• Definitions / Notation used
• Definition: fixed Shiab operator in this instantiation
  • Type-checking:
• Main technical argument: built-in gauge covariance
  • Lemma (Gauge covariance of $\bullet_{\varepsilon}$)
    • Proof
• Geometric meaning: what “Einsteinian projection” means here
• Where it lands, and why that landing space is the point
• Assumptions vs Consequences
  • Definitional
  • Ansatz
  • Consequence
• Why this matters

Once you accept that physics is written on $Y^{14}$ and only read on $X^{4}$ via $\iota^*$, you also accept a hard constraint: every operation you use to build dynamics must respect the way fields actually transform on $Y$. Curvature is $\mathrm{ad}(P_H)$-valued and rotates under $H$; “taking a Ricci trace” is not even a well-typed operation anymore. The Shiab operator is GU’s replacement: it performs an Einstein-like projection without ever identifying $\mathrm{ad}(P_H)$ with tangent tensors or invoking forbidden contractions. Operationally, it is how you extract the part of curvature that can balance torsion, in a fully gauge-covariant way.

§Definitions / Notation used

• $Y = Y^{14}$, signature $(7,7)$; $X = X^{4}$; $\iota : X \to Y$; pullback $\iota^*$.
• $TY|_X \simeq TX \oplus N_\iota$; indices $\mu,\nu$ / $a,b$ / $M,N$; $g_Y$ split with $\sigma(x)$; $\ast_X$ vs $\ast_Y$.
• $H$ gauge group; $\mathrm{ad}(P_H)$ adjoint bundle; $\Omega^k(Y, \mathrm{ad}(P_H))$ adjoint-valued $k$-forms.
• $\omega = (\varepsilon,\eta) \in G = H \ltimes N$ with $N := \Omega^1(Y, \mathrm{ad}(P_H))$; $A_0$ background; $B_\omega := A_0\cdot\varepsilon$; curvature $F_B$.
• Augmented torsion $T := \eta - \varepsilon^{-1} d_{A_0} \varepsilon \in \Omega^1(Y, \mathrm{ad}(P_H))$ (tensorial).
• Local symbol introduced here: $e \in \Omega^{11}(Y, \mathrm{ad}(P_H))$, an adjoint-valued 11-form used as the fixed “Einstein seed”; in later posts it will be specialized to $\Theta_E$.

§Definition: fixed Shiab operator in this instantiation

Define the Shiab operator $\bullet_{\varepsilon}$ as a map on adjoint-valued 2-forms: $\bullet_{\varepsilon} : \Omega^2(Y, \mathrm{ad}(P_H)) \to \Omega^1(Y, \mathrm{ad}(P_H))$

by

$$ \bullet_{\varepsilon}(F) := \ast_Y\big( e \wedge \varepsilon^{-1} F \varepsilon \big). $$

§Type-checking:

• $F$ is a 2-form on $Y$ valued in $\mathrm{ad}(P_H)$.
• $e$ is an 11-form on $Y$ valued in $\mathrm{ad}(P_H)$.
• $e \wedge (\varepsilon^{-1} F \varepsilon)$ is a 13-form valued in $\mathrm{ad}(P_H)$ (the wedge product is taken on the form degrees, with the adjoint bundle factor multiplied in the usual fiberwise way).
• $\ast_Y$ maps 13-forms to 1-forms on $Y$, giving an output that can be paired degree-for-degree with torsion $T \in \Omega^1(Y, \mathrm{ad}(P_H))$.

This is the “Einsteinian projection” in GU language: it produces the curvature object that plays the same role Ricci/G (Einstein tensor) plays in GR, but without any Ricci trace on indices.

§Main technical argument: built-in gauge covariance

§Lemma (Gauge covariance of $\bullet_{\varepsilon}$)

Let $h \in H$ be a gauge transformation. Suppose the fields transform by adjoint conjugation in the standard way: $F \mapsto F^h := h^{-1} F h$, $\varepsilon \mapsto \varepsilon^h := h^{-1} \varepsilon$, $e \mapsto e^h := h^{-1} e h$ (as appropriate for an adjoint-bundle-valued form). Then $\bullet_{\varepsilon^h}(F^h) = h^{-1} \bullet_{\varepsilon}(F) h$.

§Proof

Compute directly: $\bullet_{\varepsilon^h}(F^h) = \ast_Y\big( e^h \wedge (\varepsilon^h)^{-1} F^h \varepsilon^h \big) = \ast_Y\big( (h^{-1} e h) \wedge (\varepsilon^{-1} h) (h^{-1} F h) (h^{-1} \varepsilon) \big) = \ast_Y\big( (h^{-1} e h) \wedge (\varepsilon^{-1} F \varepsilon) \big)$.

Because conjugation by $h$ is fiberwise on $\mathrm{ad}(P_H)$ and does not act on the differential-form factor, it can be pulled out: $\ast_Y\big( (h^{-1} e h) \wedge (\varepsilon^{-1} F \varepsilon) \big) = h^{-1} \ast_Y\big( e \wedge \varepsilon^{-1} F \varepsilon \big) h = h^{-1} \bullet_{\varepsilon}(F) h$. QED.

§Geometric meaning: what “Einsteinian projection” means here

In GR, “Einsteinian projection” means: take the full curvature data and throw away the parts that do not participate in the field equations (Weyl drops out of the Ricci/scalar projection). In GU, we want the analogous outcome, but we cannot:

• interpret $F_B$ as a Riemann tensor,
• contract “internal indices” by choosing a preferred generator (that breaks gauge symmetry),
• or use any non-covariant identification between $\mathrm{ad}(P_H)$ and tangent bundles.

Instead, we proceed operationally:

1. Choose $e$ (later $\Theta_E$) so that wedge with $e$ saturates precisely the directions we want to “trace out” (in this instantiation: the 10 normal directions plus a fixed pattern leaving one visible $X$-slot).
2. Use $\ast_Y$ to turn that saturation into a 1-form—the degree that can directly balance torsion $T$ in a first-order equation.
3. Use $\varepsilon^{-1} (\cdot) \varepsilon$ so that the “block” of curvature being sampled is defined in transport terms, not by a fixed, non-transforming projector.

This makes “Einsteinian” mean: the part of curvature that survives $\bullet_{\varepsilon}$ is exactly the part that can couple linearly to the tensorial displacement field $T$ in a gauge-invariant first-order action, and the rest is annihilated by construction (the GU draft emphasizes this annihilation-of-Weyl analogy as a design goal for Shiab-type operators).

§Where it lands, and why that landing space is the point

$\bullet_{\varepsilon}(F) \in \Omega^1(Y, \mathrm{ad}(P_H))$ is not a metric Ricci tensor, and it is not a scalar curvature. It is an adjoint-valued 1-form on $Y$—exactly the same degree and bundle type as augmented torsion $T$. That is not an aesthetic choice; it is the typing constraint that makes a torsion-first, gauge-invariant action possible.

Indeed, the first-order torsion–curvature balance is written at the level of a 1-form equation: $\Upsilon_\omega := \bullet_{\varepsilon}(F_B) - \kappa T$, and the corresponding gauge-invariant action pairs $\Upsilon_\omega$ with $T$ using the $\ast_Y$-induced inner product on forms.

§Assumptions vs Consequences

§Definitional

• $\bullet_{\varepsilon}(F) := \ast_Y\big( e \wedge \varepsilon^{-1} F \varepsilon \big)$, with $e$ an adjoint-valued 11-form and $\varepsilon$ the H-component of $\omega$.

§Ansatz

• $e$ will be fixed (covariantly constant) by the gravitational selection data ($E$/$\Theta_E$) so that the operator implements the intended “gravitational trace” without forbidden identifications.
• The dynamics are torsion-first: $T$ is primary, and curvature enters through $\bullet_{\varepsilon}(F_B)$ linearly.

§Consequence

• $\bullet_{\varepsilon}$ is gauge-covariant by construction (lemma above), unlike naive Einstein contraction or fixed internal projections.
• The output lives in the correct space to couple directly to $T$, enabling a first-order, gauge-invariant action with no Ricci traces.

§Why this matters

• Toward $E$ / $\Theta_E$ selection: $e$ is where “gravity lives” inside the operator. Choosing $E$ and building $\Theta_E$ is not decoration; it is the step that makes $\bullet_{\varepsilon}$ an actual gravitational projection rather than an arbitrary linear functional.
• Toward a gauge-invariant torsion-first action: with $\bullet_{\varepsilon}(F_B)$ landing in $\Omega^1(Y, \mathrm{ad}(P_H))$, you can write the torsion–curvature interaction as a clean, gauge-consistent pairing with $T$ (recall: connections are not tensors; $T$ is).