Gauge–Gravity Spectral Synthesis

Overview

On a supplied Riemannian metric, the Gravity paper states the conditions under which the projective spectral entropy functional $S_\Pi[g,\mathcal{A}] = \tfrac{1}{2}\log\det' \mathcal{A}_{g,\mathcal{A}}$ has the Einstein tensor as its dominant horizontal $a_2$ response (a non-zero matched Einstein coefficient and small curvature relative to the renormalized higher-derivative scale; local dominance), and Q12 derives the Yang–Mills equations as its 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.

Status distinction. On supplied geometric and gauge data, 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. Identifying the supplied base with the Cosmochrony emergent metric is separately conditional on [H-L], [H-lift] and a supplied rank-four extension.

Quick facts

Author

Jérôme Beau

Work type

Preprint, version 1.6

Focus

Einstein–Yang–Mills synthesis, Yang–Mills stress tensor, renormalization matching, Seeley–DeWitt stratification

Key result

$\delta_{g,\mathcal{A}} S^{\mathrm{ren}} = 0 \implies$ conditional EYM system, with $8\pi G_N = 1/(2c_{\mathrm{EH}})$ fixed by matching

Depends on

Supplied geometric and gauge data; Gravity paper and Q12; emergent-base reading via Q11 conditional on [H-L], [H-lift] and a supplied extension

Trilogy

Gravity → Q12 → Q13

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: $\delta_g S_\Pi \supset c_{\mathrm{EH}}^{\mathrm{ren}}\,G_{\mu\nu}$, dominant for $c_{\mathrm{EH}}^{\mathrm{ren}} \neq 0$ and $R\,L_4^2 \ll 1$ (horizontal variation, $a_2$ coefficient; the coupling to matter is the conditional result of the Thermodynamics paper)
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 compact gauge group and representation $\rho$ are supplied inputs. O31 version 2.0 withdraws the proposed $\mathrm{SU}(3)$ identification and complete gauge group. Q13's spectral-order calculation is independent of that identification; the group 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 continuation is not established in the geometric sector: under [H-L] and [H-lift], Q5b gives a rank-three co-metric whose block has signature $(-,+,+)$ under [H-hyp], with stipulated signs; a four-dimensional Lorentzian metric needs a supplied extension (Q6b); and a Lorentzian counterpart of $S_\Pi$ needs an in-in construction that is open; and 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 $J^{a\nu}$.
Gauge-group identification: determine the compact group and representation independently of the withdrawn O31 chain. Until then all group-specific readings are conditional.
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