Gauge–Gravity Spectral Synthesis: Joint Einstein–Yang–Mills Equations, Born–Infeld Completion, and the Hierarchy Ratio from Projective Spectral Entropy

The companion papers Gravity 3.0 and Q12 derived, respectively, the Einstein tensor as the horizontal $a_2$ response and the Yang–Mills equations as the vertical $a_4$ response of the projective spectral entropy functional $S_\Pi[g,A] = \tfrac{1}{2}\log\det' \mathcal{A}_{g,A}$. The present paper carries out the gauge–gravity synthesis and establishes its central conceptual consequence: gravity and gauge dynamics are not unified by enlarging the symmetry group or adding fields, but are stratified by the Seeley–DeWitt level at which the same admissible operator responds to variation. The conventional question ``what symmetry unifies gravity and gauge?'' is thereby replaced by ``at what spectral order does the projection respond?'' We solve the joint variational problem $\delta_{g,A}S_\Pi = 0$ and derive the coupled Einstein–Yang–Mills system: the Yang–Mills stress tensor sources gravity through the metric dependence of the $a_4$ coefficient, with coupling constant $8\pi G_N = c_{YM}/c_{EH}$ fixed by the Seeley–DeWitt expansion alone. We compute the $a_6$ coefficient and identify the leading gauge–gravity cross-coupling $R_{\mu\nu\rho\sigma}F^{\mu\nu}F^{\rho\sigma}$. The Eddington–Born–Infeld completion of the joint functional is constructed from the same Born–Infeld saturation bound that governs the gravitational sector . The gauge–gravity hierarchy is a structural output of the spectral stratification: gravity is induced at $a_2$ (quadratic UV divergence) and gauge at $a_4$ (logarithmic UV divergence), from the same cutoff $\ell_{\mathrm{sp}}$, giving: \[ G_N g^2_{YM} \;\sim\; \frac{\ell_{\mathrm{sp}}^2}{\dim V} \cdot\Bigl[\log\frac{\Lambda}{\mu}\Bigr]^{-1} \;\ll\; 1, \] without fine-tuning. The spectral stratification $a_2 \to \text{gravity},\; a_4 \to \text{gauge},\; a_6 \to \text{mixed coupling}$ predicts that fermionic structure should appear at a dedicated spectral level carrying a Dirac-type admissible operator, which is the central open question for the continuation of the programme. The gauge group $G_\Pi$ is inherited from the spectral admissibility sub-programme ; the $\mathrm{SU}(3)$ sector is unconditional at the effective level: $[H\text{-color}]_{\mathrm{eff}}$ (equality of capacity exponents) is established analytically in .