Generative Closure in Generative Relational Being - A Preliminary Reflection on Turbulence Models

Posted on 2026-07-15 In Physics , Philosophy Views: Disqus: Word count in article: 22k Reading time ≈ 1:20

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Generative Closure in Generative Relational Being

A Preliminary Reflection on Turbulence Models

Wanhong Huang · huangwanhong@serendip.ngo

Abstract

Research on culture, norms, trust, and institutions routinely writes the macro-level as a system with dynamics of its own, an equation in which the future of a collective variable is a function of that variable alone. This paper argues that the self-containedness such writing presupposes is an assumption rather than a finding, and that a mature physical case, fully developed turbulence, shows how and why the assumption fails. The failure has a precise name in statistical physics: the closure problem. When a system with exact microscopic dynamics is projected onto the coarse variables one wishes to study, the projected dynamics are in general not closed, each order depends on a higher one, and the coarse variable is not by itself a sufficient description of its own future. The Mori-Zwanzig formalism gives the projected evolution exactly: alongside its drift it carries a memory kernel and a noise, and reduces to a self-contained macroscopic equation only when a separation of scales lets the memory decay and the noise whiten. Turbulence is the case in which that separation is absent across the range that matters, and it serves here as a calibration rather than a metaphor: it exhibits the closure problem on a fixed state space under the simultaneous loss of every feature that ordinarily rescues it, and thereby marks what a relational difficulty of merely this kind would amount to. The renormalization group, the second instrument for the passage across scales, sharpens the caution from the other side: where it succeeds, universality forbids reading a closed macroscopic law back to its microscopic mechanism; where it fails, as on the anomalous scaling of turbulence, it shows that a scaling law is no warrant for a universal macroscopic theory. The paper carries these into the study of relational emergence as a methodological caution, that a macro-level’s autonomy is to be established, not assumed, and distinguishes the closure it warns about, a closure of dynamics, from the notion the framework will require, a closure of generation. The latter is named and delimited, not established: what would ground it, an invariant of generation not underwritten by any conservation law, is set down as the open problem the reflection leaves standing.

Keywords: closure problem; Mori-Zwanzig; turbulence; relational emergence; generative closure.

A note on the standing of this paper. The paper takes one physical case, the closure problem of fully developed turbulence, and asks what it warns against in the study of how macroscopic order arises from microscopic relation. The order of exposition follows the order of the argument: the physical mechanism is set out first, in its own terms, and the transfer to relational emergence follows as a caution that mechanism licenses. One notion the framework will require, a closure of generation as against a closure of dynamics, is introduced toward the end and left as an open problem, together with the invariant that would ground it. Should the caution be shown to be either already common ground or unsupported by the physical case, the corresponding sections lose their ground. Objection, correction, and counter-evidence are welcome at huangwanhong@serendip.ngo.

§1 Introduction

A collective phenomenon studied over time invites a certain notation. Let $C$ stand for a culture, a body of norms, a level of institutional trust; one then writes

$$\frac{dC}{dt} = F(C),$$

and asks how $C$ evolves. The notation is natural, it is often useful, and it carries an assumption that is rarely examined: that the future of the macro-level is a function of the macro-level alone, that culture is determined by culture, norms by norms, the collective variable by itself. Why a macro-level should possess dynamics of its own, closed upon its own variables, is not a question the notation raises; the closedness is built in before inquiry begins.

This paper argues that the closedness is an assumption rather than a finding, and that it can fail. The argument is not made in the abstract. It is borrowed from a physical case in which the same assumption is made, is precisely stated, and is known to fail in a way that has resisted resolution for the better part of a century: fully developed turbulence, and the closure problem to which it gives rise. In that case the microscopic dynamics are known exactly, the Navier-Stokes equations, written down in the nineteenth century, and the difficulty lies entirely in passing from those dynamics to a self-contained description at the scale one wishes to study. The passage does not close. What the coarse variable does next depends on information the coarse variable does not contain.

The transfer of this difficulty to the study of relational emergence is the work of the paper, and two misreadings would each void it. The first would treat turbulence as a metaphor, a suggestive picture of complexity from which one helps oneself to conclusions; it is not so used here. Turbulence serves as a calibration: it exhibits the closure problem on a fixed state space under the loss of every feature that ordinarily rescues it, and so provides a benchmark against which a relational difficulty can be placed. The second misreading would let the borrowed notion of closure stand in for the notion the framework needs. The closure turbulence teaches is a closure of dynamics; generative relational being will require a closure of generation, and the two differ. This paper reaches that distinction and stops: the notion that would complete it is identified as an open problem, not delivered.

The account this paper offers may be stated as a single claim.

Central claim. The generation of a macro-level from microscopic relation does not by itself confer upon that macro-level a closed dynamics of its own. Whether a collective variable has dynamics in its own terms, whether $dC/dt = F(C)$ may be written at all, is a condition to be established, not a default to be assumed; and the closure problem of turbulence supplies both the precise form of the condition and a case in which the condition demonstrably does not hold. For generative relational being the consequence is a reordering of questions: prior to asking how a culture or a norm evolves stands the question of whether a culture or a norm constitutes a level at which evolution in its own terms can be written.

The claim proceeds in stages. The next section states the closure problem in its technical form, and introduces the renormalization group as the second instrument for the passage across scales, so that what follows rests on real mechanisms rather than a loose analogy (§2). The third justifies the choice of turbulence and fences it: turbulence is one case among the many that might have been chosen, singled out for its maturity, and confined to the role of a calibration (§3). The fourth gives the exact content of the failure through the Mori-Zwanzig formalism, whose memory and noise terms are what a closed macroscopic equation must be shown to be free of (§4). The fifth carries both instruments into relational emergence as four cautions, three from the failure of closure and one from its success (§5). The sixth introduces the distinction between a closure of dynamics and a closure of generation, and marks the boundary of what is here established (§6). The seventh states the limit of the turbulence case, the step it cannot take, which is the step generative relational being most needs, and the caution that the renormalization group’s own limit on turbulence yields (§7). A closing section sets down what has and has not been shown, and the work the reflection leaves open.

§2 The Closure Problem

The closure problem is a definite object in statistical physics, and the caution drawn below depends on its being that rather than a figure of speech. This section states it in its least technical form.

Take a system whose complete state $u$ evolves by known dynamics that are nonlinear. The incompressible Navier-Stokes equations [5] are the standard instance:

$$\partial_t u_i + u_j,\partial_j u_i = -\partial_i p + \nu,\partial^2 u_i, \qquad \partial_i u_i = 0,$$

in which the term $u_j\partial_j u_i$ is quadratic. Suppose that what one wishes to study is not the full field $u$ but an averaged quantity, the mean flow $U = \langle u\rangle$, say, the average taken over an ensemble of realizations. Writing $u = U + u’$ with $\langle u’\rangle = 0$ and averaging the equation, one obtains an evolution equation for $U$ that contains, through the quadratic term, a new unknown: the second moment $\langle u_i’ u_j’\rangle$, the Reynolds stress. The dynamics of the first moment depend on a second moment.

One then seeks an equation for the second moment. It is formed by applying the quadratic dynamics to the product $u_i’ u_j’$ and averaging; the quadratic term $u,\partial u$ acting on a second-order product yields a third-order correlation, so the equation for the second moment contains a third moment $\langle u_i’ u_j’ u_k’\rangle$. The equation for the third moment, formed the same way, contains a fourth. Writing $M_n$ for the $n$-th moment, itself a multi-point tensor field, not a scalar, the structure is a hierarchy

$$\partial_t M_n = \mathcal{F}!\left(M_n,, M_{n+1}\right),$$

in which the equation for each order introduces the order above it. The hierarchy does not terminate. Truncating it at a finite order $N$, retaining $M_1,\dots,M_N$ and their $N$ evolution equations, leaves $M_{N+1}$ as an unknown for which the truncation supplies no equation; the system is in this sense underdetermined, not through a mismatch of scalar counts but because the chain never closes upon itself.

The closure problem. In the moment hierarchy generated by a nonlinear dynamics, the equation for the $n$-th moment depends on the $(n{+}1)$-th; no finite truncation is self-contained. A closure is an externally supplied relation expressing the highest retained moment as a functional of lower ones, $M_{N+1} = \Phi(M_1,\dots,M_N)$, imposed to make the truncated system determinate. The relation $\Phi$ does not follow from the underlying dynamics; it is an added assumption, and the behavior of the truncated system inherits whatever the assumption puts in.

The essential point for what follows is the standing of $\Phi$. It is not derived; it is posited. The dynamics fix the hierarchy but do not fix where to cut it or how to fold the discarded orders back onto the retained ones. Every operative theory that works with a finite set of coarse variables, the eddy-viscosity models of engineering turbulence, the truncations of kinetic theory, the mean-field decouplings of statistical mechanics, is at bottom a choice of $\Phi$, licensed by whatever combination of physical argument and empirical fit the modeler can muster. Where the choice is well licensed, the closure is reliable and the coarse description earns its autonomy. Where it is not, the autonomy is borrowed, and the coarse equation reports as much about the modeler’s assumption as about the system. The closure problem is not that closures cannot be found; it is that they must be supplied from outside, and that their supply is where the substantive commitments hide.

This structure is not peculiar to fluids. The same hierarchy appears in field theory, where the equation for an $n$-point correlation function involves the $(n{+}1)$-point function, and in the kinetic theory of gases, where the equation for the one-particle distribution involves the two-particle distribution and so up the chain [9]. Perturbation theory, mean-field approximation, and the assumption of molecular chaos are, in these settings, the corresponding closures. The problem is generic to the passage from a nonlinear microscopic dynamics to a description in terms of a few averaged quantities. It is this generality that makes it worth carrying elsewhere; and it is turbulence that shows it in its least escapable form.

There is a second, and stronger, instrument for the passage across scales, and it must be introduced because it carries a caution of its own. Where the scales of a system are related not by a small parameter but by self-similarity, the systematic method is the renormalization group [3]: one coarse-grains, integrates out the shortest-wavelength degrees of freedom, and rescales, and studies the flow this induces on the space of possible descriptions. At a critical point the flow runs to a fixed point, and near that fixed point all but a few directions in the space of descriptions are irrelevant: they contract away under repeated coarse-graining, leaving a low-dimensional effective theory governed by the few relevant directions that remain. This is closure achieved not by an imposed $\Phi$ but by the dynamics of coarse-graining itself, which discards the microscopic detail for us. It is among the strongest available accounts of why a macroscopic level should have laws of its own, and it is exact where it holds.

The success has a price, and the price is the first of the two cautions this instrument yields. What the renormalization group establishes at a fixed point is that the surviving macroscopic law is reached by systems with different microscopic constitutions alike, the content of universality, that liquid-gas critical points, magnets, and binary alloys share critical exponents. Universality is the statement that the macroscopic behaviour is blind to almost everything about the microscopic: the irrelevant directions, which is to say most of the microscopic detail, have been flowed away and cannot be recovered from the macroscopic law. A successful reduction of this kind is therefore not a bridge from the macroscopic back to the microscopic but a wall: it certifies that the information needed to make the return journey has been destroyed. This bears directly on the study of macroscopic social order, and §5 states how.

§3 The Choice of Turbulence, and Its Limits as One Case

Turbulence is chosen here for a specific reason and confined to a specific role. A physical system invoked in an argument about social order is not thereby claimed to be identical to one; the reason for the choice and the limits of the role are set out so that the invocation is read as what it is.

The reason for the choice is that turbulence presents the closure problem stripped of every feature that would make it tractable. In many systems the hierarchy can be truncated because a small parameter renders the higher orders negligible: a weak coupling, a small amplitude, a ratio of scales that is large or small. Where such a parameter exists, truncation is licensed by it, the discarded orders are demonstrably smaller than the retained ones, and the closure is an approximation with a controlled error. Where scales are related by self-similarity instead, the renormalization group of §2 is the instrument; but turbulence, as §7 sets out, is the case in which that instrument reaches only part way. Fully developed turbulence has no small parameter. At high Reynolds number the nonlinear term dominates rather than perturbs the linear one; the moments do not decrease with order in a way that would justify neglecting the higher ones; and energy is transferred across the inertial range through a scale-local cascade, with no internal gap at which that range might be severed. The features usually available to rescue closure, a small parameter, an exploitable separation of scales within the range of interest, proximity to equilibrium, are the ones fully developed turbulence lacks.

This is what qualifies turbulence as a calibration. Throughout, the velocity field $u$ inhabits one and the same state space, for a fluid, the function space of admissible velocity fields, which serves here as the system’s space of states and does not itself change as the dynamics run. The difficulty is entirely that a low-dimensional projection of a dynamics on that fixed space fails to close, not that the space itself is in question. Turbulence therefore serves as a benchmark rather than a bound in any ordered sense: the point is not that no fixed-state-space system could present a harder closure problem, difficulty of this kind admits no evident total order, and the claim of a maximal element would require an argument not offered here, but that turbulence already exhibits the closure problem under the simultaneous loss of every feature that ordinarily rescues it, so that a phenomenon recalcitrant in just this way (strong coupling, no separation of scales, non-equilibrium) has an established physical counterpart and calls for no resource beyond what the study of turbulence already contains. What turbulence cannot calibrate is difficulty of a different kind, arising not on a fixed space but from the change of the space itself; that distinction is the subject of §7.

Claim 2. Turbulence exhibits the closure problem under the simultaneous loss of small parameter, scale separation, and equilibrium, all while its state space stays fixed, and so serves as a calibration benchmark: a relational difficulty consisting only in the failure of a low-dimensional projection to close on a fixed space has an established physical counterpart and requires no new apparatus. A relational difficulty that turbulence cannot calibrate must therefore differ in kind, by a feature turbulence lacks, which, as §7 argues, is the evolution of the state space itself.

The role to which turbulence is confined follows from this. It is one case, not the case. Other systems would expose other facets of the same problem: the development of an organism, in which the variables that matter are brought into being over the course of the process; the change of a language, in which the medium that constrains individual expression is itself remade by that expression; ecological succession, in which the community that selects for traits is assembled from the traits it selects. Each of these might have furnished the case, and each would show something the others do not. Turbulence is taken up first because its closure problem is the most fully worked, stated with a precision the others have not yet received. It is a first illustration, chosen for that maturity, and not privileged beyond it. Nothing in the framework requires that turbulence, rather than some other multi-scale system, be the model through which the problem is posed; the requirement is only that some such model serve, and this one is taken first.

§4 The Exact Form of the Failure

Two forms of the closure problem must be separated before the argument proceeds, since they are related in substance but distinct as objects, and the transfer of §5 depends on not confusing them. The first, of §2, is the truncation of a moment hierarchy: the closure is an algebraic constitutive relation $\Phi$ expressing a high moment through lower ones. The second, developed now, is the question whether the dynamics projected onto a coarse variable can be written in that variable alone, a closure not of a truncated hierarchy but of the reduced equation of motion. The two are two faces of one difficulty, but the objects differ: a truncation $\Phi$ is an algebraic relation among moments, whereas the obstruction to a closed reduced equation is an integral memory term, and no choice of $\Phi$ is the same object as that memory. Where the sequel speaks of an imposed closure it means whichever of the two the context fixes; the present section treats the second.

The closure problem as stated in §2 leaves the impression that the difficulty is one of bookkeeping, too many unknowns, an unclosed chain, and that a sufficiently clever truncation might dispatch it. The Mori-Zwanzig formalism [6, 8] removes this impression by giving the projected dynamics exactly, with no truncation and no approximation, and locating what a self-contained macroscopic equation would require in a term that is in general non-vanishing.

Let the microscopic dynamics be written $\dot u = \mathcal{L}u$, where $\mathcal{L}$ is the Liouville operator that generates the flow: acting on functions of the microstate it is linear, even though the underlying velocity dynamics are nonlinear, and this linear representation on the space of observables is what makes the construction below available. Let $P$ be a projection that maps an observable onto the part expressible through the coarse variable one retains, and $Q = I - P$ the complementary projection onto what is discarded. Applying the Dyson identity to the propagator $e^{t\mathcal{L}}$ split by $P$ and $Q$ yields, for the coarse variable $X = Pu$, an exact equation of motion of the form

$$\dot X(t) = \underbrace{,v(X(t)),}{\text{drift}} ;+; \underbrace{\int_0^t K(t-s),X(s),ds}{\text{memory}} ;+; \underbrace{\eta(t)}_{\text{noise}}.$$

The drift $v(X) = P\mathcal{L}X$ is the instantaneous, memoryless part; the noise $\eta(t) = e^{tQ\mathcal{L}}Q\mathcal{L}X$ is the projected-out dynamics evolving under the orthogonal propagator $e^{tQ\mathcal{L}}$; and the memory kernel $K$ is fixed by the autocorrelation of that noise, $K(t) \sim \langle \eta(t),\eta(0)\rangle$, the relation between friction and fluctuation that is the second fluctuation-dissipation theorem. Two qualifications attach to the display. First, the drift is in general a nonlinear function of $X$, a mean force, and reduces to a linear term $\Omega X$ only under the linear (Mori) projection onto $X$ itself; the general (Zwanzig) projection retains the nonlinearity. Second, the memory term is written here in its linear-response form for legibility; in the nonlinear case the convolution is against a function of the past coarse trajectory rather than against $X(s)$ directly. Subject to these, the decomposition is exact: it is the coarse dynamics, not a model of them. What the decomposition establishes is the standing of the closed equation. The clean form $\dot X = v(X)$, in which the future of the coarse variable is a function of its present value alone, is the special case in which the memory kernel and the noise are both negligible; where they are not, the coarse variable’s future depends on its own history and on the degrees of freedom the coarse description discarded.

The condition under which the exception holds is a separation of time scales. If the discarded degrees of freedom relax much faster than the retained ones, the memory kernel collapses toward a delta function and the noise toward white noise; the convolution becomes instantaneous, the fluctuating force becomes memoryless, and one recovers a closed, if now dissipative and stochastic, Markovian equation in the coarse variable. This is the circumstance under which a macro-level has dynamics of its own: when the scales below it relax fast enough to be summarized by an effective friction and an effective noise. Where the fast and slow scales are not separated, where the coarse variable and the fine ones evolve on comparable time scales and remain strongly coupled, the memory does not decay and the noise does not whiten, and the reduction to a closed equation in the coarse variable is no longer available. Turbulence is the case in which this separation is absent within the range that matters: across the inertial range the energy cascade is scale-local and self-similar, with no internal spectral gap at which the coarse and fine motions could be cleanly divided.

Claim 3. The self-contained macroscopic equation $dC/dt = v(C)$ is the special case of an exact projected dynamics in which memory and noise vanish, and it is licensed only by a separation of scales that lets the sub-macroscopic degrees of freedom relax quickly relative to the macroscopic ones. Absent that separation, the macro-level’s evolution depends on its own history and on the degrees of freedom the coarse description omits, and no closed Markovian dynamics in the macro-variable alone is licensed by the standard reduction, which is not to prove that none exists, but to withdraw the ground on which its existence is ordinarily assumed.

Two things follow that matter for the transfer to come. The first is that non-closure is not a defect of a bad choice of coarse variable that a better choice would cure; for a genuinely multi-scale system without a spectral gap, it is a feature of the passage to any low-dimensional description. The second is that history enters not as an ornament but as a term. When the memory kernel does not decay, the past of the macro-variable is part of the specification of its future, and two systems in the same present macroscopic state may have different futures because their pasts differ. What is called path dependence in other vocabularies is, in this one, a memory kernel that fails to collapse.

§5 The Caution, Carried to Relational Emergence

The transfer of the foregoing to the study of relational emergence is made now, and it is made as a caution rather than as a result. What the closure problem licenses is not a new theory of culture or norms but a set of questions that a claim about macroscopic social order must be able to answer, and a corresponding set of places at which such claims are prone to hide an unlicensed assumption. Four are set out; each takes the form of a question to put to a claim of the shape “the macro-level emerges from the micro-level and evolves in its own terms.” The first three concern the failure of closure and follow from §4; the fourth concerns the success of closure and follows from the renormalization group of §2, and cuts in the opposite direction.

The first concerns the smuggled closure. A common form of argument sets out rules for individuals, allows them to interact, and reports that a macroscopic regularity, a norm, a convention, a distribution, emerges; the macro-level is then treated as the statistical consequence of the micro-level, and its evolution studied in macroscopic terms. The closure problem locates the unstated step. Wherever the interaction among individuals is nonlinear and the relevant scales fail to separate, the conditions under which, as §4 showed, the projected dynamics carry a non-vanishing memory, the evolution of the norm level is not guaranteed to close in the norm alone: it depends on the higher-order relational information that the coarse description discarded. An account that nonetheless writes a closed dynamics for the norm has supplied a closure from outside the individual rules, whether as an algebraic decoupling of the higher-order relations or as the tacit assumption that the memory may be dropped; either way the substantive commitments are lodged in that supplied closure rather than derived. The question to put is accordingly: where, in this account, is the closure, and what licenses it? An account that cannot exhibit its closure has not dispensed with one; it has left it implicit.

The second concerns sufficiency of the macroscopic state. Where the memory kernel does not decay, the present macroscopic configuration is not a sufficient variable for its own future. Two societies with the same present distribution of a norm may diverge because their histories differ, and the divergence is not noise to be averaged away but a term in the exact dynamics. The question to put is: is the present state of the macro-variable being treated as sufficient for its evolution, and if so, what warrants ignoring the memory? A Markovian model of culture, current state plus transition rule, assumes a collapse of memory that, in a system whose scales are not separated, is exactly what cannot be assumed. History, in such a system, is not a prior condition that the present has absorbed; it is ontological, a standing part of the specification.

The third concerns the absence of scale separation itself. The license for a closed macro-dynamics was a gap between fast and slow scales. In social systems the relevant scales, individual, relational, group, institutional, cultural, do not evidently separate; they act on one another across comparable spans and remain mutually coupled, which is the condition under which no finite-dimensional coarse description closes. The question to put is: does this system exhibit a separation of scales, and if not, on what ground is a closed macro-description expected to exist at all? The force of the question is that in the no-separation case the right expectation is the opposite of the usual one. One should expect the macroscopic description to be unstable, to require repeated repair, to fail to close, and a theory that reports a clean closed macro-dynamics for such a system should be read as having imposed one, not found one.

The fourth cuts the other way, and concerns not the failure of closure but its success. Suppose that a macroscopic social regularity does close, that a level of norm or convention settles into a description governed by a few parameters, stable under the coarsening of individual detail. The renormalization group shows what such a success would cost. A macroscopic law reached by coarse-graining to a fixed point is universal: it is the same law for microscopically different systems, because the flow to the fixed point has flowed away all but a few directions, and the microscopic detail that distinguishes one constituent system from another sits in the directions it discarded. Two consequences follow, and each is a caution. The first is against reduction: where a macroscopic regularity is genuinely of this kind, deriving it from a particular microscopic mechanism is not merely hard but foreclosed, since the regularity retains none of the information that would identify which mechanism produced it, a programme that would read individual rules off the macroscopic norm is asking the norm for what the flow to the fixed point has destroyed. The second is against the inference from likeness: two societies exhibiting the same macroscopic regularity are not thereby shown to share a microscopic mechanism, for universality is precisely the statement that unlike microscopics yield like macroscopics. The question to put is accordingly: if this macroscopic regularity closes, is its closure being read as a bridge back to the mechanism that produced it, or as evidence of shared mechanism across cases, either of which universality forbids? A successful macroscopic law is a wall, not a window, and an account that treats its success as license to reason back through it has mistaken the one for the other.

Claim 4. For the study of how norms, culture, and institutions arise from individual relation, the two fixed-space instruments yield a methodological caution in four parts. From the failure of closure: that a closed macro-dynamics conceals an imposed closure, whether an algebraic decoupling of higher-order relations or a dropped memory, whose license must be exhibited; that where memory does not decay the present macro-state is an insufficient variable, and history enters as a term of the dynamics, not as background; and that where scales do not separate the default expectation should be non-closure, so that a reported clean macro-dynamics is evidence of an imposed assumption rather than of a discovered law. From the success of closure: that where a macroscopic regularity does close in the manner of a renormalization-group fixed point, its universality forbids both the derivation of the regularity from a particular microscopic mechanism and the inference of shared mechanism from shared macroscopic form.

The caution supplies no way to close the social hierarchy; it withdraws the presumption that the hierarchy closes, and converts that presumption into a question with a determinate form. Its content is the reordering stated at the outset: before “how does the macro-level evolve” stands “does the macro-level evolve in its own terms, and if so by what license.”

§6 From Dynamical Closure to Generative Closure

The caution of the previous section is stated in the vocabulary of the closure it borrows, a closure of dynamics, the existence or non-existence of $dC/dt = F(C)$. Generative relational being will require a second notion. This section introduces it, marks how it differs from the first, and states how far the present paper carries it.

The closure that Mori-Zwanzig characterizes is a closure of dynamics: the question whether the coarse variable has a self-contained equation of evolution, answered by whether memory and noise vanish. There is a second question that this one does not reach. A system may lack a closed dynamics in its coarse variable and yet sustain something across time, not a fixed macroscopic law but a persisting circuit of generation that does not break. That a structure can be maintained precisely by continual throughput rather than by rest is familiar from the study of systems held far from equilibrium [7]. A community subject to continual turnover of members, resources, and information may hold no closed dynamics at the level of any aggregate variable and nonetheless remain a community, its constitutive relation regenerated as fast as it dissipates. What persists there is not a state and not a law over states; it is a generation that continues. The closure at issue is a closure of that generation, the condition under which a generative circuit sustains itself, and it is not the same as the closure of a dynamics.

Claim 5 (proposed). I propose to distinguish generative closure from dynamical closure. A system may fail to admit a closed macroscopic dynamics, memory undecaying, noise uncolored, no reduction to an equation in the coarse variable alone, and yet, the proposal runs, exhibit a generative closure, a persistence of the circuit that generates its constitutive relation rather than of any state or law over states. The proposal is not yet a definition: the predicate that would make it one, and would keep it from collapsing into the statement that a persisting system persists, is the invariant identified as open below. It is entered here as the distinction the framework requires, pending that invariant.

To assert that generative closure is a distinct and real condition is not yet to have said what it is. Stated only through instances, the community that persists, the institution that renews, the practice that regenerates, the notion is exposed to a circularity that would empty it: if a generative closure is nothing but the fact that a system continues to generate without breaking, then to say a system has generative closure is only to say it has not yet broken, and the notion predicts nothing it was not given. Turbulence sharpens the danger: it too persists without a closed dynamics, so if generative closure cannot be told apart from what turbulence already displays, it has no content beyond the closure problem it was meant to exceed.

What would give it content is an invariant. In turbulence the persistence of the cascade is anchored by a quantity that stays constant through it: the mean rate of energy dissipation, equal across the inertial range to the flux of energy through it [4, 2]. That flux is constant because the inertial range, by definition, neither receives energy directly from the forcing nor loses it directly to viscosity, so whatever energy enters one end must pass to the next scale rather than accumulate; the constancy follows from the conservation of energy through a range with no source or sink of its own. Two features of this quantity are what a candidate for relational use must reproduce. It is defined by constancy along a definite dimension, across scale, not across time, so that the invariant is a statement about how a quantity behaves as one passes through the levels of the cascade. And it is a flux, a globally averaged rate of passage, which is why it survives the intermittency that breaks the naive self-similarity of the field [2]: the higher-order statistics fluctuate strongly from place to place while the mean flux does not. An invariant of generation would need both features, a quantity stationary along a specified dimension of the generative circuit, and a rate of passage robust to the local fluctuation of what it aggregates, in virtue of which the circuit is a sustained thing one can point to. The difficulty is that a social or relational system affords no conservation law to underwrite such a quantity. Whatever might play the role energy dissipation plays cannot be delivered by a conservation principle, and would have to be grounded, if at all, in something proper to relation, in what the parent framework terms a relational grammar, the standing constraints on how relations may combine and pass into one another, rather than in a conserved substance.

This paper does not supply that invariant. It identifies the need for it, locates the circularity its absence leaves open, and fixes three conditions on any candidate: it must be stationary along a specified dimension of the generative circuit, on the model of the flux constant across scale; it must not be underwritten by a conservation law, since relation affords none; and it must not reduce to the statement that a persisting system persists. A physical precedent for an invariant meeting the first two conditions, stationary across scale, yet fixed by global rather than local-conservation considerations, appears in the zero modes that govern anomalous scaling in turbulence, discussed in §7. Within these constraints the notion of generative closure has room to become a definite claim; within the present reflection it remains a named problem rather than a solved one.

§7 The Limit of the Turbulence Case

A case leaned on owes an account of what it cannot reach, the more so here because what turbulence cannot reach is the thing generative relational being was framed to address. This section states that limit, and states alongside it a caution that the limit of the renormalization group on turbulence yields in its own right.

Through every difficulty rehearsed above, turbulence retains one feature: its state space is fixed. The velocity field lives in a space of admissible fields that does not itself change as the dynamics proceed; the projection $P$ onto coarse variables is a fixed operator on that fixed space; the Mori-Zwanzig equation can be written because the space on which it is written does not move. Turbulence is, in this sense, a problem posed on a fixed state space. Every resource it lacks, small parameter, scale separation, equilibrium, is a resource internal to the analysis of a fixed space. The space of states itself is never in question.

The renormalization group meets its limit on turbulence at the same fixed-space boundary, and the manner of its failure carries a caution of its own. Turbulence is, on its face, the ideal object for the renormalization group: it is scale-related, it cascades, and its energy spectrum follows a power law that looks like a scaling law inviting a fixed-point treatment. The instrument does capture the scale-invariant, dimensional part of the behaviour, the Kolmogorov spectrum, but reaches only part way, for the part of turbulence that resists it is the part that matters. The higher-order structure functions do not follow the exponents dimensional analysis predicts; they scale anomalously, and this anomaly, the signature of intermittency, is not what a local fixed point delivers. Its source, in the cases where it has been computed, is not the local self-similarity a fixed point encodes but a set of zero modes: quantities invisible to dimensional analysis, fixed not within the inertial range but by the large scales and boundaries that feed it [1]. The caution this yields for the study of social order is direct. A system may display every scaling signature that invites a universal, fixed-point account, the power laws, the scale-free networks, the self-similar cascades of influence that social systems present in abundance, and still possess no clean closed macroscopic theory, because scale invariance in the observable statistics does not entail a fixed point in the dynamics. Turbulence is the standing counterexample: it holds a textbook scaling law together with a stubborn non-closure. To read a power law in a social system as a warrant for a universal macroscopic theory is to make the inference turbulence refutes.

The residue of the renormalization group’s failure bears on the invariant of §6. The zero modes are quantities held fixed not by a local conservation law but by the global conditions under which the cascade is driven, a physical instance of the invariant that account requires: one stationary across the levels of a generative process, not underwritten by a conservation of substance, and decisive for the high-order behaviour that local self-similarity misses. What the renormalization group leaves unresolved in turbulence is, in this respect, a closer model for generative relational being than what it resolves.

The distinctive claim of generative relational being is that the state space itself is in question. Where relation generates being, the space of possible states is not given in advance and held fixed while dynamics unfold upon it; it is enlarged and reshaped by the relations it hosts, so that which states are available at a later time is a function of what was generated earlier. In the notation of the framework this is the difference between writing $\dot x = F(x, R)$, which adds a relational variable $R$ to a fixed space and remains within the reach of a standard, if enlarged, dynamics, and writing an evolution of the space of states itself, $\Omega_{t+1} = H(\Omega_t, R_t)$, in which the set of available states at time $t{+}1$ is generated from the set at time $t$ together with the relations $R_t$ then obtaining. The first is the turbulence class: a larger but fixed state space, on which the Mori-Zwanzig construction still applies and closure is the familiar problem. The second lies outside it: when the state space itself evolves, the projection $P$ has no fixed space to act on, and the Mori-Zwanzig derivation loses the fixed space and fixed inner product it assumes. Whether an analogous closure problem can even be posed in this second case is not settled here; that it cannot be posed in the form turbulence gives it is what the failure of the construction shows, and the positive question is left open to §8.

Claim 6. Turbulence exhibits the closure problem of a fixed state space and, being confined to a fixed state space, can exhibit no more than that. The evolution of the state space itself, the generation of new dimensions of possibility by the relations a system hosts, lies outside the fixed-space setting the turbulence case occupies, and so outside what that case can calibrate. Turbulence therefore calibrates generative relational being without grounding it: it shows what a difficulty on a fixed space is when that difficulty is at its most severe, and thereby marks, by contrast, where the framework’s distinctive difficulty must be sought, in the evolution of the state space. Whether a difficulty so located is genuinely irreducible to a dynamics on some fixed enlarged space is not settled by this contrast; it is the open question recorded in §8. Two further points attach to the same limit: that the renormalization group’s failure on turbulence shows a scaling law to be no warrant for a universal macroscopic theory, and that the boundary-fed zero modes in which that failure resides are the nearest physical model for the invariant of generation §6 requires.

For this reason the turbulence case cannot be the foundation the framework needs, and the open problem of §6 cannot be discharged by refining the turbulence analogy. An invariant of generation, if it exists, must be an invariant of a process that remakes its own state space; turbulence, whose state space never changes, supplies the idea of an invariant but not an invariant of the required kind. The case gives the framework its calibration and its vocabulary, memory, cascade, sustained flux, but not the step past the fixed state space. That step begins where a harder dynamics on a fixed space, which is what turbulence exemplifies, leaves off: in the evolution of the space itself.

§8 The Established and the Open

This section separates what the foregoing establishes from what it leaves standing.

What is established is a reordering of questions. The passage from microscopic relation to macroscopic order does not confer on the macro-level a closed dynamics of its own; whether such a dynamics exists is a condition licensed, when it is licensed, by a separation of scales that lets the sub-macroscopic degrees of freedom be summarized as friction and noise. Where that separation is absent, as it is in turbulence, and as it appears to be across the relational scales of social life, the macroscopic description does not close, its history enters as a term rather than a background, and a reported clean macro-dynamics is to be read as an imposed closure rather than a discovered law. For the study of culture, norms, and institutions the consequence is the caution of §5: before asking how a macro-level evolves, one must ask whether it constitutes a level at which evolution in its own terms can be written, and by what license. The renormalization group adds two edges to the same caution: where a macroscopic regularity does close as a fixed point, its universality forbids reading it back to a mechanism or across cases; and where a scaling law is present without a fixed point, as in the anomalous scaling of turbulence, the scaling law is no evidence that a closed macroscopic theory exists. This much the two fixed-space instruments support, and this much the paper claims.

What is not established is generative closure. It has been distinguished from dynamical closure and shown to be the condition the framework will require, a persistence of generation rather than of state, but it has not been given content sufficient to keep it from collapsing into the statement that a system which persists has persisted. What would give it content, an invariant of generation not underwritten by any conservation law and not reducible to the fact of persistence, has been named as a requirement and left unmet. This limit follows from the case chosen: turbulence, confined by its fixed state space to the region the framework must exceed, cannot furnish the invariant of a process that remakes the space of its own states. The turbulence case calibrates the problem and supplies its language; it does not supply that invariant.

Two lines of work follow, and the paper leaves both open. The first is the invariant: to determine whether a quantity conserved or held stationary across a generative circuit can be defined in relational terms, grounded in a relational grammar rather than a conserved substance, and thereby to convert generative closure from a named distinction into an operational condition. The second is the evolving state space: to give $\Omega_{t+1} = H(\Omega_t, R_t)$ a form definite enough to be reasoned about, and in particular to settle whether such an evolution can be shown irreducible to a dynamics on any fixed enlarged space, for only if it can is the framework’s distinctive difficulty a real one rather than a fixed problem in disguise. Both lie ahead of the present paper, which goes only as far as the two open questions and the reordering that motivates them: it makes the assumption of macro-level autonomy visible as an assumption, gives through a physical case the condition under which the assumption holds and fails, and fixes what a fixed-state-space case can and cannot carry over to a framework whose subject is a changing state space.

A note on method. The moment hierarchy and the Mori-Zwanzig decomposition are used as the theorems they are; where the argument passes to relational emergence, no equation is carried across, only the structure of the question. No invariant of generation is defined here: the notion is left as the open problem of §6.

References

[1] Falkovich, Gregory, Krzysztof Gawędzki, and Massimo Vergassola (2001). Particles and Fields in Fluid Turbulence. Reviews of Modern Physics, 73(4), 913–975.

[2] Frisch, Uriel (1995). Turbulence: The Legacy of A. N. Kolmogorov. Cambridge: Cambridge University Press.

[3] Goldenfeld, Nigel (1992). Lectures on Phase Transitions and the Renormalization Group. Reading, MA: Addison-Wesley.

[4] Kolmogorov, Andrey N. (1941). The Local Structure of Turbulence in Incompressible Viscous Fluid for Very Large Reynolds Numbers. Doklady Akademii Nauk SSSR, 30, 301–305.

[5] Landau, Lev D., and Evgeny M. Lifshitz (1987). Fluid Mechanics. 2nd ed. Vol. 6 of Course of Theoretical Physics. Oxford: Pergamon Press.

[6] Mori, Hazime (1965). Transport, Collective Motion, and Brownian Motion. Progress of Theoretical Physics, 33(3), 423–455.

[7] Prigogine, Ilya (1980). From Being to Becoming: Time and Complexity in the Physical Sciences. San Francisco: W. H. Freeman.

[8] Zwanzig, Robert (1961). Memory Effects in Irreversible Thermodynamics. Physical Review, 124(4), 983–992.

[9] Zwanzig, Robert (2001). Nonequilibrium Statistical Mechanics. Oxford: Oxford University Press.

中文

生成性关系存在中的生成性闭合

关于湍流模型的一项初步反思

黄万宏 · huangwanhong@serendip.ngo

摘要

关于文化、规范、信任与制度的研究，惯常把宏观层写作一个拥有其自身动力学的系统，即一个方程，其中一个集体变量的未来是那个变量本身的函数。本文论证，这类写法所预设的自足性是一项假定而不是一项发现；而一个成熟的物理案例，即充分发展的湍流，展示了这项假定如何以及为何失效。这项失效在统计物理中有一个精确的名字：闭合问题。当一个具有精确微观动力学的系统被投影到人们希望研究的粗粒变量之上时，被投影的动力学一般而言并不闭合，每一阶都依赖于更高一阶，而粗粒变量本身并不是对其自身未来的充分描述。莫里-茨旺齐格形式给出被投影的演化的精确形式：在它的漂移项之外，它还带有一个记忆核与一个噪声，而只有当一种尺度分离让记忆衰减、让噪声白化时，它才化约为一个自足的宏观方程。湍流正是这样一种情形：在要紧的范围之内，那种分离缺席；而它在此充当一种标定，而不是一个隐喻：它在一个固定的状态空间上、在通常拯救闭合的每一项特征同时丧失的情况下，展示了闭合问题，并因而标示出一项仅仅属于这一类的关系性困难会相当于什么。重整化群，即用于跨尺度过渡的第二件工具，从另一侧加剧了这项审慎：凡它成功之处，普适性禁止把一条闭合的宏观定律读回它的微观机制；凡它失效之处，如在湍流的反常标度上，它表明一条标度律并不是一套普适宏观理论的凭据。本文把这些带入关系性涌现的研究之中，作为一项方法论上的审慎，即：一个宏观层的自主性是有待确立的，而不是被假定的；并把它所告诫的那种闭合，即动力学的闭合，与该框架将要求的那个概念，即生成的闭合，区别开来。后者是被命名并被界定的，而不是被确立的：会为它奠基的东西，即一个不由任何守恒律所担保的生成不变量，被作为本反思所留下的开放问题而记下。

关键词： 闭合问题；莫里-茨旺齐格；湍流；关系性涌现；生成性闭合。

关于本文分量的一则说明。本文取一个物理案例，即充分发展的湍流的闭合问题，并追问它在”宏观秩序如何自微观关系中生起”的研究中所告诫的是什么。论述的次序遵循论证的次序：物理机制先以其自身的术语被陈述，而向关系性涌现的转移，则作为那一机制所许可的一项审慎而随之而来。该框架将要求的一个概念，即作为对动力学闭合之对照的生成的闭合，在临近末尾处被引入，并连同会为它奠基的那个不变量一起，被作为一个开放问题而留下。倘若这项审慎被表明或者已是共识、或者不为该物理案例所支持，那么相应的诸节便失去其根据。反对、纠正与反证，欢迎寄至 huangwanhong@serendip.ngo。

§1 引论

一个随时间被研究的集体现象招致某种记法。设 $C$ 代表一种文化、一套规范、一个制度性信任的水平；人们于是写下

$$\frac{dC}{dt} = F(C),$$

并追问 $C$ 如何演化。这一记法是自然的，它常常是有用的，而它携带一项鲜少被检视的假定：宏观层的未来单是宏观层的函数，即文化由文化所决定，规范由规范所决定，集体变量由它自身所决定。一个宏观层为何应当拥有其自身的、闭合于其自身诸变量之上的动力学，这不是该记法所提出的问题；闭合性在探究开始之前就已被内建。

本文论证，这种闭合性是一项假定而不是一项发现，并且它可能失效。这一论证不是在抽象中作出的。它是从一个物理案例借来的，在那个案例中同样的假定被作出、被精确地陈述，并被知道以一种在将近一个世纪里抗拒解决的方式失效：充分发展的湍流，以及它所引起的闭合问题。在那个案例中，微观动力学是被精确知道的，即在十九世纪写下的纳维-斯托克斯方程；而困难完全在于从那套动力学过渡到人们所希望研究的尺度上的一个自足描述。这一过渡并不闭合。粗粒变量接下来做什么，取决于粗粒变量并不含有的信息。

把这一困难转移到关系性涌现的研究，是本文的工作；而有两种误读会各自使它落空。第一种会把湍流当作一个隐喻，一幅富于暗示的复杂性图景，人们从中随手取用结论；此处并不如此使用它。湍流充当一种标定：它在一个固定的状态空间上、在通常拯救闭合的每一项特征丧失的情况下，展示了闭合问题，从而提供了一个可据以安放一项关系性困难的基准。第二种误读会让借来的闭合概念替代该框架所需要的那个概念。湍流所教导的闭合是动力学的闭合；而生成性关系存在将要求生成的闭合，二者不同。本文抵达这一区分便停下：会使它完成的那个概念被指认为一个开放问题，而不是被交付。

本文所提出的说明可以陈述为一个单一的主张。

核心主张。 一个宏观层自微观关系中被生成，这本身并不赋予那个宏观层一套其自身的闭合动力学。一个集体变量是否拥有以其自身术语表达的动力学，即 $dC/dt = F(C)$ 究竟能否被写出，是一个有待确立的条件，而不是一个可被假定的默认；而湍流的闭合问题既提供了该条件的精确形式，也提供了一个该条件明显不成立的案例。对生成性关系存在而言，其后果是问题的一次重新排序：在追问一种文化或一条规范如何演化之前，先立着这个问题，即一种文化或一条规范是否构成一个可以以其自身术语写出演化的层级。

该主张分阶段推进。下一节以其技术形式陈述闭合问题，并引入重整化群作为跨尺度过渡的第二件工具，以便以下所述立足于真实的机制而不是一个松散的类比（§2）。第三节为湍流的选取作辩护并为它设界：湍流是本可选取的众多案例中的一个，因其成熟而被挑出，并被限定于标定的角色（§3）。第四节通过莫里-茨旺齐格形式给出这项失效的精确内容，其记忆项与噪声项正是一个闭合的宏观方程必须被表明其无有的东西（§4）。第五节把两件工具带入关系性涌现，作为四项审慎，其中三项出自闭合的失效，一项出自它的成功（§5）。第六节引入动力学的闭合与生成的闭合之间的区分，并标出此处所确立之物的边界（§6）。第七节陈述湍流案例的界限，即它无法迈出的那一步，而那正是生成性关系存在最需要的一步，以及重整化群在湍流上的自身界限所产出的审慎（§7）。一个结尾节写下什么已被表明、什么未被表明，以及本反思所留下的工作。

§2 闭合问题

闭合问题是统计物理中的一个确定对象，而下文所引出的审慎有赖于它是这个而不是一种修辞。本节以其最不技术的形式陈述它。

取一个系统，其完整状态 $u$ 依已知的非线性动力学而演化。不可压缩纳维-斯托克斯方程 [5] 是标准的实例：

$$\partial_t u_i + u_j,\partial_j u_i = -\partial_i p + \nu,\partial^2 u_i, \qquad \partial_i u_i = 0,$$

其中 $u_j\partial_j u_i$ 一项是二次的。假设人们所希望研究的不是完整的场 $u$，而是一个平均量，比方说平均流 $U = \langle u\rangle$，此平均取自一个实现系综。写 $u = U + u’$ 且 $\langle u’\rangle = 0$，并对该方程取平均，人们便得到一个关于 $U$ 的演化方程，它通过那个二次项含有一个新的未知量：二阶矩 $\langle u_i’ u_j’\rangle$，即雷诺应力。一阶矩的动力学依赖于一个二阶矩。

人们于是寻求一个关于二阶矩的方程。它是通过把二次动力学作用于乘积 $u_i’ u_j’$ 并取平均而形成的；二次项 $u,\partial u$ 作用于一个二阶乘积产出一个三阶关联，因此关于二阶矩的方程含有一个三阶矩 $\langle u_i’ u_j’ u_k’\rangle$。关于三阶矩的方程，以同样方式形成，含有一个四阶矩。以 $M_n$ 记第 $n$ 阶矩，它本身是一个多点张量场而不是一个标量，其结构是一个谱系

$$\partial_t M_n = \mathcal{F}!\left(M_n,, M_{n+1}\right),$$

其中关于每一阶的方程都引入它上面的那一阶。该谱系并不终止。把它在一个有限阶 $N$ 处截断，保留 $M_1,\dots,M_N$ 及其 $N$ 个演化方程，便留下 $M_{N+1}$ 作为一个未知量，而截断并不为它提供方程；系统在此意义上是欠定的，不是通过标量计数的失配，而是因为该链条从不闭合于其自身。

闭合问题。 在由一个非线性动力学所生成的矩谱系中，关于第 $n$ 阶矩的方程依赖于第 $(n{+}1)$ 阶；没有任何有限的截断是自足的。一个闭合是一个由外部供给的关系，把所保留的最高阶矩表达为较低阶矩的一个泛函，$M_{N+1} = \Phi(M_1,\dots,M_N)$，被强加以使被截断的系统成为确定的。关系 $\Phi$ 并不由底层的动力学推出；它是一项被添加的假定，而被截断系统的行为承袭了该假定放进去的任何东西。

对以下所述而言，要害之点是 $\Phi$ 的地位。它不是被推导的；它是被设定的。动力学固定了谱系，却不固定在何处切断它，也不固定如何把被丢弃的诸阶折回到所保留的诸阶之上。每一套与一组有限的粗粒变量打交道的可运作理论，即工程湍流的涡黏性模型、动理学理论的诸种截断、统计力学的诸种平均场解耦，归根结底都是对 $\Phi$ 的一个选取，由建模者所能凑集的物理论证与经验拟合的任意组合所许可。凡该选取被良好许可之处，闭合是可靠的，而粗粒描述赢得它的自主性。凡它未被许可之处，自主性是借来的，而粗粒方程所报告的关于建模者假定的东西，与关于系统的东西一样多。闭合问题不是闭合无法被找到；而是它们必须自外部被供给，并且它们的供给正是那些实质性承诺藏身之处。

这一结构并不为流体所独有。同样的谱系出现在场论中，其中关于一个 $n$ 点关联函数的方程涉及 $(n{+}1)$ 点函数；也出现在气体的动理学理论中，其中关于单粒子分布的方程涉及双粒子分布，如此沿链上行 [9]。微扰论、平均场近似以及分子混沌假定，在这些情境中是相应的诸闭合。这个问题对于”从一个非线性微观动力学过渡到一个以少数几个平均量表达的描述”是普适的。正是这种普适性使它值得被带往别处；而正是湍流以它最无从逃避的形式展示了它。

还有第二件、并且更强的用于跨尺度过渡的工具，而它必须被引入，因为它携带一项其自身的审慎。凡一个系统的诸尺度不是由一个小参数、而是由自相似性所联系之处，系统性的方法是重整化群 [3]：人们进行粗粒化，即积掉最短波长的自由度，并重新标度，进而研究这在可能描述的空间上所诱导出的流。在一个临界点上，该流奔向一个不动点；而在那个不动点附近，描述空间中除少数几个方向之外的一切方向都是无关的：它们在反复的粗粒化之下收缩消失，留下一个由所余的少数几个有关方向所支配的低维有效理论。这是不由一个被强加的 $\Phi$、而由粗粒化本身的动力学所达成的闭合，后者替我们丢弃了微观细节。它属于”一个宏观层级为何应当拥有其自身定律”的现有最强说明之列，而凡它成立之处，它是精确的。

这一成功有代价，而代价是这件工具所产出的两项审慎中的第一项。重整化群在一个不动点上所确立的是：所存留的宏观定律，是由微观构成彼此不同的诸系统一律抵达的，这正是普适性的内容，即液-气临界点、磁体与二元合金共享临界指数。普适性是这样一个陈述：宏观行为对微观的几乎一切都是盲的：无关方向，也就是说大多数微观细节，已被流走，无法自宏观定律中复得。因此，这一类成功的约化不是一座从宏观通回微观的桥，而是一堵墙：它证实了作那趟返程所需要的信息已被摧毁。这直接关乎宏观社会秩序的研究，而 §5 陈述如何关乎。

§3 湍流的选取，及其作为一个案例的界限

湍流在此因一个特定的理由被选取，并被限定于一个特定的角色。一个在关于社会秩序的论证中被援引的物理系统，并不因此就被主张为与一个社会系统相同；选取的理由与角色的界限被列出，以便该援引被读作它之所是。

选取的理由是：湍流呈现出被剥去了一切会使它可处理的特征的闭合问题。在许多系统中，谱系之所以能被截断，是因为一个小参数使得较高的诸阶可忽略：一个弱耦合、一个小振幅、一个大的或小的尺度之比。凡存在这样一个参数之处，截断由它所许可，被丢弃的诸阶明显小于所保留的诸阶，而闭合是一个带有受控误差的近似。凡诸尺度改由自相似性所联系之处，§2 的重整化群是相应的工具；但湍流，如 §7 所述，正是那件工具只抵达一部分的情形。充分发展的湍流没有小参数。在高雷诺数下，非线性项是支配、而不是微扰线性项；诸矩并不随阶数以一种会为忽略较高诸阶作辩护的方式而递减；而能量是通过一个尺度局域的级联跨越惯性区被传递的，其间没有任何可将该区截断的内部间隙。通常可用以拯救闭合的诸特征，即一个小参数、在所关切范围之内一种可资利用的尺度分离、对平衡态的邻近，正是充分发展的湍流所缺乏的。

这正是使湍流有资格充当一种标定的东西。自始至终，速度场 $u$ 栖居于同一个状态空间之中，对一个流体而言，即容许速度场的函数空间，它在此充当系统的状态空间，并且它本身并不随动力学的运行而改变。困难完全在于：一个动力学在那个固定空间上的低维投影未能闭合，而不在于那个空间本身有问题。因此，湍流在任何有序的意义上都是充当一个基准而不是一个界限：要点不是没有任何固定状态空间的系统能够呈现一个更难的闭合问题，这一类困难并不容许一个明显的全序，而”存在一个极大元”的主张会要求一个此处并未提供的论证，而是湍流已然在通常拯救闭合的每一项特征同时丧失的情况下展示了闭合问题，以致一个恰以这种方式桀骜的现象（强耦合、无尺度分离、非平衡）拥有一个确立了的物理对应物，并且不召唤任何超出湍流研究已然含有之物的资源。湍流所无法标定的，是另一种类的困难，即不生起于一个固定空间、而生起于那个空间本身之改变的困难；那一区分是 §7 的主题。

主张 2。 湍流在小参数、尺度分离与平衡态同时丧失的情况下展示了闭合问题，而其状态空间始终固定，因而充当一个标定基准：一项仅仅在于”一个低维投影未能在一个固定空间上闭合”的关系性困难，拥有一个确立了的物理对应物，并且不需要任何新的装置。因此，一项湍流所无法标定的关系性困难，必定在种类上有别，凭借某项湍流所缺乏的特征，而那，如 §7 所论，正是状态空间本身的演化。

湍流被限定于其中的那个角色，随此而来。它是一个案例，而不是那个案例。别的系统会暴露同一问题的别的侧面：一个有机体的发育，其中要紧的诸变量是在过程的进行中被带入存在的；一门语言的变迁，其中约束个体表达的那个媒介本身被那种表达所重造；生态演替，其中为诸性状进行选择的那个群落是由它所选择的诸性状所组装的。这些之中的每一个都本可提供该案例，而每一个都会显示别的所不显示的某种东西。湍流之所以被先取用，是因为它的闭合问题是被最充分地做过的，以一种其余各者尚未得到的精确性被陈述。它是一个初次的例示，因那份成熟而被选取，而不被赋予超出此的特权。该框架中没有任何东西要求由湍流、而不是由某个别的多尺度系统，来充当据以提出该问题的模型；要求仅仅是某个这样的模型充当此任，而这一个被先取用。

§4 失效的精确形式

在论证推进之前，必须把闭合问题的两种形式分开，因为它们在实质上相关而作为对象彼此有别，而 §5 的转移有赖于不把它们混淆。第一种，即 §2 的那种，是一个矩谱系的截断：闭合是一个代数的本构关系 $\Phi$，把一个高阶矩通过较低阶矩表达出来。第二种，即现在所展开的那种，是这个问题：被投影到一个粗粒变量上的动力学能否单以那个变量被写出，一种不属于被截断谱系、而属于被约化的运动方程的闭合。二者是一个困难的两个面，但对象有别：一个截断 $\Phi$ 是诸矩之间的一个代数关系，而对一个闭合的约化方程的障碍是一个积分记忆项，并且没有任何对 $\Phi$ 的选取是与那个记忆相同的对象。凡下文说到一个被强加的闭合，它指语境所固定的那两者之一；本节处理第二种。

如 §2 所陈述的闭合问题留下一种印象，仿佛困难是记账上的，即未知量太多、一条未闭合的链，仿佛一个足够巧妙的截断或可打发它。莫里-茨旺齐格形式 [6, 8] 消除了这一印象，办法是把被投影的动力学精确地给出，没有截断也没有近似，并把”一个自足的宏观方程会要求的东西”定位于一个一般而言非零的项之中。

设微观动力学被写作 $\dot u = \mathcal{L}u$，其中 $\mathcal{L}$ 是生成该流的刘维尔算子：它作用于微观态的函数之上时是线性的，尽管底层的速度动力学是非线性的，而它在可观测量空间上的这一线性表示，正是使下面的构造得以可行的东西。设 $P$ 是一个投影，把一个可观测量映到可通过人们所保留的粗粒变量表达的那一部分之上，而 $Q = I - P$ 是投到被丢弃部分之上的互补投影。把戴森恒等式应用于由 $P$ 与 $Q$ 分裂的传播子 $e^{t\mathcal{L}}$，便对粗粒变量 $X = Pu$ 产出一个如下形式的精确运动方程

$$\dot X(t) = \underbrace{,v(X(t)),}{\text{漂移}} ;+; \underbrace{\int_0^t K(t-s),X(s),ds}{\text{记忆}} ;+; \underbrace{\eta(t)}_{\text{噪声}}.$$

漂移 $v(X) = P\mathcal{L}X$ 是瞬时的、无记忆的部分；噪声 $\eta(t) = e^{tQ\mathcal{L}}Q\mathcal{L}X$ 是被投出的动力学在正交传播子 $e^{tQ\mathcal{L}}$ 之下演化；而记忆核 $K$ 由那个噪声的自关联所固定，$K(t) \sim \langle \eta(t),\eta(0)\rangle$，即摩擦与涨落之间的关系，也就是第二涨落-耗散定理。有两项限定附于这一展示式。第一，漂移一般而言是 $X$ 的一个非线性函数，即一个平均力，只有在向 $X$ 自身的线性（莫里）投影之下才化约为一个线性项 $\Omega X$；一般的（茨旺齐格）投影保留那种非线性。第二，记忆项在此为清晰起见被写作其线性响应形式；在非线性情形中，卷积是针对过去粗粒轨迹的一个函数、而不是直接针对 $X(s)$。在这些限定之下，该分解是精确的：它就是粗粒动力学，而不是它们的一个模型。该分解所确立的，是那个闭合方程的地位。干净的形式 $\dot X = v(X)$，其中粗粒变量的未来单是它当前值的函数，是记忆核与噪声二者都可忽略的那个特例；凡它们不可忽略之处，粗粒变量的未来依赖于它自身的历史以及粗粒描述所丢弃的那些自由度。

那个例外得以成立的条件是一种时间尺度的分离。如果被丢弃的诸自由度远比所保留的诸自由度更快弛豫，那么记忆核便朝一个δ函数坍缩，而噪声朝白噪声坍缩；卷积变成瞬时的，涨落力变成无记忆的，而人们复得一个闭合的、尽管如今是耗散且随机的马尔可夫方程，以粗粒变量表达。这正是一个宏观层拥有其自身动力学的情形：当它下面的诸尺度弛豫得足够快，以至可被一个有效摩擦与一个有效噪声所概括之时。凡快尺度与慢尺度不分离之处，即凡粗粒变量与精细变量在可比的时间尺度上演化并保持强耦合之处，记忆并不衰减而噪声并不白化，于是那个化约为一个以粗粒变量表达的闭合方程便不再可用。湍流正是这样一种情形：在要紧的范围之内，那种分离缺席：跨越惯性区，能量级联是尺度局域且自相似的，其间没有任何可将粗粒运动与精细运动干净地分开的内部谱隙。

主张 3。 自足的宏观方程 $dC/dt = v(C)$ 是”一个精确的被投影动力学之中记忆与噪声消失”的那个特例，而它只由一种尺度分离所许可，那种分离让亚宏观的诸自由度相对于宏观的诸自由度快速弛豫。缺了那种分离，宏观层的演化依赖于它自身的历史以及粗粒描述所略去的那些自由度，而单以宏观变量表达的闭合马尔可夫动力学并不为标准的约化所许可，这并不是要证明它不存在，而是要撤回它的存在通常被假定所立足的那个根据。

有两件对将要到来的转移要紧的事随之而来。第一是：非闭合不是一个可由更好的选取所治愈的、坏的粗粒变量选取的缺陷；对于一个真正多尺度而无谱隙的系统，它是通向任何低维描述这一过渡的一项特征。第二是：历史进入其中不是作为一件装饰而是作为一个项。当记忆核不衰减时，宏观变量的过去是对它未来之设定的一部分，而两个处于相同当前宏观状态的系统，可能因其过去有别而有不同的未来。在别的词汇中被称作路径依赖的东西，在这一词汇中，是一个未能坍缩的记忆核。

§5 那项审慎，被带往关系性涌现

把前述带往关系性涌现的研究，现在进行；而它是作为一项审慎、而不是作为一项结论被进行的。闭合问题所许可的不是一套关于文化或规范的新理论，而是一组问题，一个关于宏观社会秩序的主张必须能够回答它们，以及相应的一组地方，此类主张在那些地方易于藏起一项未被许可的假定。此处列出四项；每一项都采取一个向着形如”宏观层自微观层涌现，并以其自身术语演化”的主张所提出的问题的形式。头三项关乎闭合的失效，出自 §4；第四项关乎闭合的成功，出自 §2 的重整化群，并朝相反方向切入。

第一项关乎被夹带的闭合。一种常见的论证形式为个体列出规则，允许他们互动，并报告说一个宏观规律性，即一条规范、一个惯例、一个分布，涌现了；宏观层于是被当作微观层的统计后果，而它的演化以宏观术语被研究。闭合问题定位出那个未加陈述的步骤。凡个体之间的互动是非线性的、而相关诸尺度未能分离之处，即如 §4 所示”被投影的动力学带有一个非零记忆”的那些条件之下，规范层的演化并不被保证单以规范闭合：它依赖于粗粒描述所丢弃的那些高阶关系信息。一个尽管如此仍为规范写下一套闭合动力学的说明，已从个体规则之外供给了一个闭合，无论是作为对高阶关系的一次代数解耦，还是作为”记忆可被丢弃”这一默然假定；无论哪种方式，实质性承诺都寄寓于那个被供给的闭合之中而不是被推导。因此，所要提出的问题是：在这个说明中，闭合在哪里，什么许可它？ 一个无法展示其闭合的说明并没有免除一个闭合；它只是把它留作隐含的。

第二项关乎宏观状态的充分性。凡记忆核不衰减之处，当前的宏观格局对于它自身的未来不是一个充分的变量。两个具有相同当前规范分布的社会，可能因其历史有别而分岔，而这一分岔不是可被平均掉的噪声，而是精确动力学中的一个项。所要提出的问题是：宏观变量的当前状态是否被当作对其演化充分的，若是，什么为忽略记忆作辩护？ 一个马尔可夫的文化模型，即当前状态加上转移规则，假定了记忆的一次坍缩，而在一个诸尺度不分离的系统中，那恰恰是不可被假定的东西。历史，在这样一个系统中，不是一个当前已加以吸收的先行条件；它是本体论性的，是那设定的一个常驻部分。

第三项关乎尺度分离本身的缺席。一套闭合宏观动力学的许可曾是快尺度与慢尺度之间的一个间隙。在社会系统中，相关诸尺度，即个体的、关系的、群体的、制度的、文化的，并不明显分离；它们跨可比的跨度彼此作用并保持相互耦合，而那正是没有任何有限维粗粒描述会闭合的条件。所要提出的问题是：这个系统是否呈现一种尺度分离，若否，究竟在什么根据上期望一套闭合的宏观描述存在？ 这个问题的力量在于：在无分离的情形中，正确的期望与通常的相反。人们应当期望宏观描述是不稳定的、要求反复的修补、未能闭合，而一套为这样一个系统报告出一套干净闭合宏观动力学的理论，应当被读作已强加了一个闭合，而不是找到了一个。

第四项朝另一方向切入，关乎的不是闭合的失效而是它的成功。假设一个宏观社会规律性确实闭合了，即一个规范或惯例的层级安顿进一个由少数几个参数所支配的描述，在个体细节的粗化之下稳定。重整化群表明这样一种成功会付出什么代价。一条通过粗粒化到一个不动点而抵达的宏观定律是普适的：它对微观不同的诸系统是同一条定律，因为奔向该不动点的流已把除少数几个方向之外的一切流走，而区分一个组分系统与另一个组分系统的那些微观细节，坐落在它所丢弃的那些方向之中。有两项后果随之而来，而每一项都是一项审慎。第一项针对约化：凡一个宏观规律性真正属于这一类之处，从一个特定的微观机制推导出它不仅是难的而且是被封闭的，因为该规律性未保留任何会指认出是哪个机制产生了它的信息，一个会从宏观规范中读出个体规则的纲领，是在向那个规范索要奔向不动点的流已摧毁的东西。第二项针对由相似而来的推论：两个呈现相同宏观规律性的社会，并不因此就被表明共享一个微观机制，因为普适性恰恰是”不同的微观产出相同的宏观”这一陈述。因此，所要提出的问题是：如果这个宏观规律性闭合，它的闭合是否正被读作一座通回产生它的机制的桥，或读作跨案例共享机制的证据，而这两者都是普适性所禁止的？ 一条成功的宏观定律是一堵墙，而不是一扇窗；而一个把它的成功当作据以穿过它反向推理之许可的说明，已把这一个错当成了另一个。

主张 4。 对于”规范、文化与制度如何自个体关系中生起”的研究，两件固定空间的工具产出一项分为四部分的方法论审慎。出自闭合的失效：一套闭合的宏观动力学掩藏着一个被强加的闭合，无论是对高阶关系的一次代数解耦，还是一个被丢弃的记忆，其许可必须被展示；凡记忆不衰减之处，当前宏观状态是一个不充分的变量，而历史进入其中是作为动力学的一个项，而不是作为背景；以及凡诸尺度不分离之处，默认的期望应当是非闭合，以致一套被报告出的干净宏观动力学，是一项被强加假定的证据，而不是一条被发现定律的证据。出自闭合的成功：凡一个宏观规律性确实以一个重整化群不动点的方式闭合之处，它的普适性同时禁止从一个特定微观机制推导出该规律性，以及从共享的宏观形式推论出共享的机制。

这项审慎并不提供任何闭合社会谱系的办法；它撤回”该谱系闭合”这一预设，并把那一预设转换为一个具有确定形式的问题。它的内容是开篇所陈述的那次重新排序：在”宏观层如何演化”之前，立着”宏观层是否以其自身术语演化，若是，凭何种许可”。

§6 从动力学的闭合到生成的闭合

前一节的审慎是以它所借用的那种闭合的词汇被陈述的，即动力学的闭合，$dC/dt = F(C)$ 的存在或不存在。生成性关系存在将要求第二个概念。本节引入它，标出它如何有别于第一个，并陈述本文把它推进到何种程度。

莫里-茨旺齐格所刻画的闭合是动力学的闭合：粗粒变量是否拥有一个自足的演化方程这一问题，由记忆与噪声是否消失来回答。还有第二个问题，是这一个所不抵达的。一个系统可能在其粗粒变量中缺乏一套闭合动力学，却仍然跨时间维系着某种东西，不是一条固定的宏观定律，而是一个不断裂的、持续着的生成回路。一个结构恰恰可由持续的通量、而不是由静止来维持，这从远离平衡态被维持的系统的研究中是熟悉的 [7]。一个受制于成员、资源与信息之持续更替的共同体，可能在任何聚合变量的层级上都不持有闭合动力学，却仍然是一个共同体，其构成性关系被以与它耗散一样快的速度再生成。在那里持存的不是一个状态、也不是一条关于状态的定律；它是一个持续着的生成。所争的那种闭合是那种生成的闭合，即一个生成回路维系其自身的条件，而它与一个动力学的闭合不是同一回事。

主张 5（拟议）。 我提议把生成性闭合与动力学闭合区别开来。一个系统可能不容许一套闭合的宏观动力学，即记忆不衰减、噪声未白化、无法约化为一个单以粗粒变量表达的方程，却仍然，该提议主张，展现一种生成性闭合，即生成其构成性关系的那个回路的持存，而不是任何状态或关于状态的定律的持存。该提议尚不是一个定义：会使它成为一个定义、并会阻止它坍缩为”一个持存的系统持存了”这一陈述的那个谓词，正是下文被指认为开放的那个不变量。它在此被作为该框架所要求的那个区分而录入，以待那个不变量。

主张生成性闭合是一个独特而真实的条件，尚不等于已说出它是什么。仅通过实例被陈述，即持存的共同体、更新的制度、再生成的实践，这个概念便暴露于一种会使它落空的循环之中：如果一个生成性闭合无非就是”一个系统继续生成而不断裂”这一事实，那么说一个系统具有生成性闭合便无非是说它尚未断裂，而这个概念并不预言任何未被给予它的东西。湍流加剧了这一危险：它同样在没有闭合动力学的情况下持存，因此，如果生成性闭合无法与湍流已然展现的东西区分开来，那么它便没有超出它本欲逾越的那个闭合问题的任何内容。

会赋予它内容的是一个不变量。在湍流中，级联的持存被一个贯穿它保持恒定的量所锚定：能量耗散的平均率，它跨越惯性区等于穿过它的能量通量 [4, 2]。那个通量之所以恒定，是因为惯性区，依定义，既不直接从强迫接收能量，也不直接向黏性损失能量，因此进入一端的任何能量必须传给下一个尺度而不是积累；这一恒定性出自能量守恒穿过一个没有其自身源或汇的范围。这个量的两项特征，是一个供关系性使用的候选者必须复现的。它是由沿一个确定维度的恒定性所定义的，即跨尺度、而不是跨时间，以致该不变量是一个关于”一个量在人们穿过级联的诸层级时如何表现”的陈述。而它是一个通量，一个全局平均的通过率，这正是它为何在打破该场朴素自相似性的间歇性之下存活 [2]：高阶统计量逐处强烈涨落，而平均通量不然。一个生成不变量会需要这两项特征，即一个沿生成回路的某个指定维度平稳的量，以及一个对它所聚合之物的局部涨落稳健的通过率，凭借它，该回路是一个人们可以指点出来的被维系之物。困难在于：一个社会的或关系的系统并不提供任何守恒律来担保这样一个量。任何或可扮演能量耗散所扮演之角色的东西，都无法由一个守恒原理来交付，而只能，如果竟能的话，奠基于某种为关系所固有的东西之中，即奠基于母框架所称的一种关系语法之中，也就是关于诸关系可以如何组合、如何相互过渡的常驻约束，而不是奠基于一种被守恒的实体之中。

本文并不提供那个不变量。它指认出对它的需要，定位出它的缺席所留下的循环，并对任何候选者固定三项条件：它必须沿生成回路的某个指定维度平稳，以那个跨尺度恒定的通量为范本；它必须不由一个守恒律所担保，因为关系并不提供守恒律；而它必须不化约为”一个持存的系统持存了”这一陈述。一个满足前两项条件的不变量的物理先例，即跨尺度平稳、却由全局而非局部守恒的考量所固定的一个不变量，出现在支配湍流反常标度的零模之中，见 §7 讨论。在这些约束之内，生成性闭合这个概念有余地成为一个确定的主张；而在当前这份反思之内，它仍然是一个被命名的问题，而不是一个被解决的问题。

§7 湍流案例的界限

一个被倚重的案例欠一份关于它无法抵达之物的说明，此处尤甚，因为湍流所无法抵达的正是生成性关系存在被构造来处理的那个东西。本节陈述那个界限，并在其旁陈述重整化群在湍流上的界限本身所产出的一项审慎。

穿过上文所排演的每一项困难，湍流保留一项特征：它的状态空间是固定的。速度场活在一个容许场的空间之中，该空间本身并不随动力学的进行而改变；向粗粒变量的投影 $P$ 是那个固定空间上的一个固定算子；莫里-茨旺齐格方程之所以能被写出，是因为它被写于其上的那个空间并不移动。湍流在此意义上是一个提在一个固定状态空间上的问题。它所缺乏的每一项资源，即小参数、尺度分离、平衡态，都是一个固定空间之分析所内部的资源。状态空间本身从不成问题。

重整化群在湍流上遭遇它的界限，恰在同一道固定空间的边界上，而它失效的方式携带一项其自身的审慎。湍流表面上是重整化群的理想对象：它是尺度相关的，它级联，而它的能谱遵从一条幂律，看上去像一条邀请不动点处理的标度律。这件工具确实捕捉到该行为中标度不变的、量纲的部分，即科尔莫戈罗夫谱，但只抵达一部分，因为湍流中抗拒它的那部分正是要紧的那部分。高阶结构函数并不遵从量纲分析所预言的指数；它们反常地标度，而这一反常，即间歇性的标志，不是一个局部不动点所交付的东西。它的源头，在已被计算过的那些情形中，不是一个不动点所编码的局部自相似性，而是一组零模：对量纲分析不可见的量，不在惯性区之内、而由喂养它的大尺度与边界所固定 [1]。这为社会秩序的研究所产出的审慎是直接的。一个系统可能展现每一项邀请一套普适的、不动点式说明的标度标志，即社会系统大量呈现的那些幂律、无标度网络、影响的自相似级联，却仍然不拥有任何干净闭合的宏观理论，因为可观测统计量中的标度不变性并不蕴涵动力学中的一个不动点。湍流是那个常驻的反例：它把一条教科书式的标度律与一种顽固的非闭合持在一起。把一个社会系统中的一条幂律读作一套普适宏观理论的凭据，就是作出湍流所反驳的那个推论。

重整化群失效的残余关乎 §6 的那个不变量。零模是这样一些量，它们被固定不是由一个局部守恒律，而是由级联被驱动所处的全局条件，这是那份说明所要求的不变量的一个物理实例：一个跨越一个生成过程的诸层级平稳、不由一种实体的守恒所担保、并对局部自相似性所错失的高阶行为具有决定性的不变量。重整化群在湍流中所留下的未决之物，在这一点上，比它所解决之物是生成性关系存在的一个更贴近的模型。

生成性关系存在的独特主张是：状态空间本身成问题。凡关系生成存在之处，可能状态的空间不是预先给定、并在动力学于其上展开时保持固定的；它被它所承载的诸关系所扩大、所重塑，以致哪些状态在一个较晚的时刻可用，是较早所生成之物的一个函数。在该框架的记法中，这是写 $\dot x = F(x, R)$ 与写状态空间本身之演化二者之间的差异，前者向一个固定空间添加一个关系变量 $R$，并停留在一套标准的、尽管被扩大了的动力学的可及范围之内；而后者，即 $\Omega_{t+1} = H(\Omega_t, R_t)$，其中在时刻 $t{+}1$ 可用的状态集，是从在时刻 $t$ 的状态集连同其时成立的诸关系 $R_t$ 一起被生成的。第一种是湍流那一类：一个更大但固定的状态空间，莫里-茨旺齐格构造在其上仍然适用，而闭合是那个熟悉的问题。第二种落在它之外：当状态空间本身演化时，投影 $P$ 没有固定空间可供作用，而莫里-茨旺齐格的推导失去了它所假定的固定空间与固定内积。一个类似的闭合问题在这第二种情形中是否竟能被提出，此处并未了结；它无法以湍流所给予它的形式被提出，正是那个构造的失效所表明的，而那个肯定性的问题被留待 §8 开放。

主张 6。 湍流展示一个固定状态空间的闭合问题，而由于被限定于一个固定状态空间，它所能展示的不能超出此。状态空间本身的演化，即一个系统所承载的诸关系对可能性之新维度的生成，落在湍流案例所占据的那个固定空间设置之外，因而落在那个案例所能标定之物之外。因此，湍流标定生成性关系存在而不为它奠基：它显示一项固定空间上的困难在其最为严峻时是什么，并因而以对照的方式标示出，该框架的独特困难必须在何处被寻求，即在状态空间的演化之中。一项如此被定位的困难是否真正不可约化为某个固定的、被扩大空间上的一套动力学，这不由这一对照所了结；它是 §8 所记下的那个开放问题。有两个进一步之点附于同一界限：重整化群在湍流上的失效表明一条标度律并不是一套普适宏观理论的凭据；以及那份失效寓于其中的、由边界所喂养的零模，是 §6 所要求的那个生成不变量的最贴近的物理模型。

由于这一理由，湍流案例不可能是该框架所需要的那个基础，而 §6 的开放问题不可能通过精炼湍流类比来了结。一个生成不变量，如果它存在，必定是一个重造其自身状态空间的过程的不变量；而湍流，其状态空间从不改变，提供了一个不变量的观念，却不提供所要求那种的一个不变量。这个案例给了该框架它的标定与它的词汇，即记忆、级联、被维系的通量，却不给那越过固定状态空间的一步。那一步始于一套固定空间上的更难动力学，即湍流所例示的东西，止步之处：在那个空间本身的演化之中。

§8 已确立者与开放者

本节把前述所确立之物与它所留下之物分开。

被确立的是问题的一次重新排序。从微观关系到宏观秩序的过渡，并不赋予宏观层一套其自身的闭合动力学；这样一套动力学是否存在，是一个条件，当它被许可时，由一种尺度分离所许可，那种分离让亚宏观的诸自由度被概括为摩擦与噪声。凡那种分离缺席之处，如在湍流中，以及看来在社会生活的诸关系尺度上，宏观描述并不闭合，它的历史进入其中是作为一个项而不是一个背景，而一套被报告出的干净宏观动力学应被读作一个被强加的闭合，而不是一条被发现的定律。对文化、规范与制度的研究而言，其后果是 §5 的那项审慎：在追问一个宏观层如何演化之前，人们必须追问它是否构成一个可以以其自身术语写出演化的层级，以及凭何种许可。重整化群为同一项审慎添上两道边：凡一个宏观规律性确实作为一个不动点闭合之处，它的普适性禁止把它读回一个机制或跨案例地读；而凡一条标度律在没有一个不动点的情况下在场之处，如在湍流的反常标度中，那条标度律并不是一套闭合宏观理论存在的证据。这么多是两件固定空间的工具所支持的，而这么多是本文所主张的。

未被确立的是生成性闭合。它已被与动力学闭合区别开来，并被表明是该框架将要求的那个条件，即一种生成而非状态的持存，但它尚未被给予足以阻止它坍缩为”一个持存的系统持存了”这一陈述的内容。会赋予它内容的东西，即一个不由任何守恒律所担保、且不可约化为持存这一事实的生成不变量，已被作为一项要求而命名，并被留作未满足。这一界限出自所选取的案例：湍流，被它的固定状态空间限定于该框架必须逾越的那个区域，无法提供一个重造其自身状态空间的过程的不变量。湍流案例标定这个问题并提供它的语言；它不提供那个不变量。

有两条工作线随之而来，而本文把二者都留作开放。第一条是那个不变量：确定一个跨越一个生成回路被守恒或被保持平稳的量，能否以关系性术语被定义，即奠基于一种关系语法而不是一种被守恒的实体，并因而把生成性闭合从一个被命名的区分转换为一个可运作的条件。第二条是演化着的状态空间：给 $\Omega_{t+1} = H(\Omega_t, R_t)$ 一个确定到足以被推理的形式，特别是了结这样一种演化能否被表明不可约化为任何固定的、被扩大空间上的一套动力学，因为只有当它能够时，该框架的独特困难才是一个真实的困难，而不是一个乔装的固定问题。二者都在当前这篇论文之前方，本文只走到那两个开放问题以及推动它们的那次重新排序为止：它把宏观层自主性这一假定弄得作为一项假定而可见，通过一个物理案例给出该假定成立与失效的条件，并固定一个固定状态空间的案例能够与不能够转带给一个以变化着的状态空间为主题的框架的东西。

关于方法的一则说明。矩谱系与莫里-茨旺齐格分解是作为它们所是的定理被使用的；凡论证过渡到关系性涌现之处，没有任何方程被转带过去，只有问题的结构被转带过去。此处没有任何生成不变量被定义：这个概念被留作 §6 的那个开放问题。

参考文献