• Definitions / Notation used
• The obstruction
• GU’s move
• Why this matters
• Key takeaway
• Technical takeaway

Gravity teaches us that geometry is physics. Gauge theory teaches us that physics lives in how things are transported. Both claims are true—and fatally incompatible if taken literally. The problem is not philosophical but operational: the way gravity contracts curvature is not compatible with gauge covariance. GU begins by identifying that fault line precisely, not smoothing over it.

§Definitions / Notation used

• $X = X^4$ (observed spacetime), $Y = Y^{14}$ (ambient/observerse space); immersion $\iota: X \to Y$; pullback $\iota^*$.
• Tangent/normal split along $\iota(X)$: $TY|_X \simeq TX \oplus N_\iota,$ with $\mathrm{rank}(N_\iota)=10;$ indices $\mu,\nu$ for $TX;$ $a,b$ for $N_\iota;$ $M,N$ for $Y.$
• Metric: $g_X := \iota^* g_Y;$ metric split ansatz $g_Y \simeq g_X \oplus \sigma(x)^2 \delta_{ab} n^a \otimes n^b;$ Hodge stars $*_X$ and $*_Y.$
• Gauge structure: $H$ gauge group; $\mathcal{N} := \Omega^1(Y, \mathrm{ad}(P_H));$ inhomogeneous group $\mathcal{G} := H \ltimes \mathcal{N};$ $\omega = (\varepsilon, \phi);$ background connection $A_0;$ rotated connection $B_\omega := A_0 \cdot \varepsilon;$ curvature $F_B$.
• Shiab operator $\bullet_\varepsilon;$ $\Upsilon_\omega := \bullet_\varepsilon(F_B) - \kappa_1 T;$ action $I_1(\omega) := \int_Y \langle T,$ $*_Y \Upsilon _{\omega} \rangle.$

§The obstruction

Riemannian geometry relies on metric contraction. Given curvature, one traces indices using the metric to produce Ricci and Einstein tensors. This operation is not gauge-covariant when curvature takes values in an adjoint bundle.

If $F$ is an ad-valued curvature 2-form and $h \in H$ is a gauge transformation, then

• $F \mapsto h^{-1} F h$
• but a naïve “Ricci-type” contraction $P(F)$ does not satisfy

$$P(h^{-1} F h) = h^{-1} P(F) h$$

This is not a technical annoyance; it is a categorical mismatch. Einstein’s contraction is defined using the metric alone, whereas gauge covariance requires conjugation symmetry. Applying Riemannian projection to a gauge field breaks gauge symmetry at the level of field equations.

Ehresmann geometry avoids this by never contracting curvature without an invariant trace—but then loses the Einstein tensor entirely. That is the two-geometries problem in its sharpest form:

• Riemannian geometry gives projection but breaks gauge symmetry.
• Gauge geometry preserves symmetry but lacks a gravitational projection.

§GU’s move

GU does not attempt to reinterpret Ricci curvature as a gauge object. Instead, it replaces the contraction itself.

The Shiab operator $\bullet_\varepsilon$ is a gauge-covariant contraction, defined so that

• it acts on ad-valued curvature,
• it commutes with the $H$-action,
• and it reduces, after pullback, to an Einstein-like trace on $X$.

Everything downstream—torsion, transport, observables—exists to make this replacement meaningful.

This post does not justify $\bullet_\varepsilon$; it isolates why something like it is unavoidable.

§Why this matters

If you do not fix this obstruction, any “unification” either violates gauge symmetry or smuggles in non-covariant projections. GU’s entire torsion-first, transport-based structure exists to make a gauge-covariant Einstein-like projection possible.

§Key takeaway

Gravity and gauge theory fail to unify not because of extra dimensions or missing particles, but because Einstein’s contraction is not gauge-covariant.

§Technical takeaway

Naïve Ricci projection $P(F)$ fails:

$$P(h^{-1} F h) \neq h^{-1} P(F) h$$

GU replaces $P$ with $\bullet_\varepsilon$, a covariant contraction acting on $F_B$.