A Discrete Double-Exponential Hierarchy of the Universe: From M-Theory Roots to the String Landscape

We propose a unified generative principle for physical scales based on the discrete double exponential function Nx = exp(exp(x)), where x ∈ Z. Theoretically, we derive this scaling law from first principles, demonstrating that it emerges from the interplay between the Toda lattice potential in M-Theory, the Hagedorn density of states in string thermodynamics, and the Renormalization Group flow. This hypothesis precisely predicts 15 fundamental physical constants, encompassing positive hierarchies (x = 1 to 7) such as the Standard Model fermion count and the String Landscape bound, and negative hierarchies (x = −1 to −7) representing geometric corrections. The model re solves the Cosmological Constant problem (Λ ∼ 10−120) and derives the Fine-Structure Constant (α −1 ≈ 137.036) with high precision. We argue that the hierarchy cutoff at |x| ≤ 7 renders the universe a finite, self-consistent axiomatic system, thereby circumventing G¨odel’s incompleteness theorem.