change

Author: anmo-fish

content/publications/a_1/index.md

@@ -16,7 +16,7 @@ publication_types: ["article"]
 publication: ""
 publication_short: ""
 
-abstract: "This paper develops a structural theory of social relations showing that, under non-degenerate positional constraints, fragmentation and cohesion arise necessarily and simultaneously as dual consequences of the same bifurcation event."
+abstract: "Why do division and cohesion often intensify together? This paper develops a static structural theory of relation maintenance based on minimal positional constraints. Rather than relying on utility-based or probabilistic models, social relations are formalized as constraint satisfaction problems over an abstract position space. When a bifurcation event---such as a vote or institutional assignment---fixes agents positions, relational viability is determined solely by positional compatibility. We establish three structural results. First, under any non-degenerate positional constraint, fragmentation (relational collapse) and cohesion (condition confirmation) necessarily coexist as complementary outputs of a single compatibility function. Second, we prove a structural asymmetry of veto power: relation maintenance requires logical conjunction, while collapse requires only logical disjunction, implying that fragmentation operates as a unilateral structural veto. This yields a purely logical foundation for behavioral premises such as pairwise stability. Finally, we establish a conditional impossibility theorem: under positional plurality, avoiding relational collapse is structurally impossible, leaving coercive homogenization as the only design-level guarantee for universal cohesion. The framework isolates minimal boundary conditions and provides a formal language for analyzing polarized relational structures."
 
 tags:
 - Structural Theory
@@ -80,9 +80,9 @@ This is not a paradox. This paper shows it is a structural necessity.
 ### The Question
 
 Empirical observers of polarized societies often note a striking pattern: as groups fragment 
-along political, ideological, or institutional lines, each fragment simultaneously becomes 
-*more* internally cohesive. Standard approaches — utility maximization, probabilistic tie 
-formation, threshold models — can reproduce this pattern, but they cannot *explain* it 
+along political, ideological, or institutional lines, each fragment becomes 
+*more* internally cohesive. Standard approaches --- utility maximization, probabilistic tie 
+formation, threshold models --- can reproduce this pattern, but they cannot *explain* it 
 without importing behavioral or psychological assumptions.
 
 This paper asks a more fundamental question: is there a purely structural reason why 
@@ -102,7 +102,7 @@ The framework deliberately discards:
 - Temporal dynamics
 
 What remains is a minimal logical skeleton. A **relation** exists between two players only 
-if certain structural conditions — called *gain axes* — are satisfied. Each player $i$ has 
+if certain structural conditions --- called *gain axes* --- are satisfied. Each player $i$ has 
 a minimum condition function:
 
 $$a(i, g) \in \{0, 1\}$$
@@ -129,7 +129,7 @@ determinate position:
 $$E: P \to L$$
 
 Examples include a vote, a loyalty declaration, institutional assignment, or even an act of 
-public alignment on social media. The theory does not model *why* such events occur — only 
+public alignment on social media. The theory does not model *why* such events occur --- only 
 what *must* follow structurally once positions are fixed.
 
 Once $E$ is applied, local compatibility values propagate to the entire network. For every 
@@ -138,32 +138,52 @@ negotiation or adjustment.
 
 ---
 
-### The Main Result
+### Three Structural Results
 
 **Theorem 10.2** establishes a dichotomy. Let $g$ be a gain axis relevant to the initial 
 relation set, and let $E$ be any bifurcation event. Then exactly one of the following holds:
 
-1. **Degeneracy:** $f_g(E(i), E(j))$ is constant across all relevant pairs — the positional 
+1. **Degeneracy:** $f_g(E(i), E(j))$ is constant across all relevant pairs --- the positional 
    constraint exerts no selective pressure. This is the exceptional, empirically rare case.
 
 2. **Structural Necessity:** If non-degenerate, the post-event network *necessarily* 
    exhibits both:
-   - **Fragmentation** — relations bridging incompatible positions are structurally forced 
+   - **Fragmentation** --- relations bridging incompatible positions are structurally forced 
      to collapse ($f_g(E(i), E(j)) = 0$)
-   - **Cohesion** — relations within compatible configurations survive 
+   - **Cohesion** --- relations within compatible configurations survive 
      ($f_g(E(u), E(v)) = 1$)
 
 These are not two separate effects. They are the $0$ and $1$ outputs of the *same* 
 compatibility function applied under the *same* structural constraint.
 
+**Proposition 8.3** establishes a structural asymmetry of veto power. The logical conditions 
+governing collapse and cohesion are not symmetric:
+
+- Relational collapse requires only **unilateral** condition failure --- governed by logical 
+  disjunction ($a(i,g)=1 \lor a(j,g)=1$)
+- Cohesion confirmation requires **bilateral** match --- governed by logical conjunction 
+  ($a(i,g)=1 \land a(j,g)=1$)
+
+Fragmentation therefore operates as a unilateral structural veto, while cohesion requires 
+mutual structural alignment. This is a purely logical consequence of the axiomatic system, 
+independent of any behavioral or cultural factors. As a corollary, it provides a structural 
+foundation for the behavioral premise underlying pairwise stability in network economics: 
+the rule that severance is unilateral while formation is bilateral is not merely a modeling 
+convention --- it is a logical necessity under constraint satisfaction.
+
+**Theorem 11.3** establishes the conditional impossibility of universal cohesion. Under 
+positional plurality, there is no bifurcation event that avoids relational collapse for every 
+possible initial relation set --- except coercively homogenizing ones, which enforce 
+compatibility across all pairs by design.
+
 ---
 
 ### What This Means
 
 The paper establishes two corollaries with direct interpretive force.
 
-First, the only structural condition permitting universal cohesion — cohesion without any 
-fragmentation — is a globally existence-dependent gain axis: one where $f_g \equiv 1$ 
+First, the only structural condition permitting universal cohesion --- cohesion without any 
+fragmentation --- is a globally existence-dependent gain axis: one where $f_g \equiv 1$ 
 everywhere. Every genuinely position-dependent constraint imposes selective relational 
 mortality.
 
@@ -177,10 +197,10 @@ This is not a sociological observation. It is a theorem.
 ### Why This Matters
 
 The intensification of in-group cohesion alongside inter-group division requires no 
-additional mechanism — no radicalization dynamics, no echo chamber effects, no strategic 
+additional mechanism --- no radicalization dynamics, no echo chamber effects, no strategic 
 calculation. It is the unavoidable output of relation-maintenance constraints operating 
 under positional commitment.
 
 This provides a common structural language applicable across polarized political networks, 
-institutional sorting, and identity-based alignment — one that remains agnostic about the 
-specific processes generating any particular bifurcation event.
+institutional sorting, and identity-based alignment --- one that remains agnostic about the 
+specific processes generating any particular bifurcation event.