On the Gauge-Theoretic Foundations of Structural Mass: Derivation of the Topo-Geometric Energy Equivalence ε = m·g2, the Macroscopic Yield Potential E = mG2, and Formal Proofs

Abstract Conventional relativistic mechanics postulates mass-energy equivalence via the scalar relation \(\:E=m{c}^{2}\), assuming a continuous, structureless vacuum in which the asymptotic speed of massless bosons (\(\:c\)) acts as a universal proportionality coefficient. This treatment considers mass \(\:m\) as an opaque scalar lacking internal geometry, overlooking the fact that over 98% of visible baryonic mass originates not from the electroweak Higgs mechanism, but from dynamical chiral symmetry breaking and the confinement energy of the Quantum Chromodynamics (\(\:SU\left(3\right)\)) gluon condensate. In this paper, we formulate the analytic derivation of the Topo-Geometric Energy Equivalence \(\:\mathcal{E}=m\cdot\:{g}^{2}\) at the microscopic field-theoretic level, and the Specific Gluonic Yield Potential \(\:E=m{G}^{2}\) at the macroscopic thermodynamic scale. Through the expansion of the Bogomol’nyi-Prasad-Sommerfield (BPS) bound and the scale regularization governed by Derrick’s Theorem under the Absolute Topological Invariant constraint (\(\:{\mathcal{O}}_{topo}\approx\:1\)), mass emerges ontologically as the inertia of the topological defect modulated by the geometric impedance of the vacuum. Through six analytical lemmas, we demonstrate the volumetric phase balance on 1 gram of Plutonium-239 (\(\:{}^{239}Pu\)), revealing how the bare fraction of valence fermions (~ 1%) generates a systematic deviation from pure kinematics, quantified as a Structural Yield Gap of \(\:\varDelta\:E\approx\:2.4\times\:{10}^{11}\) J (~ 0.26%). The model is proven falsifiable by identifying the absolute minimum of lattice stress at Iron-56 (\(\:Z=26\)) and the asymptotic divergence of the coupling constant for superheavy elements (\(\:Z\ge\:118\) Oganesson), providing three concrete experimental protocols for empirical verification.