The Boundary Vacuum Law: Gradient, Boundary Failure, Topological Flow, and Pressure Capture in Social Systems

The Boundary Vacuum Law: Gradient, Boundary Failure, Topological Flow, and Pressure Capture in Social Systems Civilization Physics — Series: Boundary Laws of Self-Organizing Systems This article proposes the Boundary Vacuum Law as a structure-level law of coupled social systems. When a functional gradient remains active and the relevant boundary becomes too weak to regulate it, responsibility, authority, emotional labor, military burden, narrative control, or epistemic validation is redistributed through the remaining topology. Pressure moves toward nodes that are accessible through proximity, permeability, legitimacy, availability, coercive pull, or normative tolerance. The destination and sustainability of this flow are separate questions. A node may be highly likely to receive pressure while lacking the capacity to carry it safely. This distinction explains why reliable employees become overloaded, children are recruited into adult conflict, allies become permanent guarantors, and nominal human oversight gradually yields practical authority to automated systems. The article develops this argument through several linked mechanisms: Compact structural indices: Capture-flow intensity is summarized by F=(G×P)/BF=(G\times P)/BF=(G×P)/B, where gradient and capture accessibility increase expected flow while boundary resistance constrains it. Subject-position stability is summarized by Si=(Bi×Ai)/(Gsys×Ei)S_i=(B_i\times A_i)/(G_{\mathrm{sys}}\times E_i)Si=(Bi×Ai)/(Gsys×Ei). Gradient-regulated network flow: A weighted graph represents actors, roles, institutions, and interfaces. Boundary resistance determines edge conductance, while pressure differences generate flow across connected paths. The graph Laplacian formalizes how sources, sinks, boundaries, and topology jointly determine the distribution of pressure. Topological capture probability: Absorbing Markov chains model which available node eventually captures an unresolved function. Capture probabilities depend on routing weights, network position, accessibility, and the placement of absorbing roles. Moral desert does not enter the routing equation unless it has already changed the actor’s structural accessibility. Capacity thresholds and role instability: Queueing equations separate incoming load from sustainable processing capacity. When effective arrivals approach or exceed capacity, backlog, delay, exhaustion, ceremonial review, or functional overload follows. Reliability can increase routing toward an actor faster than formal capacity expands. Load redistribution and cascade: When a carrier or interface fails while the original gradient remains, unresolved load is rerouted through the surviving network. Newly exposed nodes may then exceed their own capacities, producing secondary failure and further redistribution. Three structural responses: Boundary repair changes resistance and conductance. Interface redesign changes routing, source placement, arrival patterns, and capture positions. Topological separation changes reachability itself. Capacity reinforcement can delay overload, although the capture path remains active until one of the underlying structural relations changes. The mathematical core applies exactly under explicit network assumptions. Intimate relations, families, organizations, collective-action systems, alliances, public narratives, and AI governance are treated as domain-specific structural realizations. Their local mechanisms differ, while the same topology recurs: persistent gradient, weak boundary, routed pressure, capture, capacity violation, and possible cascade. The law also clarifies why weak boundaries frequently create counterfeit order. A system may appear stabilized because a nearby actor has absorbed an abandoned function. The apparent repair conceals dependency, exhaustion, role drift, hidden fragility, or concentrated authority. Durable governance therefore requires structural change rather than exhortation alone. The Boundary Vacuum Law complements the Universal Inbreeding Law. Excessive internal closure causes diversity contraction and integrity decay. Insufficient regulation at an interface allows gradients to colonize adjacent positions. Together, the two laws describe how systems fail when they cannot regulate either what enters or what crosses their boundaries. Keywords: Boundary Vacuum Law · Boundary Failure · Functional Gradient · Network Flow · Graph Laplacian · Absorbing Markov Chains · Capture Probability · Queueing Networks · Capacity Threshold · Load Redistribution · Cascade Failure · Role Drift · Subject-Position Stability · Interface Redesign · Topological Separation · AI Governance · Social Systems · Civilization Physics