• Definitions / Notation used
• What “block” means when you refuse coordinates
• The lemma-level heart: “\(E\) commutes with \(A_0\)” as a covariant transport statement
  • Lemma (\(A_0\)-parallel gravitational block)
  • Why this is the right statement
• How this stabilizes the gravitational sector under transport (and why we care)
• Assumptions vs Consequences
  • Assumptions
  • Consequences
• Why this matters

Every unification attempt eventually hits the same practical question: “Which degrees of freedom are gravity?” In a torsion-first GU instantiation on a split-signature ambient space \(Y\), you cannot answer that by pointing at a 4×4 corner of a matrix and calling it “spacetime.” Coordinates are a gauge choice, and we are explicitly refusing to build the theory on gauge-breaking contractions.

So we fix the gravitational block operationally: by specifying an adjoint projector \(E\) that is stable under the one thing that matters in a transport-based theory—covariant transport by a chosen background connection \(A_0\).

§Definitions / Notation used

• \(Y\) is a 14D manifold with split signature \((7,7)\). \(X\) is a 4D manifold immersed by \(\iota: X \hookrightarrow Y\). Along \(\iota(X)\): \(TY|\_X \simeq TX \oplus N\_\iota\), with indices \(\mu,\nu\) on \(TX\); \(a,b\) on \(N\_\iota\); and \(M,N\) on \(TY\).
• \(g\_X := \iota^\* g\_Y\). We use the \(\sigma\)-split: \(g\_Y \simeq g\_X \oplus \sigma^2(x) \delta\_{ab} \hat{n}^a \hat{n}^b\), and distinguish \(\ast_X\) from \(\ast_Y\).
• \(H\) is the gauge group, \(N := \Omega^1(Y,\mathrm{ad})\) (\(\mathrm{ad} = \mathrm{ad}(P_H)\)), and \(G := H \ltimes N\). A generic gauge-affine variable is \(\omega = (\varepsilon, \eta) \in G\).
• \(A_0\) is the chosen background connection on \(Y\). From \(\omega\) we form \(B_{\omega}\) (the transported/rotated connection built from \(A_0\) and \(\varepsilon\)), its curvature \(F_B\), and the augmented torsion \(T\) (the covariant “difference” built from \(\eta\) and \(\varepsilon\) relative to \(A_0\)).
• The Shiab operator: \(\bullet_\varepsilon\).

§What “block” means when you refuse coordinates

“Block” should mean: a distinguished, gauge-covariant decomposition of the adjoint bundle \(\mathrm{ad}\) into two pieces—one that will participate in the gravitational projection and one that won’t—without ever picking a preferred basis of \(\mathrm{ad}\), and without assuming any accidental commuting subalgebras.

In this instantiation, the gravitational block is the image of a bundle endomorphism $$ E: \mathrm{ad} \to \mathrm{ad} $$ satisfying the projector axioms:

1. Idempotence: \(E^2 = E\).
2. Adjointness (with respect to the fixed fiber pairing on ad): \(E^\dagger = E\).
3. Gauge-covariance as a bundle map: \(E\) is a section of \(\mathrm{End}(\mathrm{ad})\), not a coordinate matrix.

Operationally, \(E\) defines a splitting \(\mathrm{ad} = \mathrm{Im}(E) \oplus \mathrm{Ker}(E)\), and “gravity lives in \(\mathrm{Im}(E)\)” means: whenever we build the Shiab projection, the contraction/calibration, and the first-order action, we feed them only the \(\mathrm{Im}(E)\) component (or, equivalently, we annihilate \(\mathrm{Ker}(E)\) at the first step). Nothing in that sentence required a basis.

§The lemma-level heart: “\(E\) commutes with \(A_0\)” as a covariant transport statement

Here is the key technical requirement:

\(E\) is an adjoint projector selecting the gravitational block and commuting with the chosen background \(A_0\).

To make “commuting” coordinate-free, we state it as covariant constancy of \(E\) with respect to \(A_0\):

§Lemma (\(A_0\)-parallel gravitational block)

Let \(A_0\) be the fixed background connection on \(\mathrm{ad}\) over \(Y\). The condition “\(E\) commutes with \(A_0\)” means \(D_{A_0} E = 0\) , where \(D_{A_0}\) is the covariant derivative on \(\mathrm{End}(\mathrm{ad})\) induced from \(A_0\). Equivalently, for every section \(s\) of \(\mathrm{ad}\), \(E(D_{A_0} s) = D_ {A_0} (Es)\).

Consequently, parallel transport by \(A_0\) preserves the splitting \(\mathrm{ad} = \mathrm{Im}(E) \oplus \mathrm{Ker}(E)\).

§Why this is the right statement

1. It is gauge-covariant. Under a gauge transformation \(h \in H\), both \(A_0\) and \(E\) transform, but the equation \(D_{A_0}E=0\) is meaningful in every gauge.
2. It is exactly the stability condition you want in a transport-based model: the definition of “gravity” should not drift when you move along \(Y\).
3. It is stronger than saying “\(E\) happens to commute with the local matrix \(A_{0,M}\)”: \(D_{A_0}E=0\) is a global statement about holonomy. It says \(E\) lies in the commutant of the holonomy representation induced by \(A_0\) on \(\mathrm{ad}\).

If you like to think in terms of holonomy: \(D_{A_0}E=0\) implies that for any \(A_0\)-parallel transport operator \(P_\gamma\) along a curve \(\gamma\) in \(Y\), \(P_\gamma \circ E = E \circ P_\gamma\). So “gravitational block” is not a chart artifact; it is a parallel subbundle singled out by the background transport.

§How this stabilizes the gravitational sector under transport (and why we care)

Once you declare \(E\) and require \(D_{A_0}E=0\), you get three immediate consequences that will be used in the future articles:

1. No leakage under \(A_0\)-transport.

   If you start with a gravitational-block excitation (a section \(s\) with \(s = Es\)), then transporting it by \(A_0\) keeps it in \(\mathrm{Im}(E)\). Likewise, non-gravitational components stay in \(\mathrm{Ker}(E)\). This is the cleanest way to make “gravity is a sector” a dynamical statement rather than a basis convention.

2. Covariant compatibility with the Shiab operator.

   In this instantiation, the Shiab operator \(\bullet_\varepsilon\) is fixed, and \(\varepsilon\) is taken to be this projector \(E\) (the “gravitational block selector”). Because \(E\) commutes with \(A_0\), every place where \(\bullet_\varepsilon\) relies on background-covariant constructions (via \(B_\omega\), \(F_B\), and Hodge operations tied to the \(\sigma\)-split) does not reintroduce a hidden gauge choice. Said bluntly: if \(\varepsilon\) were not \(A_0\)-parallel, the “Einsteinian” projection would drift under transport and you would lose the claim that the GR corner is stable.

3. A clean definition of “gravitational curvature” without coordinates.

   Given \(F_B \in \Omega^2(Y,\mathrm{ad})\), we can define its gravitational component as \(E F_B\) (or \(E F_B E\), depending on the adjoint conventions you adopt inside \(\bullet_\varepsilon\)). This is a genuine bundle-theoretic projection, not a contraction.

§Assumptions vs Consequences

§Assumptions

• Split-signature ambient geometry: \(Y\) carries \(\mathrm{Spin}(7,7)\) structure and the \(\sigma\)-split metric form.
• A distinguished background connection \(A_0\) is chosen on \(Y\).
• \(E\) is an adjoint projector on \(\mathrm{ad}\) selecting the gravitational block and it satisfies \(D_{A_0}E=0\).

§Consequences

• The decomposition \(\mathrm{ad} = \mathrm{Im}(E) \oplus \mathrm{Ker}(E)\) is preserved under \(A_0\)-parallel transport.
• The meaning of “gravitational” is invariant under gauge choice and under transport along \(Y\).
• The Shiab projection restricted by \(E\) defines a stable gravitational sector that can be varied without contamination from the complementary degrees of freedom.

§Why this matters

If you do not fix \(E\) as an \(A_0\)-parallel projector, you cannot honestly claim you have a well-posed “GR(+\(\Lambda\)) corner” in a gauge-covariant theory: any purported Einstein-like contraction becomes a moving target under transport, and “gravity” becomes a coordinate myth. Fixing \(E\) the way we did is the minimal, operationally meaningful step that turns “gravitational block” from storytelling into a piece of the geometry.