We present a Grand Unified Theory (TOE) framework based on the concept of Roughness Renormalization. We identify the fundamental conflict between Quantum Me- chanics (QM) and General Relativity (GR) as a mismatch in the geometric regularity of space- time: QM operates on rough paths (α = 1/2), while GR operates on smooth manifolds (α = 1). We introduce the Renormalization Operator of Roughness, Rλ, which acts as a scale transformation bridging these two regimes. Using the principle of Sakharov’s induced grav- ity, we prove that the metric tensor is not a fundamental field but an expectation value of the quadratic variation of quantum rough paths. We derive the Einstein-Hilbert action as the first-order effective action resulting from the smoothing of quantum fluctuations. This frame- work unifies the probabilistic nature of the micro-world with the deterministic geometry of the macro-world, resolving the problem of time and the non-renormalizability of gravity.
Lee Sung-gil (Wed,) studied this question.