The Impossibility of Cohesion Without Fragmentation

TL;DR

The paper develops a static axiomatic framework in which relation maintenance is governed by positional compatibility in a shared space $\mathcal{L}$ and a bifurcation event $E$ fixes positions $\ell_i$. It proves a structural impossibility: for any non-degenerate position-dependent gain axis $g$ with a compatibility function $f_g$, no configuration after $E$ can preserve all initial relations, yielding fragmentation; cohesion can occur only within compatible regions, and a degenerate or existence-dependent gain axis is required to maintain universal cohesion. The Main Theorem formalizes the dual structural outcomes—fragmentation and cohesion—as inevitable consequences of positional constraints rather than incentives or dynamics. This reframes division as a logical result of structure, with implications for polarization, identity-based sorting, and polarized networks, and points to future work integrating dynamic or strategic mechanisms atop this structural core.

Abstract

Most models in game theory and network formation implicitly assume that relations between agents are feasible whenever incentives are aligned or interaction opportunities exist. Under this premise analytical attention is directed toward equilibrium efficiency or probabilistic link formation while the possibility that a relation may be structurally infeasible is rarely examined. This paper develops a static axiomatic framework in which relation maintenance is treated as a problem of structural compatibility rather than strategic choice or stochastic realization. Agents occupy positions in an abstract space and relations are subject to minimum conditions defined over these positions. A bifurcation event such as a vote declaration or institutional assignment fixes agents positions and thereby determines which relations are compatible. We identify position dependent gain axes as the key source of structural selectivity and prove an impossibility result under any non degenerate positional constraint no bifurcation event can preserve all relations. Instead the post event network necessarily exhibits either the simultaneous emergence of fragmentation and cohesion or a degenerate trivial case in which constraints are position independent. The result is purely structural and does not rely on preferences beliefs incentives or dynamic adjustment. It establishes a fundamental limit on universally cohesive outcomes and reframes division not as a failure of design or coordination but as a logical consequence of positional constraints.

Figures (1)

• Figure 1: Structural duality between fragmentation and cohesion. Numerical simulation results for $N=200$ agents embedded in a two-dimensional position space. Relations (edges) are formed based on the positional compatibility condition $\mathrm{dist}(i,j) \le d$. (a) Weak constraint ($d=0.22$). When relation formation is weakly constrained, the network maintains a single giant connected component, resulting in high global integration; however, the internal edge density remains low. (b) Strong constraint ($d=0.05$). As the constraint is tightened, the network bifurcates into multiple connected components, decreasing global integration while increasing internal edge density. (c) Structural trajectory. As the tolerance radius $d$ decreases, integration and cohesion exhibit a non-monotonic relationship: cohesion increases in an intermediate regime despite declining integration, whereas further tightening leads to fragmentation and a subsequent decline in cohesion. Note. The network construction is formally equivalent to a random geometric graph; the focus here is on the relative effects of constraint tightening on fragmentation and cohesion.

Theorems & Definitions (57)

• Remark 1.1: Relation to Theoretical Framework
• Definition 3.1: Universal Player Set
• Definition 3.2: Gain Axes
• Definition 3.3: Dyadic Player Set
• Definition 3.4: Minimum Condition Function
• Remark 3.6: Structural Interpretation of Withdrawal
• Definition 3.7: Basic Relation
• Definition 3.8: Active Gain Axes
• Definition 3.9: Relation Matrix
• Definition 3.10: Partition of Active Gain Axes