Multiplicative Model of Life and Family: Mathematical and Philosophical Analysis of Constituent Integration and Dimensional Expansion

Abstract In conventional sociology, bioscience, and statistics, the scale of life and family has predominantly been treated as the mere summation of their constituents (the additive model: 1 + 1 = 2). However, the additive model merely aggregates individual elements as independent quantities, failing to capture the organic bonding among constituents or the emergence of a "field" as a unified system. This paper reconstructs the structure of life, couples, and families through a power-law model (1^n), grounded in the axiom that "the fundamental formula representing life is multiplication." Specifically, by defining a single human as 1 and the essence of a child as the integration of parents (Male \times Female, 1 \times 1), we demonstrate that a three-person family is precisely expressed as 1 \times 1 \times 1 \times 1 = 1^4. In this model, the relationship 1^3 = 1^4 = 1 establishes that the increase of elements (dimensional expansion) and the invariance of overall value (harmony) occur simultaneously.