Overview
The spectral gravity sub-programme (Presentation Note 4) derives the Einstein tensor as the $a_2$ infrared-dominant response of the horizontal metric variation of $S_\Pi[g]$. The gauge structure sub-programme (Presentation Note 3) identifies the gauge group $G_\Pi = \mathrm{SU}(3) \times \mathrm{SU}(2) \times \mathrm{U}(1)$ and constructs the admissible principal bundle $P_{G_\Pi}(M, G_\Pi)$. Does the same functional, extended to include the admissible gauge connection, produce Yang–Mills dynamics — and if so, at what spectral order?
The answer is yes, and the order is $a_4$. Gravity and gauge dynamics arise from the same functional by varying in orthogonal directions — horizontal (metric) and vertical (gauge connection) — at different Seeley–DeWitt orders. The conventional question "what symmetry unifies gravity and gauge?" is thereby replaced by "at what spectral order does the admissible projection respond?". This is the spectral stratification principle: $a_2 \to$ gravity, $a_4 \to$ Yang–Mills, $a_6 \to$ gauge–gravity mixing. The Yang–Mills equations $D_\mu F^{a\mu\nu} = 0$ follow from the $a_4$ vertical variation (Q12); the coupled Einstein–Yang–Mills system, the $a_6$ cross-coupling $R_{\mu\nu\rho\sigma}F^{\mu\nu}F^{\rho\sigma}$, the Eddington–Born–Infeld joint completion, and the structural hierarchy $G_N g_{\mathrm{YM}}^2 \ll 1$ all follow from Q13.
The spectral stratification chain
$\underbrace{a_2 \to G_{\mu\nu}}_{\text{horizontal } \delta_g} \;\Big|\; \underbrace{a_4 \to D_\mu F^{a\mu\nu} = 0}_{\text{vertical } \delta_A} \;\Big|\; \underbrace{a_6 \to R_{\mu\nu\rho\sigma}F^{\mu\nu}F^{\rho\sigma}}_{\text{mixed (structural)}}$
Four conceptual stages from the single functional $S_\Pi[g, A] = \tfrac{1}{2}\log\det' A_{g, A}$: extension of the operator to the gauge sector with horizontal–vertical decoupling (Q12 Lemma 1, proved); Yang–Mills equations from the vertical $a_4$ variation (Q12 Theorem 1, structural given $G_\Pi$); coupled Einstein–Yang–Mills system from the joint variation (Q13 Theorem 3.2, structural); structural hierarchy ratio $G_N g_{\mathrm{YM}}^2 \ll 1$ from the Seeley–DeWitt expansion without fine-tuning (Q13 Proposition 6.1).
Papers of the sub-programme
Yang–Mills from the vertical $a_4$ variation.
- Q12 — extends the Laplacian $A_g$ to $A_{g, A} = -(\nabla^A)^2 + E$ on the associated vector bundle of $P_{G_\Pi}(M, G_\Pi)$. Results: $a_4 \supset \tfrac{1}{12}\mathrm{Tr}(F_{\mu\nu}F^{\mu\nu})$ (heat-kernel, proved); horizontal–vertical decoupling Lemma 1 (proved); Yang–Mills equations $D_\mu F^{a\mu\nu} = 0$ from $\delta_A S_\Pi = 0$ (Theorem 1, structural given $G_\Pi$); UV hierarchy $G_N^{-1} \sim \ell_{\mathrm{sp}}^{-2}$ vs. $g_{\mathrm{YM}}^{-2} \sim \log(\Lambda/\mu)$. Q12 is shared with the gauge-structure sub-programme (Presentation Note 3), which uses its gauge-group identification component.
Joint Einstein–Yang–Mills system and hierarchy.
- Q13 — completes the trilogy (Gravity 3.0, Q12, Q13) by solving the joint variational problem $\delta_{g, A} S_\Pi = 0$. Results: coupled Einstein–Yang–Mills system $G_{\mu\nu} = 8\pi G_N T^{\mathrm{YM}}_{\mu\nu}$ with $8\pi G_N = c_{\mathrm{YM}} / c_{\mathrm{EH}}$ (Theorem 3.2, structural); $a_6$ gauge–gravity cross-coupling $R_{\mu\nu\rho\sigma}F^{\mu\nu}F^{\rho\sigma}$ (Theorem 4.1, structural, normalisation open); Eddington–Born–Infeld joint completion $\mathcal{S}^{\mathrm{EBI}} \propto \int[\sqrt{-\det(g + \ell_{\mathrm{sp}}^2 R_{\mu\nu} + \ell_{\mathrm{sp}}^2 F_{\mu\nu})} - \sqrt{-g}]$ (Theorem 5.3, conditional on [H-ext]); structural hierarchy $G_N g_{\mathrm{YM}}^2 \sim \ell_{\mathrm{sp}}^2/\dim V \cdot [\log(\Lambda/\mu)]^{-1} \ll 1$ (Proposition 6.1).
Inputs and outputs
Upstream inputs. Spectral entropy functional $S_\Pi[g]$ and its $a_2$ Einstein sector from the spectral gravity sub-programme (Presentation Note 4, Gravity 3.0); the admissible principal bundle $P_{G_\Pi}(M, G_\Pi)$ and the gauge group $G_\Pi = \mathrm{SU}(3) \times \mathrm{SU}(2) \times \mathrm{U}(1)$ from the gauge structure sub-programme (Presentation Note 3, Q6a/Q12); the effective Lorentzian base manifold $(M, g^{\mu\nu} = 2\eta^{\mu\nu})$ from the emergent geometry sub-programme (Presentation Note 2, Q5b–Q11); the spectral length scale $\ell_{\mathrm{sp}}$ and BI saturation constant $c_\chi$ (Branch I); the conditionality [H-ext] inherited from the gravitational sector (Gravity 3.0).
Outputs. Yang–Mills equations $D_\mu F^{a\mu\nu} = 0$ in the current-free sector (SM phenomenology); the coupled $G_{\mu\nu} = 8\pi G_N T^{\mathrm{YM}}_{\mu\nu}$ (gravitational-wave phenomenology, Q13); the $a_6$ cross-coupling identified (future precision tests); the joint EBI functional (astrophysical applications, conditional on [H-ext]); the structural hierarchy ratio $G_N g_{\mathrm{YM}}^2 \ll 1$ (hierarchy problem); the spectral prediction that fermionic structure should appear at a dedicated Dirac-type spectral level — the direct structural motivation for the fermionic matter sub-programme (Presentation Note 6, Q14).
Status
The bosonic dynamical sector is closed structurally. The $a_4$ heat-kernel derivation of Yang–Mills (Q12 §4) and the horizontal–vertical decoupling Lemma 1 are proved unconditionally. The Yang–Mills equations (Q12 Theorem 1), the UV hierarchy (Q12 §6), the coupled Einstein–Yang–Mills system (Q13 Theorem 3.2), the $a_6$ cross-coupling (Q13 Theorem 4.1, with normalisation open), the structural hierarchy ratio $G_N g_{\mathrm{YM}}^2 \ll 1$ (Q13 Proposition 6.1), and the spectral stratification principle are structural. The Eddington–Born–Infeld joint completion (Q13 Theorem 5.3) is conditional on Hypothesis [H-ext] (admissible coherence extensivity). Per O31 Proposition 4.23, the gauge group $G_\Pi = \mathrm{SU}(3) \times \mathrm{SU}(2) \times \mathrm{U}(1)$ is an unconditional input at the pointwise level, so the $a_4$ Yang–Mills derivation holds unconditionally for the full Standard Model gauge group. Three items remain open: the $a_6$ phenomenological normalisation, the analytical proof of [H-ext], and the coupled equations with fermionic matter currents from Note 6 (Q14).