Structural Threshold and Initial Bias：The Dual Conditions of Global Emergence——A Phenomenological, Systems-Theoretic, and Philosophical Investigation Based on a Three-State Majority Vote Model

Abstract

Conventional notions of emergence hold that, given a set of simple local rules, a system can spontaneously generate globally complex structures through self-organization, even when starting from random or disordered initial conditions. The present study challenges this view through a three-state majority vote model. Three experiments are conducted: (1) Global chessboard initialization—starting from a complete alternating ±1 structure, the system stably converges to approximately 4.5% zero states, independent of system size, with the order parameter converging to 0.4774. (2) Local chessboard block injection—starting from the all-zero state, a chessboard block is injected at the center (coverage ratio 1%–90%)—the final zero-state proportion equals the coverage ratio multiplied by 4.5%, exhibiting a perfect linear relationship (R²≈0.999) with no amplification, no synergy, and no "growth." (3) Low-structure attractor with structural seeds—the system is first stabilized at approximately 4.21%, then overlaid with chessboard blocks—it ultimately returns to the vicinity of 4.21%, failing to transition to 4.5%. Taken together, these three experiments demonstrate that global structure can only unfold from globally structured initial conditions. Local perturbations cannot substitute for globally structured initial conditions. Building on this, the study further introduces phase transition critical points—percolation transition att_c≈0.99, gravitational transition g_c≈1.2, and shear-driven bistable switching—and argues that these critical points constitute rigid constraints on the evolutionary process. The structural threshold of initial conditions and the phase transition critical points of the evolutionary dynamics together form the dual conditions of global emergence. A revised definition of emergence is proposed: "Emergence is not creation from nothing, nor growth from the local to the global, but the unfolding from potential global structure to actualized global structure."