• 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 $\mathrm{ad}$, $E(D_{A_0} s) = D_{A_0}(E s)$. 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.