• Definitions / Notation used
• The temptation: “just project/trace something”
• Main technical argument
  • Lemma (Fixed projection does not commute with gauge rotation)
    • Proof (explicit example)
• What this means: “Einstein contraction” secretly contains a projector
• A second (even harsher) obstruction: linear gauge-invariants usually don’t exist
• Assumptions vs Consequences
  • Definitional
  • Ansatz
  • Consequence
• Why this matters

General Relativity’s magic trick is that you can squeeze the full curvature tensor into something actionable: a Ricci tensor and a scalar. That squeeze is so familiar that we forget what it relies on: a very specific kind of object (a tensor) and a very specific kind of symmetry (the metric’s). In a transport-first, gauge-based picture like GU, curvature is not a tensor with free indices to trace—it is an $\mathrm{ad}(P_H)$-valued 2-form, and it rotates under $H$. If you try to “do Einstein” anyway, you quickly discover the failure is not cosmetic. It is structural.

§Definitions / Notation used

• $Y = Y^{14}$ with split signature $(7,7)$; $X = X^{4}$; immersion $\iota : X \to Y$; pullback $\iota^*$.
• Tangent/normal split along $\iota(X)$: $TY|_X \simeq TX \oplus N_\iota$; indices $\mu,\nu$ on $TX$; $a,b$ on $N_\iota$; $M,N$ on $TY$.
• $g_X := \iota^* g_Y$, and $g_Y \simeq g_X \oplus \sigma(x)^2 \delta_{ab} n^a \otimes n^b$; Hodge stars $\ast_X$ vs $\ast_Y$.
• Gauge/transport: $H$ gauge group; $N := \Omega^1(Y, \mathrm{ad}(P_H))$; $G := H \ltimes N$; $\omega = (\varepsilon,\eta)$; $A_0$ background.
• Rotated connection: $B_\omega := A_0\cdot\varepsilon$; curvature $F_B \in \Omega^2(Y, \mathrm{ad}(P_H))$.
• Augmented torsion: $T := \eta - \varepsilon^{-1} d_{A_0} \varepsilon \in \Omega^1(Y, \mathrm{ad}(P_H))$ (tensorial), while connections are not tensors.
• “Einstein contraction” on ad-valued curvature is disallowed; any Einstein-like operation is replaced by a Shiab operator $\bullet_{\varepsilon}$.

§The temptation: “just project/trace something”

In GR, the Einstein contraction (Ricci and scalar) is a projection on the tensor space in which the Riemann curvature lives. It is linear, local, and canonical because the Levi–Civita connection ties curvature rigidly to the metric.

In GU’s Ehresmannian setting, curvature is $F_B \in \Omega^2(Y, \mathrm{ad}(P_H))$. Under a gauge transformation $h \in H$, it transforms by conjugation:

$$ F_B \mapsto F_B^h := h^{-1} F_B h. $$

So if we want an “Einstein-like” contraction $C(F_B)$ that produces something usable in dynamics, at minimum we want gauge covariance: $C(F_B^h) = h^{-1} C(F_B) h$, or gauge invariance if $C$ lands in scalars.

Here is the cleanest failure mode: a fixed “gravitational block” projection.

§Main technical argument

§Lemma (Fixed projection does not commute with gauge rotation)

Let $P : \mathrm{ad}(P_H) \to \mathrm{ad}(P_H)$ be a fixed fiberwise projector (think: “keep the gravitational block; discard the rest”). Define the naive projected curvature by

$$ F_B \mapsto P(F_B) $$

(pointwise applying $P$ to the $\mathrm{ad}(P_H)$ component). Then, in general,

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

§Proof (explicit example)

Take $H = SU(2)$ for concreteness (the point is representation-theoretic, not $SU(2)$-specific). Identify $\mathrm{ad} \simeq \mathfrak{su}(2)$ with basis ${\tau_1, \tau_2, \tau_3}$. Let $P$ be the projector onto the $\tau_3$-direction: $P(\alpha_1 \tau_1 + \alpha_2 \tau_2 + \alpha_3 \tau_3) := \alpha_3 \tau_3$.

Now choose an element $h \in SU(2)$ whose adjoint action rotates $\tau_3$ into $\tau_1$ (such an $h$ exists because $\mathrm{Ad}(SU(2)) \simeq SO(3)$ acts transitively on the unit sphere in $\mathfrak{su}(2)$). Consider a curvature configuration of the form $F_B = \tau_3 \otimes f$, where $f$ is any ordinary 2-form on $Y$.

Then $P(F_B) = \tau_3 \otimes f$.

But after gauge rotation,

$$ F_B^h = h^{-1} (\tau_3 \otimes f) h = (\mathrm{Ad}_{h^{-1}} \tau_3) \otimes f = \tau_1 \otimes f, $$

so $P(F_B^h) = P(\tau_1 \otimes f) = 0$,

whereas

$$ h^{-1} P(F_B) h = h^{-1} (\tau_3 \otimes f) h = \tau_1 \otimes f \neq 0. $$

Hence $P(h^{-1} F_B h) \neq h^{-1} P(F_B) h$. QED.

§What this means: “Einstein contraction” secretly contains a projector

The Einstein contraction is not merely “contract indices”; it is a very specific projection onto a distinguished subspace (Ricci + scalar) determined by the metric structure. In a gauge setting, any analogous move that singles out a “gravitational part” of $\mathrm{ad}(P_H)$ is a choice of a projector $P$.

And the lemma says: unless that projector is $H$-invariant (equivalently: commutes with $\mathrm{Ad}_h$ for all $h$), it cannot be used inside gauge-covariant dynamics. The only universally $\mathrm{Ad}$-invariant endomorphisms of a simple adjoint representation are multiples of the identity. So any nontrivial “keep this block” operation is generically incompatible with $H$-covariance.

This is exactly the structural divide emphasized in the GU draft: contraction-like operations typically act on tangent-associated tensors, while gauge rotations act on the internal bundle factor; if you do not build the gauge rotation into the contraction/projection, the symmetry bookkeeping fails.

§A second (even harsher) obstruction: linear gauge-invariants usually don’t exist

Even if you avoid projectors and try the simplest possible “Einstein scalar” from curvature—something linear in $F_B$—there is a standard Lie-algebra fact: for a semisimple gauge algebra, there is no nonzero $\mathrm{Ad}$-invariant linear functional $\ell$ on $\mathrm{ad}$. In other words, “trace of $F_B$” is either identically zero (non-abelian case) or only sees an abelian center. This is why Yang–Mills uses $\mathrm{Tr}(F \wedge \ast F)$ (quadratic), whereas GR uses $R$ (linear).

So if you insist on an Einstein–Hilbert–like, first-order, linear-in-curvature term, you must introduce extra covariant structure to soak up the gauge rotation. In this instantiation, that extra structure is already present and tensorial: the augmented torsion $T$, built to be covariant by construction.

§Assumptions vs Consequences

§Definitional

• Curvature is $F_B \in \Omega^2(Y, \mathrm{ad}(P_H))$ transforming as $F_B \mapsto h^{-1} F_B h$.
• Any “Einstein-like” operation must be replaced by a gauge-covariant Shiab operator $\bullet_{\varepsilon}$, not a Ricci trace.

§Ansatz

• We are looking for a first-order, torsion-first dynamics in which curvature appears linearly, paired against a tensorial object ($T$).

§Consequence

• Any naive projection/trace that selects a fixed internal “gravitational block” fails to commute with gauge rotation unless it is trivial (identity/zero).
• Therefore, the contraction/projection must be engineered to be gauge-covariant by construction, not borrowed from Riemannian index gymnastics.

§Why this matters

• Toward $E$ / $\Theta_E$ selection: the “gravitational block” cannot be a fixed, non-transforming projector; it must be implemented as part of a covariant operator (this is precisely what $E$ and $\Theta_E$ will do).
• Toward a gauge-invariant action with torsion $T$: since $T$ is tensorial/covariant while connections are not, the natural first-order invariant is built from $T$ paired with a covariant image of curvature—not from tracing curvature by itself.