Yang–Mills Dynamics from Projective Spectral Entropy

Overview

The Gravity paper showed that the horizontal variation of the projective spectral entropy functional \[ S_\Pi[g] = \tfrac{1}{2}\log\det' A_g \] with respect to the base metric produces the Einstein tensor as the infrared-dominant response, via the Seeley–DeWitt coefficient $a_2$.

Q12 carries out the complementary vertical variation. The operator $A_g$ is extended to \[ A_{g,\mathcal{A}} = -(\nabla^{\mathcal{A}})^2 + E \] on the associated vector bundle of the admissible principal fibre $P_{G_\Pi}(M, G_\Pi)$, and the Seeley–DeWitt coefficient $a_4$ is computed.

The extended functional $S_\Pi[g,\mathcal{A}]$ contains the renormalized Yang–Mills term with logarithmic coupling, and its vertical variation yields the Yang–Mills equations $D_\mu F^{a\mu\nu} = 0$ in the current-free sector.

Central message. $a_2 \to \text{Einstein}$, $a_4 \to \text{Yang–Mills}$, from the single functional $S_\Pi[g,\mathcal{A}]$.

Core contributions

Extension to vector bundle: the spectral operator $A_g$ is lifted to $A_{g,\mathcal{A}}$ on the associated bundle, incorporating the admissible gauge connection $\mathcal{A}$ from Q6a/O31.
$a_4$ coefficient and Yang–Mills term: isolating the gauge component $\Omega^{\mathrm{gauge}}_{\mu\nu} = F_{\mu\nu}$ of the total bundle curvature, \[ a_4(A_{g,\mathcal{A}})\big|_{\mathrm{gauge}} = \int \frac{1}{12}\,\mathrm{tr}_\rho(F_{\mu\nu}F^{\mu\nu})\sqrt{g}\,\mathrm{d}^4x, \] with the overall $(4\pi)^{-2}$ carried once by the heat-kernel prefactor. The induced logarithmic running of $g_{\mathrm{YM}}^{-2}$ has coefficient $I_\rho/(3\cdot 16\pi^2)$ — the entropy term carries $I_\rho/(12\cdot 16\pi^2)$ for $\int F^2$, which matches $(4g_{\mathrm{YM}}^2)^{-1}$ rather than $g_{\mathrm{YM}}^{-2}$ — depending on the representation through the Dynkin index $I_\rho$, hence not universal.
Vertical variation yields Yang–Mills equations: $\delta_{\mathcal{A}} S_\Pi[g,\mathcal{A}] = 0$ gives $D_\mu F^{a\mu\nu} = 0$ (proved for the $a_4$ derivation given $G_\Pi$).
Difference in UV divergence degree: within a proper-time cutoff, the Einstein term diverges quadratically (order $a_2$) and the Yang–Mills term logarithmically (order $a_4$). This difference is structural in that scheme and is not scheme-independent — the zeta-regularized determinant carries no power divergences. The physical couplings $g_{\mathrm{YM}}$ and $G_N$ are renormalized matching data, and only the logarithmic running is computed. No predictive formula for $G_N$ and no tuning-free hierarchy is claimed (consistent with Gravity 3.3).
Vertical variation at fixed metric: at fixed base metric, the vertical variation isolates the Yang–Mills sector. Full horizontal–vertical decoupling does not hold: the metric variation of $F^2$ is the Yang–Mills energy–momentum tensor $T^{\mathrm{YM}}_{\mu\nu}$, an $a_4$-order back-reaction treated in Q13.

Spectral stratification as an organisational principle

Beyond the technical derivation, Q12 identifies a conceptual principle that distinguishes the Cosmochrony approach from conventional unification schemes.

In standard group-theoretic unification, gravity and gauge interactions are sought at the same algebraic level — different representations of a common group $G_{\mathrm{unif}}$. Their combination requires finding a symmetry large enough to accommodate both.

The present framework suggests a different picture: gravity and gauge dynamics are separated by geometric direction (horizontal vs. vertical variation on the admissible bundle) and by spectral order ($a_2$ vs. $a_4$ in the Seeley–DeWitt hierarchy). Their distinct UV behaviours — quadratic divergence for gravity, logarithmic for gauge — are structural consequences of this separation, not independent empirical inputs.

Implication for unification. The difficulty of incorporating gravity into the Standard Model framework may reflect a difference of Seeley–DeWitt order rather than the absence of a common symmetry group. The natural organisational space for unification in this framework is spectral and variational, not primarily group-theoretic.

Relation to the Cosmochrony programme

Q12 is the vertical counterpart to the Gravity paper. Each sector follows from the same functional by varying in a different direction; joining the two variations, and the conditional system that results, belongs to Q13:

Gravity paper: $\delta_g S_\Pi = 0 \implies G_{\mu\nu} = 8\pi G_N T^{(\Pi)}_{\mu\nu}$
Q12: $\delta_{\mathcal{A}} S_\Pi = 0 \implies D_\mu F^{a\mu\nu} = 0$

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 $\sigma_c(n) = \sigma_{\omega c}(n) = \sigma_{\omega^2 c}(n)$ at finite $q$) is proved analytically in O31 v1.5 (Proposition 4.23) via the single-frequency structure of the Born–Infeld fingerprint.

The joint variation — the conditional Einstein–Yang–Mills system and the $a_4$-order back-reaction $T^{\mathrm{YM}}_{\mu\nu}$ — is carried out in Q13, where the couplings likewise remain independent renormalized matching data, so no numerical value for $G_N g_{\mathrm{YM}}^2$ follows.

Open directions

SU(3) Yang–Mills uniqueness: prove that the SU(3) gauge dynamics emerge uniquely from the colour triplet co-admissibility structure, without inputting the group as data (O31 §9.2).
Matter sector: extend the vertical variation to include projected matter currents $J^{a\nu}$.
Spectral stratification beyond order 4: investigate whether higher Seeley–DeWitt coefficients $a_6$, $a_8$ produce further interaction sectors or remain suppressed.

References

Jérôme Beau. Yang–Mills Dynamics from Projective Spectral Entropy: Vertical Heat-Kernel Variation on the Admissible Fibre, 2026. doi:10.5281/zenodo.20189137