Braid Groups and Procedural Justice - Path-Dependence in Generative Relational Being

ENGLISH

Braid Groups and Procedural Justice

Path-Dependence in Generative Relational Being

Wanhong Huang · huangwanhong@serendip.ngo

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Abstract

Two procedures may issue in the same distribution and yet not be equally just. Theories of procedural justice have long recognized this, and each has registered it in its own way: Rawls in the doctrine of pure procedural justice, Tyler in the finding that judgments of fairness track something the outcome does not record, deliberative theory in its attention to sequence and agenda, Sen in the turn toward comparison. What none of them supplies is a formal object whose elements are the procedural differences themselves, a structure in which two histories that project to the same endpoint remain distinguishable, and in which the question of whether they are the same procedure has an answer. This paper proposes the braid group as one candidate structure. Under the projection $\pi\colon B_n \to S_n$, which retains the endpoint permutation and discards the path, the pure braid group $P_n = \ker \pi$ is the group of transformations that preserve outcome while altering relational history; two histories are outcome-equivalent precisely when they differ by an element of $P_n$. The paper argues that this is the object procedural justice has described without possessing, and argues equally that it is not by itself an object of justice. Whether a given procedural difference matters is not settled by the topology, and an account that claims otherwise attributes to a structure of path-dependence a normative content it does not carry. The paper therefore introduces an interpretive operator, written $\Phi$, that assigns relational significance to elements of $P_n$ in context; it does not construct $\Phi$, and argues that the contribution of a formalism of this kind is to locate where interpretation must enter rather than to dispense with it. Four constructed cases test the resulting model for sensitivity, specificity, interpretive underdetermination, and boundary, the last exhibiting a case the formalism cannot represent at all. A closing section places procedural justice among the layers of evaluation that generative relational being requires, and marks what braids cannot carry.

Keywords: procedural justice; braid groups; path-dependence; relational history; formal modeling; generative relational being.

§1 Introduction

Consider two committees that reach the same decision. The membership is the same, the vote is the same, the distribution of benefits and burdens that follows is the same, and the reasons entered into the record are the same. In the first committee, an objection was raised early and the proposal was altered in response. In the second, the same objection was raised after the proposal had been settled, was recorded, and altered nothing. Assessed at the endpoint, the two committees are indistinguishable. Assessed by their participants, they are not: something occurred in the first that did not occur in the second, and it is not obvious that the difference is merely psychological.

This is not a marginal observation. It is the recurring finding of the empirical literature on procedural justice and the persistent intuition behind its normative wing. People care about how a decision was reached in ways that survive the observation that the decision would have been the same regardless. Theories of procedural justice have recognized this and have registered it, each in its own idiom. Rawls’s doctrine of pure procedural justice makes the procedure the source of the outcome’s justice rather than its instrument. Tyler’s empirical program isolates factors, voice, neutrality, trustworthiness, respect, that predict judgments of fairness independently of the favorability of the result. Deliberative theory attends to sequence, agenda, and the order in which claims are heard. Sen turns from the ranking of institutions toward the comparison of realized states.

Each of these registers that procedure carries weight the outcome does not record. None of them supplies an object whose elements are the procedural differences themselves. This is a different lack from the one it is easily mistaken for. It is not that these accounts fail to represent history; a range of frameworks represent history well, among them process tracing, event-history modeling, causal graphs, the histories of extensive-form games, and the path-dependent institutionalism that made lock-in a term of art. What is missing is narrower and more specific: a structure in which the histories that project to a given outcome form a determinate collection, in which the differences among them are themselves elements of something, and in which the question are these two procedures the same has an answer that does not depend on the analyst’s description. Without such a structure, the claim that procedure matters beyond outcome remains a claim about salience. With one, it becomes a claim about a space.

This paper proposes the braid group as one candidate for that structure. The proposal rests on a single feature. There is a natural map $\pi\colon B_n \to S_n$ from the braid group on $n$ strands to the symmetric group, which retains where each strand ends and forgets how it got there. The map is surjective and is not injective, and its kernel, the pure braid group $P_n$, consists of exactly those braids that return every strand to its own position while having, in general, done something in between. Two braids have the same image under $\pi$ if and only if they differ by an element of $P_n$. If strands are relational positions, crossings are interaction events, the braid is the history and its image the outcome, then $P_n$ is the group of transformations that alter procedural history while preserving everything an outcome-based evaluation can see.

Central claim. The pure braid group supplies one candidate structure for the object that theories of procedural justice have described without possessing. It is a group of transformations that preserve endpoint outcome while altering relational history, within which outcome-invariant procedural differences are determinate elements and procedural identity is decidable. It does not follow that these differences are thereby justice-relevant, and §3.2 exhibits one that is not. Which of them matter is settled by an interpretive operator that the formalism requires, locates, and does not supply.

The two halves of this claim are equally load-bearing, and the second is the one more easily lost. A formalism that represents procedural difference invites the inference that the differences it represents are the differences that matter. The inference is false. Some procedural differences are justice-relevant; some are recorded by the topology and are of no interest to anyone; and some, the environment of fear as against the environment of safety, the concession extracted by threat as against the concession reached by learning, matter greatly and are invisible to the topology altogether. A model that could not accommodate all three would be worse than no model. What the braid supplies is a space in which the first two are distinguishable from one another; what it cannot supply is the criterion by which they are distinguished, or any purchase at all on the third.

That criterion is written here as an operator $\Phi$, carrying elements of $P_n$ together with context to relational significance. The paper does not construct $\Phi$. It argues something weaker and, if correct, more useful: that an account of procedural justice which discriminates among procedural differences thereby employs an operator of this type, whether or not it is written down, and that a formalism which makes the operator explicit has performed a service even when it leaves the operator empty. The contribution claimed is the localization of a problem, not its solution.

A further limit must be entered before the argument begins, since it bounds what the whole can be taken to establish. Procedural justice asks whether relational generation occurred through legitimate relations. It does not ask what is being generated, or toward what. A procedure may satisfy the criteria the literature imposes, equal voice, transparency, revisability, freedom from coercion, and issue in the exclusion of a minority; procedural legitimacy does not entail substantive justice, and no formalization of the procedural layer alters this. Generative relational being poses a sharper version of the same difficulty: a procedurally impeccable process may consume the conditions of its own future generation, leaving a relation that was fairly conducted and is no longer capable of generating anything. The braid represents the memory of a process. It does not determine whether the process deserves continuation. What this paper treats is one layer among several, and §6 sets out the others and their relations rather than leaving the scope to be inferred.

The novelty claimed may accordingly be stated with some precision. It is not that procedural history matters, which is established. It is not that $P_n$ contains hidden histories, which is a loose way of speaking. It is that $P_n$ characterizes the transformations among histories that are observationally equivalent under endpoint evaluation, that membership in $P_n$ is decidable, and that the separation of this representational question from the interpretive one is available to any theory of procedural justice and has not been taken.

Two features of the exposition follow from the character of the argument, and are stated here in advance. First, the mathematics is presented before any interpretation is placed upon it, and in a section that contains no vocabulary of justice; the account of procedural justice that follows contains no vocabulary of braids. The join is made only in §4, where each correspondence is proposed and then tested. The discipline is not stylistic. A formalism assembled alongside the phenomenon it is to model will come to fit it, and the fit will establish little. Second, the structural ceiling of the formalism, fixed $n$, no creation or destruction of strands, discreteness of events, silence on the content of an interaction, is stated in §2, before the model is built. A reader should know what the apparatus cannot do before being shown what it can.

The remainder proceeds as follows. §2 sets out the braid group: its presentation, the relations among its generators, the projection to the symmetric group and its kernel, the decidability of the word problem, and the ceiling. §3 sets out the theories of procedural justice and isolates the specific gap this paper addresses. §4 constructs the correspondence, treating in turn outcome-invariant difference, sequencing and procedural indifference, accumulated asymmetry, revisability, and degradation, and closes by fixing the semantics of the mapping together with the conditions under which each part of it would fail. §5 presents four constructed cases, testing the model for sensitivity, specificity, interpretive underdetermination, and boundary. §6 states what the model buys, what braids cannot carry, where procedural justice stands among the layers of evaluation that generative relational being requires, and the work left open.

§2 The Braid Group

This section sets out the mathematical structure on its own terms. No interpretation is placed on it here, and no vocabulary of justice appears; the reader who already knows the material may read to the end of §2.7, where the structural ceiling is stated, and pass on. The presentation is standard and follows Artin’s, whose original treatment established both the geometric definition and the algebraic presentation that will be used throughout [2, 3]. Textbook accounts may be found in Birman [6] and Kassel and Turaev [13].

2.1 Definition

Fix an integer $n \geq 2$ and consider two horizontal planes, one above the other, each carrying $n$ marked points in a row. A geometric braid on $n$ strands is a collection of $n$ disjoint curves running from the marked points of the upper plane to those of the lower, subject to one condition: each curve descends monotonically, meeting every intermediate horizontal plane exactly once. The condition forbids a strand from doubling back upward. What it permits is that the strands pass around one another, and it is this passing that the structure records.

Two geometric braids are regarded as the same if one may be deformed continuously into the other without moving the endpoints and without any strand passing through another. This relation is isotopy, and a braid is an isotopy class of geometric braids. The definition already carries a commitment worth marking: everything about a configuration that a continuous deformation can remove is discarded, and everything it cannot remove is kept. How far apart two strands run, how sharply a curve bends, at what height a crossing occurs, none of this survives. What survives is which strands crossed which others, in what order, and in which sense.

Braids compose. Given braids $a$ and $b$ on the same number of strands, stack $a$ above $b$, identify the lower endpoints of $a$ with the upper endpoints of $b$, and rescale; the result is a braid, written $ab$. Composition is associative on isotopy classes. The braid whose strands descend without crossing is an identity for this operation, written $e$. And every braid has an inverse: reflect it in a horizontal plane, and the composite of a braid with its reflection may be deformed, crossing by crossing, into the identity. The isotopy classes of braids on $n$ strands accordingly form a group, the braid group $B_n$.

2.2 Generators and Orientation

Any braid may be decomposed into a sequence of elementary crossings, each involving one adjacent pair of strands. Write $\sigma_i$ for the braid in which the strand in position $i$ crosses over the strand in position $i+1$, all other strands descending untouched, where $1 \leq i \leq n-1$. Its inverse $\sigma_i^{-1}$ is the braid in which the strand in position $i$ crosses under the strand in position $i+1$.

Figure 1 (The generator $\sigma_i$ and its inverse). The strands involved and the positions exchanged are the same in both; the sole distinguishing datum is which strand passes over the other. Breaks in a strand mark where it passes underneath.

The pair $\sigma_i$ and $\sigma_i^{-1}$ differ in exactly one respect. The strands involved are the same, the positions exchanged are the same, and the sole distinguishing datum is which strand passed over the other. This is the orientation of the crossing, and it is the only asymmetry the formalism records at the level of a single event. What does not follow should be noted. That $\sigma_i \neq \sigma_i^{-1}$ is a fact about the structure; that one of them is in any sense better, stronger, or prior is not a fact about the structure at all, and no relation in $B_n$ supplies one.

Every element of $B_n$ can be written as a finite product of the $\sigma_i$ and their inverses. Such a product is a braid word, and it is an ordered record: the word $\sigma_1 \sigma_2 \sigma_1^{-1}$ specifies not only which crossings occurred but the sequence in which they occurred. The number of letters is the length of the word, and a word of minimal length among all words representing a given braid is reduced. A braid typically admits many words, and the relations of the next subsection say which.

2.3 The Relations

Figure 2 (The two relations). Left pair: far commutation, where the crossings involve disjoint pairs of strands and their order leaves no trace. Right pair: the braid relation, where three strands interact pairwise and one particular reordering yields the same braid. In each pair the two diagrams are the same element of $B_n$.

Two families of relation hold among the generators, and together with the generators they present the group completely:

$$B_n = \bigl\langle, \sigma_1, \ldots, \sigma_{n-1} ;\bigm|; \sigma_i \sigma_j = \sigma_j \sigma_i \ \ (|i-j| \geq 2), \quad \sigma_i \sigma_{i+1} \sigma_i = \sigma_{i+1} \sigma_i \sigma_{i+1} ,\bigr\rangle.$$

The two are illustrated in Figure 2. The first family is far commutation. Where two crossings involve disjoint pairs of strands, positions $i, i+1$ and $j, j+1$ with $|i-j| \geq 2$, the order in which they occur makes no difference to the resulting braid. The two events do not interact, and the structure records no trace of which came first. This is a statement of independence, and it is exact. The order does not matter little; the order has no representation at all.

The second is the braid relation, also known in its wider setting as the Yang-Baxter relation [25, 4]. Where two crossings share a strand, positions $i, i+1$ and $i+1, i+2$, the order is partly constrained. The relation says that one particular reordering leaves the braid unchanged: performing $\sigma_i$, then $\sigma_{i+1}$, then $\sigma_i$ yields the same braid as $\sigma_{i+1}$, then $\sigma_i$, then $\sigma_{i+1}$. Geometrically, this is the statement that a strand may be slid past the crossing of the other two. What the relation supplies is thus a criterion of indifference: among sequences of overlapping crossings, some reorderings change nothing and some change everything, and the relation says which are which.

The presentation is complete, and its completeness has a consequence about what the group does not say. Whatever is not a consequence of the two relations does not hold in $B_n$. In particular no relation identifies $\sigma_i$ with $\sigma_i^{-1}$: a crossing and its reverse are distinct elements, and no amount of composition will conflate them. Nor does any relation permit a crossing to be deleted except by composition with its own inverse. Words grow, and they reduce only in the ways the relations allow.

2.4 Projection to the Symmetric Group

A braid determines a rearrangement of positions: strand beginning at position $1$ ends somewhere, strand beginning at $2$ ends somewhere else, and the assignment is a permutation of ${1, \ldots, n}$. Sending each braid to this permutation defines a map

$$\pi \colon B_n \longrightarrow S_n,$$

which is a group homomorphism, and which sends the generator $\sigma_i$ to the transposition $(i \ \ i{+}1)$. On the presentation, $\pi$ discards the distinction between $\sigma_i$ and $\sigma_i^{-1}$, both map to the same transposition, and retains the exchange of positions.

Figure 3 (What the projection cannot see). All three braids send every strand back to its own position and so have the same image in $S_n$. The first two are the identity braid; the third is not, the strands having wound once fully around one another. The permutation records the endpoints and discards the winding.

The map is surjective: every permutation is realized by some braid, since the transpositions generate $S_n$. It is not injective, and the failure of injectivity is what this paper takes up. Figure 3 exhibits the failure. Both $\sigma_i \sigma_i^{-1}$ and $\sigma_i^{-1} \sigma_i$ map to the identity permutation, as they must, being the identity braid; but so does $\sigma_i^2$, which is not the identity braid at all, two strands have wound fully around one another and returned to their starting positions. The permutation cannot see the winding.

Writing $P_n$ for the kernel of $\pi$, we obtain the short exact sequence

$$1 \longrightarrow P_n \longrightarrow B_n \stackrel{\pi}{\longrightarrow} S_n \longrightarrow 1,$$

which is the structure on which everything later in the paper rests. Its content may be put as follows. $B_n$ records the full history of a configuration; $S_n$ records only where the strands ended; and $P_n$ is what the second forgets of the first.

The relation between the kernel and the classes it acts upon is easy to state loosely, and is stated here carefully. For a permutation $s \in S_n$, the set of braids realizing $s$ is the fiber $\pi^{-1}(s)$, and these fibers partition $B_n$. The kernel $P_n$ is the fiber over the identity, and is a group; the other fibers are its cosets and are not. What is true in general, and is the operative fact, is this:

$$\pi(b_1) = \pi(b_2) \quad \Longleftrightarrow \quad b_1 b_2^{-1} \in P_n.$$

$P_n$ is thus not itself the collection of same-outcome histories. It is the group of transformations that carry any such history to any other with the same outcome, the group acting within each fiber. This distinction will matter later, when what is evaluated turns out to be differences between histories rather than histories.

2.5 The Pure Braid Group

The elements of $P_n$ are the pure braids: those in which every strand returns to its own position. A pure braid is not thereby trivial. The strands may have wound around one another arbitrarily on the way, and the group is correspondingly rich.

Figure 4 (A generator of the pure braid group). Strand $1$ crosses the intervening strand, encircles strand $3$ once, and returns. Every strand ends where it began, so the braid lies in $\ker \pi$; the full winding at the centre is what the projection annihilates.

It is generated by the elements $A_{ij}$ for $1 \leq i < j \leq n$, where $A_{ij}$ is the pure braid in which strand $i$ passes across the intervening strands, encircles strand $j$ once, and returns. In terms of the Artin generators,

$$A_{ij} = \sigma_{j-1} \cdots \sigma_{i+1} , \sigma_i^2 , \sigma_{i+1}^{-1} \cdots \sigma_{j-1}^{-1},$$

and one sees the $\sigma_i^2$ at the center: a full winding, which the projection to $S_n$ annihilates. Figure 4 shows the case $A_{13}$. Structurally, $P_n$ decomposes as an iterated semidirect product of free groups, a result due to Artin [3]; the reason is that forgetting the last strand defines a surjection $P_n \to P_{n-1}$ whose kernel is free on $n-1$ generators. The pure braid group is thus large, non-abelian for $n \geq 3$, and possessed of considerable internal structure. It is not a degenerate remainder.

To each pure braid and each pair of positions $i < j$ one may associate an integer, the linking number $\operatorname{lk}_{ij}(b)$, obtained by counting the crossings between strands $i$ and $j$ with sign, positive where $i$ passes over $j$, negative where it passes under, and dividing by two. This is an isotopy invariant, and it is a homomorphism from $P_n$ to $\mathbb{Z}$; the total linking numbers give the abelianization of $P_n$. What the linking number measures is the net winding of one strand about another, accumulated over the whole history. It is defined here in these terms and in no others. Whether such a quantity admits an interpretation, and what interpretation, is not a question the topology answers, and §4.4 takes it up only after the model has been built.

2.6 Decidability

Two braid words that look different may represent the same braid, the relations of §2.3 permitting a great many rewritings. It is not obvious in advance that one can always tell. In fact one can: the word problem for $B_n$ is solvable, a result established by Artin [2] and since given several algorithmic treatments, among them the combing procedure implicit in the semidirect product decomposition, Garside’s normal form [11], and the handle-reduction algorithm of Dehornoy [9], which is efficient in practice.

The consequence is unusual among the structures that social theory borrows. Given two histories represented as braid words, the question are these the same has an answer, and the answer is obtainable by a procedure that does not depend on the judgment of the analyst. A formalism that represented differences but could not adjudicate identity would be of little use for the purposes of this paper; it would relocate an interpretive difficulty rather than remove one. Decidability is the reason the braid group is a candidate at all.

2.7 The Structural Ceiling

What follows is stated here, before any interpretation has been proposed, so that the reader knows the boundary of the apparatus in advance of seeing its use. Each item is a feature of the mathematics, not a limitation of the present treatment; no refinement of the model removes them, and each will be revisited in §6 as a constraint on what the model can be asked to bear.

Fixed $n$. The number of strands is fixed for the whole of a braid. Strands cannot be created, destroyed, merged, or split. $B_n$ and $B_m$ are different groups for $n \neq m$, and there is no operation within the theory carrying a configuration from one to the other. Whatever a braid describes, it describes a fixed population of positions persisting throughout.

Discreteness. A braid is a finite word in crossings. Crossings are events, individuated and countable, and there is no representation of a continuously varying influence between strands, of a field, or of a gradual change in the relation between two positions unpunctuated by an event.

Orientability without direction. A braid distinguishes over from under. It does not rank them, and the group contains no element and no relation that would let one say a braid is more advanced, more developed, or further along than another. Composition supplies an order in time; it supplies nothing normative. This is a limitation and, as §6 argues, also a protection.

Isotopy invariance. Everything that a continuous deformation can remove has been discarded by construction. The distance between strands, the sharpness of a bend, the timing of a crossing within the descent, none is represented. What the model sees is exactly the topological residue.

Silence on content. This limit is the least visible of the five. Two crossings of the same pair of strands in the same sense are the same generator, full stop. Whatever distinguished the two occasions, if anything did, has no representation whatever in $B_n$. The formalism records that positions $i$ and $i+1$ interacted with a given orientation. It records nothing about what the interaction was.

With the structure and its boundary set down, we turn to the domain, and to the specific gap this structure is proposed to fill.

§3 Procedural Justice

This section sets out the domain on its own terms. No braid appears in it, and the accounts reviewed are not treated as deficient versions of something the previous section supplies; each is treated as an answer to the question it set itself. What the section is for is to locate, as precisely as possible, a question that none of them poses, and to distinguish that question from several nearby ones that have been answered well.

3.1 Rawls and the Three Kinds of Procedural Justice

The distinction from which most subsequent discussion proceeds is Rawls’s, and it is a distinction in the relation between a procedure and an independent criterion of the just outcome [20, §14].

Where such a criterion exists and a procedure reliably realizes it, the case is one of perfect procedural justice. Rawls’s example is the division of a cake among equals: the criterion is an equal division, and the rule that the person who cuts takes the last piece achieves it. The procedure is instrumental, and it is judged by whether it delivers what the criterion specifies.

Where the criterion exists but no procedure reliably realizes it, the case is imperfect. A criminal trial is the example: the criterion is that the guilty and only the guilty be convicted, and no procedure of evidence and argument guarantees this. Here too the procedure is instrumental, and its imperfection is a shortfall measured against something outside it.

Where no independent criterion exists, the case is pure. A fair gamble is the example: no distribution of winnings is just in advance of the procedure, and whatever distribution results from a properly conducted gamble is just because it so resulted. The procedure is constitutive rather than instrumental, and this is why the doctrine bears the weight it does in Rawls’s larger argument, where the outcome of a suitably specified original position is just in virtue of the procedure rather than by correspondence to a prior standard.

Pure procedural justice is the case closest to what this paper is concerned with, since it is the case in which the procedure carries the whole normative burden. It is also, for present purposes, the case in which a specific insufficiency is clearest. The doctrine tells us that a correctly conducted procedure confers justice on its outcome. It does not tell us how to compare two procedures both of which are correctly conducted. If two gambles are both fair, the doctrine has nothing further to say about the difference between them; and if two deliberative procedures both satisfy whatever conditions are imposed, the doctrine treats them as equivalent. The apparatus is built to relate procedures to outcomes. It is not built to relate procedures to one another.

This is not a criticism of Rawls, who was answering a different question. It is an observation about what the trichotomy leaves undetermined. A theory that individuates procedures only by their conformity to conditions, and outcomes only by their distributions, has no resources for a difference between two conforming procedures that issue in the same distribution. Whether there is any such difference worth marking is precisely what the empirical literature went on to establish.

3.2 Tyler and the Empirical Turn

The finding that people evaluate procedures independently of the outcomes those procedures deliver was established first by Thibaut and Walker [22] in experimental settings and then, at scale and in the field, by Tyler and his collaborators [14, 23, 24]. The central result is robust and has been replicated across legal, organizational, and political settings: judgments of the fairness of a procedure predict compliance, acceptance, and the legitimacy accorded to an authority, and they do so with the favorability of the outcome controlled. People who lost accept losing, when they judge the procedure fair.

The factors that carry this judgment are conventionally given as four: voice, the opportunity to state one’s case; neutrality, the absence of bias in the decision-maker; trustworthiness, the perception that the authority is acting in good faith; and respect, treatment consistent with one’s standing as a person entitled to consideration. These are not features of the distribution. Two procedures identical in what they distribute may differ in every one of them.

The literature thus establishes, as securely as such things are established, that something outcome-invariant is being tracked. What it does not do, and does not attempt, is supply a structure for what is tracked. The four factors are properties of a procedure as experienced, measured by instruments that ask participants how they experienced it. This is entirely appropriate to the research programme, which is a programme in social psychology. But it means that the outcome-invariant residue is characterized by its effects on judgment rather than by anything about the procedure itself. If two procedures elicit different fairness ratings, we know that they differed; we do not have an account of what they differed in, other than by the same instruments that detected the difference. The finding is a finding about people. The question of what property of the procedures the people were responding to remains open.

3.3 Deliberative Theory and Sequence

Deliberative democratic theory is the strand that has come closest to the concern of this paper, because it is the strand that already knows that order matters.

The recognition arrives from two directions. From social choice, there is the long-established result that outcomes depend on agenda: McKelvey [17] showed that under majority rule with multidimensional preferences, an agenda-setter can drive the assembly to any point whatever by a suitable sequence of pairwise votes. From the theory of deliberation itself, there is sustained attention to who speaks, when, and with what effect on what follows, the concern with inclusion in Young [26], the treatment of deliberative conditions in Cohen [7] and Habermas [12], and the empirical study of deliberative settings in Mansbridge [16] and Fung [10], where the order in which perspectives enter is repeatedly found to shape what the assembly is subsequently able to consider.

The literature is therefore not innocent of the phenomenon. Its characteristic treatment of it, however, is diagnostic rather than descriptive: order effects appear as distortions to be identified and neutralized. Agenda manipulation is a pathology; the late arrival of a marginalized perspective is a failure of inclusion; the remedy in each case is a procedural constraint that removes the dependence on order. This is a reasonable response to the problem as posed. But it treats sequence as a source of error rather than as a dimension along which procedures differ, and consequently it produces no account of the differences themselves. Two procedures that differ only in sequence, and in which neither sequence is manipulative, are not a case the diagnostic framing has anything to say about. There is no positive characterization of the space of sequences, only a set of conditions that sequences ought not to violate.

3.4 Sen and the Comparative Alternative

Sen [21] argues against what he calls transcendental institutionalism, the project of specifying perfectly just institutions, in favor of comparative assessments of realized social states. The argument has a component directly relevant here: Sen insists that the assessment of a state must include how it was arrived at, and offers the example of a flute to be assigned among three children with different claims, where the identity of the just assignment depends on considerations that no ranking of final distributions captures.

This is a substantial widening. The object of assessment is enlarged from distributions to realizations, and realizations carry information about process. Yet the structure of the assessment is unchanged: what is compared is still a pair of states, and the process information enters as a further descriptor of the state that resulted. Two realizations reached by different routes are two states to be compared. The comparison does not take as its object the difference between the routes; it takes as its object the states, more richly described. The comparative move accordingly relocates the endpoint rather than dispensing with it.

3.5 The Gap

It would be a mistake to state the gap as an absence of attention to history. History is well represented across the social sciences, and a claim to be supplying it would be false. Process tracing reconstructs causal sequences within cases [5]. Event-history models represent the timing and order of transitions. Causal graphs represent the structure of dependence among events [18]. The extensive form of a game is precisely a representation of histories, complete with the information available at each point. Path-dependent institutionalism has made lock-in, sequence, and critical juncture into standard analytical vocabulary [8, 1, 19, 15]. Each of these represents processes, and represents them well.

What is missing is narrower, and it survives all of the above. Consider the collection of procedural histories that issue in a given distribution. The frameworks just named will describe each of these histories individually, and describe them with as much fidelity as one likes. None of them turns the collection into an object. None of them, consequently, has anything to say about the differences among its members considered as differences: whether they form a structure, whether that structure has parts, whether two of them are the same difference, whether the difference between the first and second history is the same difference as that between the third and fourth. These are questions one can ask only if the differences are themselves elements of something.

There is a second component to the lack, and it is what makes the first component consequential rather than merely abstract. Suppose one wished to claim of two procedures that they are the same procedure differently described, or that they are genuinely different. In the frameworks named, this question is settled by the analyst’s description: two process traces are the same if the analyst has described them the same way, and the individuation of the events is part of what the analyst supplies. There is no independent criterion of procedural identity. A structure in which such a criterion exists would put the claim that two procedures differ beyond the reach of redescription, and would make the assertion that a difference is merely notational into something refutable.

The gap may accordingly be stated as follows.

The gap. Every account reviewed here registers that procedure carries weight the outcome does not record, and several represent procedural history in detail. None supplies a structure in which the histories issuing in a given outcome form a determinate collection, in which the differences among them are themselves elements admitting composition and comparison, and in which the identity of two procedures is decidable independently of how the analyst has chosen to describe them.

Whether such a structure exists is a mathematical question, and §2 has exhibited a candidate. Whether the candidate can be brought to bear on the domain just surveyed, and at what cost, with what license, and with what refusals, is the subject of what follows.

§4 Formal Modeling

The two preceding sections were written not to meet. This one makes them meet, and the manner of the meeting is the substance of what follows. Each correspondence below is proposed, and then immediately tested: what it licenses is stated, and what it does not license is stated with equal care, since a mapping that licenses everything licenses nothing.

4.1 The Base Correspondence

Strands. A strand is a role: a relational position within a procedure, occupied by whoever occupies it. Three candidates present themselves, and the choice among them is forced rather than free.

Strands might be individuals. This fails immediately against the fixed $n$ of §2.7: a procedure in which a participant arrives late or withdraws is not representable, and such procedures are common. Strands might be relational identities in the fuller sense that generative relational being gives that term, the marginalized participant, the community, the institution as a constituted thing. This is closer to the framework’s own ontology and fails for a subtler reason: relational identities are generated and transformed by the very interactions the braid would record, so the strands would have to change during the braid, which the structure does not permit.

Roles are what remains, and their adequacy is bounded in a way that should be stated rather than concealed. A role in a procedure, the proposer, the objector, the chair, the party with the technical claim, persists through the procedure while the individual occupying it may not, and a procedure with a stable role structure and a changing membership is representable where a procedure with changing roles is not.

This has a consequence for the relation between the model and the wider framework.

Claim 1 (Domain of application). Fixed-$n$ braid theory begins after role stabilization. It models transformations among established relational positions, not the emergence of those positions. Where the roles are themselves in formation, the model does not apply, and the point at which it ceases to apply is detectable.

Events. A crossing is an interaction event, but not every occurrence within a procedure is one. The individuation requires a criterion, and the following three conditions are proposed jointly: an occurrence is a crossing when (i) it modifies the relational state of at least two roles; (ii) it is recognized within the procedure as an interaction, rather than being incidental to it; and (iii) it alters the space of transitions subsequently available.

The third condition is the one that does the work, and it is supplied by the framework rather than by the mathematics. An interaction, on this account, is contact that changes what can happen next. Two participants who are simultaneously present in a room have not thereby crossed. Two participants one of whom has advanced a claim the other must now answer have.

The condition carries an implication that should be entered explicitly.

Claim 2 (Inherited ontology). Event individuation is performed before topological encoding, and by criteria the braid does not contain. Condition (iii) refers to a space of future possibility of which the braid has no representation whatever. The formalism therefore inherits an event ontology it does not generate.

The shape of this dependence is familiar: a coordinate system does not generate the geometry it describes, and a notation does not generate the meanings it records. What follows for the model is that its outputs are only as well-founded as the individuation supplied to it, and that disagreement about a case may be disagreement about the encoding rather than about the topology. Such disagreement is at least locatable, which is more than the alternative offers.

Orientation. The generator $\sigma_i$ and its inverse $\sigma_i^{-1}$ differ in which role passed over the other. This is the orientation of the interaction, and it is precisely that: which of the two roles was, in the event, the one whose position prevailed. Figure 1 shows the two. It is not a measure of power, of domination, of advantage, or of injustice, and the mapping does not license the inference from orientation to any of these. Two parties may exchange positions in a hundred ways of which the topology sees only one bit, and a role that prevails in an interaction may be the weaker party prevailing, the stronger party conceding, or neither. Asymmetry, where the paper needs it, is defined separately in §4.4 and is not read off the generator.

History and outcome. A braid word is the ordered record of the interaction events, and the braid is the equivalence class of such records under the relations of §2.3. Its image $\pi(b)$ is the terminal configuration of roles: the rearrangement of relational positions that the procedure effected.

It is tempting to call this the distributional outcome, and the temptation should be resisted. The two coincide only where the distribution is a function of terminal relational position, where, that is, what each party receives is determined by where each party ends. This condition is often satisfied and is not trivial, and §5.2 exhibits a procedure in which it fails: two assemblies may approve the same allocation while leaving the roles in different terminal arrangements, or leave the roles as they were while altering what is allocated. Where the condition fails, outcome-equivalence in the model’s sense and outcome-equivalence in the distributional sense come apart, and the model tracks the former.

What the braid does not contain. The correspondence must be stated with a negative half, or it will be read as claiming more than it does. The braid is not the procedure. It is the relational event history generated by a procedure. A procedure comprises rules, authorizations, the information available to each party, the intentions with which parties act, the meanings the parties attach to what occurs, and an institutional setting that constrains all of these. None of this is in the braid. The procedure generates the braid; the braid does not recover the procedure, and two procedures differing in every respect just enumerated may generate the same braid.

4.2 Outcome-Invariant Procedural Difference

We may now state what the structure supplies against the gap of §3.5.

Let a procedure generate a braid $b \in B_n$. Its outcome is $\pi(b) \in S_n$, and the procedures issuing in a given outcome $s$ are those whose braids lie in the fiber $\pi^{-1}(s)$. By the criterion above, two procedures are outcome-equivalent exactly when their braids differ by an element of $P_n$. The differences among outcome-equivalent procedures are therefore not a heterogeneous collection of dissimilarities; they are the elements of a group, they compose, they invert, and two of them may be compared and found equal or unequal by a decidable procedure.

The first consequence concerns Rawls. Pure procedural justice, as §3.1 observed, has no resources for comparing two correctly conducted procedures. In the present terms, the doctrine evaluates in $S_n$. This is a statement of where its instruments look, and no slight against them. And it makes precise what was earlier said loosely: an evaluation conducted in $S_n$ is structurally blind. It is not incomplete in the way a coarse measurement is incomplete. There is no refinement of outcome evaluation that recovers $P_n$, because $P_n$ is by construction what the projection annihilates.

The second consequence concerns identity. The claim that two procedures differ, in the frameworks of §3.5, is settled by the analyst’s description. Here it is settled by the criterion above together with the decidability of §2.6: given encodings, whether $b_1 b_2^{-1} = e$ has an answer, and the answer does not depend on who is asking. The encoding remains the analyst’s, by Claim 2; but the analyst who has supplied an encoding has thereby given up the freedom to redescribe.

Claim 3 (The object). The pure braid group is a structure of the kind §3.5 found missing: the histories issuing in a given outcome form the fiber $\pi^{-1}(s)$; the differences among them are the elements of $P_n$, acting within the fiber; and the identity of two procedures is decidable independently of description. Outcome-based evaluation is blind to $P_n$ by construction, not by coarseness.

What does not follow must be stated in the same breath. It does not follow that the elements of $P_n$ are justice-relevant, singly or collectively. $P_n$ is a space of candidate differences, and the claim that a candidate matters is a claim of a different kind, requiring an argument the topology cannot supply. A reader who takes Claim 3 to have established that procedural differences matter has read into it the conclusion §4.7 exists to withhold.

4.3 Sequence and Procedural Indifference

Deliberative theory, by §3.3, knows that order matters and treats order effects as distortions. It has no positive characterization of which reorderings are substantive. The relations of §2.3 supply one, and it has two parts.

Far commutation says that crossings involving disjoint pairs of roles may be reordered freely. Where the objection of one party to a second and the objection of a third party to a fourth are the events in question, the sequence between them is not represented at all: the two orders generate the same braid, and no evaluation sensitive only to the relational history can distinguish them. This is a strong statement, and it is falsifiable. If participants reliably distinguish such orders, and the distinction is not attributable to something outside the relational history, then either the encoding has mis-individuated the events, the two objections were not in fact disjoint, or the model is wrong at this point.

The braid relation says that among overlapping crossings, one specific reordering is likewise indifferent: $\sigma_i \sigma_{i+1} \sigma_i$ and $\sigma_{i+1} \sigma_i \sigma_{i+1}$ are the same braid. Where three roles interact pairwise in sequence, there are two orders that produce the same relational history and others that do not. The relation thus divides reorderings of overlapping events into the indifferent and the substantive, which is exactly the division §3.3 found the literature unable to draw.

The limit of this test is as important as the test. That two procedures are the same braid establishes that they are the same relational history. It does not establish that they are the same in any other respect, and the respects in which they may differ are unbounded, since by §4.1 the braid contains no rules, intentions, information, or meanings. Two Yang-Baxter-equivalent sequences may differ in whether the interactions were conducted with respect or with humiliation. The braid sees one object; the participants may see two. The model’s verdict of indifference is a verdict about relational topology and about nothing else.

4.4 Asymmetry and Persistence

The linking number was defined in §2.5 as the signed count of crossings between two strands. Its interpretation is now proposed, and the proposal is deliberately weak.

Claim 4 (Persistence observable). For a pure braid $b$ and roles $i, j$, the quantity $\operatorname{lk}_{ij}(b)$ is an observable of relational persistence: the extent to which the interaction between two roles has accumulated in a consistent orientation over the history rather than cancelling. It is not a measure of dominance, inequality, advantage, or injustice, and no such reading is licensed by the topology.

The reason for the weakness is that a stronger reading fails on cases. A high linking number between two roles is consistent with sustained domination of one by the other; it is equally consistent with a productive and repeated engagement in which one party consistently carries a particular function. The topology does not distinguish these, and a definition asserting that it does would be false.

The connection to the framework’s account of asymmetry and absorption may now be made. Generativity, in that account, absorbs asymmetry rather than accumulating it: a relation in which asymmetries arise and are worked through differs from one in which they arise and persist. In braid terms, absorption appears as reduction, a history whose word, however long, reduces; whose crossings meet their inverses; whose linking numbers return toward zero. Persistence appears as its failure. The correspondence is exact enough to be useful and must be handled carefully, because reduction is a property of the braid and absorption is a claim about the relation, and the second does not follow from the first.

A further limit applies here, and it constrains the model more than any other. Winding has no direction. The braid $\sigma_i^{k}$ records that two roles wound about one another $k$ times; it does not record whether the winding was a deepening or a deadlock. A negotiation that returns twenty times to the same issue because each return teaches the parties something, and a negotiation that returns twenty times because neither party can move, generate the same topological signature. The distinction between them is real and consequential, and the braid does not carry it.

The situation admits a precise statement. What a braid supplies is the kinematics of a relational history, the structure of the trajectory, and not its dynamics. A trajectory does not determine the forces that produced it, and no amount of refinement of the trajectory’s description will supply them. Anything the model is asked to say about whether a history was generative or degenerative therefore requires a state variable the braid does not carry, together with an interpretation of the sort §4.7 formalizes.

4.5 Revisability and Reversal

That $\sigma_i$ and $\sigma_i^{-1}$ are distinct and mutually inverse gives the model a natural image of correction: an interaction that reverses an earlier one. Appeal, redress, the reconsideration of a decision, the retraction of a claim, each is an event whose orientation is opposite to one that preceded it.

Figure 5 (Reversal and residue). Left: a crossing immediately followed by its inverse cancels, and the relational history is as if neither had occurred. Right: the same reversal separated from its original by an interaction touching one of the same roles does not cancel, and the history retains a trace of an event formally undone.

The structure then says something that is not obvious and is not trivial. Where a crossing is immediately followed by its inverse, the two cancel: $\sigma_i \sigma_i^{-1} = e$, and the relational history is as if neither had occurred. Where they are separated in the word by other crossings, they need not cancel, and in general do not. Reversal at a distance leaves residue.

Figure 5 contrasts the two cases. The reason is structural. Between the crossing and its reversal, other interactions have occurred, and the relations of §2.3 permit the reversal to be brought back to meet its original only when the intervening crossings commute past it, which, by far commutation, requires that they involve roles disjoint from the pair in question. Where the intervening interactions touched the same roles, the cancellation is blocked, and the history retains a trace of an event that has been formally undone.

The reading for appeals procedures is direct, and it is a genuine output of the model rather than a restatement of its input. A decision reversed on appeal is not thereby a decision that was never taken, and whether the reversal restores the prior relational state depends on what occurred in between and on whom it involved. The model does not merely permit this observation; it predicts when the residue occurs and when it does not.

4.6 Degradation

The concept treated here does not come from the literature of §3. It is supplied by generative relational being, where degradation names the failure of a relation to continue generating, and it is entered here because the braid structure sharpens a distinction that framework draws informally. It is treated only to the depth the present argument requires.

The framework distinguishes two levels. At the first, generation is blocked while the grammar of the relation remains intact: the parties continue to interact in the terms they have, and those terms cease to produce anything new. At the second, the grammar itself is rewritten: what counts as an interaction, who the parties are, what moves are available, these change, and the relation that results is a different relation, rather than a degraded version of the earlier one.

In the model, the second level is unambiguous. A rewriting of the grammar is an alteration of the generators or the relations among them, and a configuration so altered is not an element of $B_n$ at all. Level 2 degradation is exit from the group.

The first level is not unambiguous, and the previous subsection has already shown why. The braid signature of blocked generation, a word that grows without reducing, winding accumulating between a fixed pair of roles, is exactly the signature of productive repetition. The signature is necessary and it is not sufficient. To assert Level 1 degradation of a history exhibiting it, one requires in addition a judgment about whether the space of relational possibility contracted or expanded, and the braid contains no such space.

The asymmetry between the levels may be stated as follows.

Claim 5 (Detectability asymmetry). Level 2 degradation is detectable as failure of the model: the encoding breaks, and the breaking is the finding. Level 1 degradation is not detectable within the model at all, its topological signature being shared with generative repetition; it requires an interpretation supplied from outside.

4.7 The Interpretive Operator

Everything above has been withheld from the same conclusion, and the withholding may now be given a form.

The model represents outcome-invariant procedural differences as elements of $P_n$. It does not evaluate them. Some of these differences are justice-relevant; some are not; and the assignment of significance is made on grounds that are not topological. Write

$$\Phi \colon P_n \times C \longrightarrow \mathcal{R}$$

for the operator effecting that assignment, where $\mathcal{R}$ is a space of relational significance and $C$ carries what the braid does not: role meanings, institutional setting, the information available to the parties, the content of the interactions.

The context argument is not a formality. The same element of $P_n$ may signify differently in different settings, and the case that shows it most economically is the reversal of §4.5. A crossing followed by its inverse is one element of the group. Occurring in one setting it is a healthy correction; in another the public humiliation of the party reversed; in a third a forced retreat under pressure. The topology is identical in all three. Any operator assigning significance to the topology alone would have to give them the same value, and would be wrong twice.

For readability the paper writes $\Phi$ as acting on $P_n$ where the context is fixed and understood. The full form is above, and the abbreviation should not be mistaken for a claim that significance is recoverable from topology.

This paper does not construct $\Phi$. The claim entered is weaker and is not, on that account, empty.

Claim 6 (Necessity and localization). An account of procedural justice that distinguishes the procedural differences that matter from those that do not thereby employs an operator of the type above, whether or not it is written down. A formalism that separates the representation of differences from their evaluation has therefore performed a service even where the evaluative half is left empty: it converts a difficulty distributed throughout an account into a single identified object.

The objection that the hard part has been moved into $\Phi$ is correct, and is the point. What has been gained is that it has been moved somewhere, and can now be examined.

4.8 Semantics and Failure Conditions

The table below fixes the mapping. The third and fourth columns are the operative ones: a correspondence that licenses nothing specific is decoration, and one whose failure conditions are unstated cannot be wrong.

Domain	Braid object	Licenses	Does not license
Relational position	Strand	Fixed role structure through a procedure	Individuals; emerging or dissolving identities
Interaction event	Crossing	Countable, ordered events meeting the three conditions	Continuous influence; incidental co-presence
Orientation of interaction	$\sigma_i$ vs. $\sigma_i^{-1}$	Which role’s position prevailed	Power, dominance, advantage, injustice
Procedural history	Braid word	Ordered record; comparison; decidable identity	The procedure itself, with its rules and meanings
Distributional outcome	$\pi(b) \in S_n$	Terminal configuration of roles	The distribution, except where it is a function of that configuration
Procedural difference	Element of $P_n$	Composition, inversion, comparison of differences	That the difference is justice-relevant
Accumulated interaction	$\operatorname{lk}_{ij}(b)$	Persistence of orientation across a history	Domination; inequality; injustice
Absorption of asymmetry	Word reduction	Formal image of asymmetry worked through	That reduction was generative
Correction, appeal	$\sigma_i^{-1}$ after $\sigma_i$	Cancellation when adjacent; residue at distance	That the correction was adequate or fair
Level 1 degradation	Unreduced growth	A necessary signature	Sufficiency; generative repetition shares it
Level 2 degradation	Exit from $B_n$	Detection as encoding failure	Representation of what replaced the grammar
Relational significance	$\Phi(p, c)$	Location of the evaluative step	Any particular assignment of value

Three failure conditions apply to the mapping as a whole, and each is testable in principle. First, if participants in procedures reliably distinguish far-commuting reorderings, and the distinction is not traceable to mis-individuated events or to content outside the relational history, the treatment of independence in §4.3 is wrong. Second, if procedures whose braids are equal are found to differ in respects the model attributes to the braid, the encoding is under-specified or the correspondence is too coarse. Third, if the elements of $P_n$ arising in practice turn out to be uniformly justice-relevant, then $\Phi$ is constant and the elaborate apparatus of representation without evaluation is unmotivated, the model would then be claiming a distinction that does no work. §5 takes up the third directly, since a model that cannot produce an insignificant difference is a machine for confirming itself.

§5 Constructed Cases

5.1 Method

The four cases below are constructed rather than observed. The choice is deliberate and follows the practice of modeling in general: a constructed case can be specified so that the quantity under test is the only thing varying, which no observed case permits. Historical instances carry uncontrolled variation in the dimensions the model claims to be sensitive to and in those it claims to be blind to, and a model tested against them fails or succeeds for reasons that cannot be isolated. The cases here are accordingly fictional, and are built to be realistic in the sense that each could occur, not in the sense that each did.

The cases are built for discrimination rather than illustration. A model that produces a difference wherever a difference exists has established nothing, since by Claim 3 nontrivial elements of $P_n$ are guaranteed and their exhibition is arithmetic rather than evidence. The four cases accordingly test four distinct things. Case A tests sensitivity: whether the difference the model isolates is the difference participants contest. Case B tests specificity: whether the model can produce a procedural difference that carries no significance, which by §4.8 it must be able to do or be unfalsifiable. Case C tests underdetermination: whether procedures the model identifies may differ in ways that matter. Case D tests the boundary: whether the model fails where §2.7 says it must, and whether the failure is legible when it comes.

The encoding protocol is uniform. In each case the roles are named and fixed; the events are individuated by the three conditions of §4.1; the orientation of each crossing is stated; and the resulting word is given. Where two variants are compared, their images under $\pi$ are computed and the separating element of $P_n$ is exhibited.

5.2 Case A: The Village Water System

Two villages decide whether to build a community water system. In both, the decision is taken by an assembly; in both, the same system is approved, with the same budget, the same allocation of access, and approval by the same margin.

Three roles are relevant, and they are stable throughout in each village: the council, $1$; the technical advisor, $2$; and the households whose water access is at stake, $3$. Individuals vary and the roles do not, which is the condition Claim 1 requires.

Village A. The council raises the proposal, and the advisor’s assessment modifies it: $\sigma_1$. The households then object that the proposed siting will disadvantage the outlying settlements, and the advisor revises the design in response: $\sigma_2$. The revised design returns to the council, which accepts the revision: $\sigma_1^{-1}$. The assembly approves. The word is

$$b_A = \sigma_1 , \sigma_2 , \sigma_1^{-1}.$$

Village B. The council raises the proposal and the advisor’s assessment modifies it: $\sigma_1$. The council then settles the design: the advisor’s revision is accepted before consultation: $\sigma_1^{-1}$. The households are consulted afterward, raise the same objection about the outlying settlements, and it is recorded: $\sigma_2$. The assembly approves. The word is

$$b_B = \sigma_1 , \sigma_1^{-1} , \sigma_2 = \sigma_2 .$$

Outcome. $\pi(b_A) = (1,2)(2,3)(1,2) = (1,3)$ and $\pi(b_B) = (2,3)$. These are not equal, and the case as stated does not yet exhibit what it was built to exhibit.

This is a finding about the encoding rather than an oversight to be repaired quietly. Two procedures with the same distributional outcome must have the same permutation of roles, and the permutation records which role ends in which position. The difficulty is that “the same decision was taken” is not the same claim as “the roles ended in the same relation.” In Village B, the households did not exchange position with the advisor in the course of the decision; their objection was recorded after the design was fixed. In Village A they did.

The repair is to encode the recording of an objection that alters nothing as what it is: an event that fails condition (iii) of §4.1, since it does not alter the space of transitions subsequently available. It is not a crossing. Village B’s word is then

$$b_B = \sigma_1 , \sigma_1^{-1} = e,$$

and the two procedures now have different outcomes for a reason that is substantive rather than technical: in Village B, no relational rearrangement occurred at all.

What the case establishes. Not what it was designed to establish, and something more useful. The case was constructed to show two procedures with the same outcome differing by a pure braid. It shows instead that the difference between deliberation-before-decision and consultation-after-decision is not an outcome-invariant difference at all. It is a difference in outcome, in the model’s sense of outcome, which is the rearrangement of relational positions, and not the distribution of water. The households who altered the design occupy a different terminal position from the households whose objection was filed.

This is a result about the model’s sense of outcome, and it sharpens §4.2. The projection $\pi$ does not record the distribution. It records the terminal configuration of roles, and a procedure may leave the distribution untouched while rearranging the roles, or leave the roles untouched while altering the distribution. The identification of $\pi(b)$ with “what outcome evaluation sees” in §4.1 was too quick, and the honest statement is narrower: $\pi(b)$ is what an evaluation of terminal relational position sees, which coincides with distributional evaluation only where the distribution is a function of that position.

Sensitivity, retested.

Figure 6 (Case A, retested for sensitivity). Both procedures leave the roles in the same terminal arrangement, so no evaluation of outcome distinguishes them. In the second, council and advisor exchange and re-exchange position before the households are heard. The procedures differ by $\sigma_1^{2}$, a pure braid of linking number one.

A case that does test sensitivity may be built from the same material. Suppose in both villages the households do alter the design, so that $\pi(b_A) = \pi(b_{A’})$, but in Village $A’$ the alteration comes after the advisor and the council have twice exchanged and re-exchanged positions over the siting question:

$$b_{A’} = \sigma_1^{2} , \sigma_1 , \sigma_2 , \sigma_1^{-1} , \qquad b_A = \sigma_1 , \sigma_2 , \sigma_1^{-1} .$$

The two are drawn in Figure 6. Then $b_{A’} b_A^{-1} = \sigma_1^{2} \in P_3$, a pure braid of linking number one between council and advisor. The procedures are outcome-equivalent and differ by a determinate element of the kernel. Whether participants contest this difference is the substantive question, and the answer is not obvious: an extended exchange between council and advisor before the households are heard may be experienced as due diligence or as the settling of the matter in advance. The model isolates the difference and hands the question on. That is the whole of what Claim 3 promised, and the case shows the promise kept and its narrowness.

5.3 Case B: Two Sittings of a Committee

A committee of three roles, chair $1$, first member $2$, second member $3$, considers two motions in a single sitting. The motions are unrelated: the first concerns the maintenance schedule, the second the wording of an annual report. In each, one member raises a point and the chair rules on it.

In sitting $X$, the maintenance point is raised and ruled on, then the report point: $b_X = \sigma_1 \sigma_2$. In sitting $Y$, the order is reversed: $b_Y = \sigma_2 \sigma_1$. Both sittings dispose of both motions identically.

Here $\pi(\sigma_1 \sigma_2)$ and $\pi(\sigma_2 \sigma_1)$ are the two distinct three-cycles, so this pair is not outcome-equivalent, and the far commutation of §4.3 does not apply either, the two crossings sharing the middle strand. Construct instead the comparison the test requires. Let each sitting comprise the same three substantive exchanges, and let sitting $Y$ open with a preliminary consultation between the two members in which each defers to the other in turn and the matter is left where it stood:

$$b_X = \sigma_1 , \sigma_2 , \sigma_1 , \qquad b_Y = \sigma_2^{2} , \sigma_1 , \sigma_2 , \sigma_1 .$$

Then $\pi(b_X) = \pi(b_Y) = (1,3)$, and

$$b_Y , b_X^{-1} = \sigma_2^{2} \in P_3 ,$$

a pure braid of linking number one between the two members. Figure 7 shows the two sittings and the element separating them. The element is nontrivial: sitting $Y$ contains a full winding between the members that sitting $X$ does not, and by §2.6 the fact that the two sittings are different procedures is decidable rather than a matter of the analyst’s description.

Figure 7 (Case B). The two sittings leave the roles identically arranged and differ by the pure braid on the right: a full winding between the two members, arising from a preliminary consultation on a matter about which they agreed. The model reports a difference; no participant has reason to care about it.

Assessment. No participant regards the sittings as differing in fairness, and there is no evident ground on which one should. The additional exchange in sitting $Y$ was a consultation between two members about a matter on which they agreed, conducted in the ordinary course, affecting no one’s standing and no one’s opportunity to be heard. The model reports a difference; the difference is of no interest.

Claim 7 (Specificity). The case exhibits a nontrivial $p \in P_n$ for which $\Phi(p, c) = 0$ on any reasonable assignment. The model is therefore not a machine for converting topological difference into normative difference, and the third failure condition of §4.8 is not met here.

Omitting this case would be a serious fault. Without it, every difference the model isolates would be available for treatment as significant, and the apparatus of §4.7, the insistence that representation and evaluation are separate, would be unmotivated. A model that cannot produce an insignificant difference has no need of an operator to filter differences, and its author’s protestations that topology carries no normative weight would be belied by a practice in which topology always did.

Toward a criterion. What distinguishes Case A’s contested pure braid from Case B’s uncontested one? A conjecture, offered as such: the winding in Case A occurred between roles whose relative position bore on the matter under decision, and altered the sequence in which a third role could enter; the winding in Case B occurred between roles whose exchange bore on nothing at issue and foreclosed no subsequent move. The suggestion is that the relevant feature is the pure braid together with its relation to the possibility space, which is to say, precisely the content that condition (iii) of §4.1 appeals to and the braid does not contain. If this is right, $\Phi$ draws on the same information that event individuation draws on, and the two problems are one problem. The conjecture is not established here.

5.4 Case C: The River Treaty

Two states share a river and negotiate its allocation. Three roles: the upstream state $1$, the downstream state $2$, and the technical commission $3$ that certifies the hydrological basis of any agreement. In both versions the outcome is a fifty-fifty allocation, certified, on the same schedule.

Version I. The upstream state opens with a demand for the greater share, which the downstream state refuses: $\sigma_1$. The commission’s assessment establishes the downstream state’s agricultural dependency, and the upstream state revises: $\sigma_2 \sigma_1^{-1}$. The downstream state, learning the upstream state’s energy constraint, concedes on timing: $\sigma_1 \sigma_2^{-1}$. The parties converge. The word is

$$b_{\mathrm{I}} = \sigma_1 , \sigma_2 , \sigma_1^{-1} , \sigma_1 , \sigma_2^{-1} = \sigma_1 .$$

Version II. The upstream state opens with the same demand and the same refusal follows: $\sigma_1$. The upstream state signals that a trade agreement under separate negotiation may be affected; the commission’s assessment is entered and the downstream state’s position shifts: $\sigma_2 \sigma_1^{-1}$. The downstream state accepts the timing terms: $\sigma_1 \sigma_2^{-1}$. The word is

$$b_{\mathrm{II}} = \sigma_1 , \sigma_2 , \sigma_1^{-1} , \sigma_1 , \sigma_2^{-1} = \sigma_1 .$$

Figure 8 (Case C). The two panels are identical, and that is the finding. Each crossing is annotated with what occurred on that occasion; the annotations differ throughout, and the braid records none of them. Both words reduce to the single crossing at right. The difference between agreement reached by learning and agreement reached under duress is not an element of $P_n$ awaiting evaluation, it is absent from the representation.

Assessment. The braids are equal. Not outcome-equivalent, not differing by a pure braid: equal. The model reports that these are the same procedure, and by the criterion above together with the decidability of §2.6, the report is not a matter of judgment.

Figure 8 draws the two encodings side by side, annotating each crossing with what took place on that occasion. The diagrams are indistinguishable and the annotations are not. The procedures are not the same. In Version I the downstream state’s shift followed from information about the river; in Version II it followed from a threat concerning something else entirely. The parties know the difference, any observer knows it, and the relation the two treaties leave behind is not the same relation. A treaty reached by mutual learning and a treaty reached under duress may be word-for-word identical and are not the same achievement.

Nothing in the encoding is wrong. The roles are correctly individuated, the events meet all three conditions, the orientations are correct. The model has done what it does, and what it does not do is carry the content of an interaction. By the last item of §2.7, two crossings of the same pair in the same sense are the same generator, whatever occurred on the two occasions. Version I’s third event and Version II’s third event are both the downstream state shifting its position in response to something the upstream state did. The topology sees a shift.

What would be required. The distinction demands a state variable the braid does not carry: minimally, whether the shift was produced by a change in the shifting party’s information about the object of the negotiation or by a change in its expectations about matters outside it. This is not a refinement of the topology and cannot be obtained by encoding more carefully. It is the content of the crossing, and the model’s silence on content is structural.

The case accordingly confirms §4.1’s negative half in a strong form. Not that the braid omits some of the procedure, but that it omits enough of it that two procedures differing in what is arguably the most important respect available, whether agreement was reached through understanding or through pressure, are formally indistinguishable. The winding observation of §4.4 is a corollary rather than the point: here the words do not merely wind alike, they are the same word.

5.5 Case D: The Union

A firm negotiates terms with its workers. Initially each worker negotiates individually, and the relevant roles are the firm $1$, worker $A$ as $2$, and worker $B$ as $3$. The firm makes an offer to $A$, who accepts a variation: $\sigma_1$. The firm makes a different offer to $B$: $\sigma_2$, and $B$ refuses: $\sigma_2^{-1}$. So far the encoding proceeds.

The workers then form a union. Subsequent negotiation is between the firm and the union, and the union bargains for both workers together.

The point of failure. The encoding cannot continue, and where it stops can be stated exactly. A report of general breakdown would conceal the structure of the failure. Three distinct things have occurred, and §2.7 treats them differently.

Two strands have merged into one. The union is the role in which the several positions of $A$ and $B$ are now jointly occupied, and not a fourth role alongside them. The configuration after formation has fewer relational positions than before, and $B_3$ and $B_2$ are different groups with no operation of the theory carrying an element of one to the other. This is the fixed-$n$ ceiling, and it is reached exactly at the formation.

A role has been created. The union did not exist before and exists after, and its existence is a product of the interactions the braid was recording. By Claim 1 the model applies after role stabilization; here the procedure destabilized the roles, doing the thing the model’s domain excludes.

The grammar has changed. Before formation, a refusal by one worker left the other’s negotiation untouched, and the two were independent in the sense far commutation formalizes. After formation there is no such independence, because there is no longer a pair of positions to be independent. What counts as an interaction, and which interactions are available, have both been altered. This is Level 2 degradation in the sense of §4.6. The term formally denotes a rewriting of the grammar, and carries no implication of deterioration: union formation may be an improvement in the respects that matter.

The finding. These three are separable, and separating them is the case’s contribution. Strand merging, strand creation, and relation redefinition are distinct failures requiring distinct extensions: a structure permitting merges, a structure permitting births, and a structure in which the relations among generators may themselves vary. A single gesture at “the model breaks here” would conceal that three different repairs are needed. §6 takes up what the repairs would involve.

The legibility of the failure is itself a result. The model does not degrade gracefully into wrongness; it stops, at an identifiable event, for reasons that name themselves. A formalism whose domain boundary is visible from inside is more useful than one that continues to produce output after its assumptions have lapsed.

5.6 Cross-Case Comparison

Case A tested sensitivity and returned, first, a correction: the model’s outcome is the terminal configuration of roles, not the distribution, and the two coincide only under conditions the paper had not stated. With the correction in hand, the case exhibits a contested pure braid and hands the question of its significance on, which is what the model claims to do and the limit of what it claims.

Case B tested specificity and returned a nontrivial element of $P_n$ that no one has reason to care about. This is the case without which the model would be unfalsifiable, and it also produced the beginning of a criterion for $\Phi$: the difference between the two windings appears to lie in their relation to the space of subsequent possibility, which is content the braid does not contain and event individuation already requires.

Case C tested underdetermination and returned a negative result stronger than the one it was built for. Two procedures differing in whether agreement was reached by learning or under duress generate the same word, not merely words differing by an element of the kernel. The model’s silence on content is a wall, and not a residual imprecision.

Case D tested the boundary and returned a legible failure decomposing into three distinct ceiling violations. The model stopped where §2.7 said it would, at an identifiable event, and the manner of stopping distinguished three different extensions that would be required.

Taken together the four establish the necessity of $\Phi$ from two directions at once. Case B shows that topological difference does not suffice for significance: there are differences $\Phi$ must send to zero. Case C shows that topological identity does not suffice for insignificance: there are significant differences $\Phi$ cannot see at all, because they are not in $P_n$ to be evaluated. An operator is required, and it must draw on information the braid does not carry, which is why the operator carries a context argument, and why the conjecture of §5.3 locates that information in the same place event individuation finds it.

The boundary the cases jointly draw may then be stated. Within its domain the model represents procedural differences determinately and adjudicates procedural identity decidably. It does not evaluate what it represents, it does not represent the content of what it records, and it ceases to apply when the roles it presupposes are themselves in formation. Each of these three is a limit of a different kind, and §6 treats them in turn.

§6 Discussion

6.1 Yield

Five things, of unequal weight.

A decidable criterion of procedural identity. By §2.6 and the criterion above, whether two encoded procedures are the same is answerable by a procedure independent of the analyst. This is the component that answers the second half of the gap in §3.5, and it is what distinguishes the proposal from the frameworks that already represent history well. Encoding remains the analyst’s, by Claim 2; but an analyst who has encoded has forfeited the freedom to redescribe.

A criterion of procedural indifference. Far commutation and the braid relation divide reorderings into those that change the relational history and those that do not. Deliberative theory wanted this division, as §3.3 argued, and its diagnostic framing could not supply it. What the model gives is narrow, indifference of relational topology, not of experience, and it is a positive characterization where none existed.

A prediction about reversal. §4.5 is the one place the formalism generates a result rather than restating an intuition in new notation. Cancellation of a crossing by its inverse is complete when they are adjacent and incomplete when separated by interactions touching the same roles; and this is derived, not stipulated. Appeal restores the prior relational state under a condition the model specifies. Whether the condition is empirically right is testable and is not tested here.

An observable of persistence. The linking number, under the deliberately weak reading of Claim 4, measures accumulation of orientation across a history. That it is not a measure of injustice is what makes it usable: a quantity defined as injustice would have prejudged the questions worth asking.

A location for the evaluative step. Before the operator was written, the interpretive move in accounts of procedural justice is distributed, some of it in the individuation of what counts as a procedural feature, some in the selection of which features to measure, some in the normative argument proper. The model does not remove the move. It collects it.

6.2 Direction and Its Absence

A formal structure imported into normative theory carries a risk of smuggling a direction of travel. The risk bears naming because the framework this paper belongs to has committed itself against it, and a formalism that reintroduced it through the back door would undo the commitment more effectively than an argument could.

The braid group supplies no direction. There is no partial order on $B_n$ under which some braids are more advanced than others; length is not such an order, since a longer word may represent a shorter braid; and composition orders events in time without ranking configurations. No braid is progress. Two procedures related by an element of $P_n$ stand in no relation of priority whatever, and the model has no resources for saying otherwise.

This limitation of the formalism is simultaneously a protection. Because no normative ordering is available in $B_n$, none can enter the account by way of $B_n$; whatever normative content the account carries must be carried by $\Phi$, where it is visible and can be argued with. The conatus back door, the move by which persistence or elaboration or increase becomes self-justifying, is closed here by absence rather than by prohibition. There is nothing in the structure that could be mistaken for a direction.

6.3 Limits

Six limits, each traced to §2.7 and each now furnished with a case or an argument.

Emergence of roles. Fixed $n$ forbids strand creation, and Case D reached the prohibition at an identifiable event. The framework’s account of subject production, relational positions generated by the interactions they then participate in, is thus outside the model, not partially represented within it.

Merging and dissolution. Distinct from creation, and requiring a distinct repair. Case D exhibited two workers becoming one bargaining position; nothing in $B_n$ carries $B_3$ to $B_2$.

Alteration of the grammar. Distinct again. Where the relations among generators change, where what counts as an interaction, or which interactions are available, is itself rewritten, the configuration is not an element of the group. This is Level 2 degradation in the sense of §4.6, and its detectability is entirely negative: the model reports it by failing.

Continuous influence. Crossings are discrete events. A relation that shifts gradually without punctuating events has no representation, and the three conditions of §4.1 will find no crossings to individuate.

The content of an interaction. Case C is the demonstration. Two procedures identical in topology may differ in whether agreement was reached by learning or under duress, and the difference is substantial and is not recoverable by more careful encoding. The braid records that an interaction occurred with an orientation. It records nothing of what the interaction was.

The direction of winding. By §4.4, repetition that deepens and repetition that deadlocks share a signature. Any claim about degradation at Level 1 accordingly requires a state variable the model does not contain. A concept central to the framework is therefore one the topology cannot adjudicate.

6.4 Layers of Evaluation

The scope declaration of §1 may now be developed, since the model is in hand and its boundary is known. This subsection states where the contribution sits and is not itself the contribution, entered so that the reader does not take a treatment of one layer for a treatment of the whole.

The architecture runs: a relational process generates a history; the history is represented as a braid; the projection $\pi$ discards the path and retains the terminal configuration; what it discards is characterized by $P_n$; and $\Phi$ assigns significance to elements of $P_n$ in context. Everything this paper has done lies at or below $\Phi$. Above it lie questions of evaluation, and they are several questions of different kinds.

Generativity ($G$) asks whether an interaction expands the space of relational possibility available afterward. This is descriptive, not normative, a fact about a process, ascertainable in principle without any judgment about whether the expansion is welcome. It is also, and this is the awkward part, presupposed by the model already: condition (iii) of §4.1 individuates events by whether they alter the space of subsequent transitions, which is to say that the encoding requires a notion of possibility space that the braid does not represent. The gap is load-bearing. A formal account of $G$ would be a precondition for a fully specified encoding, not merely an extension of it.

Procedural justice ($J_p$) asks whether generation occurred through legitimate relations. This is the layer treated here, and it is a question about legitimacy rather than about capacity or about value.

Generative justice ($J_g$) asks whether a process preserves and expands the capacity of its participants to generate. This is the distinctively framework-specific question, and it is independent of $J_p$ in both directions. A procedure impeccable by the criteria of §3 may leave its participants less able to generate than it found them, consuming trust, foreclosing positions, exhausting the conditions on which future interaction depends, and the case is neither exotic nor rare. That $J_p = 1$ is compatible with $J_g = 0$ is the reason procedural justice cannot be the whole of what the framework requires.

Normative orientation ($V$) asks toward what. Neither generativity nor procedural legitimacy answers it, and two examples establish it. A hurricane is a highly generative dissipative structure, producing organization across scales; nothing follows about its value. A community may fairly, transparently, and with full participation design a more efficient system of exploitation; nothing about the fairness of the design redeems what was designed. Generativity is not value, and legitimate procedure is not value, and an account that lets either stand in for the third commits the error this paper’s guardrail was erected against.

These four are not four species of one thing, and the differences among them are differences of kind: $G$ descriptive, $J_p$ a matter of legitimacy, $J_g$ a matter of sustained capacity, $V$ a matter of direction. Collapsing them is the characteristic failure. Collapse $J_p$ into $V$ and legitimate procedure becomes self-justifying, which is the teleology §6.2 excludes. Collapse $G$ into $V$ and generation becomes its own warrant, which is the conatus back door the framework has consistently refused. The separation is the same guardrail, positioned where the pressure to breach it is greatest.

As to how the layers might be investigated: $G$ invites treatment in terms of reachability and the size of the option set available after an interaction as against before. $J_g$ admits a candidate formal criterion, whether the set of relational states reachable after a procedure contains the set reachable before, which is offered here as a conjecture and is not developed. $V$ is not, on the present evidence, amenable to formal treatment of the kind attempted here, and naming it as such is preferable to gesturing at a formalism that does not exist.

6.5 Open Problems

Constructing $\Phi$. The central one. The paper has argued that an account discriminating among procedural differences thereby employs an operator of the type above, and has declined to build it. What Case B produced is a starting point and no more: the conjecture that a pure braid’s significance turns on its relation to the space of subsequent possibility, and that $\Phi$ therefore draws on the same information that event individuation draws on. If that is right, the two problems are one, and a treatment of $G$ would supply both.

Structures permitting birth, death, merging, and splitting. Cobordism is the natural direction. Where a braid is a collection of strands running between two fixed configurations, a cobordism is a manifold between two boundaries whose components may join, divide, appear, and vanish; the four failures Case D separated correspond to the four operations a cobordism permits and a braid does not. The framework’s account of subject production requires exactly these, which suggests that the extension is a precondition for treating emergence formally at all.

Compositional settings. Braided monoidal categories generalize the braid group to a setting where objects may be composed, and would be the appropriate home for procedures built from sub-procedures. Higher categorical structures allow the relations themselves to vary, which is what Level 2 degradation would require to be represented rather than merely detected as failure.

Other candidate structures. The braid group is one candidate and the paper has claimed no more. Fundamental groups and groupoids of configuration spaces, mapping class groups, and path spaces more generally are alternatives with different strengths, and a comparison would be informative. It is possible that some other structure supplies decidable procedural identity at lower cost in assumptions; if so, the argument of §3.5 survives and the candidate changes.

Operationalization. The persistence observable of Claim 4 is defined and unmeasured. Whether it can be computed from records of actual procedures, minutes, transcripts, negotiation logs, is an empirical question, and the encoding problem of Claim 2 is where the difficulty would concentrate.

Degradation at length. The two-level distinction of §4.6 has been treated only as far as the present argument required. The asymmetry of Claim 5, one level detectable only as model failure, the other not detectable at all without interpretation, calls for a fuller treatment than the present argument affords it.

6.6 Conclusion

Procedural justice has long described a class of differences that lacked a formal object adequate to represent them. Rawls’s doctrine of pure procedural justice made the procedure constitutive of the justice of its outcome and gave no way to compare two procedures both correctly conducted. Tyler’s programme established that something outcome-invariant is tracked and characterized it by its effects on judgment. Deliberative theory knew that sequence mattered and treated the knowledge diagnostically. Sen widened the object of assessment and left it an object still assessed at its terminus.

The pure braid group supplies one candidate structure for what these accounts described. It is a group of transformations preserving terminal configuration while altering relational history, within which outcome-invariant differences are determinate elements and procedural identity is decidable. That is the positive claim, and the four cases have shown its reach and where it stops. It does not evaluate what it represents. It does not represent the content of what it records, Case C being the demonstration. It ceases to apply where the roles it presupposes are still in formation, which is where the framework’s own deepest questions begin.

What remains, then, is a structural substrate on which evaluations may operate, together with a specification of where each of them must enter. The braid carries the memory of a process. Whether the process was legitimate, whether it left its participants able to continue generating, and whether what it generated was worth generating are three further questions, and the value of a formalism that answers none of them is that it makes the asking of each unavoidable and separate.

Acknowledgments

The argument was refined through successive rounds of critical reading, and several of its central corrections, the status of $P_n$ as a transformation group rather than a collection, the context-dependence of $\Phi$, the directionlessness of winding, and the necessity of a case in which the model produces a difference of no significance, originated as objections. Errors that remain are the author’s.

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中文

辫群与程序正义

生成性关系存在中的路径依赖

黄万宏 · huangwanhong@serendip.ngo

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摘要

两个程序可能发为同一个分配，却仍然并不同等地正义。程序正义诸理论早已认识到这一点，而每一个都以它自己的方式登记了它：罗尔斯以纯粹程序正义的学说，泰勒以”关于公平的判断追踪着某种结果所不记录之物”这一发现，协商理论以它对次序与议程的关注，森以向比较的转向。它们之中没有一个所提供的，是一个其元素就是那些程序性差异本身的形式对象，一个其中两个投影到同一终点的历史仍然可分辨、并且其中”它们是否是同一个程序”这一问题有一个答案的结构。本文提出辫群作为一个候选结构。在投影 $\pi\colon B_n \to S_n$ 之下，该投影保留终点置换而丢弃路径，纯辫群 $P_n = \ker \pi$ 是保持结果而改变关系性历史的变换之群；两个历史是结果等价的，恰当它们相差 $P_n$ 的一个元素之时。本文论证这就是程序正义已经描述、却未曾拥有的那个对象，并同等地论证它本身并不是一个正义的对象。一个给定的程序性差异是否要紧，不由拓扑所裁定，而一个主张相反的说明，把一种它并不承载的规范性内容归给了一个路径依赖的结构。因此，本文引入一个解释性算子，写作 $\Phi$，它在语境中把关系性意义赋予 $P_n$ 的诸元素；它并不构造 $\Phi$，并论证这一类形式体系的贡献是定位解释必须于何处进入，而不是免除解释。四个被构造的案例，就敏感性、特异性、解释上的欠决定性以及边界，检验所得的模型，最后一个展示一个该形式体系根本无法表示的案例。一个结尾节把程序正义置于生成性关系存在所要求的诸评估层级之间，并标出辫所不能承载之物。

关键词： 程序正义；辫群；路径依赖；关系性历史；形式建模；生成性关系存在。

§1 引论

考虑两个抵达同一决定的委员会。成员构成是相同的，表决是相同的，随之而来的收益与负担的分配是相同的，而记入记录的理由也是相同的。在第一个委员会中，一项反对被早早提出，而提案作出回应而被更改。在第二个委员会中，同样的反对是在提案已被敲定之后才被提出，被记录在案，并且什么也没有更改。在终点处评估，这两个委员会无从分辨。由它们的参与者来评估，则它们并非如此：某种东西在第一个中发生了，而在第二个中没有发生，而这一差异仅仅是心理上的，并不显然。

这不是一个边缘的观察。它是关于程序正义的经验文献的反复发现，也是它规范一翼背后那个执着的直觉。人们在意一个决定是如何抵达的，其方式经受得住”该决定无论如何都会是相同的”这一观察。程序正义诸理论已经认识到这一点，并已登记了它，各以它自己的语汇。罗尔斯的纯粹程序正义学说使程序成为其结果之正义的源头、而不是它的工具。泰勒的经验纲领分离出若干因素，即发言、中立、可信赖、尊重，它们独立于结果的有利与否而预测关于公平的判断。协商理论关注次序、议程，以及诸主张被听取的顺序。森从对诸制度的排序转向对已实现诸状态的比较。

这些之中的每一个都登记了程序承载着结果所不记录的分量。它们之中没有一个提供一个其元素就是那些程序性差异本身的对象。这是一种不同于它容易被误当作的那种匮乏。并不是这些说明未能表示历史；一系列框架都很好地表示历史，其中有过程追踪、事件史建模、因果图、扩展式博弈的诸历史，以及那个把锁定弄成一个术语的路径依赖制度主义。所缺的是更狭窄、更具体的东西：一个结构，其中投影到一个给定结果的诸历史构成一个确定的集合，其中它们之间的差异本身就是某种东西的元素，而其中”这两个程序是否相同”这一问题有一个不依赖于分析者之描述的答案。没有这样一个结构，”程序在结果之外要紧”这一主张仍然是一个关于显著性的主张。有了这样一个结构，它便成为一个关于一个空间的主张。

本文提出辫群作为那个结构的一个候选者。这一提议立足于单一一项特征。有一个自然的映射 $\pi\colon B_n \to S_n$，从 $n$ 股上的辫群到对称群，它保留每一股在何处结束，而遗忘它是如何到达那里的。该映射是满的，并且不是单的，而它的核，即纯辫群 $P_n$，恰恰由那些使每一股都回到它自己位置、而一般而言在其间做了某种事的辫所构成。两个辫在 $\pi$ 之下有相同的像，当且仅当它们相差 $P_n$ 的一个元素。如果诸股是关系性位置，诸交叉是互动事件，辫是历史而它的像是结果，那么 $P_n$ 就是那个改变程序性历史、而保持一个基于结果的评估所能看到的一切之变换的群。

核心主张。 纯辫群为程序正义诸理论已经描述、却未曾拥有的那个对象提供一个候选结构。它是一个保持终点结果而改变关系性历史的变换之群，在其中结果不变的程序性差异是确定的元素，而程序的同一性是可判定的。由此并不能推出这些差异因此就是与正义相关的，而 §3.2 展示一个并非如此的差异。它们之中哪些要紧，由一个该形式体系所要求、所定位、而不提供的解释性算子所裁定。

这一主张的两个半部同等地承重，而第二个是更容易被丢失的那个。一个表示程序性差异的形式体系，招致这样一个推论：它所表示的差异就是那些要紧的差异。这个推论是假的。有些程序性差异是与正义相关的；有些被拓扑所记录、而不为任何人所关心；还有些，即恐惧的环境相对于安全的环境、由威胁所榨取的让步相对于由学习所抵达的让步，极为要紧、而对拓扑全然不可见。一个无法容纳全部三者的模型，会比没有模型更糟。辫所提供的，是一个其中前两者彼此可分辨的空间；它所不能提供的，是据以把它们区别开来的判准，或对第三者的任何着力之处。

那个判准在此被写作一个算子 $\Phi$，它把 $P_n$ 的诸元素连同语境一起，携带到关系性意义。本文并不构造 $\Phi$。它论证某种更弱、而若正确则更有用的东西：一个在诸程序性差异之间进行区分的程序正义说明，因此就使用一个这种类型的算子，无论它是否被写下来；而一个使该算子显明的形式体系，即便在它把该算子留空之时，也已执行了一项服务。所主张的贡献是一个问题的定位，而不是它的解决。

在论证开始之前，还须录入一项进一步的界限，因为它界定了整体所能被认为确立之物。程序正义追问关系性生成是否通过合法的关系而发生。它并不追问所生成的是什么，或朝向什么。一个程序可能满足该文献所强加的诸判准，即平等的发言、透明、可修正、免于强制，却发为对一个少数群体的排斥；程序上的合法性并不蕴涵实质上的正义，而程序层的任何形式化都不改变这一点。生成性关系存在提出同一困难的一个更尖锐的版本：一个程序上无可挑剔的过程，可能消耗它自身未来生成的诸条件，留下一段被公平地举行、而不再能够生成任何东西的关系。辫表示一个过程的记忆。它并不决定那个过程是否值得延续。本文所处理的是若干层级之中的一层，而 §6 列出其余各层及其诸关系，而不是把范围留待被推断。

因此，所主张的新颖性可以被相当精确地陈述。不是程序性历史要紧，那已被确立。不是 $P_n$ 含有隐藏的历史，那是一种松散的说法。而是 $P_n$ 刻画了那些在终点评估之下观察上等价的历史之间的诸变换，是 $P_n$ 中的成员资格是可判定的，以及是把这个表示性的问题与那个解释性的问题分开，对任何程序正义理论都是可用的、而未曾被采取。

论述的两项特征出自该论证的性质，并在此预先陈述。第一，数学是在任何解释被置于它之上之前被呈现的，并且是在一个不含任何正义语汇的节中；随后的程序正义说明不含任何辫的语汇。二者的接合只在 §4 中作出，其中每一个对应都被提出、然后被检验。这一纪律不是文体上的。一个与它所要建模的现象一同被组装的形式体系，会逐渐贴合它，而那种贴合将确立不了什么。第二，该形式体系的结构性天花板，即固定的 $n$、股的无创生或毁灭、事件的离散性、对一次互动之内容的沉默，是在 §2 中、在模型被建构之前被陈述的。一个读者应当在被展示该装置所能做什么之前，就知道它所不能做什么。

其余部分如下推进。§2 列出辫群：它的呈示、它的诸生成元之间的诸关系、向对称群的投影及其核、字问题的可判定性，以及那个天花板。§3 列出程序正义诸理论，并分离出本文所处理的那个具体缺口。§4 建构那个对应，依次处理结果不变的差异、排序与程序上的无差别、被累积的不对称、可修正性以及退化，并以固定该映射的语义连同其每一部分会失败的诸条件作结。§5 呈现四个被构造的案例，就敏感性、特异性、解释上的欠决定性以及边界检验该模型。§6 陈述该模型所买到的东西、辫所不能承载之物、程序正义在生成性关系存在所要求的诸评估层级之间所处的位置，以及所留下的工作。

§2 辫群

本节以其自身的术语列出该数学结构。此处没有任何解释被置于它之上，也没有任何正义的语汇出现；已然知晓这一材料的读者可以读到 §2.7 的末尾，那里陈述了结构性天花板，然后越过。这一呈现是标准的，并遵循阿廷的呈现，他的原初处理确立了将被贯穿使用的几何定义与代数呈示二者 [2, 3]。教科书式的说明可见于伯曼 [6] 以及卡塞尔与图拉耶夫 [13]。

2.1 定义

固定一个整数 $n \geq 2$，并考虑两个水平平面，一个在另一个之上，各自承载一排 $n$ 个标记点。$n$ 股上的一个几何辫，是一组 $n$ 条不相交的曲线，从上平面的诸标记点通向下平面的诸标记点，服从一个条件：每条曲线单调地下降，与每一个中间水平平面恰好相遇一次。这个条件禁止一股向上折回。它所允许的，是诸股彼此绕过，而正是这种绕过，是该结构所记录的东西。

两个几何辫被视为相同的，如果一个可以被连续地形变为另一个，而不移动端点、也不让任何一股穿过另一股。这个关系是同痕，而一个辫是几何辫的一个同痕类。这个定义已经携带一项值得标出的承诺：一个格局中一切一个连续形变所能移除的东西都被丢弃，而一切它所不能移除的东西都被保留。两股相隔多远、一条曲线弯得多急、一个交叉在何高度发生，这些都不存活。所存活的，是哪些股交叉了哪些别的股，以何种次序，以及在何种意义上。

辫可以复合。给定同样股数上的辫 $a$ 与 $b$，把 $a$ 叠在 $b$ 之上，把 $a$ 的下端点与 $b$ 的上端点相认同，并重新标度；其结果是一个辫，写作 $ab$。复合在同痕类上是可结合的。诸股不交叉地下降的那个辫，对这一运算是一个单位元，写作 $e$。而每一个辫都有一个逆：在一个水平平面中反射它，则一个辫与它的反射的复合可以被逐个交叉地形变为单位元。因此，$n$ 股上诸辫的同痕类构成一个群，即辫群 $B_n$。

2.2 生成元与朝向

任何辫都可以被分解为一系列初等交叉，每一个都涉及一个相邻的股对。写 $\sigma_i$ 为这样的辫：位置 $i$ 上的股越过位置 $i+1$ 上的股，所有其他股不受触动地下降，其中 $1 \leq i \leq n-1$。它的逆 $\sigma_i^{-1}$ 是这样的辫：位置 $i$ 上的股从位置 $i+1$ 上的股下方穿过。

图 1（生成元 $\sigma_i$ 及其逆）。 所涉及的诸股与被交换的诸位置在二者中都相同；唯一的区别性数据是哪一股越过另一股。一股上的断口标出它从下方穿过之处。

$\sigma_i$ 与 $\sigma_i^{-1}$ 这一对，恰在一个方面有别。所涉及的诸股是相同的，被交换的诸位置是相同的，而唯一的区别性数据是哪一股越过了另一股。这是该交叉的朝向，而它是该形式体系在单一事件的层面上所记录的唯一不对称。所不能推出的应被注意。$\sigma_i \neq \sigma_i^{-1}$ 是一个关于该结构的事实；而它们之中的一个在任何意义上更好、更强、或更在先，则根本不是一个关于该结构的事实，而 $B_n$ 中没有任何关系提供一个这样的事实。

$B_n$ 的每一个元素都可以被写作诸 $\sigma_i$ 及其逆的一个有限乘积。这样一个乘积是一个辫字，而它是一个有序的记录：字 $\sigma_1 \sigma_2 \sigma_1^{-1}$ 不仅规定了哪些交叉发生了，而且规定了它们发生的次序。字母的个数是该字的长度，而在表示一个给定辫的所有字之中一个长度最小的字是既约的。一个辫通常容许许多字，而下一小节的诸关系说出是哪些。

2.3 诸关系

图 2（那两个关系）。 左对：远交换，其中诸交叉涉及不相交的股对，而它们的次序不留下痕迹。右对：辫关系，其中三股成对地互动，而一个特定的重新排序产出同一个辫。在每一对中，两个图是 $B_n$ 的同一个元素。

诸生成元之间成立两族关系，而它们连同诸生成元一起完全地呈示该群：

$$B_n = \bigl\langle, \sigma_1, \ldots, \sigma_{n-1} ;\bigm|; \sigma_i \sigma_j = \sigma_j \sigma_i \ \ (|i-j| \geq 2), \quad \sigma_i \sigma_{i+1} \sigma_i = \sigma_{i+1} \sigma_i \sigma_{i+1} ,\bigr\rangle.$$

二者由图 2 所例示。第一族是远交换。凡两个交叉涉及不相交的股对之处，即位置 $i, i+1$ 与 $j, j+1$ 且 $|i-j| \geq 2$，它们发生的次序对所得的辫没有区别。这两个事件不互动，而该结构不记录哪一个先来的任何痕迹。这是一个关于独立性的陈述，而它是精确的。次序不是要紧得少；次序根本没有表示。

第二个是辫关系，在它更广的设置中也被称为杨-巴克斯特关系 [25, 4]。凡两个交叉共享一股之处，即位置 $i, i+1$ 与 $i+1, i+2$，次序被部分地约束。该关系说，一个特定的重新排序使辫不变：施行 $\sigma_i$，然后 $\sigma_{i+1}$，然后 $\sigma_i$，产出与 $\sigma_{i+1}$，然后 $\sigma_i$，然后 $\sigma_{i+1}$ 同一个辫。在几何上，这是”一股可以被滑过另外两股的交叉”这一陈述。因此，该关系所提供的是一个无差别的判准：在诸重叠交叉的序列之中，有些重新排序什么也不改变，有些改变一切，而该关系说出哪些是哪些。

这个呈示是完全的，而它的完全性有一个关于该群所不说之物的后果。凡不是那两个关系之后承的东西，在 $B_n$ 中都不成立。特别地，没有任何关系把 $\sigma_i$ 与 $\sigma_i^{-1}$ 相认同：一个交叉与它的反向是彼此有别的元素，而再多的复合也不会把它们混同。也没有任何关系允许一个交叉被删除，除非通过与它自己的逆复合。字会增长，而它们只以诸关系所允许的方式约化。

2.4 向对称群的投影

一个辫决定诸位置的一次重新排列：起始于位置 $1$ 的股在某处结束，起始于 $2$ 的股在别处结束，而这一指派是 ${1, \ldots, n}$ 的一个置换。把每一个辫送到这个置换，定义一个映射

$$\pi \colon B_n \longrightarrow S_n,$$

它是一个群同态，并把生成元 $\sigma_i$ 送到对换 $(i \ \ i{+}1)$。在该呈示上，$\pi$ 丢弃 $\sigma_i$ 与 $\sigma_i^{-1}$ 之间的区别，二者都映到同一个对换，而保留诸位置的交换。

图 3（投影所不能看见的东西）。 全部三个辫都把每一股送回它自己的位置，因而在 $S_n$ 中有相同的像。前两个是单位辫；第三个不是，诸股曾彼此完整地缠绕一次。置换记录端点而丢弃缠绕。

该映射是满的：每一个置换都由某个辫所实现，因为诸对换生成 $S_n$。它不是单的，而单性的失败正是本文所接过的东西。图 3 展示这一失败。$\sigma_i \sigma_i^{-1}$ 与 $\sigma_i^{-1} \sigma_i$ 都映到单位置换，正如它们必然如此，二者是单位辫；但 $\sigma_i^2$ 也如此，而它根本不是单位辫，两股曾彼此完整地缠绕并回到它们的起始位置。置换不能看见那个缠绕。

写 $P_n$ 为 $\pi$ 的核，我们得到短正合序列

$$1 \longrightarrow P_n \longrightarrow B_n \stackrel{\pi}{\longrightarrow} S_n \longrightarrow 1,$$

它是本文后面的一切所立于其上的结构。它的内容可以被表述如下。$B_n$ 记录一个格局的完整历史；$S_n$ 只记录诸股在何处结束；而 $P_n$ 是第二者对第一者所遗忘之物。

核与它所作用于其上的诸类之间的关系，容易被松散地陈述，而在此被审慎地陈述。对一个置换 $s \in S_n$，实现 $s$ 的诸辫之集是纤维 $\pi^{-1}(s)$，而这些纤维划分 $B_n$。核 $P_n$ 是单位元之上的纤维，并且是一个群；其他诸纤维是它的诸陪集，并且不是。一般而言为真、并且是那个起作用的事实的，是这个：

$$\pi(b_1) = \pi(b_2) \quad \Longleftrightarrow \quad b_1 b_2^{-1} \in P_n.$$

因此，$P_n$ 本身并不是同结果诸历史的集合。它是把任何这样一个历史携带到任何别的同结果历史的变换之群，即在每一个纤维之内作用的那个群。这一区别在后面将要紧，那时被评估的东西将原来是诸历史之间的差异、而不是诸历史。

2.5 纯辫群

$P_n$ 的诸元素是纯辫：那些其中每一股都回到它自己位置的辫。一个纯辫并不因此就是平凡的。诸股可能在途中彼此任意地缠绕过，而该群相应地丰富。

图 4（纯辫群的一个生成元）。 股 $1$ 交叉那个居间的股，环绕股 $3$ 一次，并返回。每一股都在它开始之处结束，因而该辫落在 $\ker \pi$ 之中；中心处那个完整的缠绕正是投影所湮灭的东西。

它由诸元素 $A_{ij}$ 所生成，其中 $1 \leq i < j \leq n$，而 $A_{ij}$ 是这样的纯辫：股 $i$ 越过那些居间的股，环绕股 $j$ 一次，并返回。以阿廷生成元表达，

$$A_{ij} = \sigma_{j-1} \cdots \sigma_{i+1} , \sigma_i^2 , \sigma_{i+1}^{-1} \cdots \sigma_{j-1}^{-1},$$

而人们看到中心处的 $\sigma_i^2$：一个完整的缠绕，它被向 $S_n$ 的投影所湮灭。图 4 展示 $A_{13}$ 的情形。在结构上，$P_n$ 分解为诸自由群的一个迭代半直积，这是阿廷 [3] 的一个结果；其理由是，遗忘最后一股定义一个满射 $P_n \to P_{n-1}$，其核在 $n-1$ 个生成元上自由。因此，纯辫群是大的，对 $n \geq 3$ 是非阿贝尔的，并且拥有相当的内部结构。它不是一个退化的余项。

对每一个纯辫和每一个位置对 $i < j$，人们可以关联一个整数，即环绕数 $\operatorname{lk}_{ij}(b)$，它通过带符号地计数股 $i$ 与股 $j$ 之间的诸交叉，即在 $i$ 越过 $j$ 之处为正、在它从下方穿过之处为负，并除以二而得到。这是一个同痕不变量，并且是一个从 $P_n$ 到 $\mathbb{Z}$ 的同态；诸总环绕数给出 $P_n$ 的阿贝尔化。环绕数所测量的，是一股绕另一股的净缠绕，累积于整个历史之上。它在此以这些术语、而不以任何别的术语被定义。这样一个量是否容许一个解释、以及何种解释，不是拓扑所回答的一个问题，而 §4.4 只在模型被建构之后才接过它。

2.6 可判定性

两个看上去不同的辫字可能表示同一个辫，§2.3 的诸关系允许极多的改写。人们总能分辨，这一点事先并不显然。事实上人们能够：$B_n$ 的字问题是可解的，这是阿廷 [2] 所确立的一个结果，此后被给予了若干算法上的处理，其中有那个隐含于半直积分解之中的梳理程序、加赛德的正规形式 [11]，以及德奥努瓦 [9] 的把手约化算法，后者在实践中是高效的。

其后果在社会理论所借用的诸结构之中是不寻常的。给定两个被表示为辫字的历史，”这些是否相同”这一问题有一个答案，而该答案可由一个不依赖于分析者判断的程序得到。一个表示诸差异、却无法裁定同一性的形式体系，对本文的目的将没有多少用处；它会重新安置一个解释性的困难，而不是移除一个。可判定性是辫群竟成为一个候选者的理由。

2.7 结构性天花板

以下所述在此被陈述，在任何解释被提出之前，以便读者在看到该装置的使用之前就知道它的边界。每一项都是该数学的一个特征，而不是当前处理的一个局限；模型的任何精炼都不移除它们，而每一个都将在 §6 中作为”该模型所能被要求承担之物”的一项约束而被重访。

固定的 $n$。 股数对一个辫的整体是固定的。诸股不能被创生、毁灭、合并或分裂。$B_n$ 与 $B_m$ 对 $n \neq m$ 是不同的群，而理论之内没有任何运算把一个格局从一者携带到另一者。无论一个辫描述什么，它描述一个自始至终持续的、固定的位置总体。

离散性。 一个辫是诸交叉中的一个有限的字。诸交叉是事件，被个别化并可计数，而没有任何对诸股之间一种连续变化的影响、对一个场、或对两个位置之间一种不被事件所标点的渐进变化的表示。

可定向而无方向。 一个辫区分越过与穿下。它并不对它们排序，而该群不含任何会让人说一个辫比另一个更先进、更发展、或更靠前的元素或关系。复合提供一种在时间中的次序；它不提供任何规范性的东西。这是一个局限，而如 §6 所论，也是一种保护。

同痕不变性。 一切一个连续形变所能移除的东西，都已由构造被丢弃。诸股之间的距离、一个弯的急缓、一个交叉在下降之中的时机，都没有被表示。该模型所看见的，恰恰是那个拓扑残余。

对内容的沉默。 这一界限是五者之中最不可见的。同一股对以同一意义的两个交叉，是同一个生成元，句号。无论什么区别了那两个场合，如果确有什么区别了它们，在 $B_n$ 中都没有任何表示。该形式体系记录位置 $i$ 与 $i+1$ 以一个给定的朝向互动了。它不记录任何关于那次互动是什么的东西。

结构及其边界既已写下，我们转向那个领域，以及这个结构被提议来填补的那个具体缺口。

§3 程序正义

本节以其自身的术语列出那个领域。其中没有任何辫出现，而所评述的诸说明并不被当作某种前一节所提供之物的欠缺版本；每一个都被当作对它为自己所设定之问题的一个回答。本节所为的，是尽可能精确地定位一个它们之中没有一个所提出的问题，并把那个问题与几个已被很好地回答的邻近问题区别开来。

3.1 罗尔斯与三种程序正义

大多数后续讨论所由之出发的那个区分是罗尔斯的，而它是程序与一个关于正义结果的独立判准之间的关系中的一个区分 [20, §14]。

凡这样一个判准存在、而一个程序可靠地实现它之处，那个案例是完善的程序正义的一个案例。罗尔斯的例子是在诸平等者之间分蛋糕：判准是一个平等的分割，而”切的人取最后一块”这一规则达成它。程序是工具性的，而它由它是否交付判准所规定之物来评判。

凡判准存在、而没有程序可靠地实现它之处，那个案例是不完善的。一场刑事审判是那个例子：判准是有罪者、并且只有有罪者被定罪，而没有任何证据与论辩的程序保证这一点。此处程序也是工具性的，而它的不完善是对照某种在它之外的东西所测得的一个短缺。

凡没有独立判准存在之处，那个案例是纯粹的。一场公平的赌局是那个例子：没有任何彩金的分配在程序之先就是正义的，而无论什么分配由一场被恰当地举行的赌局所产生，都因它如此产生而是正义的。程序是构成性的、而不是工具性的，而这正是为何该学说在罗尔斯更大的论证中承载它所承载的分量，在那里，一个被恰当地规定的原初位置的结果，是凭借程序、而不是通过与一个在先标准的对应而正义的。

纯粹程序正义是最接近本文所关切之物的那个案例，因为它是程序承载整个规范性负担的那个案例。就当前目的而言，它也是一个具体的不足最为清晰的那个案例。该学说告诉我们，一个被正确地举行的程序把正义赋予它的结果。它并不告诉我们如何比较两个都被正确地举行的程序。如果两场赌局都是公平的，该学说对它们之间的差异没有任何进一步可说；而如果两个协商程序都满足无论所强加的什么条件，该学说把它们当作等价的。这个装置被造来把诸程序关联到诸结果。它不是被造来把诸程序彼此关联的。

这不是对罗尔斯的一个批评，他在回答一个不同的问题。它是一个关于那个三分法所留待未决之物的观察。一个只以诸程序对诸条件的符合来个别化诸程序、只以诸结果的诸分配来个别化诸结果的理论，对于两个发为同一个分配的、相符合的程序之间的差异，没有任何资源。是否有任何这样一个值得标出的差异，恰恰是那个经验文献接着去确立的东西。

3.2 泰勒与经验的转向

“人们独立于诸程序所交付的结果而评估诸程序”这一发现，先由蒂博与沃克 [22] 在实验设置中确立，然后，大规模地并且在田野中，由泰勒及其合作者所确立 [14, 23, 24]。那个核心结果是稳健的，并已跨法律的、组织的与政治的诸设置被复现：关于一个程序之公平的判断，预测服从、接受，以及给予一个权威的合法性，而它们在结果的有利与否被控制的情况下如此。输了的人接受输，当他们判断该程序是公平的之时。

承载这一判断的诸因素，惯常被给为四个：发言，即陈述自己一方的机会；中立，即决定者身上偏见的不在场；可信赖，即”该权威在善意地行动”这一知觉；以及尊重，即与一个人作为一个有权得到考量者的地位相一致的对待。这些不是分配的诸特征。两个在它们所分配之物上相同的程序，可能在它们之中的每一个上都有别。

因此，该文献确立了，一如此类事物之被确立那般稳固地，某种结果不变的东西正在被追踪。它所不做、并且不试图做的，是为那个被追踪之物提供一个结构。那四个因素是一个如所经验的程序的诸属性，由询问参与者他们如何经验它的诸工具来测量。这完全适合于那个研究纲领，它是一个社会心理学中的纲领。但它意味着，那个结果不变的残余，是由它对判断的诸效果、而不是由任何关于程序本身的东西来刻画的。如果两个程序引出不同的公平评分，我们知道它们有别；我们并不拥有一个关于它们在何处有别的说明，除了通过那些检测到该差异的同样的工具。那个发现是一个关于人的发现。人们所回应的是诸程序的什么属性，这一问题仍然开放。

3.3 协商理论与次序

协商民主理论是最接近本文之关切的那条脉络，因为它是那条已然知道次序要紧的脉络。

这一认识从两个方向到来。从社会选择，有那个早已确立的结果，即诸结果依赖于议程：麦克尔维 [17] 表明，在多维偏好的多数规则之下，一个议程设定者能够通过一个合适的成对表决序列，把议会驱向任何一点。从协商理论本身，有对”谁发言、何时、以及对随后之物有何效果”的持续关注，即扬 [26] 中对包容的关切、科恩 [7] 与哈贝马斯 [12] 中对协商诸条件的处理，以及曼斯布里奇 [16] 与冯 [10] 中对协商诸设置的经验研究，其中诸视角进入的顺序被反复发现塑造议会随后所能考量之物。

因此，该文献对这一现象并非无知。然而，它对它的特有处理是诊断性的、而不是描述性的：次序效应作为有待被识别与被中和的诸扭曲而出现。议程操纵是一种病理；一个被边缘化的视角的迟到是包容的一次失败；每一种情形中的补救都是一个移除对次序之依赖的程序约束。这是对如此提出的那个问题的一个合理回应。但它把次序当作错误的一个源头、而不是当作诸程序据以有别的一个维度，从而它产出不了对差异本身的任何说明。两个只在次序上有别、而其中两个次序都不是操纵性的程序，不是那个诊断性框架有任何可说的一个案例。没有对诸次序之空间的任何肯定性刻画，只有一组诸次序不应违反的条件。

3.4 森与比较的替代方案

森 [21] 反对他所称的超越性制度主义，即规定完善地正义的诸制度这一工程，而支持对已实现诸社会状态的比较性评估。该论证有一个直接相关于此处的组成部分：森坚持，对一个状态的评估必须包括它是如何被抵达的，并提供了一支笛子有待在三个具有不同诉求的孩子之间被指派的例子，其中正义的指派之身份，依赖于没有任何对诸最终分配的排序所捕获的诸考量。

这是一次实质性的拓宽。评估的对象从诸分配被扩大到诸实现，而诸实现承载关于过程的信息。然而评估的结构不变：所比较的仍然是一对状态，而过程信息作为对那个所得状态的一个进一步的描述项而进入。经由不同路线所抵达的两个实现，是有待被比较的两个状态。那个比较并不以诸路线之间的差异为它的对象；它以诸状态、被更丰富地描述，为它的对象。因此，那个比较性的举动重新安置那个终点、而不是免除它。

3.5 那个缺口

把那个缺口陈述为对历史之关注的一种不在场，会是一个错误。历史在诸社会科学之间被很好地表示，而一个”要提供它”的主张会是假的。过程追踪重建诸案例之内的因果序列 [5]。事件史模型表示诸转变的时机与次序。因果图表示诸事件之间依赖的结构 [18]。一个博弈的扩展式恰恰是诸历史的一个表示，连同在每一点上可用的信息一起完备。路径依赖制度主义已把锁定、次序与关键节点弄成了标准的分析语汇 [8, 1, 19, 15]。这些之中的每一个都表示诸过程，并且很好地表示它们。

所缺的是更狭窄的东西，而它经受得住以上一切。考虑发为一个给定分配的诸程序性历史之集合。刚被点名的诸框架将各别地描述这些历史之中的每一个，并以人们所愿的那般多的忠实性描述它们。它们之中没有一个把那个集合弄成一个对象。因此，它们之中没有一个对它的诸成员之间被当作差异来考量的诸差异有任何可说：它们是否构成一个结构，那个结构是否有诸部分，它们之中的两个是否是同一个差异，第一个与第二个历史之间的差异是否是第三个与第四个之间的同一个差异。这些是人们只有当那些差异本身就是某种东西的元素之时才能提出的问题。

这一匮乏有第二个组成部分，而它是那个使第一个组成部分有后果、而不仅仅是抽象的东西。假设人们希望主张两个程序是被不同地描述的同一个程序，或是真正不同的。在所点名的诸框架中，这一问题由分析者的描述所裁定：两个过程追踪是相同的，如果分析者已把它们描述得相同，而诸事件的个别化是分析者所提供之物的一部分。没有对程序同一性的独立判准。一个其中这样一个判准存在的结构，会把”两个程序有别”这一主张置于重新描述的够不着之处，并会使”一个差异仅仅是记号上的”这一断言成为可被反驳的东西。

因此，那个缺口可以被陈述如下。

那个缺口。 此处所评述的每一个说明都登记了程序承载着结果所不记录的分量，而其中数个详细地表示程序性历史。它们之中没有一个提供一个结构，其中发为一个给定结果的诸历史构成一个确定的集合，其中它们之间的差异本身就是容许复合与比较的元素，而其中两个程序的同一性是独立于分析者所选择的描述方式而可判定的。

这样一个结构是否存在，是一个数学问题，而 §2 已经展示了一个候选者。那个候选者能否被带来施加于刚被概览的那个领域，并且以何种代价、以何种许可、以何种拒绝，是随后所述的主题。

§4 形式建模

前两节被写得并不相遇。这一节使它们相遇，而相遇的方式正是随后所述的实质。以下每一个对应都被提出、然后被立即检验：它所许可的东西被陈述，而它所不许可的东西被同等审慎地陈述，因为一个许可一切的映射什么也不许可。

4.1 基础对应

诸股。 一股是一个角色：一个程序之内的关系性位置，由无论谁占据它者所占据。有三个候选者自荐，而它们之间的选择是被迫的、而不是自由的。

诸股可能是诸个体。这立刻败于 §2.7 的固定 $n$：一个其中一位参与者迟到或退出的程序是不可表示的，而这样的程序是常见的。诸股可能是更充分意义上的诸关系性身份，即生成性关系存在给予那个术语的意义，即被边缘化的参与者、共同体、作为一个被构成之物的制度。这更接近该框架自身的本体论，而它败于一个更微妙的理由：诸关系性身份是由辫会记录的那些互动本身所生成与所转变的，因此诸股会不得不在辫的进行之中改变，而该结构并不允许这一点。

诸角色是所余下的，而它们的适足性以一种应被陈述、而不是被隐瞒的方式被界定。一个程序之中的一个角色，即提议者、反对者、主席、持有技术性诉求的一方，在程序的进行之中持续，而占据它的个体可能不持续；而一个具有稳定角色结构与变化中的成员构成的程序是可表示的，而一个具有变化中的诸角色的程序则不是。

这对该模型与那个更广框架之间的关系有一个后果。

主张 1（应用的领域）。 固定 $n$ 的辫理论在角色稳定化之后开始。它对已确立的诸关系性位置之间的诸变换建模，而不是对那些位置的涌现建模。凡诸角色本身尚在形成之处，该模型不适用，而它停止适用的那一点是可检测的。

诸事件。 一个交叉是一个互动事件，但一个程序之内并非每一个发生都是一个互动事件。这一个别化要求一个判准，而以下三个条件被联合地提出：一个发生是一个交叉，当（i）它更改至少两个角色的关系性状态；（ii）它在程序之内被认作一次互动，而不是对它是附带的；以及（iii）它更改随后可用的诸转变之空间。

第三个条件是那个起作用的，而它是由该框架、而不是由那个数学所提供的。依此说明，一次互动是改变接下来能发生什么的接触。同时在场于一个房间中的两位参与者，并不因此就交叉了。两位参与者，其中一位已提出一个另一位如今必须回答的诉求，则交叉了。

这个条件携带一个应被显明地录入的蕴涵。

主张 2（被承袭的本体论）。 事件的个别化是在拓扑编码之前、并且由辫所不含的诸判准而被执行的。条件（iii）指涉一个未来可能性的空间，而辫对之没有任何表示。因此，该形式体系承袭一个它并不生成的事件本体论。

这种依赖的形状是熟悉的：一个坐标系并不生成它所描述的几何，而一个记号并不生成它所记录的诸意义。对该模型所随之而来的，是它的诸输出只与被提供给它的那个个别化一样有根基，以及关于一个案例的分歧可能是关于那个编码、而不是关于那个拓扑的分歧。这样的分歧至少是可定位的，而这比那个替代方案所提供的更多。

朝向。 生成元 $\sigma_i$ 与它的逆 $\sigma_i^{-1}$ 在哪一个角色越过另一个上有别。这是那次互动的朝向，而它恰恰是那个：那两个角色之中哪一个，在那个事件中，是其位置占了上风的那一个。图 1 展示这两者。它不是权力、支配、优势或不义的一个度量，而那个映射并不许可从朝向到这些之中任何一个的推论。两方可能以一百种方式交换诸位置，而拓扑只看见其中一个比特，而一个在一次互动中占上风的角色，可能是较弱的一方占上风、较强的一方让步，或二者皆非。不对称，凡本文需要它之处，在 §4.4 中被分开地定义，而不是从那个生成元上被读出。

历史与结果。 一个辫字是那些互动事件的有序记录，而辫是这样一些记录在 §2.3 的诸关系之下的等价类。它的像 $\pi(b)$ 是诸角色的终点格局：程序所施行的诸关系性位置的重新排列。

把这个称作分配性结果是诱人的，而这个诱惑应被抵制。二者只在分配是终点关系性位置的一个函数之处重合，即在每一方所接收之物由每一方在何处结束所决定之处。这个条件常常被满足，并且不是平凡的，而 §5.2 展示一个其中它失败的程序：两个议会可能批准同一个分派，而把诸角色留在不同的终点排列中，或把诸角色留如它们所是，而更改所分派之物。凡这个条件失败之处，模型意义上的结果等价与分配意义上的结果等价分开，而该模型追踪前者。

辫所不含之物。 这一对应必须以一个否定的半部被陈述，否则它将被读作主张多于它所主张之物。辫不是程序。它是一个程序所生成的关系性事件历史。一个程序包括诸规则、诸授权、每一方可用的信息、诸方据以行动的诸意图、诸方附于所发生之事的诸意义，以及一个约束所有这些的制度性设置。这些之中没有一个在辫之中。程序生成辫；辫不复原程序，而两个在刚被列举的每一个方面都有别的程序，可能生成同一个辫。

4.2 结果不变的程序性差异

我们现在可以陈述那个结构针对 §3.5 的缺口所提供之物。

设一个程序生成一个辫 $b \in B_n$。它的结果是 $\pi(b) \in S_n$，而发为一个给定结果 $s$ 的诸程序，是那些其辫落在纤维 $\pi^{-1}(s)$ 之中者。由上面那个判准，两个程序是结果等价的，恰当它们的辫相差 $P_n$ 的一个元素之时。因此，诸结果等价程序之间的差异不是一个异质的诸相异之集合；它们是一个群的诸元素，它们复合，它们求逆，而它们之中的两个可以由一个可判定的程序被比较并被发现相等或不相等。

第一个后果关乎罗尔斯。纯粹程序正义，如 §3.1 所观察到的，对于比较两个被正确地举行的程序没有任何资源。以当前的术语言之，该学说在 $S_n$ 中评估。这是一个关于它的诸工具看向何处的陈述，而不是对它们的贬低。而它把此前被松散地说出之物弄精确：一个在 $S_n$ 中被举行的评估是结构上盲的。它不是以一个粗糙的测量之为不完备的那种方式不完备。没有对结果评估的任何精炼能够复得 $P_n$，因为 $P_n$ 由构造正是那个投影所湮灭之物。

第二个后果关乎同一性。”两个程序有别”这一主张，在 §3.5 的诸框架中，由分析者的描述所裁定。此处它由上面那个判准连同 §2.6 的可判定性所裁定：给定诸编码，$b_1 b_2^{-1} = e$ 是否成立有一个答案，而那个答案不依赖于是谁在问。那个编码仍然是分析者的，由主张 2；但那个已提供一个编码的分析者，因此已放弃了重新描述的自由。

主张 3（那个对象）。 纯辫群是 §3.5 发现所缺的那一类结构：发为一个给定结果的诸历史构成纤维 $\pi^{-1}(s)$；它们之间的差异是 $P_n$ 的诸元素，在那个纤维之内作用；而两个程序的同一性是独立于描述而可判定的。基于结果的评估由构造、而不是由粗糙，对 $P_n$ 是盲的。

所不能推出之物必须在同一口气中被陈述。并不能推出 $P_n$ 的诸元素是与正义相关的，无论是单个地还是集体地。$P_n$ 是一个候选差异的空间，而”一个候选者要紧”这一主张是一个不同种类的主张，要求一个拓扑所不能提供的论证。一个把主张 3 认作已确立了”程序性差异要紧”的读者，已把 §4.7 为之而存在、要保留的那个结论读入了它。

4.3 排序与程序上的无差别

协商理论，由 §3.3，知道次序要紧，并把次序效应当作诸扭曲来处理。它对哪些重新排序是实质性的没有任何肯定性刻画。§2.3 的诸关系提供一个，而它有两个部分。

远交换说，涉及不相交的角色对的诸交叉可以被自由地重新排序。凡一方对第二方的反对与第三方对第四方的反对是所论的诸事件之处，它们之间的次序根本没有被表示：那两个次序生成同一个辫，而没有任何只对关系性历史敏感的评估能够分辨它们。这是一个强的陈述，而它是可证伪的。如果诸参与者可靠地分辨这样的次序，而那个分辨不可归于关系性历史之外的某种东西，那么或者那个编码已错误地个别化了诸事件，即那两个反对事实上并不不相交，或者该模型在这一点上是错的。

辫关系说，在诸重叠交叉之中，一个特定的重新排序同样是无差别的：$\sigma_i \sigma_{i+1} \sigma_i$ 与 $\sigma_{i+1} \sigma_i \sigma_{i+1}$ 是同一个辫。凡三个角色成对地依序互动之处，有两个次序产出同一个关系性历史，而别的次序则不。因此，该关系把诸重叠事件的诸重新排序划分为无差别的与实质性的，而这恰恰是 §3.3 发现该文献无法划出的那个划分。

这个检验的界限与这个检验同等重要。两个程序是同一个辫，这确立它们是同一个关系性历史。它并不确立它们在任何别的方面相同，而它们可能有别的诸方面是无界的，因为由 §4.1，辫不含任何规则、意图、信息或意义。两个杨-巴克斯特等价的序列，可能在诸互动是带着尊重还是带着羞辱而被举行上有别。辫看见一个对象；诸参与者可能看见两个。该模型的无差别之裁决，是一个关于关系性拓扑、而不关于任何别的东西的裁决。

4.4 不对称与持续

环绕数在 §2.5 中被定义为两股之间诸交叉的带符号计数。它的解释现在被提出，而这一提议是刻意地弱的。

主张 4（持续可观测量）。 对一个纯辫 $b$ 与诸角色 $i, j$，量 $\operatorname{lk}_{ij}(b)$ 是关系性持续的一个可观测量：两个角色之间的互动在历史之上以一个一贯的朝向累积、而不是相消的程度。它不是支配、不平等、优势或不义的一个度量，而拓扑并不许可任何这样的读法。

弱的理由是，一个更强的读法败于诸案例。两个角色之间一个高的环绕数，与一者对另一者的持续支配相容；它同等地与一种富有产出的、反复的交往相容，在其中一方一贯地承载一个特定的功能。拓扑并不分辨这些，而一个断言它分辨的定义会是假的。

到该框架关于不对称与吸收的说明的连接现在可以被作出。生成性，在那个说明中，吸收不对称、而不是累积它：一段其中诸不对称生起并被工作透彻的关系，有别于一段其中它们生起并持续的关系。以辫的术语言之，吸收作为约化而出现，即一个其字，无论多长，都约化的历史；其诸交叉与它们的逆相遇；其诸环绕数朝零返回。持续作为它的失败而出现。这一对应足够精确以有用，而必须被审慎地处理，因为约化是辫的一个属性，而吸收是一个关于关系的主张，而第二者并不由第一者推出。

一个进一步的界限在此适用，而它比任何别的都更约束该模型。缠绕没有方向。辫 $\sigma_i^{k}$ 记录两个角色彼此缠绕了 $k$ 次；它不记录那个缠绕是一次深化还是一个僵局。一场因每一次返回都教给诸方某种东西而二十次返回同一议题的谈判，与一场因任何一方都无法移动而二十次返回的谈判，生成同一个拓扑签名。它们之间的区别是真实而有后果的，而辫不承载它。

这一情形容许一个精确的陈述。一个辫所提供的，是一个关系性历史的运动学，即那个轨迹的结构，而不是它的动力学。一个轨迹并不决定产生它的诸力，而对那个轨迹之描述的任何精炼都不会提供它们。因此，该模型被要求就”一个历史是否是生成性的还是退化性的”所说的任何东西，都要求一个辫所不承载的状态变量，连同一种 §4.7 所形式化的那类解释。

4.5 可修正性与反转

$\sigma_i$ 与 $\sigma_i^{-1}$ 是彼此有别且互逆的，这给了该模型一个纠正的自然形象：一次反转一个较早互动的互动。上诉、救济、一个决定的重新考量、一个诉求的撤回，每一个都是一个其朝向与一个在先者相反的事件。

图 5（反转与残余）。 左：一个交叉紧接着被它的逆所跟随，则相消，而关系性历史一如二者都未曾发生。右：同一个反转，由一次触及同样角色之一的互动与它的原初者相隔，则不相消，而历史保留一个形式上被撤销的事件的一道痕迹。

那个结构于是说出某种不显然、且不平凡的东西。凡一个交叉紧接着被它的逆所跟随之处，二者相消：$\sigma_i \sigma_i^{-1} = e$，而关系性历史一如二者都未曾发生。凡它们在字中被别的交叉所相隔之处，它们不必相消，并且一般而言不相消。远处的反转留下残余。

图 5 对照这两种情形。理由是结构性的。在那个交叉与它的反转之间，别的互动已经发生，而 §2.3 的诸关系只在那些居间的交叉与它交换而过之时，才允许那个反转被带回来与它的原初者相遇，而这，由远交换，要求它们涉及与所论的那一对不相交的角色。凡那些居间的互动触及同样的角色之处，那个相消被阻断，而历史保留一个已被形式上撤销的事件的一道痕迹。

对上诉程序的读法是直接的，而它是该模型的一个真正的输出、而不是对它的输入的一个重述。一个在上诉中被反转的决定，并不因此就是一个从未被作出的决定，而那个反转是否复原先前的关系性状态，依赖于在其间发生了什么、以及它涉及谁。该模型不仅仅允许这个观察；它预言那个残余何时发生、何时不发生。

4.6 退化

此处所处理的概念不来自 §3 的文献。它是由生成性关系存在所提供的，在那里退化命名一段关系未能继续生成，而它在此被录入，是因为辫结构使那个框架非形式地划出的一个区分变得尖锐。它只被处理到当前论证所要求的深度。

该框架区分两个层级。在第一个，生成被阻断，而关系的语法保持完好：诸方继续以它们所有的诸术语互动，而那些术语停止产出任何新东西。在第二个，语法本身被改写：什么算作一次互动、诸方是谁、哪些动作可用，这些都改变，而所得的关系是一段不同的关系，而不是那段较早关系的一个退化了的版本。

在该模型中，第二个层级是无歧义的。语法的一次改写是诸生成元或它们之间诸关系的一次更改，而一个如此被更改的格局根本不是 $B_n$ 的一个元素。层级 2 退化是对该群的退出。

第一个层级不是无歧义的，而前一小节已经表明了为何。被阻断的生成的辫签名，即一个增长而不约化的字、缠绕在一个固定的角色对之间累积，恰恰是富有产出的重复的签名。那个签名是必要的，而它不是充分的。要断言一个展现它的历史的层级 1 退化，人们额外要求一个关于关系性可能之空间是收缩还是扩张的判断，而辫不含任何这样的空间。

诸层级之间的不对称可以被陈述如下。

主张 5（可检测性的不对称）。 层级 2 退化作为该模型的失败而可检测：编码破裂，而那个破裂就是那个发现。层级 1 退化根本无法在该模型之内被检测，它的拓扑签名与生成性重复所共享；它要求一个自外部所提供的解释。

4.7 那个解释性算子

以上一切都被从同一个结论那里保留下来，而那个保留现在可以被给予一个形式。

该模型把结果不变的程序性差异表示为 $P_n$ 的诸元素。它并不评估它们。这些差异之中有些是与正义相关的；有些不是；而意义的赋予是基于不是拓扑性的诸根据而作出的。写

$$\Phi \colon P_n \times C \longrightarrow \mathcal{R}$$

为施行那一赋予的算子，其中 $\mathcal{R}$ 是一个关系性意义的空间，而 $C$ 承载辫所不承载之物：角色的诸意义、制度性设置、诸方可用的信息、诸互动的内容。

那个语境论元不是一个形式上的客套。$P_n$ 的同一个元素可能在不同的设置中有不同的意义，而最经济地表明它的那个案例是 §4.5 的反转。一个交叉后跟它的逆是那个群的一个元素。发生在一个设置中它是一次健康的纠正；在另一个中是那个被反转之方的公开羞辱；在第三个中是一次压力之下被迫的退却。拓扑在全部三者中都相同。任何单向拓扑赋予意义的算子，都会不得不给它们同一个值，而会错两次。

为可读性起见，本文把 $\Phi$ 写作作用于 $P_n$，在语境被固定并被理解之处。完整的形式是上面那个，而这一缩写不应被误当作”意义可从拓扑复得”这一主张。

本文并不构造 $\Phi$。所录入的主张是更弱的，而它并不因此就是空的。

主张 6（必要性与定位）。 一个把要紧的程序性差异与不要紧的程序性差异区别开来的程序正义说明，因此就使用一个上述类型的算子，无论它是否被写下来。因此，一个把差异的表示与它们的评估分开的形式体系，即便在那个评估性的半部被留空之时，也已执行了一项服务：它把一个分布于一个说明各处的困难转换为一个单一的、被指认的对象。

“难的部分已被移入 $\Phi$” 这一反对是正确的，而它正是那个要点。所获得的，是它已被移到某处，而如今可以被考察。

4.8 语义与失败条件

下面的表格固定那个映射。第三与第四列是那两个起作用的：一个不许可任何具体之物的对应是装饰，而一个其失败条件未被陈述的对应不能是错的。

领域	辫对象	许可	不许可
关系性位置	股	贯穿一个程序的固定角色结构	诸个体；涌现中或消解中的诸身份
互动事件	交叉	满足那三个条件的、可计数的、有序的事件	连续的影响；附带的共同在场
互动的朝向	$\sigma_i$ 对 $\sigma_i^{-1}$	哪一个角色的位置占了上风	权力、支配、优势、不义
程序性历史	辫字	有序记录；比较；可判定的同一性	程序本身，连同它的诸规则与诸意义
分配性结果	$\pi(b) \in S_n$	诸角色的终点格局	分配，除非它是那个格局的一个函数
程序性差异	$P_n$ 的元素	差异的复合、求逆、比较	那个差异是与正义相关的
被累积的互动	$\operatorname{lk}_{ij}(b)$	朝向在一个历史之上的持续	支配；不平等；不义
不对称的吸收	字的约化	不对称被工作透彻的形式形象	那个约化是生成性的
纠正、上诉	$\sigma_i$ 之后的 $\sigma_i^{-1}$	相邻时相消；远处的残余	那个纠正是充分的或公平的
层级 1 退化	未约化的增长	一个必要的签名	充分性；生成性重复共享它
层级 2 退化	对 $B_n$ 的退出	作为编码失败的检测	对取代那个语法之物的表示
关系性意义	$\Phi(p, c)$	那个评估性步骤的定位	任何特定的价值赋予

三个失败条件适用于作为整体的那个映射，而每一个在原则上是可检验的。第一，如果诸程序中的参与者可靠地分辨诸远交换的重新排序，而那个分辨不可追溯到被错误个别化的诸事件、或到关系性历史之外的内容，那么 §4.3 中对独立性的处理是错的。第二，如果其辫相等的诸程序被发现在该模型归于辫的诸方面有别，那么那个编码是欠规定的，或那个对应是太粗糙的。第三，如果实践中所生起的 $P_n$ 的诸元素原来是一律与正义相关的，那么 $\Phi$ 是常值的，而那个”表示而不评估”的精细装置就是没有动机的，该模型于是会是在主张一个不做任何工作的区分。§5 直接接过第三个，因为一个无法产出一个不重要差异的模型，是一台确认它自己的机器。

§5 被构造的案例

5.1 方法

以下四个案例是被构造的、而不是被观察的。这一选择是刻意的，并遵循一般建模的实践：一个被构造的案例可以被规定得使受检验的那个量是唯一变化中的东西，而这是没有任何被观察的案例所允许的。历史实例在该模型声称对之敏感的诸维度、以及在它声称对之盲的诸维度上，都携带不受控的变异，而一个对照它们被检验的模型，因不能被分离的诸理由而失败或成功。因此，此处的诸案例是虚构的，并被造得在”每一个都可能发生”、而不是在”每一个都确实发生”的意义上是现实的。

这些案例是为区分、而不是为例示而造的。一个在差异存在之处到处都产出一个差异的模型什么也没有确立，因为由主张 3，$P_n$ 的非平凡元素是被保证的，而它们的展示是算术、而不是证据。因此，这四个案例检验四个彼此有别的东西。案例 A 检验敏感性：该模型所分离的差异是否是诸参与者所争的差异。案例 B 检验特异性：该模型能否产出一个不承载任何意义的程序性差异，而由 §4.8，它必须能够这样做、否则就是不可证伪的。案例 C 检验欠决定性：该模型所认同的诸程序是否可能在要紧的诸方面有别。案例 D 检验边界：该模型是否在 §2.7 说它必然失败之处失败，以及那个失败到来时是否是可辨读的。

编码规程是一律的。在每一个案例中，诸角色被命名并被固定；诸事件由 §4.1 的三个条件被个别化；每一个交叉的朝向被陈述；而所得的字被给出。凡两个变体被比较之处，它们在 $\pi$ 之下的诸像被计算，而那个分开它们的 $P_n$ 的元素被展示。

5.2 案例 A：乡村供水系统

两个村庄决定是否建造一个社区供水系统。在两者中，决定都由一个议会作出；在两者中，同一个系统都被批准，以同样的预算、同样的取水分派，并以同样的票数差批准。

三个角色是相关的，而它们在每个村庄中自始至终是稳定的：议会，$1$；技术顾问，$2$；以及其取水权利攸关的诸家户，$3$。诸个体变化而诸角色不变，这正是主张 1 所要求的条件。

村庄 A。 议会提出提案，而顾问的评估更改它：$\sigma_1$。诸家户随后反对说，所提议的选址将使外围定居点处于不利，而顾问作出回应而修订设计：$\sigma_2$。被修订的设计返回议会，议会接受那个修订：$\sigma_1^{-1}$。议会批准。那个字是

$$b_A = \sigma_1 , \sigma_2 , \sigma_1^{-1}.$$

村庄 B。 议会提出提案，而顾问的评估更改它：$\sigma_1$。议会随后敲定设计：顾问的修订在磋商之前被接受：$\sigma_1^{-1}$。诸家户此后被磋商，提出关于外围定居点的同一个反对，而它被记录在案：$\sigma_2$。议会批准。那个字是

$$b_B = \sigma_1 , \sigma_1^{-1} , \sigma_2 = \sigma_2 .$$

结果。 $\pi(b_A) = (1,2)(2,3)(1,2) = (1,3)$ 而 $\pi(b_B) = (2,3)$。这些并不相等，而如此陈述的案例尚未展示它被造来展示之物。

这是一个关于那个编码的发现、而不是一个有待被悄悄修补的疏忽。两个具有同一个分配性结果的程序必须有诸角色的同一个置换，而那个置换记录哪一个角色在哪一个位置结束。困难在于，”同一个决定被作出”不是与”诸角色在同一个关系中结束”相同的主张。在村庄 B 中，诸家户在那个决定的过程中并没有与顾问交换位置；他们的反对是在设计被固定之后才被记录的。在村庄 A 中他们交换了。

那个修补是把”一个什么也不更改的反对之被记录”编码为它之所是：一个未能满足 §4.1 之条件（iii）的事件，因为它不更改随后可用的诸转变之空间。它不是一个交叉。村庄 B 的字于是是

$$b_B = \sigma_1 , \sigma_1^{-1} = e,$$

而那两个程序现在有不同的结果，其理由是实质性的、而不是技术性的：在村庄 B 中，根本没有关系性的重新排列发生。

这个案例所确立之物。 不是它被设计来确立之物，而是某种更有用的东西。这个案例被构造来表明两个具有同一结果的程序相差一个纯辫。它反而表明，”决定之前的协商”与”决定之后的磋商”之间的差异根本不是一个结果不变的差异。它是一个结果上的差异，以该模型的结果之意义言之，即诸关系性位置的重新排列，而不是水的分配。那些更改了设计的家户，占据一个不同于那些其反对被存档的家户的终点位置。

这是一个关于该模型的结果之意义的结果，而它使 §4.2 变得尖锐。投影 $\pi$ 并不记录分配。它记录诸角色的终点格局，而一个程序可能把分配留如不动而重新排列诸角色，或把诸角色留如不动而更改分配。§4.1 中把 $\pi(b)$ 与”结果评估所看见之物”相认同，是太快了，而那个诚实的陈述是更狭窄的：$\pi(b)$ 是一个对终点关系性位置的评估所看见之物，而它只在分配是那个位置的一个函数之处与分配性评估重合。

敏感性，重新受检。

图 6（案例 A，重新受检以查敏感性）。 两个程序都把诸角色留在同一个终点排列中，因此没有对结果的评估分辨它们。在第二个中，议会与顾问在诸家户被听取之前交换并重新交换位置。这两个程序相差 $\sigma_1^{2}$，一个环绕数为一的纯辫。

一个确实检验敏感性的案例可以由同样的材料造出。假设在两个村庄中诸家户都确实更改了设计，使得 $\pi(b_A) = \pi(b_{A’})$，但在村庄 $A’$ 中，那个更改是在顾问与议会已就选址问题两次交换并重新交换位置之后才到来：

$$b_{A’} = \sigma_1^{2} , \sigma_1 , \sigma_2 , \sigma_1^{-1} , \qquad b_A = \sigma_1 , \sigma_2 , \sigma_1^{-1} .$$

这两者由图 6 所绘。那么 $b_{A’} b_A^{-1} = \sigma_1^{2} \in P_3$，一个议会与顾问之间环绕数为一的纯辫。这两个程序是结果等价的，并相差核的一个确定元素。诸参与者是否争这个差异，是那个实质性的问题，而答案并不显然：议会与顾问之间在诸家户被听取之前一次延长的交换，可能被经验为尽职调查、或被经验为此事的预先敲定。该模型分离出那个差异，并把那个问题递交出去。那就是主张 3 所允诺之物的全部，而这个案例表明那个允诺被守住、以及它的狭窄。

5.3 案例 B：一个委员会的两次会议

一个具有三个角色的委员会，即主席 $1$、第一委员 $2$、第二委员 $3$，在单一一次会议中考虑两项动议。这两项动议不相关：第一项关乎维护日程，第二项关乎一份年度报告的措辞。在每一项中，一位委员提出一点，而主席对它作出裁定。

在会议 $X$ 中，维护那一点被提出并被裁定，然后是报告那一点：$b_X = \sigma_1 \sigma_2$。在会议 $Y$ 中，次序被反转：$b_Y = \sigma_2 \sigma_1$。两次会议都同样地处置了两项动议。

此处 $\pi(\sigma_1 \sigma_2)$ 与 $\pi(\sigma_2 \sigma_1)$ 是两个彼此有别的三轮换，因此这一对不是结果等价的，而 §4.3 的远交换也不适用，那两个交叉共享中间那股。改为构造那个检验所要求的比较。设每一次会议都包括同样的三个实质性交换，并设会议 $Y$ 以两位委员之间的一次预备磋商开场，在其中每一位依次向另一位让步，而事情被留如它所立之处：

$$b_X = \sigma_1 , \sigma_2 , \sigma_1 , \qquad b_Y = \sigma_2^{2} , \sigma_1 , \sigma_2 , \sigma_1 .$$

那么 $\pi(b_X) = \pi(b_Y) = (1,3)$，而

$$b_Y , b_X^{-1} = \sigma_2^{2} \in P_3 ,$$

一个两位委员之间环绕数为一的纯辫。图 7 展示这两次会议以及那个分开它们的元素。那个元素是非平凡的：会议 $Y$ 含有一个两位委员之间的完整缠绕，而会议 $X$ 不含，而由 §2.6，”这两次会议是不同的程序”这一事实是可判定的、而不是分析者之描述的一件事。

图 7（案例 B）。 这两次会议把诸角色同样地排列，而相差右边那个纯辫：两位委员之间的一个完整缠绕，生自一次关于他们所同意之事项的预备磋商。该模型报告一个差异；没有参与者有理由在意它。

评估。 没有参与者认为这两次会议在公平上有别，而也没有明显的根据据以有一次应当如此。会议 $Y$ 中那个额外的交换，是两位委员之间关于一个他们所同意之事项的一次磋商，在寻常的过程中被举行，不影响任何人的地位、不影响任何人被听取的机会。该模型报告一个差异；那个差异是无人关心的。

主张 7（特异性）。 这个案例展示一个非平凡的 $p \in P_n$，对它在任何合理的赋予之下 $\Phi(p, c) = 0$。因此，该模型不是一台把拓扑差异转换为规范差异的机器，而 §4.8 的第三个失败条件在此处不被满足。

省略这个案例会是一个严重的过错。没有它，该模型所分离的每一个差异都会可供被当作重要的来处理，而 §4.7 的装置，即”表示与评估是分开的”这一坚持，就会是没有动机的。一个无法产出一个不重要差异的模型，不需要一个算子来过滤诸差异，而它的作者关于”拓扑不承载任何规范分量”的抗辩，会被一种其中拓扑总是承载规范分量的实践所背弃。

朝向一个判准。 什么把案例 A 那个被争的纯辫与案例 B 那个不被争的纯辫区别开来？一个猜想，作为如此而被提出：案例 A 中的缠绕发生在其相对位置关涉所决定之事项的诸角色之间，并更改了一个第三角色能够进入的次序；案例 B 中的缠绕发生在其交换关涉不到任何争议之事的诸角色之间，并没有封闭任何随后的动作。所提示的是，那个相关的特征是那个纯辫连同它与可能之空间的关系，也就是说，恰恰是 §4.1 之条件（iii）所诉诸、而辫所不含的那个内容。如果这是对的，那么 $\Phi$ 汲取事件个别化所汲取的同样的信息，而这两个问题是一个问题。这个猜想在此未被确立。

5.4 案例 C：河流条约

两个国家共享一条河流，并谈判它的分派。三个角色：上游国 $1$、下游国 $2$，以及为任何协议证明水文基础的技术委员会 $3$。在两个版本中，结果都是一个五五分派，被证明，依同样的日程。

版本 I。 上游国以一个对更大份额的要求开场，下游国拒绝之：$\sigma_1$。委员会的评估确立下游国的农业依赖性，而上游国修订：$\sigma_2 \sigma_1^{-1}$。下游国，得知上游国的能源约束，在时间安排上让步：$\sigma_1 \sigma_2^{-1}$。诸方趋同。那个字是

$$b_{\mathrm{I}} = \sigma_1 , \sigma_2 , \sigma_1^{-1} , \sigma_1 , \sigma_2^{-1} = \sigma_1 .$$

版本 II。 上游国以同样的要求开场，而同样的拒绝随之而来：$\sigma_1$。上游国示意，一个在另行谈判之中的贸易协议可能受影响；委员会的评估被录入，而下游国的立场移动：$\sigma_2 \sigma_1^{-1}$。下游国接受那些时间安排条款：$\sigma_1 \sigma_2^{-1}$。那个字是

$$b_{\mathrm{II}} = \sigma_1 , \sigma_2 , \sigma_1^{-1} , \sigma_1 , \sigma_2^{-1} = \sigma_1 .$$

图 8（案例 C）。 这两个面板是相同的，而那就是那个发现。每一个交叉都被注以那个场合上所发生之事；那些注解通篇有别，而辫不记录它们之中的任何一个。两个字都约化为右边那个单一的交叉。”由学习所抵达的协议”与”在胁迫之下所抵达的协议”之间的差异，不是一个等待被评估的 $P_n$ 的元素，它自那个表示中缺席。

评估。 这两个辫是相等的。不是结果等价，不是相差一个纯辫：相等。该模型报告这些是同一个程序，而由上面那个判准连同 §2.6 的可判定性，那个报告不是一件判断的事。

图 8 把这两个编码并排绘出，把每一个交叉注以那个场合上所发生之事。那些图无从分辨，而那些注解则不。这两个程序并不相同。在版本 I 中，下游国的移动出自关于那条河的信息；在版本 II 中，它出自一个关乎全然别的东西的威胁。诸方知道那个区别，任何观察者都知道它，而这两个条约所留下的关系不是同一段关系。一个由相互学习所抵达的条约与一个在胁迫之下所抵达的条约，可能逐字相同，而不是同一个成就。

那个编码中没有任何东西是错的。诸角色被正确地个别化，诸事件满足全部三个条件，诸朝向是正确的。该模型做了它所做之事，而它所不做之事是承载一次互动的内容。由 §2.7 的最后一项，同一股对以同一意义的两个交叉，是同一个生成元，无论那两个场合上发生了什么。版本 I 的第三个事件与版本 II 的第三个事件，都是下游国回应上游国所做之事而移动它的立场。拓扑看见一次移动。

所会要求之物。 那个区别要求一个辫所不承载的状态变量：至少，那次移动是由那个移动之方关于谈判对象的信息之改变所产生的，还是由它关于它之外诸事项的期望之改变所产生的。这不是那个拓扑的一个精炼，而无法通过更审慎地编码而得到。它是那个交叉的内容，而该模型对内容的沉默是结构性的。

因此，这个案例以一种强的形式确认 §4.1 的否定半部。不是辫略去了程序的一些，而是它略去了它的足够多，以致两个在可以说是最重要的可用方面有别的程序，即协议是通过理解还是通过压力而被抵达的，是形式上无从分辨的。§4.4 的缠绕观察是一个推论、而不是那个要点：此处诸字不仅缠绕得相像，它们是同一个字。

5.5 案例 D：工会

一家公司与它的诸工人谈判条款。起初每一位工人个别地谈判，而相关的诸角色是公司 $1$、作为 $2$ 的工人 $A$、以及作为 $3$ 的工人 $B$。公司向 $A$ 作出一个报价，$A$ 接受一个变体：$\sigma_1$。公司向 $B$ 作出一个不同的报价：$\sigma_2$，而 $B$ 拒绝：$\sigma_2^{-1}$。到目前为止那个编码进行下去。

诸工人随后组成一个工会。随后的谈判是在公司与工会之间，而工会为两位工人一起讨价还价。

失败之点。 那个编码无法继续，而它在何处停止可以被精确地陈述。一个关于总体崩溃的报告会隐瞒那个失败的结构。三个彼此有别的东西已经发生，而 §2.7 不同地处理它们。

两股已合并为一股。工会是这样一个角色，其中 $A$ 与 $B$ 的若干位置如今被联合地占据，而不是它们旁边的一个第四角色。组成之后的格局比之前有更少的关系性位置，而 $B_3$ 与 $B_2$ 是不同的群，理论没有任何运算把一者的元素携带到另一者。这是那个固定 $n$ 的天花板，而它恰在那个组成处被抵达。

一个角色已被创生。工会之前不存在而之后存在，而它的存在是辫正在记录的那些互动的一个产物。由主张 1，该模型在角色稳定化之后适用；此处那个程序使诸角色去稳定化，做那件该模型的领域所排除之事。

那个语法已改变。组成之前，一位工人的一次拒绝把另一位的谈判留如不动，而这两者在远交换所形式化的意义上是独立的。组成之后没有这样的独立，因为不再有一对位置来独立。什么算作一次互动、以及哪些互动可用，二者都已被更改。这是 §4.6 意义上的层级 2 退化。这个术语形式上指一次语法的改写，而不携带任何恶化的蕴涵：工会的组成可能在要紧的诸方面是一次改善。

那个发现。 这三者是可分开的，而把它们分开正是这个案例的贡献。股的合并、股的创生与关系的重新定义，是要求彼此有别的诸扩展的、彼此有别的失败：一个允许诸合并的结构、一个允许诸诞生的结构，以及一个其中诸生成元之间的诸关系本身可以变化的结构。一个对”该模型在此破裂”的单一姿态，会隐瞒需要三个不同的修补。§6 接过那些修补会涉及什么。

那个失败的可辨读性本身是一个结果。该模型并不优雅地退化为错误；它停止，在一个可指认的事件处，因诸自我命名的理由。一个其领域边界从内部可见的形式体系，比一个在它的诸假定失效之后仍继续产出输出的形式体系更有用。

5.6 跨案例比较

案例 A 检验敏感性，并首先返回一个纠正：该模型的结果是诸角色的终点格局、而不是分配，而这二者只在本文尚未陈述的诸条件之下重合。有了那个纠正在手，这个案例展示一个被争的纯辫，并把关于它的意义的问题递交出去，而这正是该模型声称要做之事、以及它所声称之物的界限。

案例 B 检验特异性，并返回一个无人有理由在意的、非平凡的 $P_n$ 的元素。这是那个没有它该模型就会是不可证伪的案例，而它也产出了一个对 $\Phi$ 的判准的开端：那两个缠绕之间的差异，似乎在于它们与随后可能之空间的关系，而那是辫所不含、且事件个别化已然要求的内容。

案例 C 检验欠决定性，并返回一个比它被造来所针对的更强的否定结果。两个在”协议是由学习还是在胁迫之下所抵达的”上有别的程序生成同一个字，而不仅仅是相差核的一个元素的诸字。该模型对内容的沉默是一堵墙，而不是一个残余的不精确。

案例 D 检验边界，并返回一个分解为三个彼此有别的天花板违反的可辨读失败。该模型在 §2.7 说它会停止之处停止，在一个可指认的事件处，而那个停止的方式区分了会被要求的三个不同的扩展。

四者合在一起，同时从两个方向确立 $\Phi$ 的必要性。案例 B 表明，拓扑差异对于意义并不充分：有诸差异是 $\Phi$ 必须送到零的。案例 C 表明，拓扑同一性对于不重要并不充分：有诸重要差异是 $\Phi$ 根本无法看见的，因为它们不在 $P_n$ 之中以待被评估。一个算子是被要求的，而它必须汲取辫所不承载的信息，而这正是为何那个算子携带一个语境论元，以及为何 §5.3 的猜想把那个信息定位在事件个别化找到它的同一个地方。

诸案例共同划出的那个边界于是可以被陈述。在它的领域之内，该模型确定地表示诸程序性差异，并可判定地裁定程序的同一性。它并不评估它所表示之物，它并不表示它所记录之物的内容，而它在它所预设的诸角色本身尚在形成之时停止适用。这三者中的每一个都是一个不同种类的界限，而 §6 依次处理它们。

§6 讨论

6.1 收获

五样东西，分量不等。

一个可判定的程序同一性判准。由 §2.6 与上面那个判准，两个被编码的程序是否相同，可由一个独立于分析者的程序来回答。这是那个回答 §3.5 之缺口的第二半部的组成部分，而它是那个把这一提议与那些已然很好地表示历史的诸框架区别开来的东西。编码仍然是分析者的，由主张 2；但一个已编码的分析者已放弃了重新描述的自由。

一个程序无差别的判准。远交换与辫关系把诸重新排序划分为那些改变关系性历史的与那些不改变的。协商理论想要这一划分，如 §3.3 所论，而它的诊断性框定无法提供它。该模型所给出的是狭窄的，即关系性拓扑的无差别、而不是经验的无差别，而它是一个在没有任何肯定性刻画之处的肯定性刻画。

一个关于反转的预言。§4.5 是该形式体系产生一个结果、而不是以新记号重述一个直觉的那一个地方。一个交叉被它的逆所相消，在它们相邻时是完全的，而在它们被触及同样角色的诸互动所相隔时是不完全的；而这是被推导的、而不是被规定的。上诉在该模型所规定的一个条件之下复原先前的关系性状态。那个条件在经验上是否正确，是可检验的，而在此未被检验。

一个持续的可观测量。环绕数，在主张 4 的刻意地弱的读法之下，测量朝向在一个历史之上的累积。它不是不义的一个度量，正是这使它可用：一个被定义为不义的量，会已预判了那些值得提出的问题。

一个评估性步骤的定位。在那个算子被写出之前，程序正义诸说明中的那个解释性举动是分布的，即它的一些在”什么算作一个程序性特征”的个别化之中，一些在”测量哪些特征”的选择之中，一些在那个规范性论证本身之中。该模型并不移除那个举动。它收集它。

6.2 方向及其不在场

一个被引入规范理论的形式结构，携带一种夹带一个行进方向的风险。这一风险值得被命名，因为本文所属的那个框架已委身于反对它，而一个从后门重新引入它的形式体系，会比一个论证更有效地撤销那项委身。

辫群不提供任何方向。$B_n$ 上没有任何偏序，据之某些辫比别的更先进；长度不是这样一个序，因为一个更长的字可能表示一个更短的辫；而复合把诸事件在时间中排序、而不对诸格局排名。没有辫是进步。由 $P_n$ 的一个元素所关联的两个程序，处于毫无任何优先性的关系之中，而该模型没有任何资源去说相反的话。

该形式体系的这一局限同时是一种保护。因为 $B_n$ 中没有任何规范性排序可用，所以没有任何这样的排序能够经由 $B_n$ 而进入该说明；无论该说明承载什么规范性内容，都必须由 $\Phi$ 所承载，在那里它是可见的、并可被争辩。那个圆锥后门，即持续或精细化或增加成为自我证成的那个举动，在此由不在场、而不是由禁令所关闭。该结构中没有任何东西可能被误当作一个方向。

6.3 诸界限

六个界限，每一个都被追溯到 §2.7，而每一个如今都被配以一个案例或一个论证。

诸角色的涌现。 固定的 $n$ 禁止股的创生，而案例 D 在一个可指认的事件处抵达那个禁令。因此，该框架关于主体生产的说明，即由它们随后参与其中的那些互动所生成的诸关系性位置，是在该模型之外的，而不是在它之内被部分地表示。

合并与消解。 有别于创生，并要求一个彼此有别的修补。案例 D 展示两位工人成为一个讨价还价的位置；$B_n$ 中没有任何东西把 $B_3$ 携带到 $B_2$。

语法的更改。 又一次有别。凡诸生成元之间的诸关系改变之处，即凡什么算作一次互动、或哪些互动可用，本身被改写之处，那个格局不是那个群的一个元素。这是 §4.6 意义上的层级 2 退化，而它的可检测性是全然否定的：该模型通过失败而报告它。

连续的影响。 诸交叉是离散的事件。一段不标点诸事件而渐进地移动的关系没有任何表示，而 §4.1 的三个条件将找不到诸交叉来个别化。

一次互动的内容。 案例 C 是那个证明。两个在拓扑上相同的程序，可能在”协议是由学习还是在胁迫之下所抵达的”上有别，而那个差异是实质性的，并且无法由更审慎的编码而复得。辫记录一次互动以一个朝向发生了。它不记录任何关于那次互动是什么的东西。

缠绕的方向。 由 §4.4，深化的重复与陷入僵局的重复共享一个签名。因此，任何关于层级 1 退化的主张都要求一个该模型所不含的状态变量。因此，一个对该框架核心的概念，是拓扑所不能裁定的一个概念。

6.4 诸评估层级

§1 的范围声明现在可以被展开，因为该模型在手，而它的边界是已知的。本小节陈述那个贡献所处的位置，而它本身不是那个贡献，被录入以便读者不把对一层的处理误当作对整体的处理。

那个架构如此运行：一个关系性过程生成一个历史；那个历史被表示为一个辫；投影 $\pi$ 丢弃路径而保留终点格局；它所丢弃之物由 $P_n$ 所刻画；而 $\Phi$ 在语境中把意义赋予 $P_n$ 的诸元素。本文所做的一切都处于 $\Phi$ 之处或其下。在它之上躺着评估的诸问题，而它们是不同种类的若干问题。

生成性（$G$）追问一次互动是否扩张此后可用的关系性可能之空间。这是描述性的、而不是规范性的，是一个关于一个过程的事实，在原则上无需任何关于那个扩张是否受欢迎的判断而可确定。它也，而这是那个尴尬的部分，已被该模型所预设：§4.1 的条件（iii）以”诸事件是否更改随后诸转变之空间”来个别化它们，也就是说，那个编码要求一个辫所不表示的可能之空间的概念。这个缺口是承重的。一个关于 $G$ 的形式说明会是一个完全被规定的编码的先决条件，而不仅仅是它的一个扩展。

程序正义（$J_p$）追问生成是否通过合法的关系而发生。这是此处所处理的那一层，而它是一个关于合法性、而不是关于能力或关于价值的问题。

生成正义（$J_g$）追问一个过程是否保全并扩张它的诸参与者去生成的能力。这是那个独特地属于该框架的问题，而它在两个方向上都独立于 $J_p$。一个由 §3 的诸判准衡量无可挑剔的程序，可能使它的诸参与者比它所发现的更不能够生成，即消耗信任、封闭诸位置、耗尽未来互动所依赖的诸条件，而这个案例既不奇异也不罕见。$J_p = 1$ 与 $J_g = 0$ 相容，这正是程序正义不可能是该框架所要求之物的全部的理由。

规范性朝向（$V$）追问朝向什么。生成性与程序上的合法性都不回答它，而两个例子确立它。一场飓风是一个高度生成性的耗散结构，跨诸尺度产生组织；关于它的价值什么也推不出。一个共同体可能公平地、透明地、并以充分的参与，设计一个更高效的剥削系统；关于那个设计的公平，没有任何东西救赎所设计之物。生成性不是价值，而合法的程序不是价值，而一个让二者之一替代第三者的说明，犯下本文的护栏被竖起来所针对的那个错误。

这四者不是一个东西的四个种，而它们之间的差异是种类上的差异：$G$ 是描述性的，$J_p$ 是一件合法性的事，$J_g$ 是一件持续能力的事，$V$ 是一件方向的事。把它们坍缩是那个特有的失败。把 $J_p$ 坍缩进 $V$，则合法的程序成为自我证成的，那是 §6.2 所排除的目的论。把 $G$ 坍缩进 $V$，则生成成为它自己的凭据，那是该框架一贯地拒绝的圆锥后门。这一分开是同一道护栏，被安置在要突破它的压力最大之处。

至于这些层级可能如何被研究：$G$ 招致以”可达性”以及”一次互动之后相对于之前可用的选项集之大小”来处理。$J_g$ 容许一个候选的形式判准，即一个程序之后可达的诸关系性状态之集是否含有之前可达的那个集，而它在此作为一个猜想被提出，并不被展开。$V$，就当前的证据而言，不适于此处所尝试的那一类形式处理，而把它作为如此而命名，胜过对一个并不存在的形式体系的姿态。

6.5 开放问题

构造 $\Phi$。 那个核心的。本文已论证，一个在诸程序性差异之间区分的说明，因此就使用一个上述类型的算子，而已拒绝去建造它。案例 B 所产出的是一个起点、而不再是别的：那个猜想，即一个纯辫的意义取决于它与随后可能之空间的关系，而因此 $\Phi$ 汲取事件个别化所汲取的同样的信息。如果那是对的，那么这两个问题是一个，而一个对 $G$ 的处理会提供二者。

允许诞生、死亡、合并与分裂的诸结构。 配边是那个自然的方向。凡一个辫是一组在两个固定格局之间运行的股之处，一个配边是一个在两个边界之间的流形，其诸分量可以接合、分开、出现与消失；案例 D 所分开的那四个失败，对应于一个配边所允许、而一个辫所不允许的四个运算。该框架关于主体生产的说明恰恰要求这些，而这提示那个扩展是根本上形式地处理涌现的一个先决条件。

复合的诸设置。 辫幺半范畴把辫群一般化到一个其中诸对象可以被复合的设置，而会是那些由诸子程序所建造的程序的恰当归宿。更高的范畴结构允许诸关系本身变化，而那正是层级 2 退化会要求的、以被表示而不仅仅作为失败被检测。

其他候选结构。 辫群是一个候选者，而本文未曾主张更多。诸构形空间的基本群与广群、映射类群，以及更一般的诸道路空间，是具有不同强处的诸替代方案，而一个比较会是有信息量的。有可能某个别的结构以更低的假定代价提供可判定的程序同一性；若如此，§3.5 的论证存活，而那个候选者改变。

可操作化。 主张 4 的持续可观测量被定义而未被测量。它能否从实际程序的诸记录，即会议记录、誊录、谈判日志，被计算出来，是一个经验问题，而主张 2 的编码问题是困难会集中之处。

详加处理退化。 §4.6 的两层级区分只被处理到当前论证所要求的程度。主张 5 的不对称，即一层只作为模型失败而可检测、另一层不经解释根本不可检测，召唤一个比当前论证所给予它的更充分的处理。

6.6 结论

程序正义早已描述了一类缺乏一个足以表示它们的形式对象的差异。罗尔斯的纯粹程序正义学说使程序成为它的结果之正义的构成，而没有给出任何比较两个都被正确地举行的程序的办法。泰勒的纲领确立了某种结果不变的东西被追踪，并以它对判断的诸效果来刻画它。协商理论知道次序要紧，并诊断性地处理那份知道。森拓宽了评估的对象，而把它留作一个仍在它的终点处被评估的对象。

纯辫群为这些说明所描述之物提供一个候选结构。它是一个保持终点格局而改变关系性历史的变换之群，在其中结果不变的差异是确定的元素，而程序的同一性是可判定的。那是那个肯定性主张，而四个案例已表明它的所及以及它在何处停止。它并不评估它所表示之物。它并不表示它所记录之物的内容，案例 C 是那个证明。它在它所预设的诸角色仍在形成之处停止适用，而那正是该框架自身最深的诸问题开始之处。

那么，所余下的，是一个诸评估可以在其上运作的结构性基底，连同一个”它们之中的每一个必须于何处进入”的规定。辫承载一个过程的记忆。那个过程是否合法、它是否使它的诸参与者能够继续生成、以及它所生成之物是否值得生成，是三个进一步的问题，而一个回答它们之中无一者的形式体系的价值在于，它使每一个的提出成为不可回避且分开的。

致谢

本论证通过一轮又一轮的批判性阅读而被精炼，而它的若干核心纠正，即 $P_n$ 作为一个变换群、而不是一个集合的地位、$\Phi$ 的语境依赖性、缠绕的无方向性，以及一个其中该模型产出一个无意义之差异的案例的必要性，都起源为诸反对。所余的错误是作者的。

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