• 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).