Yang–Mills Dynamics from Projective Spectral Entropy: Vertical Heat-Kernel Variation on the Admissible Fibre

The companion Gravity paper showed that the horizontal variation of the projective spectral entropy functional $S_\Pi[g] = \tfrac{1}{2}\log\det' A_g$ with respect to the base metric produces the Einstein tensor as the infrared-dominant response, through the Seeley–DeWitt coefficient $a_2$. The present paper carries out the complementary vertical variation. Extending $A_g$ to a Laplace-type operator $A_{g,\mathcal{A}} = -(\nabla^{\mathcal{A}})^2 + E$ on the associated vector bundle of the admissible principal fibre, with the connection and the fibre representation $\rho$ held fixed, we compute the local logarithmic part of the Seeley–DeWitt coefficient $a_4$ and isolate its gauge component, the Yang–Mills density $\tfrac{1}{12}\,\mathrm{tr}_\rho(F_{\mu\nu}F^{\mu\nu})$. Varying $S_\Pi[g, \mathcal{A}]$ with respect to the admissible connection at fixed base metric then yields the source-free Yang–Mills equations $D_\mu F^{a\mu\nu} = 0$. This local kinetic Yang–Mills sector, and its variation, are the robust result of the paper. The coefficient of the induced logarithm depends on the fibre representation content through the Dynkin index $I_\rho$ and is therefore not universal. The physical gauge coupling $g_{\mathrm{YM}}$, like Newton's constant, remains a renormalized matching datum, of which only the logarithmic running is computed here; a single logarithmic divergence does not by itself establish asymptotic freedom, which depends on the full projected field content and is not claimed. What is structural, within the proper-time scheme used here, is the difference in ultraviolet divergence degree between the two sectors — quadratic at $a_2$ for gravity, logarithmic at $a_4$ for gauge — and not any predicted numerical hierarchy of the physical couplings; the contrast is not scheme-independent, the zeta-regularized determinant carrying no power divergences. The gauge group $G_\Pi = \mathrm{SU}(3)\times\mathrm{SU}(2)\times \mathrm{U}(1)$ is taken as input from the spectral admissibility sub-programme, whose $\mathrm{SU}(3)$ sector is established there. Interpretation. The result suggests that, within this framework, the natural organisational space for unification may be spectral and variational rather than purely group-theoretic: gravity and gauge dynamics are separated by geometric direction (horizontal versus vertical) and by heat-kernel order. This reading is offered as an interpretive outlook, not as a theorem.