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

The companion 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, via 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, we compute the Seeley–DeWitt coefficient $a_4$ for the total operator and extract the Yang–Mills term $c_{\mathrm{YM}}\int \mathrm{Tr}(F_{\mu\nu}F^{\mu\nu})\sqrt{g}\, \mathrm{d}^4 x$. Variation of $S_\Pi[g, \mathcal{A}]$ with respect to the admissible connection $\mathcal{A}$ then yields the Yang–Mills equations $D_\mu F^{a\mu\nu} = 0$ in the current-free sector. The induced gauge coupling $g_{\mathrm{YM}}^{-2} \sim (12 \cdot 16\pi^2)^{-1}\log(\Lambda/\mu)$ is logarithmic in the UV cutoff, while Newton's constant $G_N^{-1} \sim \ell_{\mathrm{sp}}^{-2}$ is quadratic. This spectral hierarchy is a structural consequence of the Seeley–DeWitt expansion and does not require a separate input. The gauge group $G_\Pi$ is taken as input from the spectral admissibility programme ; the SU(3) sector is unconditional at the effective level: $[H\text{-color}]_{\mathrm{eff}}$ (equality of capacity exponents $\delta_c = \delta_{\omega c} = \delta_{\omega^2c}$) is established analytically in . The central message is: $a_2 \to \text{Einstein}$, $a_4 \to \text{Yang--Mills}$, from the single functional $S_\Pi[g, \mathcal{A}]$. This spectral stratification suggests that the natural organisational space for unification is spectral and variational rather than purely group-theoretic: gravity and gauge dynamics are separated by geometric direction and heat-kernel order, not by a missing symmetry group.