The Convergence of an Infinite Domain, the Establishment of an Origin Reference Position, and the Generation of Elements

This work develops an abstract framework for rethinking the classic question “Why is there something rather than nothing?” not from already-given discrete objects, but from a prior infinite domain and the structures that make “elements” possible. Instead of assuming that the world is fundamentally a collection of independent things to which relations are later added, the paper starts with an undivided infinite domain Ω and shows how an origin reference position and discrete elements can be generated step by step.

First, a family of convergence operators {C_ε} is introduced to model multi-scale “compress-and-redraw” behaviors on Ω. Under a suitable topology, this family converges to a global contraction point o, interpreted as an origin reference position: a multi-scale superposition center where, under finite observational resolution, many differences are flattened into the experiential impression that “everything begins here”. A referentialization map κ: Ω → ℝⁿ then encodes deviations with respect to this origin, so that each finite reference vector corresponds to a large and complex preimage in Ω. Finally, a partial designation–response operator E: D ⊂ ℝⁿ → 𝒰 maps selected, structurally “open” reference values into a universe of elements 𝒰.

In this framework, elements are not primitive building blocks but emergent outcomes of the structural chain Ω ⟶ {C_ε} ⟶ o ⟶ κ ⟶ ℝⁿ ⟶ E ⟶ 𝒰. The paper argues that, in this precise sense, “the finite contains the infinite” is not merely a poetic slogan but a structural and ontological necessity.