_{ij} \in [0,1]$ between two domains is their separation in $\Theta$, normalized so that $\Delta_{ij}=0$ where the modes coincide and $\Delta_{ij}=1$ where they are maximally distinct.

Definition 4 (Contact graph). The contact graph $\Gamma$ has the domains as vertices, with $(i,j) \in \Gamma$ where the two domains stand in a relation such that determinations in one bear on the other. Domains not in contact have no boundary, and no condition below applies to them.

Definition 5 (Exchange and asymmetry). For $(i,j) \in \Gamma$, the exchange intensity $e_{ij} \in [0,1]$ is the volume of traffic across the boundary, and is symmetric. The asymmetry $\sigma_{ij} \in [-1,1]$ records the extent to which one domain’s terms govern the exchange, with $\sigma_{ij}>0$ where $i$’s terms govern, $\sigma_{ij} = -\sigma_{ji}$, and $|\sigma_{ij}|=1$ where one domain’s terms govern entirely.

Definition 6 (Boundary viability). The viability of the boundary $(i,j)$ is $v_{ij} = e_{ij},(1-|\sigma_{ij}|)$. A boundary is viable where $v_{ij}>0$, which holds precisely where exchange occurs and neither domain’s terms govern it entirely.

Two features of the last definition should be marked. It is a stipulation chosen because it makes the two ways of failing visible in one expression. It renders, second, the two failures distinguishable: $v_{ij}=0$ with $e_{ij}=0$ is separation, and $v_{ij}=0$ with $|\sigma_{ij}|=1$ is domination. The vanishing of viability is therefore the meeting point of two distinct conditions.

Definition 7 (Imposition load). The load borne by domain $i$ is $\Lambda_i = \sum_{j:(i,j)\in\Gamma} e_{ij},\max(0,;\sigma_{ji})$, the summed traffic across boundaries at which another domain’s terms govern.

§9 The Two Conditions and the Four Failure Modes

9.1 The Conditions

Proposed conditions. A plural relational arrangement is defensible when both hold:
$$ \text{(C1)}\quad c_i>0 \ \ \text{for every } i, \qquad\qquad \text{(C2)}\quad v_{ij}>0 \ \ \text{for every } (i,j)\in\Gamma. $$
Condition C1 requires that every domain retain the capacity to reproduce its own mode of articulation. Condition C2 requires that every boundary permit exchange without the terms of either side governing entirely.

The form of the conditions carries the normative content. Both are universally quantified over domains and boundaries, and neither is an aggregate. No sum, average, or weighted total appears, and nothing in the conditions permits a deficiency at one domain to be offset by a surplus at another.

9.2 The Independence of the Conditions

Proposition 1 (Independence). Neither condition implies the other. There exist arrangements satisfying C1 and violating C2, and arrangements satisfying C2 and violating C1.

The proposition is established by the two constructions in §11. Scenario B has every domain at maximal coherence with every boundary dead, satisfying C1 and violating C2. Scenario A has every boundary viable with one domain extinguished, satisfying C2 and violating C1. The two conditions are therefore doing separate work, and an account stating only one of them would miss the failures the other catches.

The first construction deserves emphasis. Under separation, coherence attains its maximum for every domain: a domain subject to no imposition reproduces itself perfectly. Isolation is optimal by the measure of C1 alone. A theory of plural governance built on the protection of domains, with no condition on what passes between them, therefore recommends the arrangement that the critique of parallel lives was directed against.

9.3 The Four Failure Modes

Definition 8 (Co-creation). Co-creation occurs at a boundary where a determination arises there that belongs to neither adjoining domain’s prior mode of articulation. The rate of co-creation at $(i,j)$ is stipulated as $n_{ij} = \eta, v_{ij}, \Delta_{ij}, \min(c_i,c_j)$, with $\eta>0$.

The stipulation carries the crystallographic transfer of §4: formation occurs preferentially at boundaries, requires a difference across the boundary, and requires that both sides be capable of contributing.

Proposition 2 (The four failure modes). $n_{ij}>0$ holds if and only if all four of the following hold:
$$ e_{ij}>0, \qquad |\sigma_{ij}|<1, \qquad \Delta_{ij}>0, \qquad \min(c_i,c_j)>0. $$
Co-creation therefore fails in exactly four ways: by separation $(e_{ij}=0)$, by domination $(|\sigma_{ij}|=1)$, by homogenization $(\Delta_{ij}=0)$, and by extinction $(\min(c_i,c_j)=0)$.

Proof. $n_{ij}$ is a product of $\eta>0$ with $v_{ij}$, $\Delta_{ij}$, and $\min(c_i,c_j)$, each non-negative. A product of non-negative factors is positive if and only if every factor is positive. By Definition 6, $v_{ij}>0$ if and only if $e_{ij}>0$ and $|\sigma_{ij}|<1$. $\blacksquare$

Figure 3. The four ways in which co-creation fails, shown schematically. Shading and internal rulings represent a domain’s orientation. Under homogenization the two orientations coincide and the boundary is no longer a meeting of anything. Under separation the domains are intact and unconnected. Under domination one domain’s terms govern the exchange, drawn as a single arrow. Under extinction one domain has lost the capacity to reproduce its mode of articulation.

The proposition earns the paper’s title. Homogenization and separation are the two failures named there, and the derivation places them alongside two others as consequences of one requirement. A polity avoiding fragmentation by pursuing unity moves from the second failure to the first, and the model registers no improvement.

9.4 Findings and Residue

Two conditions are stated, in a form that quantifies over domains and boundaries and aggregates nothing. The conditions are logically independent, and the isolation result shows why both are required. Four failure modes follow from a single requirement by a two-line argument, and two of them are the extremes the paper is named against.

§10 The Formal Model

The conditions of §9 are static. This part supplies dynamics, so that the loss of a condition may be described as a process as well as a state.

10.1 The Dynamics

Definition 9 (Coherence dynamics). For each domain $i$, with intrinsic reproduction rate $\rho_i>0$ and susceptibility $\delta>0$,
$$ \dot c_i ;=; \rho_i, c_i,(1-c_i);-;\delta, c_i, \Lambda_i. \tag{12} $$

The first term is logistic: a domain reproduces its mode of articulation at a rate proportional to the coherence already attained and to the room remaining. The second term is the cost of rendering: a domain obliged to state itself in another’s terms transmits its own less effectively, at a rate proportional to its coherence and to the load it bears.

The model is deliberately minimal. Nothing is optimized, no agent chooses, and the controls $e$ and $\sigma$ enter only through $\Lambda$.

10.2 The Threshold

Proposition 3 (Coherence threshold). Fix $\Lambda_i$. If $\delta\Lambda_i \geq \rho_i$ then $c_i(t)\to 0$ from any $c_i(0)\in(0,1]$. If $\delta\Lambda_i < \rho_i$ then $c_i(t)\to c_i^{} = 1-\delta\Lambda_i/\rho_i$, which is asymptotically stable on $(0,1]$.*

Proof. Write $\dot c_i = c_i\bigl(\rho_i-\delta\Lambda_i-\rho_i c_i\bigr)$. On $c_i>0$ the sign of $\dot c_i$ is the sign of $\rho_i-\delta\Lambda_i-\rho_i c_i$. If $\rho_i-\delta\Lambda_i\leq 0$ this is negative for all $c_i>0$, so $c_i$ decreases monotonically and is bounded below by $0$; the only equilibrium in $[0,1]$ is $c_i=0$. If $\rho_i-\delta\Lambda_i>0$ the expression vanishes at $c_i^{}=(\rho_i-\delta\Lambda_i)/\rho_i \in (0,1]$, is positive below it and negative above it, so $c_i^{}$ attracts every trajectory from $(0,1]$. $\blacksquare$

Figure 4. The coherence threshold, with $\rho_i=\delta=1$. Panel (a) gives equilibrium coherence as a function of imposition load: coherence declines linearly and reaches zero at $\Lambda_i=\rho_i/\delta$, beyond which the domain cannot reproduce itself. Panel (b) shows trajectories either side of the threshold from a common initial condition.

Figure 4 displays the result. Two features bear on the argument. The approach to extinction is continuous in the load, so a domain under increasing imposition loses coherence gradually and without any threshold being visibly crossed until it is. Extinction is reached, second, at a finite load, so imposition need not be total to be fatal: it suffices that $\Lambda_i$ exceed $\rho_i/\delta$.

Definition 10 (Degenerative transformation). A degenerative transformation is a change in the arrangement that raises some $\Lambda_i$, lowering $c_i^{}$. It is extinguishing where the resulting load satisfies $\delta\Lambda_i\geq\rho_i$.*

The definition supplies the model’s account of injustice. Linguistic suppression, epistemic exclusion, obligatory translation into an official register, the administration of a category that a domain does not use, and the absence of any procedure in which a domain’s terms are receivable are each descriptions of a raised $\Lambda_i$.

10.3 The Refusal of Aggregation

Proposition 4 (Non-aggregability). Condition C1 is not implied by the maximization of any weighted aggregate $\sum_i w_i c_i$ with $w_i>0$. There exist states violating C1 that attain a strictly greater aggregate than states satisfying it.

Proof. Take three domains and $w_i=1$. The state $(0.34,0.34,0.34)$ satisfies C1 with aggregate $1.02$. The state $(0,0.75,0.75)$ violates C1 with aggregate $1.50$. The aggregate ranks the second above the first while the second extinguishes a domain. $\blacksquare$

The proposition is elementary and it is the formal statement of the position inherited in §7. A maximizing criterion permits the extinction of a domain to be purchased by gains elsewhere, and the universally quantified condition does not.

10.4 Underdetermination

An objection arises at this point and should be met. Constraining a system to remain within a set may look like optimization in another notation, with the set playing the part of an objective. The reply is that a maximum selects and a constraint does not.

Proposition 5 (Underdetermination). Consider three domains in a triangle, $\rho_i=\delta=1$, uniform exchange $e$ on every boundary, and asymmetry $\sigma\ge 0$ oriented so that domain $0$’s terms govern at both its boundaries and domain $1$’s govern at its boundary with domain $2$. Then $\Lambda_0=0$, $\Lambda_1=e\sigma$, $\Lambda_2=2e\sigma$, and conditions C1 and C2 hold jointly if and only if $e>0$, $\sigma<1$, and $2e\sigma<1$. This set has non-empty interior in $(e,\sigma)$, and the conditions therefore do not determine the arrangement.

Proof. By Proposition 3, $c_i>0$ if and only if $\Lambda_i<1$; the binding case is $\Lambda_2=2e\sigma$. By Definition 6, $v>0$ if and only if $e>0$ and $\sigma<1$. The conjunction is the stated region, which contains the open neighbourhood of $(e,\sigma)=(1/2,1/2)$. $\blacksquare$

Figure 5. The admissible region in the control plane for the arrangement of Proposition 5. The shaded set is bounded on the right by total imposition $(\sigma=1)$, below by the absence of exchange $(e=0)$, and above by the extinction locus $2e\sigma=1$. The three scenarios of §11 are marked. The region is an open set with interior, so the conditions admit a continuum of arrangements and single out none.

Figure 5 displays the region. Its shape carries the philosophical point of §7 in operational form. A criterion of this kind excludes arrangements without prescribing one, and the question of which admissible arrangement a polity adopts is left open for reasons the criterion does not supply. A polity is bounded, and the direction it takes within those bounds is its own.

10.5 Capture of Orientation

The model as stated has domination reduce coherence, so that a dominated domain approaches extinction. That representation is incomplete, and correcting it yields the model’s least obvious result.

A domain subjected to sustained imposition need not cease to reproduce itself. It may continue to reproduce, and to do so in terms it has taken from the domain imposing on it, so that what is lost is the distinctness of its articulation, with its capacity to articulate preserved. Let orientation accordingly move under the same pressure that generates the imposition load:

$$ \dot\theta_{i} ;=; \kappa \sum_{j,:,(i,j)\in\Gamma} e_{ij},\max(0,\sigma_{ji}),\bigl(\theta_{j}-\theta_{i}\bigr), \tag{13} $$

so that a domain drifts toward the orientation of whichever domain imposes on it, at a rate set by the intensity and the asymmetry of the boundary between them, and does not drift at all where the boundary is symmetric.

Figure 6. Capture of orientation under sustained asymmetric exchange, computed from equations (12) and (13). In (a) every coherence remains positive, the three domains settling at $1.00$, $0.62$ and $0.23$. In (b) the orientations of the two subordinated domains are drawn to that of the third. In (c) the misorientations across all three boundaries fall to zero. Co-creation therefore ceases although no domain has been extinguished and every boundary has remained open.

Proposition 6 (Capture). Under equation (13) with $\sigma_{ij}$ of fixed sign and $e_{ij}>0$, the orientations of dominated domains converge on that of the dominating domain, so that $\Delta_{ij}\to0$ across the affected boundaries. Where the imposition load remains below the threshold of Proposition 3, this occurs with every coherence positive and every boundary viable.

The simulation reported in Figure 6 exhibits it: the coherences settle at $1.00$, $0.62$ and $0.23$, all positive; every boundary remains open; and every misorientation falls to zero, so that the co-creation rate of §9 vanishes through its third factor.

Three consequences follow.

Capture is a path from domination to homogenization. The four failure modes of §9 were established as logically independent, and equation (13) supplies a dynamical route from one to another: a polity that imposes need not extinguish anything, since sustained imposition converts difference into sameness and sameness ends co-creation without any domain being lost.

Capture is invisible to a test on coherence alone. An observer monitoring whether each domain continues to reproduce itself would record no failure whatever along the trajectory of Figure 6. Every domain is intact, every channel is open, and the capacity that has been lost is registered by neither measurement.

Capture is, third, the form in which the account represents a claim the framework makes elsewhere: that pressure on a relational subject characteristically redirects what it can generate, leaving the generating itself intact. A domain under capture is generating as much as before and generating in another’s terms. The distinction between a condition on generation and a condition on the distinctness of what is generated is the distinction between the first of the two conditions and the third factor of the co-creation rate, and §9 requires both for that reason.

A caution belongs with this, and it is a caution about the source and leaves the model untouched. In the physical case a domain’s orientation is ordinarily fixed, and the microstructure evolves by boundaries migrating so that one orientation replaces another. Continuous reorientation of a domain does occur, by rotation coupled to shear, but it is prominent only in fine-grained and high-temperature regimes. Equation (13) therefore has a physical counterpart in a restricted regime, falling short of a general feature of the source, and it is offered here as a stipulation about relational domains for which the physical case supplies a suggestive precedent, falling short of a warrant.

§11 A Case Study

The model is exhibited on three illustrations, chosen on a stated principle, with the arrangement varied within the first and the consequences traced.

11.1 Why These Illustrations

Illustrations in a paper of this kind are chosen for what they discriminate, and the principle of selection is stated below in place of leaving a reader to infer it. The four failure modes of §9 are distinguished by which factor of the co-creation rate vanishes, and an illustration earns its place by exhibiting a failure that the others do not, under conditions that differ in the respects the model treats as relevant: whether the domains are territorially separated, whether one of them administers the arrangement, and whether the boundaries are institutionally recognized at all.

Three illustrations are accordingly used. The first is a resource decision among domains sharing a territory and answering to an authority that is itself one of them, which exhibits domination and separation and the passage between them. The second is a scientific collaboration among domains with no shared authority and boundaries that are recognized and staffed, which exhibits the productive case and the special-boundary difficulty of §4. The third is a professional standard-setting body in which the boundaries are unrecognized, which exhibits capture, the failure that no coherence test detects.

They are illustrations. Their function is to display what the propositions assert, in settings whose institutional features differ enough that a reader may judge whether the assertions are worth taking to a case where measurement might be attempted.

11.2 The First Setting

Three domains bear on the disposition of a watershed. The first is customary, sustained by inherited practice and articulating the watershed in terms of obligation and standing. The second is regulatory and scientific, articulating it in terms of measured condition against stated criteria. The third is commercial, articulating it in terms of valuation and return. Each is in contact with both others, so the contact graph is a triangle.

Orientations are placed at $\theta = (0.15\pi,, 0.55\pi,, 0.85\pi)$ on a circle of period $\pi$, giving misorientations $\Delta_{01}=0.8$, $\Delta_{02}=0.6$, $\Delta_{12}=0.6$. Reproduction rates and susceptibility are set to one, and every domain begins at $c_i=0.6$. The labels carry no empirical claim, and the numbers are chosen for legibility.

11.3 The Arrangements Compared

Scenario A, domination. Exchange is intense across every boundary, $e_{ij}=1$, and the terms of the customary domain are largely displaced: $\sigma_{01}=0.85$, $\sigma_{02}=0.75$, $\sigma_{12}=0.50$, with the regulatory and commercial domains governing the exchange. Loads follow as $\Lambda = (0,,0.85,,1.25)$.

Scenario B, separation. No exchange occurs, $e_{ij}=0$, and asymmetry is undefined and set to zero. Loads are $\Lambda=(0,0,0)$.

Scenario C, viable exchange. Exchange is moderate, $e_{ij}=0.55$, and asymmetry is small: $\sigma_{01}=0.15$, $\sigma_{02}=0.05$, $\sigma_{12}=0.10$. Loads are $\Lambda \approx (0,,0.082,,0.083)$.

11.4 Outcomes

Figure 7. Coherence trajectories under the three arrangements, with aggregate co-creation at right. Under domination the commercial domain crosses the extinction threshold and the regulatory domain is heavily suppressed, while the domain whose terms govern rises to full coherence. Under separation every domain attains maximal coherence and no co-creation occurs. Under viable exchange all domains persist near full coherence and co-creation is an order of magnitude greater than under either failure.

Scenario	Coherence at rest	C1 holds	C2 holds	Aggregate co-creation
A. domination	$(1.00,,0.15,,0.00)$	no	yes	$0.018$
B. separation	$(1.00,,1.00,,1.00)$	yes	no	$0.000$
C. viable exchange	$(1.00,,0.92,,0.92)$	yes	yes	$0.903$

Four observations follow from the table and from Figure 7.

First, the two failures are distinguished by different conditions, which establishes Proposition 1. Domination preserves every boundary in a viable state while extinguishing a domain. Separation preserves every domain while leaving every boundary dead.

Second, separation attains the highest coherence available. Every domain reaches $c_i=1$, and by the measure of domain integrity alone the arrangement is optimal. Co-creation is nevertheless zero, and the arrangement is a set of intact domains with nothing between them.

Third, the domain whose terms govern gains under domination. Its load is zero, and its coherence rises to the maximum while its neighbours decline. The model reproduces without further assumption the observation that imposition is not costly to the imposer and is invisible from that position.

Fourth, the viable arrangement is not the one that maximizes any single quantity in the table. Coherence is lower than under separation, and the coherence of the governing domain is no higher than under domination. What distinguishes it is that both conditions hold, and co-creation follows.

11.5 The Path from C to A

A final observation concerns how an arrangement moves between these states. Holding exchange at $e=0.55$ and raising asymmetry continuously from $\sigma=0.05$, the load on the most burdened domain rises continuously, its equilibrium coherence falls linearly, and nothing discontinuous occurs until the threshold of Proposition 3 is reached. Before that point the arrangement satisfies both conditions and displays a domain in decline. A polity monitoring only whether its domains persist would observe no failure until the failure was complete.

11.6 The Second Setting: Recognized and Staffed Boundaries

The second illustration differs from the first in three respects the model treats as relevant. The domains occupy no common territory, no one of them administers the arrangement, and the boundaries between them are recognized and provided with people whose work is conducted there.

Consider three research communities bearing on a common problem, one experimental, one theoretical, and one computational, joined by shared instruments, joint appointments, and a literature that each reads. Exchange is high, asymmetry is low because no community can compel another, and misorientation is substantial because each articulates the problem in terms the others do not use. On the model this is the productive configuration: with $e$ large, $|\sigma|$ small, $\Delta$ substantial and every $c_i$ positive, all four factors of the co-creation rate are bounded away from zero, and the arrangement generates what §13 calls new domains at the boundaries, which is what a shared instrument or a joint method in fact is.

The illustration is included for a second reason, which is that it displays the difficulty entered in §4. Two of these communities may develop an accommodation so practised that exchange between them becomes cheap and unremarkable: a shared formalism that each can use without translation. On the physical side such a relation is a special boundary, low in energy and correspondingly poor as a site at which new order forms. The model as written does not represent it, since a low-energy accommodation between widely differing domains would require boundary property to depend on more than misorientation, and §16 records the gap. What the illustration shows is that the gap is not artificial: relations of this kind are common, and an account that predicted high novelty wherever difference is high would misdescribe them.

11.7 The Third Setting: Unrecognized Boundaries

The third illustration is chosen because it exhibits capture, and because capture is the failure that the first two settings would not reveal.

Consider a professional body setting a standard, whose members are drawn from several practices with different formations. The body has procedures, its deliberations are open, no practice is excluded, and no participant is silenced. The arrangement would satisfy any test framed in terms of participation. What the body lacks is any recognition that the practices articulate the matter differently: it treats its members as individuals holding views, and the differences among their formations appear within it only as disagreement to be resolved.

Under that description the boundaries carry exchange and are asymmetric, since one formation supplies the vocabulary in which the standard is written and the others must render their concerns in it to be heard. Equation (13) then applies, and the trajectory is the one shown in Figure 6. Each practice continues to reproduce itself; none is extinguished; the body’s records show continuous participation by all. What falls is the misorientation, as the practices come to articulate the matter in the terms the standard uses, and with it the rate at which the arrangement generates anything the standard did not already contain.

The illustration turns on the mismatch between what such a body would monitor and what would be occurring. Participation is recorded and is intact. Coherence is intact. The quantity that has gone is the one for which the body has no measure and, having no recognition of its boundaries, no reason to seek one.

11.8 Findings and Residue

The three arrangements separate the two conditions, exhibit the isolation result, and show co-creation depending on both. Separation maximizes coherence and produces nothing. Domination is costless to the domain whose terms govern. The approach to extinction is, finally, continuous, so the loss of a domain is not signalled in advance by the condition that its coherence be positive.

§12 The Wall between Description and Justification

An account written in the vocabulary of generativity is exposed to a specific failure, and the exposure is greatest at exactly the point this paper has now reached. The failure is set out here, together with what in the account prevents it.

12.1 The Sentence That Must Remain Refusable

Consider the proposition that destroying a domain may release the generative capacity of the rest. In a coupled system this is not rhetoric; it is frequently true, and it is observable. Where a domain is removed, coordination costs among the survivors fall, contested boundaries disappear, and what remains may generate more freely than before.

An account that took the free unfolding of generativity as its end could not refuse that proposition, since the proposition would be true in the account’s own terms. It would have replaced the question whether a party is useful with the question whether a party contributes to the system’s generative capacity, and those are the same question in different vocabulary. The account offered here must therefore be able to assert the following.

The sentence. Even where extinguishing a domain would in fact release the generative capacity of the remaining configuration, extinguishing it is impermissible on the conditions stated.

12.2 Why the Conditions Permit That Assertion

Three features of the conditions of §9, taken together, secure it.

The conditions are universally quantified and are not aggregates. Proposition 4 establishes that $\forall i,(c_i>0)$ is not implied by the maximization of any weighted sum, and exhibits a configuration in which a domain is extinct and the aggregate is nonetheless higher. An arrangement that extinguished a domain would fail the condition whatever happened to the total, because there is no total in the condition. The refusal of aggregation is therefore a property of the form in which the conditions are written, holding independently of any preference of the author.

The conditions are constraints, and they function as such throughout. Proposition 5 establishes that they admit an open region of arrangements and select none. A constraint of that kind cannot be traded against, since there is no scale on which the trade would be conducted, and it cannot be optimized, since it is satisfied or unsatisfied and does not admit of degree at the point where it binds.

The normative force of the conditions is, third, imported and never derived. Nothing in §3 entails that a domain ought to persist; a physical account entails nothing of the kind about anything. The reasons for holding that a domain’s capacity to reproduce its own terms may not be extinguished are the reasons given in the tradition surveyed in §7, and they are reasons about the standing of parties, with the productivity of configurations set aside. The model states those reasons in a form that permits their consequences to be traced; it does not supply them.

12.3 Three Statements the Account Does Not Make

That generativity is a purpose. The relations described here exhibit a tendency to sustain the conditions of their own continued generation. That is a description of what is observed, and this paper treats it as a primitive of the framework, declining to explain it. To explain it would be to name whatever it serves, and whatever were named would become the more fundamental good, with generativity reduced to its means. The tendency is accordingly descriptive throughout, and the arguments below draw their reasons from §7.

That what conduces to generativity is thereby justified. §3 records that coarsening proceeds without an agent and that plurality is metastable. Neither observation is a reason. That a configuration tends toward the loss of its domains explains why sustaining plurality requires something and does not establish that the tendency is legitimate, nor that resisting it is.

That a domain’s claim rests on what it contributes. A domain that generated nothing of value to any other would stand under the conditions exactly as one that generated much. The conditions are indifferent to output by construction, and a version of them sensitive to output would be the utilitarianism this paper set out to avoid, reconstructed one level up.

12.4 The Point at Which the Wall Is Thinnest

The weakest point may now be named. The co-creation rate of §9 is a quantity, it is defined so as to be greater or smaller, and the case study reports it as a number. Nothing prevents a reader from treating that number as a maximand, and an arrangement chosen to maximize it would be an arrangement chosen on precisely the grounds this section rejects.

The defence available is structural. The co-creation rate is a diagnostic and appears in no condition. The conditions are the two positivity requirements, and an arrangement satisfying them is admissible whether its co-creation rate is high or low. Should a reader nonetheless wish to maximize the rate, the conditions constrain that maximization exactly as they constrain any other objective, and an arrangement that raised the rate by extinguishing a domain would be inadmissible for the reason given above. The rate is reported in §11 because it makes the failure modes legible, and it carries no normative weight anywhere in the account.

12.5 Findings and Residue

The conditions permit the refusal of the sentence that an account of this kind is most likely to be unable to refuse, and they do so because they are universally quantified, because they function as constraints, and because their normative content is imported. The account treats the tendency of relations to sustain their own conditions of generation as a primitive that justifies nothing. The thinnest point is the co-creation rate, which is a quantity and might be mistaken for a maximand; it enters no condition, and §15 records the risk.

§13 The Contribution of the Crystallographic Source

§5 through §7 conceded that interfaces are productive, that plural centres without an apex are established, and that the refusal of maximization is settled. This part states what the source supplies beyond them.

13.1 The Boundary as a Structure

Polycentricity theorizes centres and treats what lies between them as relation or adjustment. The trading zone literature theorizes what develops at an interface and treats it as a language. The crystallographic source treats the boundary as a structure whose characteristics are determined by the orientations meeting at it, and it distinguishes boundaries by those characteristics.

Two distinctions carry over as hypotheses. Boundaries between domains of small misorientation differ in kind from boundaries between domains of large misorientation, and the two behave differently in every respect that matters physically. Boundaries differ, second, in whether the two orders meeting at them are accommodated by a common arrangement or whether the meeting is abrupt, which bears on whether the boundary strengthens or weakens the material.

Hypothesis on boundary typology. Boundaries between relational domains admit of classification by the misorientation across them and by the degree to which the arrangement at the boundary accommodates both modes of articulation. The classification bears on whether the boundary transmits, resists, or generates, and the classes are not ordered by a single scale.

The hypothesis states what a typology would require, and the typology itself is outstanding. §16 enters it as a question.

13.2 The Triple as the Unit of Analysis

A second consequence concerns what is analysed. Where boundaries are structures, the unit of analysis is neither the domain nor the pair of domains, and is the triple consisting of two domains together with the boundary between them. The triple is not decomposable: the properties of the boundary follow from both orientations and belong to neither domain.

The proposal is a departure from network analysis, in which an edge carries a weight and the structure resides in the vertices and the pattern of connection. An edge carrying a structure of its own, with its own conditions of persistence, is a different object.

13.3 Nucleation at Boundaries

The third contribution carries the most weight in the argument above. In materials, the formation of a new phase begins preferentially at boundaries, because the energy already invested in the boundary lowers the barrier to forming something new there. Formation in the interior of a domain requires more.

Transferred, the proposition is that novelty arises preferentially at boundaries and requires both that a difference exist across the boundary and that the boundary permit exchange. Definition 8 states this, and Proposition 2 draws out its consequences. The transferred proposition is a hypothesis, and §4 governs the standing of the physical fact behind it.

The formulation supplies a reason, absent from an account resting on coexistence, why plurality is worth sustaining. Domains held apart produce nothing between them, and domains dissolved into one another have no boundary at which anything could form. Plurality is a condition of novelty on this account, and the two failures the paper is named against are the two ways of removing the condition.

13.4 Findings and Residue

The crystallographic source supplies a treatment of the boundary as a structure, a unit of analysis that is a triple, and a mechanism locating novelty at boundaries. The first is a hypothesis awaiting a typology, the second is a proposal about method, and the third is stipulated in the model of §10 and carries its central result.

§14 Relation to Viability Theory and to Systems Theory

Two existing bodies of work supply the mathematics and the vocabulary the account requires, and neither was reached for in constructing it.

14.1 Viability Theory

Aubin (1991) develops the study of systems required to remain within a constraint set, with no optimization performed. The viability kernel of a set is the collection of states from which some admissible evolution remains within it indefinitely, and the theory concerns the computation of such kernels and the characterization of the evolutions that maintain them (Aubin, Bayen, and Saint-Pierre 2011).

The correspondence is exact enough to be adopted. Conditions C1 and C2 define a constraint set. The controls are the arrangements $e$ and $\sigma$. The region computed in Proposition 5 is a section through the set of admissible controls. The theory’s characteristic result, that viable controls are generically multiple, is the formal statement of the underdetermination that §10 demonstrates by construction.

Adopting the vocabulary strengthens the account, since it supplies a developed mathematics for a normative form that political philosophy has left in prose. The programme it suggests is stated among the open questions.

Two further frameworks share the form. Ecological resilience distinguishes the capacity of a system to persist within a domain of attraction from its speed of return to a single equilibrium (Holling 1996), and the distinction is the one required here. The framing of environmental governance by boundaries within which operation is admissible (Rockström et al. 2009), developed normatively as a space bounded by floors and ceilings (Raworth 2017), is an existing example of a criterion of the present form applied at scale.

14.2 Self-Reproduction

Condition C1 requires that a domain reproduce its own mode of articulation, and the requirement has an established name. Luhmann (1995), developing the account of self-producing systems given by Maturana and Varela (1980), treats a social system as reproducing its own elements through its own operations, so that the system’s continuation is an achievement of those operations. Coherence as defined here is operational closure under another description, and the correspondence should be recorded.

One difference bears on the argument. The systems-theoretic account emphasizes closure, and treats what crosses a system’s boundary as perturbation to be processed internally. The account here requires that boundaries be viable, which is a requirement about what passes across them, and Proposition 2 makes novelty depend on it. Closure is a condition of a domain’s persistence, and the arrangement requires more to be defensible.

14.3 The Precedent for the Term

Mueller and Hechter (2021) develop an account of social order in the European Union under the name of crystallization, in which cooperation at lower levels of organization sustains order at a higher level. The account is set in a plural and multilingual polity and connects to the literature on the commons. The present paper’s use of the term is accordingly not a coinage in an empty field, and the difference lies in the level of the mechanism: the prior account concerns the emergence of order from lower-level cooperation, and the account here concerns the structure of boundaries between domains once order has emerged.

An older usage runs the other way. Cultural crystallization, in the sense given by Gehlen (1961), names a condition of exhaustion in which the possibilities of a culture have been worked through and nothing further arises. The sense used here is opposed to that one: crystallization is where structure is generated, and the failure modes of Proposition 2 are the conditions under which generation ceases.

14.4 Findings and Residue

Viability theory supplies the mathematics for a non-optimizing criterion, ecological resilience and the boundary framings of environmental governance supply precedents for its normative form, and systems theory supplies the established name for condition C1. The term crystallization has a prior use in a closely related setting, and an older use with the opposite valence.

§15 The Limits of the Account

The analogy is disciplined by declaration. §4 states what transfers and what does not, and the statement is a declaration of method. Nothing establishes that the four transferred features hold of relational domains, and a reader who denies the transfer is left with a stipulated model bearing a crystallographic vocabulary.

The propositions are true of the model. Propositions 1 through 6 are consequences of Definitions 1 through 10. They establish nothing about any polity, and their interest depends entirely on whether the definitions capture anything.

The quantities are not measurable. Coherence, orientation, exchange, and asymmetry have no measurement procedures. A model whose variables cannot be measured cannot be tested, and the account is accordingly a proposal for work.

Domains are treated as discrete and given. Parties belong to many domains at once, domains overlap and nest, and their boundaries are contested. The model treats them as a fixed finite set with a fixed contact graph, and every result depends on that simplification.

The dynamics are stipulated. Logistic reproduction with a linear cost of rendering was chosen for tractability and for the sharpness of its threshold. Other dynamics would yield other thresholds, and nothing here shows the choice to be the right one.

Nucleation is asserted, not derived. Definition 8 makes co-creation a product of four factors, and Proposition 2 follows from that form alone. The proposition is therefore a consequence of how co-creation was defined, and its force depends on whether the definition is a defensible reconstruction of what co-creation is.

The core intuitions are inherited. As §5 through §7 concede, the productivity of interfaces, plural centres without an apex, and the refusal of maximization are all established elsewhere.

Two claims are entered against this list. The derivation of four failure modes from a single requirement places unity and fragmentation in one frame with two further failures, which the surveyed accounts do not do. The treatment of the boundary as a structure with a typology of its own addresses an asymmetry that the polycentric literature leaves open.

§16 Open Questions

The first group concerns the model.

Q1. What would a measurement procedure for coherence look like? The condition C1 turns on a domain’s capacity to reproduce its mode of articulation across changes of membership, and an operationalization would convert the account from a proposal into a framework.

Q2. Are the failure modes of Proposition 2 exhaustive of the ways co-creation fails, or exhaustive only of the ways permitted by Definition 8? The proposition is a consequence of a stipulated product form, and a different reconstruction would yield a different list.

Q3. Does the threshold of Proposition 3 survive more realistic dynamics? Reproduction that is not logistic, costs of rendering that are not linear in load, and domains that adapt their own reproduction rates under pressure would each alter the result.

The second group concerns the boundary, which is where the account claims most.

Q4. What is the typology of relational boundaries? The hypothesis of §13 proposes classification by misorientation and by accommodation. Constructing the classification, and establishing that its classes behave differently, is the work the account most requires.

Q5. Which boundaries strengthen and which weaken what they join? In materials the answer depends on structure and is well studied. The corresponding question for relational domains is open and appears answerable.

Q6. Can a boundary become a domain? The trading zone literature describes interlanguages that stabilize into practices of their own. The model has no mechanism by which a boundary acquires coherence and enters the contact graph as a vertex, and supplying one would close the account’s principal structural gap.

The third group concerns the normative content, and the questions here are sharpest because the claims are strongest.

Q7. Do the two conditions exhaust what a defensible plural arrangement requires? An arrangement may satisfy both while distributing the benefits of co-creation in a manner no account of justice would accept, and the conditions are silent about distribution.

Q8. How is the contact graph itself to be assessed? The conditions apply to boundaries that exist. An arrangement that never brings two domains into contact violates nothing, and the question of which domains ought to be in contact is not posed by the account.

Q9. What follows where the conditions cannot be jointly satisfied? Proposition 5 exhibits a case where the admissible region is non-empty. Where the region is empty, the account states that every available arrangement is indefensible and offers no guidance on choosing among them.

The last question is the one the account is directed toward.

Q10. Under what conditions does a plural arrangement resist coarsening? Materials coarsen because boundaries carry energy and their reduction is favoured; polities exhibit a corresponding tendency, since uniform procedure is cheaper to administer than plural procedure and the pressure toward it is continuous. An arrangement satisfying both conditions when established may fail them a generation later with no act of imposition by anyone. What sustains a plurality of domains against that pressure is the question on which the practical value of the account depends, and it is entirely open.

A closing observation about the shape of the list. The questions of the first group could be settled by anyone willing to work the model. The questions of the second are where the account claims novelty and where its rivals leave room. The questions of the third concern whether the conditions are the right ones, and a reader persuaded of the survey and unpersuaded of the conditions may take the first and leave the second.

---

A note on method. The formal model is stipulated and its propositions are elementary. The propositions were verified numerically before being stated, and Proposition 3 was checked against integration of the dynamics across randomized parameters, with all apparent discrepancies falling within a small neighbourhood of the bifurcation and attributable to slow convergence at criticality. Proposition 5 was checked against the closed-form equilibrium across randomized controls with no discrepancies. The figures are generated from the model and from a growing-front schematic, and their sources accompany the paper.

A note on the analogy. The physical facts reported in §3 are stated as they are understood in materials science and are used as a source of structure. No physical result is offered as evidence for any claim about a polity, and §4 enumerates the points at which the transfer fails. A reader who judges the enumeration incomplete is invited to extend it, since the value of the exercise depends on the enumeration being honest.

References

Anderson, Elizabeth S. (1999). What Is the Point of Equality? Ethics, 109(2), 287–337.

Aubin, Jean-Pierre (1991). Viability Theory. Boston: Birkhäuser.

Aubin, Jean-Pierre, Alexandre M. Bayen, and Patrick Saint-Pierre (2011). Viability Theory: New Directions. 2nd ed. Berlin: Springer.

Berlin, Isaiah (1969). Two Concepts of Liberty. In Four Essays on Liberty, 118–172. Oxford: Oxford University Press.

Bhabha, Homi K. (1994). The Location of Culture. London: Routledge.

Black, Max (1962). Models and Metaphors: Studies in Language and Philosophy. Ithaca: Cornell University Press.

Bouchard, Gérard (2012). L’interculturalisme: un point de vue québécois. Montreal: Boréal.

Cahn, John W. (1956). The Kinetics of Grain Boundary Nucleated Reactions. Acta Metallurgica, 4(5), 449–459.

Cantle, Ted (2005). Community Cohesion: A New Framework for Race and Diversity. Basingstoke: Palgrave Macmillan.

Cantle, Ted (2012). Interculturalism: The New Era of Cohesion and Diversity. Basingstoke: Palgrave Macmillan.

Clemm, P. J., and J. C. Fisher (1955). The Influence of Grain Boundaries on the Nucleation of Secondary Phases. Acta Metallurgica, 3(1), 70–73.

Frankfurt, Harry (1987). Equality as a Moral Ideal. Ethics, 98(1), 21–43.

Galison, Peter (1997). Image and Logic: A Material Culture of Microphysics. Chicago: University of Chicago Press.

Gehlen, Arnold (1961). Über kulturelle Kristallisation. In Studien zur Anthropologie und Soziologie. Neuwied: Luchterhand.

Hesse, Mary B. (1966). Models and Analogies in Science. Notre Dame: University of Notre Dame Press.

Holling, C. S. (1996). Engineering Resilience versus Ecological Resilience. In Peter C. Schulze, ed., Engineering Within Ecological Constraints, 31–44. Washington, DC: National Academy Press.

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

Maturana, Humberto R., and Francisco J. Varela (1980). Autopoiesis and Cognition: The Realization of the Living. Dordrecht: Reidel.

Mueller, Sean, and Michael Hechter (2021). Centralization through Decentralization? The Crystallization of Social Order in the European Union. Territory, Politics, Governance, 9(1), 133–152.

Nussbaum, Martha C. (2011). Creating Capabilities: The Human Development Approach. Cambridge, MA: Harvard University Press.

Ostrom, Elinor (2010). Beyond Markets and States: Polycentric Governance of Complex Economic Systems. American Economic Review, 100(3), 641–672.

Ostrom, Vincent, Charles M. Tiebout, and Robert Warren (1961). The Organization of Government in Metropolitan Areas: A Theoretical Inquiry. American Political Science Review, 55(4), 831–842.

Parekh, Bhikhu (2000). Rethinking Multiculturalism: Cultural Diversity and Political Theory. Basingstoke: Palgrave.

Pettit, Philip (1997). Republicanism: A Theory of Freedom and Government. Oxford: Clarendon Press.

Polanyi, Michael (1951). The Logic of Liberty: Reflections and Rejoinders. London: Routledge & Kegan Paul.

Pratt, Mary Louise (1992). Imperial Eyes: Travel Writing and Transculturation. London: Routledge.

Rawls, John (1971). A Theory of Justice. Cambridge, MA: Harvard University Press.

Raworth, Kate (2017). Doughnut Economics: Seven Ways to Think Like a 21st-Century Economist. White River Junction: Chelsea Green.

Rockström, Johan, et al. (2009). A Safe Operating Space for Humanity. Nature, 461, 472–475.

Sen, Amartya (1999). Development as Freedom. Oxford: Oxford University Press.

Sokal, Alan, and Jean Bricmont (1998). Fashionable Nonsense: Postmodern Intellectuals’ Abuse of Science. New York: Picador.

Star, Susan Leigh, and James R. Griesemer (1989). Institutional Ecology, ‘Translations’ and Boundary Objects. Social Studies of Science, 19(3), 387–420.

Tsing, Anna Lowenhaupt (2005). Friction: An Ethnography of Global Connection. Princeton: Princeton University Press.

中文

摘要

一个持有若干相干传统的政体面对两个相反的危险。它可能把它们消解进一个单一秩序，而它可能容许它们分离直到没有任何东西在它们之间通过。治理通常被理论化为对那第二个的避免，而那第一个被当作它趋近的理想。本文审查一个图景，在其中两者都是同一种类的失败。多晶材料通过许多取向不同的域持有它们的形态，在它们自己权利中是结构的诸边界处被接合，而这样的材料既非单晶也非粉末。本文问什么从那个图景转移到诸复数关系域的治理、谨慎地陈述什么未能转移，并在幸存的部分之上建造一个形式模型。那个模型是刻意非最优化的：它陈述两个正性条件，一个要求每个域保有再生产它自己意义系统的能力，一个要求每个边界许可无强加的交换，而它把不义界定为两者之一的失去。四个失败模式作为一个定理、而非一个清单随之而来，其中统一与碎片化是两个。一个三域案例研究被贯穿，带动态在每一步被显示。那次综述让步界面的生产性在科学亚文化的研究中被确立、没有顶点的复数中心在多中心治理的研究中被确立，而最大化的拒绝在政治哲学中被安定。显得保持开放之物是那个边界本身，被当作一个带它自己诸性质的结构，以及新颖在边界处的成核，那是那个晶体学来源供出某样它诸对手所不供之物之点。那个说明的诸命题成立于生成它们的那个模型，而它们会关乎一个政体所处的诸条件被铺陈，连同会不得不首先被安定的诸问题。那些问题被持为所供出之物中更有价值的一半。

关键词： 关系复数性；晶界；可行性条件；共同创造；生成性。

关于本文站位的一则附记。 这是一篇讨论文，而是一个系列中的第四篇。它辨认一个结构、概览既存学术对它所作之物、标记每一个说明显得用尽之处，并提议一个工作的方向。它不完成一个理论。两个特征在一开始要求注意。本文从一门自然科学借用，而§4陈述什么许可那个借用以及它在何处失败，因为一个不受纪律的物理语汇向社会理论的转移是一个已知且应得的批评标靶。而本文供出一个形式模型，它的诸命题是被规定之动态的诸后果；它们对那个模型为真，而它们对任何政体的关系是一个本文不安定的进一步问题。在一个此处作为开放被引入的主张事实上已被占据之处，或在一个来源被读错之处，改正被欢迎至 huangwanhong@serendip.ngo。

§1 引论

一个含有若干相干实践传统的政体可能以两个相反的方式失败。它的诸传统可能被消解进一个单一秩序，无论凭同化、凭把一个传统的诸范畴强加于其余，还是凭那个对统一程序的寻常行政偏好。或它的诸传统可能分离，直到每一个内部完好而没有任何有后果之物在它们之间通过。第一个失败留下一个轻易裁定、而已失去它可能裁定于其间之物的政体。第二个留下一个其诸部分幸存、而无法作为一个行动的政体。

政治理论已不对称地处理这些。碎片化是那个被承认的危险，而关于凝聚、整合与共享身份的诸文献被针对它。统一，相比之下，倾向于作为那个极限理想出现，在诸情形容许之处被趋近。一个达成完美一致的政体会被算作幸运。

此处所审查的图景把两者都当作失败，而它从一类既非其一发生的材料取它的结构。大多数实践上重要的固体是多晶的。它们由许多域组成，每一个内部有序，其取向不同，沿两个秩序相遇之处的诸边界被接合。一个多晶不是一个单晶，因为没有取向延伸穿过整体。它不是一个粉末，因为那些域被接合而那个材料聚在一起并承载。它的诸性质既取决于那些边界也取决于那些域，而在若干方面那些边界是那个有旨趣行为发生之处。

下面所追寻的暗示是这个结构供出一个在两个既得替代中皆不可得的复数治理图景。一个政体可局部地被排序而不全局地被排序。取向的差异可能是它诸性质的一个条件。两个域之间的边界可能是一个带它自己特征的结构，以及某样既非任一域所含之物在其处生起的场所。

三个困难推动那个探究。第一个关乎那个规范问题的形式。一个通过最大化一个善之度量而进行的复数治理理论输入那个复数性被引入以抵抗的诸承诺：有一个可度量的善、诸安排是可排名的、一个域中的诸损失可被另一个中的诸收益补偿。第二个关乎诸边界的处理。在诸共同体之间的边界被理论化之处，它们倾向于作为待被减少的诸障碍或待被管理的诸缺陷出现，而一个边界是一个生产性结构的可能性鲜少被形式地追寻。第三个关乎新颖的诸条件。停在共存的复数治理诸说明描述一个其中没有新东西从那个复数性生起的安排，而那一种类的安排难以同一个精巧的分离区分。

五个工作之体关乎这些困难而每一个在下面被处理。科学亚文化的研究已确立没有一个共享全局框架的诸群体在界面处协调，在那里新的中间结构发展（加利森 1997；斯塔与格里泽默 1989）。后殖民与民族志工作已论证在差异地带的相遇产生不可还原为任一方的诸形式（巴巴 1994；普拉特 1992；秦 2005）。多中心治理的研究已确立秩序可被许多带交叠管辖且无顶点的决定中心所维系（波兰尼 1951；奥斯特罗姆、蒂布与沃伦 1961；奥斯特罗姆 2010）。政治哲学自聚合功利主义的拒绝以来，已发展作为门槛、可行能力与关系条件、以替代最大值的诸正义设想（罗尔斯 1971；法兰克福 1987；安德森 1999；森 1999；佩迪特 1997；伯林 1969）。可行性理论最终已发展一门被约束保持在一个集合之内、无最优化被施行之诸系统的数学（奥班 1991；奥班、巴延与圣皮埃尔 2011）。这些文献安定这个主题的一个可观比例，而本文的一个目的是记录多少；下面的综述相应地通过先让步、而后定位剩下之物来进行。

那个说明在一个作者已在别处铺陈的框架之内被发展，而它可被以一句话陈述：

生成性关系存在。 一个关于主体、意义、价值、创造与规范性如何通过生成性关系过程共同涌现的理论。

那个框架的三个特征在此关乎。诸价值被当作在诸关系之内被生成，以致关系结构的一个复数性产出种类上不同的诸价值。诸关系被当作生成性的，即一个处于运作秩序中的关系产生意义的、价值的，以及那个关系本身所要求之物的诸进一步裁定。那个框架第三携带一个支配它诸主张如何被推进的方法论承诺，它的基调是历史辩证法，以致诸命题作为在一个历史框架内可修订的被引入、知识与实践的有限被承认为一个常在状况，而哲学论证在那个主题容许它之处被连到可操作的表述。一篇在那个承诺之下被书写的论文陈述它的诸主张以致它们可被放弃。

四个贡献被供出，以不同程度的信心被主张。第一个是那个类比的一个受纪律陈述，在§4被给出，它列举什么从晶体学转移与什么未能。第二个是§9中所陈述的那对条件，它们把复数治理表达为约束满足，跨诸域无聚合。第三个是§10的形式模型，它的诸命题包括从一个单一要求对四个失败模式的一个推导，以及一个那些条件把那个安排留下欠定的证明。第四个是§13中所发展的那个边界处理，在那里那个晶体学来源供出一个被概览诸对手留得单薄的结构。在§16所记下的诸问题被持为所供出之物中更有价值的一半。

§2 范围与方法

2.1 那个容纳手法

三个问题组织那次讨论，而材料通过履行它们之一赢得它的位置：

（i）那个转移。 什么可从多晶材料的行为被承载到诸复数关系域的治理，而在什么许可上。

（ii）那些条件。 什么必须成立于一个复数安排以使它可辩护，鉴于最大化不可得。

（iii）那个边界。 一个诸域之间的边界是什么，而在什么条件下它产生某样既非任一域所含之物。

2.2 那个让步的语域

一个其诸组件在个别上不具原创性的理论仍可通过它连接它们的方式作出贡献，而那个连接仅在那些组件被正确地归属之处变得可见。此处的论证是建筑性的，而一个建筑性的论证被任何采过它自己石料的主张所削弱。三节大体由诸让步组成并出于那个缘由被置于那个建构性材料之前。

2.3 那项承担的诸边界

三个事项躺在那个探究之外，而标记它们固定§9的诸条件是关于什么的诸条件。

那个对象是一个条件，把诸安排搁置一旁。一个条件规定一个安排必须留下可能之物；哪些安排会在一个给定政体中确保它是一个制度设计的问题，而§16如此引入它。

那个对象是一个模型，而这样一个模型可能描述的政体是一个进一步的事项。§11的案例研究贯穿那个模型在变化诸参数之下的行为，而它的诸标签为可辨识性被选择。

诸域的个体化被当作给定。一个政体包含哪些关系结构，而凭什么程序它们被辨认，是每一个下面的条件所停靠之预设，而§16把它当作那个说明的第一个未了问题处理。

2.4 关于来源的一则附记

那次综述汲取材料科学、科学与技术研究、政治理论、科学哲学与应用数学。跨这样一个范围的诸概念转移是本文的首要方法论风险，而§4被书写以容纳它。

§3 结晶作为一个源现象

那个来源在任何东西从它被汲取之前被铺陈，因为一个从一个被误描来源的转移毫无价值。那个说明以物理文献所用的诸术语被给出，带那些支配关系被陈述，因为一个定性的意译会恰恰隐藏每一个关系成立所处的诸条件，而那些条件关乎什么可被转移。

3.1 成核与一个壁垒的存在

一个被冷却到它凝固点以下的液体不立即使自己有序。对于半径 $r$ 的有序相的一个球形区域，相对于那个无序母体的自由能变化是一个偏爱那个新相之体积项与一个反对它之表面项之间的竞争：

$$ \Delta G(r) ;=; -\tfrac{4}{3}\pi r^{3},\Delta g_{v} ;+; 4\pi r^{2}\gamma , \tag{1} $$

其中 $\Delta g_{v}>0$ 是那个新相每单位体积的体自由能减少的量级而 $\gamma>0$ 是每单位面积的界面自由能。那个表面项在小 $r$ 处占主导而那个体积项在大 $r$ 处，故 $\Delta G$ 上升、达到一个最大值，并下落。小区域因而通过消失降低它们的自由能，而唯有一个被涨落带过那个最大值的区域能通过增长降低它的自由能。令 $\mathrm{d}\Delta G/\mathrm{d}r=0$，

$$ r^{}=\frac{2\gamma}{\Delta g_{v}}, \qquad \Delta G^{}=\frac{16\pi\gamma^{3}}{3,\Delta g_{v}^{2}} . \tag{2} $$

图1的面板（a）绘制方程（1）连同它的两个组分；那个被标记的最大值同方程（2）符合到那个绘图的分辨率。两个特征在下面要紧。秩序在分散的诸点开始，把那个场域的其余留得不变，而一个门槛必须被跨过一个新安排才能根本持续。

图1。 §3的三个后来承载分量的关系，从文中所给的诸表达式计算。在（a）中一个成形区域的自由能，方程（1），带体积与表面贡献被点线与虚线绘制而那个临界点被标记。在（b）中方程（3）的因子 $f(\theta)$，一个预先存在的表面凭之减少那个壁垒。在（c）中边界能对着取向差：低角度极限之下的里德–肖克利形式（5）、它之上的一个近似平台，以及一个特殊取向差处的尖点。面板（c）在它的竖直标度上是示意的，被绘制以显示那个形状，特别是§4转在其上的那个非单调性。

3.2 为什么新秩序在边界处出现

一个在一个预先存在表面上成形的区域无需重新创造它的整个界面。对于一个在接触角 $\theta$ 处遇到一个平坦基底的球冠，

$$ \Delta G^{}_{\text{het}} ;=; f(\theta),\Delta G^{}_{\text{hom}}, \qquad f(\theta)=\frac{2-3\cos\theta+\cos^{3}\theta}{4}=\tfrac{1}{4}(1-\cos\theta)^{2}(2+\cos\theta), \tag{3} $$

带 $f(180^{\circ})=1$、$f(90^{\circ})=\tfrac{1}{2}$，而当 $\theta\to0$ 时 $f\to0$。图1的面板（b）绘制它。一个精确之点应被记录，因为它常被弄错：$f(\theta)$ 减少那个壁垒而把方程（2）的临界半径 $r^{*}$ 留得不变。

在此要紧的情形是在两个现存域之间一个边界上的成形，被克莱姆与费舍尔（1955）与卡恩（1956）处理。一个在那里成形的区域摧毁那个边界的一块，而那个被摧毁边界的能量是对着那个必须被创造之界面的一个抵免。以 $k=\gamma_{\alpha\alpha}/2\gamma_{\alpha\beta}$，即那个现存边界之能量对那个新界面之能量两倍的比率，那个壁垒随那个抵免上升而下落，而它在每单位成形体积摧毁更多边界面积的诸交界处进一步下落。那里所确立的排序是

$$ \Delta G^{}_{\text{hom}} ;>; \Delta G^{}{\text{face}} ;>; \Delta G^{*}{\text{edge}} ;>; \Delta G^{*}_{\text{corner}}, \tag{4} $$

带一个给定类别之位点上的成形仅在那个类别的一个临界 $k$ 之下发生。新秩序因而优先地在现存域相遇之处出现，而最容易在若干一次相遇之处。这是§13从中汲取最多的那个单一关系，而它在此以它诸作者给它的形式被陈述。

3.3 边界结构与取向差

一个诸域之取向相差一个小角度 $\theta$ 的边界可被描述为一个间距 $b/\theta$ 的位错阵列。求和它们的弹性能给出那个里德–肖克利关系

$$ \gamma(\theta) ;=; \gamma_{0},\theta,(A-\ln\theta), \qquad \gamma_{0}=\frac{Gb}{4\pi(1-\nu)}, \tag{5} $$

带 $G$ 是剪切模量、$b$ 是伯格斯矢量的量级、$\nu$ 是泊松比，而 $A$ 是一个吸收那个位错核的常数。那个关系仅对低角度边界成立，惯常 $\theta\lesssim15^{\circ}$；在那之上那个描述失败，而边界能近似地饱和。

关于 $\gamma$ 作为取向差之函数的两个事实对本文是决定性的并在此在正文之体中被记录。边界能在取向差上不单调：它在小角度处陡峭地上升、在大角度处饱和，而在特定取向差处拥有深的局部极小，在那些取向差处两个晶格共享一个高比例的位点。相干孪晶是那个标准实例，在铜中带一个大致低于一个一般边界一个数量级的能量。一个边界第二被五个宏观参数所规定，三个为那个相对取向而两个为那个边界所躺的平面，以致取向差欠定边界诸性质。图1的面板（c）显示那个形状。§4陈述这所禁止之物。

3.4 边界作为障碍与作为弱点

诸边界阻碍变形的传播，而一个多晶的屈服应力随域尺寸 $d$ 下落而上升：

$$ \sigma_{y} ;=; \sigma_{0} + k,d^{-1/2}. \tag{6} $$

一个许多小域的材料比一个少数大域的材料更强，那是诸域之复数性不是一个缺陷的物理意义。那个关系在一个交叉尺寸之下反转，在那里变形过渡到边界中介的诸机制而进一步的细化削弱那个材料；铜的诸模拟把那个最大值置于近十到十五纳米。

诸边界同时是增强输运的路径，沿它们的扩散超过穿过域内部的扩散数个数量级，而它们是杂质聚集处以及断裂在它们如此做时传播的位点。诸边界相应地既非缺陷也非本身的优点：它们所做之物取决于成问题的性质以及在它们处已聚集之物。

3.5 粗化与复数性的地位

边界迁移被曲率驱动。以迁移率 $M$ 与驱动压 $P$，

$$ v = M P, \qquad P = \gamma\kappa = \frac{2\gamma}{r}, \tag{7} $$

故一个边界朝它的曲率中心移动而边界面积下落。在二维中冯·诺伊曼–马林斯关系为一个 $n$ 边的各向同性域使那个后果精确：

$$ \frac{\mathrm{d}A}{\mathrm{d}t} ;=; \frac{\pi M\gamma}{3},(n-6), \tag{8} $$

以致带多于六个邻居的域增长、带更少的收缩，而六是中性的。诸域被失去、诸幸存者扩大，而那个平均尺寸如下推进

$$ \bar D^{,n} - \bar D_{0}^{,n} = \kappa t , \tag{9} $$

带在理想各向同性情形中 $n=2$ 而在真实材料中通常在二与四之间的诸值。

图2。 一个900个域的被模拟场域在一个局部曲率减少规则之下的粗化。在（a）中那个幸存域的数目单调地下落；在（b）中那些幸存者的平均尺寸上升；在（c）中那个运行结束时剩下的边界网络。那个被模拟规则比曲率流粗化得稍快，在方程（9）中给出一个近 $1.5$ 的最佳拟合指数而理想值是 $2$；那个指数在下面不承载分量。那个模拟被用于诸域的单调且非受迫的失去，那对局部规则的选择是稳健的。

图2以模拟显示那个过程。两个后果随之而来，而那第二个直接关乎那个来源的任何规范读法。

粗化不要求行动者。没有域意图消费它的诸邻居；每一个边界回应它自己的曲率，而复数性的失去从到处运作的局部规则随之而来。

复数性第二不是那个被偏爱的状态。每一个边界携带正的过剩自由能，故那个单晶是那个热力学基态而一个多晶是亚稳的，被动力学阻碍持在存在中：低迁移率，以及钉扎边界的诸障碍。半径 $r$ 与体积分数 $f$ 的第二相粒子的一个弥散施加一个钉扎压，它在一个数量级 $R_{\text{lim}} \sim r/f$ 的极限尺寸处遏止增长，

$$ R_{\text{lim}} ;\sim; \frac{r}{f}, \tag{10} $$

带标准估计 $R_{\text{lim}}=4r/3f$ 而前因子随惯例变化。在这样的阻碍缺席或被移除之处，粗化恢复；而在正常粗化被钉扎但少数边界逃脱之处，一小数目的域可能增长以消费其余，一个被称为反常增长的状况，那是一个多晶被刻意地转换为某样趋近一个单晶之物所经由的路线。

一则限定与此属于一起，因为它将在§15要紧。复数性的亚稳性不是绝对的。倘若一个偏析物种在诸边界处积累它降低它们的能量，在那个最简单的处理中

$$ \gamma ;=; \gamma_{0} - \Gamma_{s}!\left(\Delta H_{\text{seg}} + k_{B}T\ln X\right), \tag{11} $$

而在 $\gamma$ 被驱向零之处那个粗化的驱动力消失而一个有限域尺寸的复数状态成为一个真正的平衡、以替代一个延迟。这条路线被预测并被模拟；它不是实验上被安定的，而它在此作为一个其确认未了的可能性被报告。

3.6 诸发现与残余

秩序在分散诸点并唯有过一个门槛生起、凭局部规则延伸而无全局计算且无物被最优化，并在独立取向的诸区域相遇之处产生诸边界。新秩序优先地在那些边界处成形，而最容易在若干相遇之处，因为在那里成形摧毁现存边界并回收它的能量。边界能在小角度处随取向差上升、在大角度处饱和，而在特殊取向差处下沉，以致差异与边界性质不单调地相关。诸边界既阻碍变形又供出输运与失效沿之行进的诸路径。诸域的复数性是亚稳的并凭粗化衰减，除非某样东西阻碍它，而原则上可被一个在诸边界本身处积累的物种所稳定。

§4 那个类比的纪律

物理语汇向社会理论的诸转移有一个糟糕的记录，而它们所招致的批评大体是应得的（索卡尔与布里克蒙 1998）。负责任地进行这样一个转移的装置可得，而本节应用它。

4.1 所主张的地位

赫西（1966）在一个类比之内区分被知道是共享的诸性质、被知道不同的诸性质，以及其地位未定的诸性质。推理从第一个进行、被第二个阻断，而在第三个中作为猜想被许可。布莱克（1962）把一个模型当作一个通过输入一个结构而组织一个主题的工具，并主张它的价值在于它所使可问的诸问题。

那个类比的地位。 晶体学在此作为一个结构的来源与作为一个假说的生成器被使用。它的角色被那两个功能穷尽：下面所推进的诸主张关乎关系域及它们的诸边界，而它们对政治理论内部的诸根据与对§10的模型负责。§10的形式模型是一个被规定的系统，它的诸命题对它为真，而它对任何政体的关系是猜想的。

4.2 那个正类比

四个结构特征被向前承载。

产生延伸秩序的局部规则。 相干的诸安排可通过每一个都是局部的诸互动跨一个人口延伸，没有参与者拥有那个整体的结构。

产生不同取向的独立成核。 在秩序独立地在若干点开始之处，那些由此产生的域将理所当然地在取向上不同，而它们的差异要求没有超出它们诸起源之独立性的解释。

产生边界的碰撞。 在两个相干域相遇之处，那个相遇的表面是一个由两者判定且与任一者不同一的结构。

边界作为成形的位点。 一个边界处的诸条件不同于任一域之内的诸条件，而某样新东西的成形优先地在那里发生。

4.3 那个负类比

八个不类比被引入。每一个标记那个来源的一个从那个目标缺席的特征，而下面的诸论证被建构以致转在两者共同的诸特征上。它们中的两个改正那个来源将不许可的一个转移并被首先给出。

一个材料的诸单位没有利益、目的，或解释。 一个原子不理解它的诸邻居，而物理情形中没有任何东西对应于一方对另一方在做什么的读法。下面每一个关乎强加与渲染的论证没有物理对应物。

晶界不可评价。 一个物理边界既非正义也非不义。下面每一个条件的规范内容被政治理论中的论证供出并被输入那个模型，从不从那个物理来源被推导。

没有社会自由能。 那个物理说明有使某些演化确定的热力学势。没有那种量在此被主张，而下面的模型是被规定的，没有从任何这样的势的推导。

取向不可测量。 一个晶体学取向是一个物理量。下面所用的一个关系域的取向是一个表述模式的占位符，而当前容许没有测量程序。

诸域不被锐利地界定。 一个晶粒有一个确定的范围。一个关系域没有，它的成员资格是被争议的，而诸方一次属于若干。那个模型把诸域当作离散的，因为一个可处理的模型要求它，而§15记录那个代价。

粗化不是刻意的；均质化常常是。 那个物理过程没有行动者。那个对应的社会过程频繁地有一个，而那个差异在责任成问题的每一点要紧。

差异不与它所产生之边界的性质单调地变化。 这是最可能被假定掉的不类比，而§3记录了禁止它的那个物理事实。边界能仅在小角度处随取向差上升、此后饱和，而在两个秩序恰好共享结构的特殊取向差处陡峭地下落。两个后果为§10的模型随之而来。在共同创造被弄得随取向差增加之处，那是那个模型的一个规定并仅在那个低角度区被那个物理来源所许可。那个物理来源第二积极地暗示某样那个模型不表征之物：特定的对广泛不同的域可能立于一个不寻常地包容的关系，维持起来便宜而相应地不生产任何新东西，那会是方程（5）及它诸例外意义上的一个特殊边界。这样诸边界的关系类似物是否存在被引入诸开放问题之中。

边界诸性质被取向差欠定。 一个物理边界要求五个参数而取向差供出三个。下面的模型以一个单一差异与两个进一步的量刻画一个边界，而如此做是为了可处理性；那个物理来源中没有任何东西暗示如此粗糙的一个描述是恰当的。

4.4 诸发现与残余

四个结构特征转移而六个不。那些被转移的特征关乎复数相干域的涌现、躺在它们之间之物的结构，以及新颖在那里的优先成形。那个说明的规范内容从政治理论被输入，而那个物理来源仅供出结构与语汇。

§5 界面作为生产性场所

本文最大的让步被亏欠于一个于数十年前抵达它中心观察的工作。

5.1 交易区

加利森（1997）审查持有不相容诸承诺且共享没有全局框架的科学亚文化的协调。理论家、实验家与仪器建造者协调，在他的说明上，在一个交易区之内，在那里一个局部的间语言为交换的目的发展。那个间语言作为一个恰当于一个特定交易的受限行话开始、随那个交换变得例行而增厚为一个皮钦语，而可能稳定为一个丰富到足以支持一个它自己实践的克里奥尔语。他的概括表述，即所达成之物是无全局意义的局部协调，陈述本文的图景从另一个方向抵达的那个立场。

那个对当下说明的后果应被无保留地陈述。缺乏一个共同框架的相干域在它们的界面处生成一个属于任一者、且能够在它自己权利中成为一个域之结构这个命题不是新的。它是那个交易区文献的中心发现，而一篇把它呈现为一个发现的论文会是错的。

5.2 边界对象

斯塔与格里泽默（1989）辨认一次栖居若干共同体、可塑到足以被适应于局部要求且稳健到足以跨诸场所保全一个共同身份的诸对象。它们的处理确立一个界面可被带它们自己特征的稳定结构所栖居，而它发展这样诸结构的一个类型学。

5.3 接触、混杂、摩擦

三个进一步的处理从人文学科占据同一地形。普拉特（1992）命名那个接触区，其中先前分离的诸方在不对称的诸条件下相遇。巴巴（1994）论证文化诸陈述在一个陈说的第三空间中被建构，而从相遇涌现之物不可还原为任一来源。秦（2005）把差异地带的摩擦当作生产新安排的。

5.4 共存与共同创造

一个平行的辩论已以政策术语被进行。由坎特尔（2005, 2012）发展的对多元文化主义的批评主张确保诸共同体之站位的诸安排可能产生平行生活，其中诸群体以很少接触占据同一领土并分开地发展。跨文化主义，由布沙尔（2012）与其他人发展，提议多样被当作一个生成某样新东西的资源，而帕雷克（2000）论证诸文化通过对话被转变。

从共存到共同创造的举动相应地是一个既存立场的构成性举动，而本文继承它。

5.5 那个残余

两样东西显得剩下。

那个文献理论化诸间语言、诸边界对象与诸第三空间而不供出一个按结构对诸边界的类型学。一个边界加强还是削弱它所接合之物、而什么把那两个情形区分开来，不是这些说明被建来回答的问题。

那个文献第二大体是描述性的。它确立诸界面是生产性的并审查协调成功所处的诸条件。一个把一个可辩护界面同一个不可辩护的区分开来的判据不在它诸产物之中。

5.6 诸发现与残余

诸界面的生产性是被确立的、从共存到共同创造的举动是跨文化主义的构成性举动，而间语言、边界对象、接触区与第三空间的语汇可得。那个残余在于诸边界的结构类型学与一个为它们的规范判据。

§6 没有顶点的复数中心

那个多晶图景在治理理论中有一个近亲，而那个关系必须被直白地陈述。

6.1 多中心性

波兰尼（1951）为许多参与者在一般规则之下相互调整而无中央指导的诸安排引入多中心秩序。奥斯特罗姆、蒂布与沃伦（1961）把那个术语带入政治科学，把大都市治理描述为许多带交叠管辖且无单一顶点的形式独立决定中心。奥斯特罗姆（2010）为共池资源发展多中心治理，确立这一种类的诸安排维系它们自己并解释一个市场与国家的二分所不解释的诸结果。

6.2 那个让步

那个多晶图景的结构主张，即一个稳定秩序要求诸域之内的局部相干连同它们之间可行的诸关系而要求没有延伸穿过整体的单一秩序，是多中心性的主张。那两个词描述同一个结构，而一个熟悉那个多中心文献的读者会发现那个图景熟悉。

6.3 那个残余

那两个文献之间的一个不对称留下空间，而它是本文工作于其中的那个不对称。

多中心性理论化诸中心。它的单位是那个决定中心，它的诸问题关乎诸中心的自主、诸管辖的交叠，以及诸中心相互调整所处的诸规则。躺在诸中心之间之物被当作关系、调整，或合同，没有它自己的结构。

那个晶体学来源如域一般密集地理论化诸边界。一个边界有一个由在它处相遇之诸取向所判定的结构，那个结构变化，而那个变化说明那个材料行为的甚多。§13把这个不对称发展为本文的首要主张。

6.4 诸发现与残余

没有顶点的复数中心是被确立的地面，而那个多晶图景对那个主张什么也不添加。那个残余在于对立于诸中心之间之物的处理，多中心性把它留得单薄而那个晶体学来源把它当作一个结构。

§7 无最大化的正义

把正义写为一个最优化问题的拒绝是本文的规范起点并在政治哲学中是一个被安定的立场。

7.1 那个既定的拒绝

罗尔斯（1971）拒绝聚合功利主义，以那个根据，即它未能认真对待诸人之间的区分，因为一个最大化判据许可对一个人的一个损失被对另一个人的一个收益补偿，仿佛那两者发生在一个单一生命之内。那个拒绝对随后的文献是根基性的。

四个发展把它进一步承载。法兰克福（1987）论证要紧之物是每一个是否有足够，以致那个判据是一个门槛而没有最大值被要求。森（1999）与努斯鲍姆（2011）以被确保的诸可行能力构架正义，那是一个关于一个人能够做什么与是什么的条件。安德森（1999）把平等奠基于诸人立于其中之关系的性质，以致那个关切对象在性质上是关系的。佩迪特（1997）把自由当作非支配，那是一个关系的一个性质并容许没有自然的极大化对象。伯林（1969）最终论证根本诸价值是复数且不可通约的，以致没有单一标度可得、诸安排能在其上被排名。

7.2 那个让步

一篇为拒绝最大化主张功劳的论文会是在为一门学科的被安定共识主张功劳。那个拒绝在此被继承，而上面的诸来源是它的作者。

7.3 那个残余

相对较不发达之物是那个拒绝的形式表达。一个作为一个门槛或作为一个关系条件被陈述的判据通常被留在散文中，而在建模工作中通常被伸手够到的数学是最优化，它重新引入那个散文所拒绝之物。§10以一个许可一个模型而不重新引入一个极大化对象的形式陈述那些条件，而§14记录所要求的数学已经存在。

7.4 诸发现与残余

最大化的拒绝是被安定的并在此被继承。那个残余在于形式地表达诸复数条件而不把一个目标函数偷运回去。

§8 那个关系场域

那个装置在此被陈述。那些定义是规定性的并为可处理性被选择。

定义1（关系域）。 一个关系域 $D_i$ 是一个诸方的集合连同它们之间的诸关系，以致那些关系维系一个共同的表述模式：一个陈述什么成问题、什么算作一个理由，以及什么会安定一个问题的方式。

定义2（相干性）。 一个域的相干性 $c_i \in [0,1]$ 是它跨诸场合并跨成员资格之诸变化再生产它自己表述模式的能力。一个带 $c_i = 0$ 的域已失去那个能力：它的表述模式不再被传递，而它之内生起的诸问题被以某个其他域的诸术语安定。

定义3（取向与取向差）。 一个域的取向 $\theta_i$ 是它的表述模式，被表示为一个空间 $\Theta$ 中的一个点。两个域之间的取向差 $\Delta_{ij} \in [0,1]$ 是它们在 $\Theta$ 中的分离，被归一化以致在诸模式重合之处 $\Delta_{ij}=0$ 而在它们最大限度不同之处 $\Delta_{ij}=1$。

定义4（接触图）。 那个接触图 $\Gamma$ 以诸域为顶点，带 $(i,j) \in \Gamma$ 在那两个域立于一个关系、以致一个中的诸裁定关系着另一个之处。不在接触中的诸域没有边界，而下面没有条件适用于它们。

定义5（交换与不对称）。 对于 $(i,j) \in \Gamma$，交换强度 $e_{ij} \in [0,1]$ 是跨那个边界的流量之体量，并是对称的。不对称 $\sigma_{ij} \in [-1,1]$ 记录一个域的诸术语支配那个交换的程度，带 $\sigma_{ij}>0$ 在 $i$ 的诸术语支配之处、$\sigma_{ij} = -\sigma_{ji}$，而 $|\sigma_{ij}|=1$ 在一个域的诸术语完全支配之处。

定义6（边界可行性）。 边界 $(i,j)$ 的可行性是 $v_{ij} = e_{ij},(1-|\sigma_{ij}|)$。一个边界在 $v_{ij}>0$ 之处可行，那恰恰在交换发生且任一域的诸术语都不完全支配它之处成立。

那最后一个定义的两个特征应被标记。它是一个被选择的规定，因为它使那两个失败方式在一个表达式中可见。它第二使那两个失败可区分：$v_{ij}=0$ 带 $e_{ij}=0$ 是分离，而 $v_{ij}=0$ 带 $|\sigma_{ij}|=1$ 是支配。可行性的消失因而是两个不同条件的相遇点。

定义7（强加负荷）。 域 $i$ 所承担的负荷是 $\Lambda_i = \sum_{j:(i,j)\in\Gamma} e_{ij},\max(0,;\sigma_{ji})$，即跨那些另一个域的诸术语支配之边界的被求和流量。

§9 两个条件与四个失败模式

9.1 那些条件

被提议的诸条件。 一个复数关系安排在两者都成立时可辩护：
$$ \text{(C1)}\quad c_i>0 \ \ \text{对每个 } i, \qquad\qquad \text{(C2)}\quad v_{ij}>0 \ \ \text{对每个 } (i,j)\in\Gamma. $$
条件 C1 要求每个域保有再生产它自己表述模式的能力。条件 C2 要求每个边界许可交换而任一侧的诸术语都不完全支配。

那些条件的形式承载那个规范内容。两者都被全称量化于诸域与诸边界之上，而两者都不是一个聚合。没有和、平均，或加权总量出现，而那些条件中没有任何东西许可一个域的一个欠缺被另一个的一个盈余抵销。

9.2 那些条件的独立性

命题1（独立性）。 两个条件都不蕴含另一个。存在满足 C1 而违反 C2 的诸安排，以及满足 C2 而违反 C1 的诸安排。

那个命题被§11中的两个构造所确立。情景 B 有每个域处于最大相干性而每个边界死亡，满足 C1 而违反 C2。情景 A 有每个边界可行而一个域被消亡，满足 C2 而违反 C1。那两个条件因而在做分开的工作，而一个仅陈述它们之一的说明会错过另一个所捕捉的诸失败。

那第一个构造配得上强调。在分离之下，相干性为每个域达到它的最大值：一个不受制于强加的域完美地再生产它自己。孤立按 C1 独自的度量是最优的。一个建立在诸域之保护之上、对什么在它们之间通过没有条件的复数治理理论，因而推荐那个平行生活之批评被针对的安排。

9.3 那四个失败模式

定义8（共同创造）。 共同创造发生于一个边界处，在那里一个属于任一相邻域之先前表述模式的裁定在那里生起。$(i,j)$ 处的共同创造率被规定为 $n_{ij} = \eta, v_{ij}, \Delta_{ij}, \min(c_i,c_j)$，带 $\eta>0$。

那个规定承载§4的那个晶体学转移：成形优先地在诸边界处发生、要求一个跨那个边界的差异，并要求两侧都能够贡献。

命题2（那四个失败模式）。 $n_{ij}>0$ 成立当且仅当下列全部四个成立：
$$ e_{ij}>0, \qquad |\sigma_{ij}|<1, \qquad \Delta_{ij}>0, \qquad \min(c_i,c_j)>0. $$
共同创造因而恰恰以四种方式失败：凭分离 $(e_{ij}=0)$、凭支配 $(|\sigma_{ij}|=1)$、凭均质化 $(\Delta_{ij}=0)$，而凭消亡 $(\min(c_i,c_j)=0)$。

证明。 $n_{ij}$ 是 $\eta>0$ 同 $v_{ij}$、$\Delta_{ij}$ 与 $\min(c_i,c_j)$ 的一个乘积，每一个非负。一个非负因子的乘积为正当且仅当每一个因子为正。由定义6，$v_{ij}>0$ 当且仅当 $e_{ij}>0$ 且 $|\sigma_{ij}|<1$。$\blacksquare$

图3。 共同创造失败的四种方式，被示意地显示。阴影与内部划线表示一个域的取向。在均质化之下那两个取向重合而那个边界不再是任何东西的一个相遇。在分离之下那些域完好且不相连。在支配之下一个域的诸术语支配那个交换，被绘制为一个单一箭头。在消亡之下一个域已失去再生产它表述模式的能力。

那个命题赢得本文的标题。均质化与分离是那里所命名的两个失败，而那个推导把它们同两个其他一起放置为一个要求的诸后果。一个通过追求统一避免碎片化的政体从那第二个失败移到那第一个，而那个模型登记没有改善。

9.4 诸发现与残余

两个条件被陈述，以一个量化于诸域与诸边界之上而聚合无物的形式。那些条件是逻辑上独立的，而那个孤立结果显示为什么两者都被要求。四个失败模式从一个单一要求由一个两行论证随之而来，而它们中的两个是本文被对着命名的诸极端。

§10 那个形式模型

§9的诸条件是静态的。这一部分供出动态，以致一个条件的失去可作为一个过程一如作为一个状态被描述。

10.1 那个动态

定义9（相干性动态）。 对每个域 $i$，带内在再生产率 $\rho_i>0$ 与易感性 $\delta>0$，
$$ \dot c_i ;=; \rho_i, c_i,(1-c_i);-;\delta, c_i, \Lambda_i. \tag{12} $$

那第一项是逻辑斯蒂的：一个域以一个正比于已达到之相干性与所剩空间的率再生产它的表述模式。那第二项是渲染的代价：一个被迫以另一个的诸术语陈述自己的域较不有效地传递它自己的，以一个正比于它的相干性与它所承担之负荷的率。

那个模型是刻意最小的。无物被最优化、没有行动者选择，而那些控制 $e$ 与 $\sigma$ 仅通过 $\Lambda$ 进入。

10.2 那个门槛

命题3（相干性门槛）。 固定 $\Lambda_i$。若 $\delta\Lambda_i \geq \rho_i$ 则 $c_i(t)\to 0$ 从任何 $c_i(0)\in(0,1]$。若 $\delta\Lambda_i < \rho_i$ 则 $c_i(t)\to c_i^{} = 1-\delta\Lambda_i/\rho_i$，它在 $(0,1]$ 上渐近稳定。*

证明。 写 $\dot c_i = c_i\bigl(\rho_i-\delta\Lambda_i-\rho_i c_i\bigr)$。在 $c_i>0$ 上 $\dot c_i$ 的符号是 $\rho_i-\delta\Lambda_i-\rho_i c_i$ 的符号。若 $\rho_i-\delta\Lambda_i\leq 0$ 这对所有 $c_i>0$ 为负，故 $c_i$ 单调地减少并被 $0$ 从下界定；$[0,1]$ 中唯一的平衡是 $c_i=0$。若 $\rho_i-\delta\Lambda_i>0$ 那个表达式在 $c_i^{}=(\rho_i-\delta\Lambda_i)/\rho_i \in (0,1]$ 处消失、在它之下为正而在它之上为负，故 $c_i^{}$ 吸引每一个从 $(0,1]$ 出发的轨迹。$\blacksquare$

图4。 那个相干性门槛，带 $\rho_i=\delta=1$。面板（a）给出平衡相干性作为强加负荷的一个函数：相干性线性地下降并在 $\Lambda_i=\rho_i/\delta$ 处达到零，越过它那个域无法再生产它自己。面板（b）显示那个门槛两侧从一个共同初始条件的诸轨迹。

图4显示那个结果。两个特征关乎那个论证。对消亡的趋近在那个负荷上是连续的，故一个在增加强加之下的域逐渐地并在任何门槛被可见地跨过、直到它被跨过之前失去相干性。消亡第二在一个有限负荷处被达到，故强加无需是全部的才致命：$\Lambda_i$ 超过 $\rho_i/\delta$ 就够。

定义10（退化性变换）。 一个退化性变换是那个安排中一个抬高某个 $\Lambda_i$、降低 $c_i^{}$ 的变化。它在那个由此产生的负荷满足 $\delta\Lambda_i\geq\rho_i$ 之处是消亡性的。*

那个定义供出那个模型对不义的说明。语言压制、认识排除、义务性地翻译进一个官方语域、对一个域不使用之范畴的施行，以及任何一个域的诸术语在其中可被接收之程序的缺席，各是一个被抬高之 $\Lambda_i$ 的诸描述。

10.3 聚合的拒绝

命题4（不可聚合性）。 条件 C1 不被任何带 $w_i>0$ 的加权聚合 $\sum_i w_i c_i$ 的最大化所蕴含。存在违反 C1 的诸状态，它们达到一个严格大于满足它之诸状态的聚合。

证明。 取三个域与 $w_i=1$。状态 $(0.34,0.34,0.34)$ 满足 C1，带聚合 $1.02$。状态 $(0,0.75,0.75)$ 违反 C1，带聚合 $1.50$。那个聚合把第二个排在第一个之上而第二个消亡一个域。$\blacksquare$

那个命题是初等的而它是§7中所继承之立场的形式陈述。一个最大化判据许可一个域的消亡被别处的诸收益购得，而那个全称量化的条件不。

10.4 欠定

一个反对在这一点生起并应被迎击。约束一个系统保持在一个集合之内可能看起来像另一记号中的最优化，那个集合扮演一个目标的角色。那个回复是一个最大值选择而一个约束不。

命题5（欠定）。 考量一个三角形中的三个域，$\rho_i=\delta=1$、每个边界上的均匀交换 $e$，以及被取向以致域 $0$ 的诸术语在它两个边界处支配而域 $1$ 的在它同域 $2$ 的边界处支配的不对称 $\sigma\ge 0$。则 $\Lambda_0=0$、$\Lambda_1=e\sigma$、$\Lambda_2=2e\sigma$，而条件 C1 与 C2 共同成立当且仅当 $e>0$、$\sigma<1$，且 $2e\sigma<1$。这个集合在 $(e,\sigma)$ 中有非空内部，而那些条件因而不判定那个安排。

证明。 由命题3，$c_i>0$ 当且仅当 $\Lambda_i<1$；那个约束情形是 $\Lambda_2=2e\sigma$。由定义6，$v>0$ 当且仅当 $e>0$ 且 $\sigma<1$。那个合取是所陈述的区域，它含有 $(e,\sigma)=(1/2,1/2)$ 的开邻域。$\blacksquare$

图5。 命题5之安排的控制平面中的可容许区域。那个阴影集合在右侧被全部强加 $(\sigma=1)$、在下侧被交换的缺席 $(e=0)$，而在上侧被消亡轨迹 $2e\sigma=1$ 所界定。§11的三个情景被标记。那个区域是一个带内部的开集，故那些条件容许一个安排的连续统而挑出无一。

图5显示那个区域。它的形状以操作形式承载§7的哲学要点。这一种类的一个判据排除诸安排而不规定一个，而一个政体采纳哪个可容许安排的问题被留下开放，出于那个判据不供出的诸缘由。一个政体被界定，而它在那些界限之内所取的方向是它自己的。

10.5 取向的俘获

如所陈述的那个模型有支配减少相干性，以致一个被支配的域趋近消亡。那个表征是不完整的，而改正它产出那个模型最不明显的结果。

一个受制于持续强加的域无需停止再生产它自己。它可继续再生产，而以它已从对它强加之域取来的诸术语如此做，以致所失去之物是它表述的独特，而它表述的能力被保全。相应地让取向在生成那个强加负荷的同一压力之下移动：

$$ \dot\theta_{i} ;=; \kappa \sum_{j,:,(i,j)\in\Gamma} e_{ij},\max(0,\sigma_{ji}),\bigl(\theta_{j}-\theta_{i}\bigr), \tag{13} $$

以致一个域朝无论哪个对它强加之域的取向漂移，以一个由它们之间边界的强度与不对称所设定的率，而在那个边界对称之处根本不漂移。

图6。 持续不对称交换之下取向的俘获，从方程（12）与（13）计算。在（a）中每个相干性保持为正，那三个域在 $1.00$、$0.62$ 与 $0.23$ 处安定。在（b）中那两个被从属域的取向被拉向那第三个的。在（c）中跨所有三个边界的取向差落到零。共同创造因而停止，尽管没有域被消亡而每个边界都保持开放。

命题6（俘获）。 在方程（13）之下，带固定符号的 $\sigma_{ij}$ 与 $e_{ij}>0$，被支配诸域的取向收敛于那个支配域的取向，以致跨那些受影响边界 $\Delta_{ij}\to0$。在那个强加负荷保持低于命题3之门槛之处，这带每个相干性为正而每个边界可行地发生。

图6中所报告的模拟展示它：那些相干性在 $1.00$、$0.62$ 与 $0.23$ 处安定，全部为正；每个边界保持开放；而每个取向差落到零，以致§9的共同创造率通过它的第三个因子消失。

三个后果随之而来。

俘获是一条从支配到均质化的路径。§9的四个失败模式被确立为逻辑上独立的，而方程（13）供出一条从一个到另一个的动态路线：一个强加的政体无需消亡任何东西，因为持续的强加把差异转换为相同而相同结束共同创造而没有任何域被失去。

俘获对一个仅在相干性上的检验不可见。一个监测每个域是否继续再生产它自己的观察者会沿图6的轨迹记录完全没有失败。每个域完好、每个渠道开放，而那个已被失去的能力被两个测量都不登记。

俘获第三是那个说明表征一个那个框架在别处所作之主张所采取的形式：对一个关系主体的压力特征性地重定向它所能生成之物，把那个生成本身留得完好。一个在俘获之下的域正生成同以前一样多并以另一个的诸术语生成。一个生成上的条件与一个所生成之物之独特上的条件之间的区分是那两个条件中第一个与那个共同创造率之第三个因子之间的区分，而§9出于那个缘由要求两者。

一则告诫与此属于一起，而它是一则关于那个来源的告诫并把那个模型留得未触动。在那个物理情形中一个域的取向通常是固定的，而那个微观结构通过诸边界迁移以致一个取向替代另一个而演化。一个域的连续再取向确实发生，凭耦合于剪切的旋转，但它仅在细晶粒与高温区中突出。方程（13）因而在一个受限区中有一个物理对应物，达不到那个来源的一个一般特征，而它在此作为一个关于关系域的规定被供出，那个物理情形为它供出一个暗示性的先例，达不到一个凭据。

§11 一个案例研究

那个模型在三个例示上被展示，按一个被陈述的原则被选择，带那个安排在第一个之内被变化而诸后果被追溯。

11.1 为什么这些例示

这一种类论文中的诸例示为它们所区别之物被选择，而那个选择原则在下面被陈述、以替代留给一个读者去推断它。§9的四个失败模式被那个共同创造率的哪个因子消失所区分，而一个例示通过展示一个其他所不的失败赢得它的位置，在那些以那个模型当作相关之诸方面不同的条件之下：那些域是否在领土上分离、它们之一是否施行那个安排，以及那些边界是否根本被制度地承认。

三个例示相应地被使用。第一个是一个在共享一个领土并对着一个本身是它们之一之权威负责的诸域之间的资源决定，它展示支配与分离以及它们之间的过渡。第二个是一个在没有共享权威且诸边界被承认并被配备人员之诸域之间的科学协作，它展示那个生产性情形与§4的那个特殊边界困难。第三个是一个其中诸边界未被承认的专业标准制定机体，它展示俘获，那个没有相干性检验探测的失败。

它们是例示。它们的功能是显示那些命题所断言之物，在其制度特征不同到足以使一个读者可判断那些断言是否值得带到一个测量可能被尝试之情形的诸设定中。

11.2 那第一个设定

三个域关系着一个流域的处置。第一个是习惯性的，被继承的实践所维系并以义务与站位的诸术语表述那个流域。第二个是监管性与科学性的，以对着被陈述判据的被测量状况的诸术语表述它。第三个是商业性的，以估值与回报的诸术语表述它。每一个都同两个其他在接触中，故那个接触图是一个三角形。

诸取向被置于一个周期 $\pi$ 的圆上的 $\theta = (0.15\pi,, 0.55\pi,, 0.85\pi)$，给出诸取向差 $\Delta_{01}=0.8$、$\Delta_{02}=0.6$、$\Delta_{12}=0.6$。再生产率与易感性被设为一，而每个域从 $c_i=0.6$ 开始。那些标签不携带经验主张，而那些数字为可辨识性被选择。

11.3 被比较的诸安排

情景 A，支配。 交换跨每个边界强烈，$e_{ij}=1$，而那个习惯域的诸术语大体被取代：$\sigma_{01}=0.85$、$\sigma_{02}=0.75$、$\sigma_{12}=0.50$，带那个监管与商业域支配那个交换。诸负荷随之而来为 $\Lambda = (0,,0.85,,1.25)$。

情景 B，分离。 没有交换发生，$e_{ij}=0$，而不对称未被界定并被设为零。诸负荷是 $\Lambda=(0,0,0)$。

情景 C，可行交换。 交换适中，$e_{ij}=0.55$，而不对称小：$\sigma_{01}=0.15$、$\sigma_{02}=0.05$、$\sigma_{12}=0.10$。诸负荷是 $\Lambda \approx (0,,0.082,,0.083)$。

11.4 诸结果

图7。 三个安排之下的相干性轨迹，带聚合共同创造在右。在支配之下那个商业域跨过那个消亡门槛而那个监管域被沉重地压制，而那个其诸术语支配之域上升到全部相干性。在分离之下每个域达到最大相干性而没有共同创造发生。在可行交换之下所有域近全部相干性地持续而共同创造比在任一失败之下大一个数量级。

情景	安歇处的相干性	C1 成立	C2 成立	聚合共同创造
A. 支配	$(1.00,,0.15,,0.00)$	否	是	$0.018$
B. 分离	$(1.00,,1.00,,1.00)$	是	否	$0.000$
C. 可行交换	$(1.00,,0.92,,0.92)$	是	是	$0.903$

四个观察从那个表与从图7随之而来。

第一，那两个失败被不同的条件区分，那确立命题1。支配把每个边界保全于一个可行状态而消亡一个域。分离把每个域保全而把每个边界留得死亡。

第二，分离达到那个可得的最高相干性。每个域达到 $c_i=1$，而按域完整独自的度量那个安排是最优的。共同创造尽管如此是零，而那个安排是一个带它们之间无物的完好诸域之集。

第三，那个其诸术语支配之域在支配之下获益。它的负荷是零，而它的相干性上升到那个最大值而它的诸邻居下降。那个模型无进一步假定地再生产那个观察，即强加对那个强加者不昂贵并从那个位置不可见。

第四，那个可行安排不是那个最大化那个表中任何单一量的。相干性低于在分离之下，而那个支配域的相干性不高于在支配之下。区别它之物是两个条件都成立，而共同创造随之而来。

11.5 从 C 到 A 的路径

一个最后的观察关乎一个安排如何在这些状态之间移动。把交换持在 $e=0.55$ 并从 $\sigma=0.05$ 连续地抬高不对称，那个最被负担域上的负荷连续地上升、它的平衡相干性线性地下落，而没有不连续之物发生直到命题3的门槛被达到。在那一点之前那个安排满足两个条件并显示一个在下降中的域。一个仅监测它诸域是否持续的政体会观察不到失败直到那个失败完成。

11.6 那第二个设定：被承认并被配备人员的边界

那第二个例示在那个模型当作相关的三个方面不同于第一个。那些域占据没有共同领土、它们中没有一个施行那个安排，而它们之间的边界被承认并被供以其工作在那里被进行的诸人。

考量关系着一个共同问题的三个研究共同体，一个实验的、一个理论的，而一个计算的，被共享仪器、联合任命，以及一个每一个都读的文献所接合。交换高、不对称低因为没有共同体能强迫另一个，而取向差实质因为每一个以其他不使用的诸术语表述那个问题。在那个模型上这是那个生产性构型：带 $e$ 大、$|\sigma|$ 小、$\Delta$ 实质而每个 $c_i$ 为正，那个共同创造率的所有四个因子都从零被界开，而那个安排生成§13所称的在诸边界处的新域，那正是一个共享仪器或一个联合方法事实上所是之物。

那个例示被包括是为了一个第二缘由，那是它显示§4中所引入的困难。这些共同体中的两个可能发展一个如此熟练的调停以致它们之间的交换变得便宜且不足为奇：一个每一个都能不带翻译使用的共享形式体系。在物理一侧这样一个关系是一个特殊边界，能量低而相应地作为一个新秩序在其处成形的位点差。如所书写的那个模型不表征它，因为一个广泛不同域之间的低能量调停会要求边界性质取决于多于取向差之物，而§16记录那个缺口。那个例示所显示之物是那个缺口不是人为的：这一种类的诸关系是常见的，而一个在差异高之处到处预测高新颖的说明会误描它们。

11.7 那第三个设定：未被承认的边界

那第三个例示被选择因为它展示俘获，而因为俘获是那两个第一设定不会揭示的失败。

考量一个设定一个标准的专业机体，它的成员从若干带不同形构的实践被抽取。那个机体有诸程序、它的诸审议开放、没有实践被排除，而没有参与者被噤声。那个安排会满足任何以参与的诸术语构架的检验。那个机体所缺者是对那些实践不同地表述那个事项的任何承认：它把它的成员当作持有诸看法的个体，而它们诸形构之间的差异仅作为待被解决的不一致在它之内出现。

在那个描述之下那些边界携带交换并是不对称的，因为一个形构供出那个标准被书写所用的词汇而其他必须以它渲染它们的诸关切才能被听闻。方程（13）于是适用，而那个轨迹是图6中所显示的一个。每个实践继续再生产它自己；无一被消亡；那个机体的记录显示由所有的连续参与。所下落之物是那个取向差，随那些实践前来以那个标准所用的诸术语表述那个事项，而随它那个安排生成任何那个标准不曾已含有之物的率。

那个例示转在这样一个机体会监测之物与会正在发生之物之间的失配上。参与被记录并是完好的。相干性是完好的。已消失的那个量是那个机体没有度量的一个，而它没有对它诸边界的承认、没有理由寻求一个。

11.8 诸发现与残余

那三个安排分开那两个条件、展示那个孤立结果，并显示共同创造取决于两者。分离最大化相干性并生产无物。支配对那个其诸术语支配之域无代价。对消亡的趋近最终是连续的，故一个域的失去不被它相干性为正这个条件事先信号。

§12 描述与辩护之间的墙

一个以生成性之语汇书写的说明被暴露于一个特定的失败，而那个暴露恰恰在本文现在所抵达之点最大。那个失败在此被铺陈，连同那个说明中防止它之物。

12.1 那个必须保持可拒绝的句子

考量摧毁一个域可能释放其余生成能力这个命题。在一个耦合系统中这不是修辞；它频繁地为真，而它是可观察的。在一个域被移除之处，诸幸存者之间的协调代价下落、被争议的诸边界消失，而剩下之物可能比以前更自由地生成。

一个把生成性的自由展开当作它目的的说明无法拒绝那个命题，因为那个命题会在那个说明自己的诸术语中为真。它会已把一方是否有用的问题以一方是否对那个系统的生成能力作出贡献的问题替换，而那些是不同语汇中的同一个问题。此处所供出的说明因而必须能够断言下列。

那个句子。 即便消亡一个域事实上会释放那个剩余构型的生成能力，消亡它在所陈述的诸条件上是不可容许的。

12.2 为什么那些条件许可那个断言

§9之诸条件的三个特征，一起来看，确保它。

那些条件被全称量化并不是聚合。命题4确立 $\forall i,(c_i>0)$ 不被任何加权和的最大化所蕴含，并展示一个其中一个域消亡而那个聚合尽管如此更高的构型。一个消亡一个域的安排会失败于那个条件，无论那个总量发生什么，因为那个条件中没有总量。聚合的拒绝因而是那些条件被书写所用之形式的一个性质，独立于那个作者的任何偏好而成立。

那些条件是约束，而它们自始至终作为约束运作。命题5确立它们容许一个开放的诸安排区域并挑选无一。那一种类的一个约束无法被交易，因为没有那个交易会被进行所处的标度，而它无法被最优化，因为它被满足或未被满足而在它约束之点不容许程度。

那些条件的规范力量第三是被输入而从不被推导的。§3中没有任何东西蕴含一个域应当持续；一个物理说明关于任何东西不蕴含那种的任何东西。持一个域再生产它自己诸术语的能力不可被消亡的诸缘由是§7中所概览之传统里所给的诸缘由，而它们是关于诸方之站位的诸缘由，把诸构型的生产性搁置一旁。那个模型以一个许可它们诸后果被追溯的形式陈述那些缘由；它不供出它们。

12.3 那个说明所不作的三个陈述

生成性是一个目的。 此处所描述的诸关系展示一个维系它们自己继续生成之诸条件的倾向。那是一个对所观察之物的描述，而本文把它当作那个框架的一个原始概念，谢绝解释它。解释它会是命名无论它服务什么，而无论什么被命名会成为那个更根本的善，带生成性被还原为它的手段。那个倾向相应地自始至终是描述性的，而下面的诸论证从§7汲取它们的诸缘由。

凡有助于生成性者因此被辩护。 §3记录粗化无一行动者地进行而复数性是亚稳的。两个观察都不是一个缘由。一个构型倾向于它诸域的失去这解释为什么维系复数性要求某样东西而不确立那个倾向是正当的，也不确立抵抗它是。

一个域的主张停靠于它所贡献之物。 一个对任何其他不生成有价值之物的域会恰如一个生成甚多的一样立于那些条件之下。那些条件凭构造对输出漠不关心，而它们的一个对输出敏感的版本会是本文着手避免的那个功利主义，在一个层次之上被重构。

12.4 那墙最薄之点

那个最弱之点现在可被命名。§9的共同创造率是一个量、它被界定以致更大或更小，而那个案例研究把它作为一个数字报告。没有任何东西阻止一个读者把那个数字当作一个极大化对象，而一个被选择以最大化它的安排会是一个恰恰在本节所拒绝之根据上被选择的安排。

那个可得的辩护是结构性的。那个共同创造率是一个诊断并在没有条件中出现。那些条件是那两个正性要求，而一个满足它们的安排是可容许的，无论它的共同创造率高还是低。倘若一个读者尽管如此希望最大化那个率，那些条件约束那个最大化恰如它们约束任何其他目标，而一个通过消亡一个域抬高那个率的安排会因上面所给的缘由不可容许。那个率在§11被报告因为它使那些失败模式可辨识，而它在那个说明的任何地方不承载规范分量。

12.5 诸发现与残余

那些条件许可对那个这一种类说明最可能无法拒绝之句子的拒绝，而它们如此做因为它们被全称量化、因为它们作为约束运作，而因为它们的规范内容是被输入的。那个说明把诸关系维系它们自己生成之诸条件的倾向当作一个不辩护任何东西的原始概念。那最薄之点是那个共同创造率，它是一个量而可能被误认为一个极大化对象；它进入没有条件，而§15记录那个风险。

§13 晶体学来源的贡献

§5到§7让步诸界面是生产性的、没有顶点的复数中心是被确立的，而最大化的拒绝是被安定的。这一部分陈述那个来源在它们之外所供出之物。

13.1 那个边界作为一个结构

多中心性理论化诸中心并把躺在它们之间之物当作关系或调整。那个交易区文献理论化在一个界面处所发展之物并把它当作一门语言。那个晶体学来源把那个边界当作一个其特征由在它处相遇之诸取向所判定的结构，而它按那些特征区别诸边界。

两个区分作为假说转移过来。小取向差之域之间的边界在种类上不同于大取向差之域之间的边界，而那两个在每一个物理上要紧的方面行为不同。诸边界第二在于在它们处相遇之两个秩序是否被一个共同安排所容纳，还是那个相遇是突兀的，那关乎那个边界加强还是削弱那个材料。

关于边界类型学的假说。 关系域之间的边界容许按跨它们的取向差、并按那个边界处的安排容纳两个表述模式的程度分类。那个分类关乎那个边界传输、抵抗，还是生成，而那些类别不被一个单一标度排序。

那个假说陈述一个类型学会要求之物，而那个类型学本身是未了的。§16如一个问题引入它。

13.2 那个三元组作为分析的单位

一个第二后果关乎什么被分析。在诸边界是结构之处，那个分析的单位既非那个域也非那对域，而是那个由两个域连同它们之间的边界所组成的三元组。那个三元组不可分解：那个边界的诸性质从两个取向随之而来并属于任一域都不。

那个提议是一个对网络分析的背离，在其中一条边携带一个权重而那个结构驻留在诸顶点与那个连接模式中。一条携带一个它自己结构、带它自己持续诸条件的边，是一个不同的对象。

13.3 边界处的成核

那第三个贡献在上面的论证中承载最多分量。在材料中，一个新相的成形优先地在诸边界处开始，因为已投资于那个边界的能量降低在那里成形某样新东西的壁垒。一个域内部的成形要求更多。

被转移，那个命题是新颖优先地在诸边界处生起并要求两者，即一个差异跨那个边界存在而那个边界许可交换。定义8陈述这个，而命题2引出它的诸后果。那个被转移的命题是一个假说，而§4支配它背后那个物理事实的地位。

那个表述供出一个缘由，从一个停靠于共存的说明缺席，为什么复数性值得维系。被持分开的诸域在它们之间生产无物，而被消解进彼此的诸域没有任何东西能在其处成形的边界。复数性在这个说明上是新颖的一个条件，而本文被对着命名的那两个失败是移除那个条件的两种方式。

13.4 诸发现与残余

那个晶体学来源供出一个把边界当作一个结构的处理、一个是三元组的分析单位，以及一个把新颖定位于诸边界处的机制。第一个是一个等待一个类型学的假说、第二个是一个关于方法的提议，而第三个在§10的模型中被规定并承载它的中心结果。

§14 同可行性理论与系统理论的关系

两个既存工作之体供出那个说明所要求的数学与语汇，而两者在建构它时都不曾被伸手够到。

14.1 可行性理论

奥班（1991）发展被要求保持在一个约束集之内、无最优化被施行之诸系统的研究。一个集合的可行性核是那个某个可容许演化从中无限地保持在它之内之诸状态的汇集，而那个理论关乎这样诸核的计算与维持它们之诸演化的刻画（奥班、巴延与圣皮埃尔 2011）。

那个对应精确到足以被采纳。条件 C1 与 C2 界定一个约束集。那些控制是那些安排 $e$ 与 $\sigma$。命题5中所计算的那个区域是穿过那个可容许控制集的一个截面。那个理论的特征结果，即可行控制一般地是多重的，是§10凭构造所示范的那个欠定的形式陈述。

采纳那个语汇加强那个说明，因为它为一个政治哲学已留在散文中的规范形式供出一门发达的数学。它所暗示的纲领在诸开放问题之中被陈述。

两个进一步的框架共享那个形式。生态韧性把一个系统持续在一个吸引域之内的能力同它返回一个单一平衡的速度区分开来（霍林 1996），而那个区分是此处所要求的一个。环境治理凭运作在其之内可容许之诸边界的构架（罗克斯特伦 et al. 2009），作为一个被下限与上限所界定的空间被规范地发展（雷沃斯 2017），是一个当下形式之判据在规模上被应用的既存例子。

14.2 自我再生产

条件 C1 要求一个域再生产它自己的表述模式，而那个要求有一个既定的名字。卢曼（1995），发展马图拉纳与瓦雷拉（1980）所给的自我生产系统之说明，把一个社会系统当作通过它自己的诸运作再生产它自己的诸要素，以致那个系统的延续是那些运作的一个成就。此处所界定的相干性是另一描述之下的操作闭合，而那个对应应被记录。

一个差异关乎那个论证。那个系统论说明强调闭合，并把跨一个系统边界之物当作待被内部处理的扰动。此处的说明要求诸边界可行，那是一个关于什么跨它们通过的要求，而命题2使新颖取决于它。闭合是一个域之持续的一个条件，而那个安排要求更多才可辩护。

14.3 那个术语的先例

米勒与赫克特（2021）以结晶的名字发展一个欧洲联盟中社会秩序的说明，其中较低组织层级的合作维系一个较高层级的秩序。那个说明被设在一个复数且多语言政体中并连到关于公地的文献。当下这篇论文对那个术语的使用相应地不是在一个空场域中的一个杜撰，而那个差异在于那个机制的层级：那个在先说明关乎秩序从较低层级合作的涌现，而此处的说明关乎秩序已涌现之后诸域之间边界的结构。

一个更老的用法反着跑。文化结晶，在盖伦（1961）所给的意义上，命名一个枯竭的状况，其中一个文化的诸可能性已被穷尽而没有进一步之物生起。此处所用的意义对立于那一个：结晶是结构被生成之处，而命题2的诸失败模式是生成停止所处的诸条件。

14.4 诸发现与残余

可行性理论为一个非最优化判据供出那个数学、生态韧性与环境治理的诸边界构架为它的规范形式供出诸先例，而系统理论为条件 C1 供出那个既定的名字。结晶这个术语在一个密切相关的设定中有一个在先使用，以及一个带相反价数的更老使用。

§15 该说明的界限

那个类比被声明所规训。 §4陈述什么转移与什么不，而那个陈述是一个方法的声明。没有任何东西确立那四个被转移的特征成立于关系域，而一个否认那个转移的读者被留下一个带一个晶体学语汇的被规定模型。

那些命题对那个模型为真。 命题1到6是定义1到10的诸后果。它们关于任何政体什么也不确立，而它们的旨趣完全取决于那些定义是否捕捉任何东西。

那些量不可测量。 相干性、取向、交换与不对称没有测量程序。一个其变量无法被测量的模型无法被检验，而那个说明相应地是一个工作的提议。

诸域被当作离散且给定。 诸方一次属于许多域、诸域交叠并嵌套，而它们的诸边界是被争议的。那个模型把它们当作一个带一个固定接触图的固定有限集，而每一个结果取决于那个简化。

那个动态是被规定的。 带一个线性渲染代价的逻辑斯蒂再生产为可处理性并为它门槛的锐利被选择。其他动态会产出其他门槛，而此处没有任何东西显示那个选择是正确的一个。

成核被断言，而非被推导。 定义8使共同创造成为四个因子的一个乘积，而命题2仅从那个形式随之而来。那个命题因而是共同创造如何被界定的一个后果，而它的力量取决于那个定义是否是对共同创造所是之物的一个可辩护重构。

那些核心直觉是被继承的。 如§5到§7所让步，诸界面的生产性、没有顶点的复数中心，以及最大化的拒绝都在别处被确立。

两个主张被对着这个清单引入。从一个单一要求对四个失败模式的推导把统一与碎片化同两个进一步的失败放在一个框架中，那是那些被概览的说明所不做的。把那个边界当作一个带它自己类型学之结构的处理致意一个那个多中心文献留下开放的不对称。

§16 开放的问题

那第一组关乎那个模型。

Q1。 一个为相干性的测量程序会像什么样？条件 C1 转在一个域跨成员资格之诸变化再生产它表述模式的能力上，而一个操作化会把那个说明从一个提议转换为一个框架。

Q2。 命题2的诸失败模式穷尽共同创造失败的诸方式，还是仅穷尽被定义8所许可的诸方式？那个命题是一个被规定乘积形式的后果，而一个不同的重构会产出一个不同的清单。

Q3。 命题3的门槛在更现实的动态之下幸存吗？不是逻辑斯蒂的再生产、在负荷上不线性的渲染代价，以及在压力之下适应它们自己再生产率的诸域，会各改变那个结果。

那第二组关乎那个边界，那是那个说明主张最多之处。

Q4。 关系边界的类型学是什么？§13的假说提议按取向差并按容纳的分类。建构那个分类，并确立它的诸类别行为不同，是那个说明最要求的工作。

Q5。 哪些边界加强而哪些削弱它们所接合之物？在材料中那个答案取决于结构并被良好研究。为关系域的对应问题是开放的并显得可回答。

Q6。 一个边界能成为一个域吗？那个交易区文献描述稳定为它们自己实践的诸间语言。那个模型没有一个边界凭之获得相干性并作为一个顶点进入那个接触图的机制，而供出一个会闭合那个说明的首要结构缺口。

那第三组关乎那个规范内容，而诸问题在此最尖锐，因为那些主张最强。

Q7。 那两个条件穷尽一个可辩护复数安排所要求之物吗？一个安排可满足两者而以一个没有正义说明会接受的方式分配共同创造的诸益处，而那些条件就分配沉默。

Q8。 那个接触图本身如何被评估？那些条件适用于存在的诸边界。一个从不把两个域带入接触的安排违反无物，而哪些域应当在接触中的问题不被那个说明提出。

Q9。 在那些条件无法被共同满足之处什么随之而来？命题5展示一个其中那个可容许区域非空的情形。在那个区域为空之处，那个说明陈述每一个可得的安排是不可辩护的并供出没有关于在它们之间选择的指引。

那最后一个问题是那个说明被导向的一个。

Q10。 在什么条件下一个复数安排抵抗粗化？材料粗化因为诸边界携带能量而它们的减少被偏爱；诸政体展示一个对应的倾向，因为统一程序比复数程序更便宜施行而朝向它的压力是连续的。一个在被确立时满足两个条件的安排可能在一代之后失败于它们，而没有任何人的任何强加行为。什么维系一个诸域之复数性对着那个压力是那个说明之实践价值所依赖的问题，而它完全是开放的。

一个关于那个清单形状的收尾观察。第一组的诸问题能被任何愿意工作那个模型者安定。第二组的诸问题是那个说明主张新颖且它诸对手留下空间之处。第三组的诸问题关乎那些条件是否是正确的诸个，而一个被那次综述说服而未被那些条件说服的读者可取那第一个而留那第二个。

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关于方法的一则附记。 那个形式模型是被规定的而它的诸命题是初等的。那些命题在被陈述之前被数值地验证，而命题3被对着那个动态跨随机化诸参数的积分核查，带所有表面的诸差异落于那个分岔的一个小邻域之内并可归于临界处的慢收敛。命题5被对着那个闭式平衡跨随机化诸控制核查而无诸差异。那些图从那个模型并从一个增长前沿示意图被生成，而它们的诸来源伴随本文。

关于那个类比的一则附记。 §3中所报告的诸物理事实被如它们在材料科学中被理解地陈述并被用作一个结构的来源。没有物理结果作为任何关于一个政体之主张的证据被供出，而§4列举那个转移失败的诸点。一个判断那个列举不完整的读者被邀请去延伸它，因为那个练习的价值取决于那个列举是诚实的。

References

Anderson, Elizabeth S. (1999). What Is the Point of Equality? Ethics, 109(2), 287–337.

Aubin, Jean-Pierre (1991). Viability Theory. Boston: Birkhäuser.

Aubin, Jean-Pierre, Alexandre M. Bayen, and Patrick Saint-Pierre (2011). Viability Theory: New Directions. 2nd ed. Berlin: Springer.

Berlin, Isaiah (1969). Two Concepts of Liberty. In Four Essays on Liberty, 118–172. Oxford: Oxford University Press.

Bhabha, Homi K. (1994). The Location of Culture. London: Routledge.

Black, Max (1962). Models and Metaphors: Studies in Language and Philosophy. Ithaca: Cornell University Press.

Bouchard, Gérard (2012). L’interculturalisme: un point de vue québécois. Montreal: Boréal.

Cahn, John W. (1956). The Kinetics of Grain Boundary Nucleated Reactions. Acta Metallurgica, 4(5), 449–459.

Cantle, Ted (2005). Community Cohesion: A New Framework for Race and Diversity. Basingstoke: Palgrave Macmillan.

Cantle, Ted (2012). Interculturalism: The New Era of Cohesion and Diversity. Basingstoke: Palgrave Macmillan.

Clemm, P. J., and J. C. Fisher (1955). The Influence of Grain Boundaries on the Nucleation of Secondary Phases. Acta Metallurgica, 3(1), 70–73.

Frankfurt, Harry (1987). Equality as a Moral Ideal. Ethics, 98(1), 21–43.

Galison, Peter (1997). Image and Logic: A Material Culture of Microphysics. Chicago: University of Chicago Press.

Gehlen, Arnold (1961). Über kulturelle Kristallisation. In Studien zur Anthropologie und Soziologie. Neuwied: Luchterhand.

Hesse, Mary B. (1966). Models and Analogies in Science. Notre Dame: University of Notre Dame Press.

Holling, C. S. (1996). Engineering Resilience versus Ecological Resilience. In Peter C. Schulze, ed., Engineering Within Ecological Constraints, 31–44. Washington, DC: National Academy Press.

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

Maturana, Humberto R., and Francisco J. Varela (1980). Autopoiesis and Cognition: The Realization of the Living. Dordrecht: Reidel.

Mueller, Sean, and Michael Hechter (2021). Centralization through Decentralization? The Crystallization of Social Order in the European Union. Territory, Politics, Governance, 9(1), 133–152.

Nussbaum, Martha C. (2011). Creating Capabilities: The Human Development Approach. Cambridge, MA: Harvard University Press.

Ostrom, Elinor (2010). Beyond Markets and States: Polycentric Governance of Complex Economic Systems. American Economic Review, 100(3), 641–672.

Ostrom, Vincent, Charles M. Tiebout, and Robert Warren (1961). The Organization of Government in Metropolitan Areas: A Theoretical Inquiry. American Political Science Review, 55(4), 831–842.

Parekh, Bhikhu (2000). Rethinking Multiculturalism: Cultural Diversity and Political Theory. Basingstoke: Palgrave.

Pettit, Philip (1997). Republicanism: A Theory of Freedom and Government. Oxford: Clarendon Press.

Polanyi, Michael (1951). The Logic of Liberty: Reflections and Rejoinders. London: Routledge & Kegan Paul.

Pratt, Mary Louise (1992). Imperial Eyes: Travel Writing and Transculturation. London: Routledge.

Rawls, John (1971). A Theory of Justice. Cambridge, MA: Harvard University Press.

Raworth, Kate (2017). Doughnut Economics: Seven Ways to Think Like a 21st-Century Economist. White River Junction: Chelsea Green.

Rockström, Johan, et al. (2009). A Safe Operating Space for Humanity. Nature, 461, 472–475.

Sen, Amartya (1999). Development as Freedom. Oxford: Oxford University Press.

Sokal, Alan, and Jean Bricmont (1998). Fashionable Nonsense: Postmodern Intellectuals’ Abuse of Science. New York: Picador.

Star, Susan Leigh, and James R. Griesemer (1989). Institutional Ecology, ‘Translations’ and Boundary Objects. Social Studies of Science, 19(3), 387–420.

Tsing, Anna Lowenhaupt (2005). Friction: An Ethnography of Global Connection. Princeton: Princeton University Press.