• 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.