Reinterpreting UV Divergence and the Hierarchy Problem via k-Foam Geometry: A Discrete Graph Theory Approach

Modern quantum field theory struggles with two major issues originating from the continuum assumption: 1) UV divergence (infinities at r → 0) 2) The Hierarchy Problem (the 17-order gap between the Higgs mass and the Planck scale) This working paper, based on the k-Foam Theory (a discrete geometric framework), proposes a natural resolution to these problems without relying on artificial renormalization or extra dimensions. --------------------------------------------------Key Proposals & Derivations-------------------------------------------------- 1 Natural Resolution of UV Divergence By redefining the universe as a non-empty connected graph: Graph ≠ ∅ (Axiom 0) and prohibiting complete determination: P ≠ 1 (Axiom 3) the topological state where r → 0 becomes physically unreachable. As a result, infinite momentum states never appear, eliminating UV divergence at the structural level. 2 Reduction of the Hierarchy Problem Instead of defining the fundamental scale of space using the gravitationally-dependent Planck length, the theory shifts the baseline to the pure QCD scale: ~ 1 fm Under this assumption, the 17-order magnitude gap between the Higgs scale and the Planck scale is reduced to a much smaller topological compression ratio of roughly 3–4 orders. 3 Geometric Derivation of Strong Force Scales The physical limits of quark confinement and nuclear force range are derived purely from integer ratios of grid k-values: Quark confinement limit: 1. 25 fm = 5/4 ratio Nuclear force range: 2. 0 fm = 6/3 ratio 4 Geometric Derivation of the Higgs Mass (125 GeV) The Higgs resonance is interpreted as the simultaneous yielding of the minimum spatial block: k = 4 tetrahedron Using an information-theoretic scaling relation: 0. 2 GeV × 5⁴ = 125 GeV This provides a geometric interpretation of the observed Higgs mass. --------------------------------------------------Conceptual Position-------------------------------------------------- This paper does not reject existing physics. Instead, it attempts to debug the underlying assumptions — the "OS" of the universe — by replacing the continuum assumption with a discrete geometric foundation.