This paper presents a revised and significantly strengthened foundational formulation of Scale-Gauge Theory (SGT), superseding the framework first proposed in the author's earlier preprint Formulation of Scale-Gauge Theory and Its Cosmological Consequences, Part I: Foundational Framework (Zenodo, 2025, DOI: 10.5281/zenodo.17688698). Local scale symmetry, originating in Weyl's 1918 attempt to unify gravity and electromagnetism, has recently received renewed attention as a candidate resolution of contemporary puzzles such as the cosmological-constant problem and the unnaturalness of the Higgs mass. However, any formulation based on Weyl geometry faces two obstacles: the non-metricity condition predicts a second clock effect (already excluded experimentally), and a theorem proved by Oda (2022) shows that implementing the Weyl transformation as a spacetime symmetry on Riemannian geometry makes the associated Noether current identically zero. This paper proposes a realization of local scale symmetry that avoids both obstacles simultaneously, by detaching the scale-gauge field Wμ from the Weyl connection and defining it as an independent internal gauge field on a Riemannian manifold. The main results consist of two theorems and two propositions: Theorem 1: the Lagrangian is locally scale invariant up to a total divergence under the conformal coupling ξ = 1/6;Theorem 2: metric compatibility holds in every gauge, so no second clock effect arises;Proposition 1: the transformation law Wμ → Wμ − ∂μα produces a non-trivial contribution to Noether's second theorem that is absent under Oda's theorem;Proposition 2: the scale-centroid condition ∑i ciβi = 0 is characterized as the consequence of Noether's second theorem for local scale symmetry, i.e. as a classical Noether identity. A key clarification in this revision is that βi = 4 − Δi is the classical scale-dimension offset of an operator, and is explicitly not identified with the quantum-theoretic renormalization-group beta function. This corrects a looser association made in the author's earlier preprint and cleanly separates classical (kinematical) from quantum (dynamical) aspects of the theory. The present construction is closely related to the non-metric gauge theories of Ghilencea and collaborators, with which a detailed comparison is given; the two approaches are shown to be complementary realizations that differ essentially in their underlying geometry (Riemannian vs. non-metric) and their mass-generation mechanism (spontaneous symmetry breaking vs. non-metric geometry). This paper establishes the classical and kinematical foundation for the series; mass generation via spontaneous symmetry breaking and the resulting two-phase structure of the universe, together with observational consequences for galactic dynamics, are developed in subsequent papers of the series. Keywords: local scale symmetry; Riemannian geometry; non-metricity; conformal coupling; Noether's second theorem; scale-centroid condition; internal gauge field; Weyl geometry
Susumu Goto (Mon,) studied this question.