What question did this study set out to answer?

This research aims to explore the vertical variation of projective spectral entropy and its relationship with Yang–Mills dynamics.

June 2, 2026Open Access

Yang–Mills Dynamics from Projective Spectral Entropy: Vertical Heat-Kernel Variation on the Admissible Fibre

Key Points

This research aims to explore the vertical variation of projective spectral entropy and its relationship with Yang–Mills dynamics.
Extended the operator $A_g$ to a Laplace-type operator $A_{g,\mathcal{A}}$ on a vector bundle.
Computed the Seeley–DeWitt coefficient $a_4$ and extracted the Yang–Mills term from $S_\Pi[g, \mathcal{A}]$.
Derived the Yang–Mills equations from the variation with respect to the admissible connection $\mathcal{A}$.
Extracted the Yang–Mills term $c_{\mathrm{YM}}\int \mathrm{Tr}(F_{\mu u}F^{\mu u})\sqrt{g}\, \mathrm{d}^4 x$.
Induced gauge coupling $g_{\mathrm{YM}}^{-2} \sim (12 \cdot 16\pi^2)^{-1}\log(\Lambda/\mu)$, showing logarithmic dependence on the UV cutoff.
Showed that gravity and gauge dynamics are organized by heat-kernel order and geometric direction, rather than by symmetry group missing.

Abstract

The companion paper showed that the horizontal variation of the projective spectral entropy functional S_ = 12' Ag with respect to the base metric produces the Einstein tensor as the infrared-dominant response, via the Seeley–DeWitt coefficient a₂. The present paper carries out the complementary vertical variation: extending Ag to a Laplace-type operator A₆, ₀ = - (^A) ² + E on the associated vector bundle of the admissible principal fibre, we compute the Seeley–DeWitt coefficient a₄ for the total operator and extract the Yang–Mills term cₘ₌ Tr (F_F^) g\, d⁴ x. Variation of S_, A with respect to the admissible connection A then yields the Yang–Mills equations D_ F^a = 0 in the current-free sector. The induced gauge coupling gₘ₌^-2 (12 16²) ^-1 (/) is logarithmic in the UV cutoff, while Newton's constant GN^-1 ₒ₏^-2 is quadratic. This spectral hierarchy is a structural consequence of the Seeley–DeWitt expansion and does not require a separate input. The gauge group G_ is taken as input from the spectral admissibility programme ; the SU (3) sector is unconditional: H-color₄₅₅ (equality of capacity exponents c = ₂ = ℂ₂) and H-color₏₎₈₍ₓₖ₈ₒ₄ (exact equality at finite q) are both established analytically in. The central message is: a₂ Einstein, a₄ Yang--Mills, from the single functional S_, A. This spectral stratification suggests that the natural organisational space for unification is spectral and variational rather than purely group-theoretic: gravity and gauge dynamics are separated by geometric direction and heat-kernel order, not by a missing symmetry group.

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