Gauge–Gravity Spectral Synthesis

Overview

The Gravity paper and Q12 established, independently, that the projective spectral entropy functional $S_\Pi[g,\mathcal{A}] = \tfrac{1}{2}\log\det' \mathcal{A}_{g,\mathcal{A}}$ produces the Einstein tensor as the horizontal $a_2$ response and the Yang–Mills equations as the vertical $a_4$ response.

Q13 joins the two variations within a single renormalized local action. Varying it with respect to the connection at fixed metric gives $D_\mu F^{a\mu\nu} = 0$; varying the same $\mathrm{tr}_\rho(F^2)$ term with respect to the metric gives its energy–momentum tensor, through $(1/\sqrt{g})\,\delta\!\int\!\sqrt{g}\,\mathrm{tr}_\rho(F^2)/\delta g^{\mu\nu} = 2\tau_{\mu\nu}$.

The resulting Einstein–Yang–Mills system is conditional: since $\mathrm{tr}_\rho(F^2) = I_\rho F^a F^a$, canonical matching fixes the products $c_{\mathrm{EH}} = 1/(16\pi G_N)$ and $c_F I_\rho = 1/(4g_{\mathrm{YM}}^2)$, and the equation then holds by definition of $T^{\mathrm{YM}}$, with $8\pi G_N = 1/(2c_{\mathrm{EH}})$ depending on the Einstein coefficient alone. Nothing predicts a coupling: $G_N$, $g_{\mathrm{YM}}$ and the cosmological coefficient are independent renormalized matching data. The derivation is Euclidean.

Central message. One matched local action supplies both field equations. Within a proper-time cutoff the spectral expansion fixes the degree of ultraviolet divergence of each sector — quadratic at $a_2$, logarithmic at $a_4$ — and not the finite couplings. This divergence contrast is scheme-dependent: the zeta-regularized determinant carries no power divergences.

Core contributions

Metric variation of the gauge kinetic term (Lemma): with $\tau_{\mu\nu} = \mathrm{tr}_\rho(F_{\mu\rho}F_\nu{}^\rho - \tfrac14 g_{\mu\nu}F^2)$, \[ \frac{1}{\sqrt{g}}\frac{\delta}{\delta g^{\mu\nu}} \int\sqrt{g}\,\mathrm{tr}_\rho(F^2)\,d^4x \;=\; 2\,\tau_{\mu\nu}, \] using only that $F_{\mu\nu}$ with indices down is metric-independent.
Conditional Einstein–Yang–Mills system (Theorem): joint stationarity of the matched action gives \begin{align*} G_{\mu\nu} + \Lambda_{\mathrm{eff}}\,g_{\mu\nu} &= 8\pi G_N\, T^{\mathrm{YM}}_{\mu\nu},\\ D_\mu F^{a\mu\nu} &= 0, \end{align*} the second from the fixed-metric connection variation. The cosmological $a_0$ term does contribute to the metric equation and is retained as $\Lambda_{\mathrm{ren}}$.
The normalisation is matching, not prediction: $T^{\mathrm{YM}}$ already carries $g_{\mathrm{YM}}^{-2}$ through $c_F I_\rho$, so the gravitational coupling enters only via $8\pi G_N = 1/(2c_{\mathrm{EH}})$. $G_N$, $g_{\mathrm{YM}}$, $\Lambda_{\mathrm{ren}}$ and the dimension-six coefficients are independent renormalized data, so the ratio $G_N g_{\mathrm{YM}}^2$ is not an output of the construction.
$a_6$ inventory (not a computation): a representative spanning list of the gauge–gravity invariants that can occur at order $a_6$, modulo integrations by parts and Bianchi identities. In four dimensions $a_6$ produces no ultraviolet divergence; the pure-gauge invariant at this order is $\mathrm{tr}_\rho(F^3)$; the listed structures are interrelated, so no unique cross-term is singled out and no $a_6$ correction to the equations of motion is asserted.

Spectral stratification as an organisational principle

The central conceptual contribution of Q13 — and of the trilogy as a whole — is the identification of spectral stratification as the correct organisational principle for the unification of gravity and gauge interactions.

In conventional group-theoretic unification, one asks: what symmetry group $G_{\mathrm{unif}}$ contains both gauge and gravitational degrees of freedom? The difficulty of finding such a group is treated as the central obstacle.

The present framework replaces this question with: at what Seeley–DeWitt order does the admissible operator $\mathcal{A}_{g,\mathcal{A}}$ respond to variation? Gravity and gauge dynamics arise from the same operator at different spectral levels, not from different representations of a common group.

Spectral stratification principle (interpretive outlook, not a theorem). $a_2 \to \text{Einstein gravity}$, $a_4 \to \text{Yang–Mills gauge dynamics}$. The type of a fundamental interaction is determined by the Seeley–DeWitt order at which the admissible projection responds. This reading rests on the qualitative separation of spectral orders, not on any quantitative claim about the couplings: no numerical value for $G_N g_{\mathrm{YM}}^2$ follows from it.

Relation to the Cosmochrony programme

Q13 closes the gauge–gravity synthesis trilogy:

Gravity paper: $\delta_g S_\Pi = 0 \implies G_{\mu\nu} = 8\pi G_N T^{(\Pi)}_{\mu\nu}$ (horizontal variation, $a_2$ coefficient)
Q12: $\delta_{\mathcal{A}} S_\Pi = 0 \implies D_\mu F^{a\mu\nu} = 0$ (vertical variation, $a_4$ coefficient)
Q13: $\delta_{g,\mathcal{A}} S^{\mathrm{ren}} = 0 \implies$ conditional coupled EYM system, with the couplings as matching data

The gauge group $G_\Pi$ is inherited from Q6a/O31. The SU(3) sector is unconditional at the pointwise level: $[H\text{-color}]_{\mathrm{pointwise}}$ (exact equality at finite $q$) is proved analytically in O31 v1.5 (Proposition 4.23) via the single-frequency BI fingerprint structure. All main results of Q13 are independent of this result; it enters only through the representation $\rho$ in which $\mathrm{tr}_\rho$ and the Dynkin index $I_\rho$ are taken.

The natural continuation of the programme is the treatment of fermionic matter and chirality, where the spectral stratification principle predicts the existence of a dedicated spectral level carrying spinor structure.

Open directions

Non-linear completion: whether the admissibility saturation bound induces a well-defined non-linear completion of the joint functional is open. Any construction must be checked for dimensional homogeneity of its matrix argument and for the semidefinite sign structure of $F_{\mu\rho}F^\rho{}_\nu$, both of which constrain the admissible determinant forms severely.
Quantitative $a_6$ coefficients: not computed here; they require an explicit Seeley–DeWitt computation for the admissible operator, or independent matching, after reduction of the spanning list to a minimal independent set.
Matching data and the coupling ratio: whether the admissibility structure constrains any combination of $G_N$, $g_{\mathrm{YM}}$, $\Lambda_{\mathrm{ren}}$ — in particular the ratio $G_N g_{\mathrm{YM}}^2$ — is open.
Lorentzian continuation: the derivation is Euclidean and $T^{\mathrm{YM}}$ is defined by a variational convention; the gauge-sector continuation, including the relative sign between curvature and source and the positivity of the gauge energy density, remains open.
Matter currents: extension of the joint variational system to include projected matter currents $J^{a\nu}$.
SU(3) Yang–Mills uniqueness: show that the SU(3) gauge dynamics emerge uniquely from the colour triplet co-admissibility structure, without inputting the group as data (O31 §9.2; $[H\text{-color}]_{\mathrm{eff}}$ is proved, full uniqueness open).
Fermionic matter: identify the dedicated Seeley–DeWitt level carrying spinor structure and derive Dirac-type dynamics from the admissible projection.

References

Jérôme Beau. Gauge–Gravity Spectral Synthesis: A Conditional Einstein–Yang–Mills System from Projective Spectral Entropy, 2026. doi:10.5281/zenodo.20209859