• 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).