Formulation of Scale-Gauge Theory and Its Cosmological Consequences Part I (Revised): Local Scale Symmetry on Riemannian Geometry Free from Non-Metricity

<p>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 <em>Formulation of Scale-Gauge Theory and Its Cosmological Consequences, Part I: Foundational Framework</em> (Zenodo, 2025, DOI: 10.5281/zenodo.17688698).</p>

<p>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.</p>

<p>This paper proposes a realization of local scale symmetry that avoids both obstacles simultaneously, by detaching the scale-gauge field <em>W</em>_&mu; from the Weyl connection and defining it as an <strong>independent internal gauge field</strong> on a Riemannian manifold. The main results consist of two theorems and two propositions:</p>

<ul>
<li><strong>Theorem 1:</strong> the Lagrangian is locally scale invariant up to a total divergence under the conformal coupling &xi; = 1/6;</li>
<li><strong>Theorem 2:</strong> metric compatibility holds in every gauge, so no second clock effect arises;</li>
<li><strong>Proposition 1:</strong> the transformation law <em>W</em>_&mu; &rarr; <em>W</em>_&mu; &minus; &part;_&mu;&alpha; produces a non-trivial contribution to Noether's second theorem that is absent under Oda's theorem;</li>
<li><strong>Proposition 2:</strong> the scale-centroid condition &sum;_i c_i&beta;_i = 0 is characterized as the consequence of Noether's second theorem for local scale symmetry, i.e. as a classical Noether identity.</li>
</ul>

<p>A key clarification in this revision is that &beta;_i = 4 &minus; &Delta;_i is the <strong>classical scale-dimension offset</strong> of an operator, and is explicitly <em>not identified</em> 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.</p>

<p>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).</p>

<p>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.</p>

<p><strong>Keywords:</strong> local scale symmetry; Riemannian geometry; non-metricity; conformal coupling; Noether's second theorem; scale-centroid condition; internal gauge field; Weyl geometry</p>