The Discrete Origin of Quantization

Previous work (Chen, 2026b) established the uniqueness of cosmic presentness and stipulated that the single-step state difference is an indivisible atomic token (Axiom 6), but it did not prove why the magnitudes of these variations must be discrete. This paper rigorously fills that gap. First, Lemma 0 is proved: the finiteness of the universe's total information capacity N (bits) implies that the reachable value space V of any given lattice point is a finite set (cardinality κ < ∞). This leads to Theorem 1: the set of single-step variation magnitudes Λ_step is a finite discrete set, with a non-zero infimum ϵ_min > 0. Any variation smaller than ϵ_min would require a continuum state space, violating finite information capacity. Theorem 2 further proves that Λ_step lacks additive closure; therefore, numerical multiples (e.g., ||Δq|| = 3ϵ_min) do not represent physical sub-steps; the single-step change is an indivisible atomic token. Finally, it is proved that the three states of "positive/negative/zero" constitute the complete spectrum of quantized variation. Conclusion: the magnitudes of all physical variations in the universe are necessarily discrete (quantized), with a non-zero minimum scale ϵ_min, and macroscopic continuity arises from coarse-grained averaging.