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 carries out the gauge–gravity synthesis: it solves the joint variational problem $\delta_{g,\mathcal{A}} S_\Pi = 0$, coupling the two sectors into a single dynamical system. The metric variation of the $a_4$ coefficient yields the Yang–Mills stress tensor as a source for gravity; the gauge variation of $a_2$ is zero by horizontal–vertical decoupling.

The result is the Einstein–Yang–Mills system derived from a single functional, with the gauge–gravity hierarchy appearing as an inescapable structural consequence of the difference in Seeley–DeWitt order between the two sectors.

Central message. Gravity and gauge dynamics are not unified by enlarging the symmetry group, but are stratified by Seeley–DeWitt order within the single operator $\mathcal{A}_{g,\mathcal{A}}$: $a_2 \to$ gravity, $a_4 \to$ gauge, $a_6 \to$ mixed coupling.

Core contributions

Joint Einstein–Yang–Mills equations (Theorem): the joint variational principle $\delta_{g,\mathcal{A}} S_\Pi = 0$ yields the coupled system: \begin{align*} G_{\mu\nu} &= 8\pi G_N\, T^{\mathrm{YM}}_{\mu\nu} + \mathcal{O}(a_6),\\ D_\mu F^{a\mu\nu} &= 0 + \mathcal{O}(a_6), \end{align*} where $T^{\mathrm{YM}}_{\mu\nu}$ arises from the metric dependence of the $a_4$ coefficient. The two sectors are decoupled at leading order by the horizontal–vertical decoupling lemma.
$a_6$ gauge–gravity cross-coupling (Theorem): the $a_6$ Seeley–DeWitt coefficient contains the mixed term $R_{\mu\nu\rho\sigma}F^{\mu\nu}F^{\rho\sigma}$, suppressed by $\ell_{\mathrm{sp}}^2 \ll 1$ and relevant only near the spectral cutoff.
Eddington–Born–Infeld joint completion (Theorem): the same $c_\chi$ saturation bound that regulates the gravitational sector extends to the joint functional, producing a unified non-linear regulator that recovers both sector-specific BI completions as limits. A residual tensorial ambiguity in $\Xi_{\mu\nu}$ affects only subleading terms.
Gauge–gravity hierarchy ratio (Proposition): \[ G_N g_{\mathrm{YM}}^2 \;\sim\; \frac{\ell_{\mathrm{sp}}^2}{\dim V} \cdot\Bigl[\log\frac{\Lambda}{\mu}\Bigr]^{-1} \;\ll\; 1, \] without fine-tuning. The suppression has two independent origins: the UV scale $\ell_{\mathrm{sp}}^2$, and the difference between quadratic ($a_2$) and logarithmic ($a_4$) UV divergences.

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. The hierarchy $G_N g_{\mathrm{YM}}^2 \ll 1$ is a direct output of the spectral stratification, not an input requiring fine-tuning.

Spectral stratification principle. $a_2 \to \text{Einstein gravity (quadratic UV)}$, $a_4 \to \text{Yang–Mills gauge dynamics (logarithmic UV)}$, $a_6 \to \text{gauge–gravity cross-coupling (suppressed by } \ell_{\mathrm{sp}}^2)$. The type of a fundamental interaction is determined by the Seeley–DeWitt order at which the admissible projection responds.

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: $\delta_{g,\mathcal{A}} S_\Pi = 0 \implies$ coupled EYM system + hierarchy ratio + EBI joint completion

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; the only sensitivity is in the value of $\dim V$ in the hierarchy ratio.

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

EBI tensor ambiguity: the precise form of $\Xi_{\mu\nu}$ in the joint EBI functional is not uniquely fixed at the level of Q13 and affects only subleading corrections.
Numerical $a_6$ coefficients: the values $\gamma_1, \gamma_2, \ldots$ in the $a_6$ coefficient depend on the specific admissible operator and require explicit computation.
Lorentzian continuation of the joint EBI functional: the geometric sector is extended in Q5b; the gauge sector extension 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: Joint Einstein–Yang–Mills Equations, Born–Infeld Completion, and the Hierarchy Ratio from Projective Spectral Entropy, 2026. doi:10.5281/zenodo.20262079