• Definitions / Notation used
• What $\Theta_E$ is
• The lemma-level heart: $\Theta_E$ implements “normal saturation → $X$-visible contraction”
  • Lemma (Normal saturation projector)
  • “One diagram in words” (how the saturation works)
• Why $D\Theta_E = 0$ is non-negotiable here
  • Operational consequences of $D\Theta_E=0$
• How $\ast_Y(\Theta_E \wedge (\cdot))$ induces $\sigma$-scaling
• Assumptions vs Consequences
  • Assumptions
  • Consequences
• Why this matters

We still haven’t solved the real geometric problem: curvature lives on $Y$, but our observer lives on $X$. The phrase “take the Ricci trace” is illegal here unless you can express it as something intrinsic—something that (i) respects the split signature $\mathrm{Spin}(7,7)$, (ii) does not pick coordinates, and (iii) knows how to ignore the ten normal directions while keeping precisely the $X$-information you want.

$\Theta_E$ is that device. It is the mechanism that turns a $Y$-form into an $X$-visible contraction by saturating the normal bundle.

§Definitions / Notation used

• $Y$ ($\mathrm{Spin}(7,7)$), $X$, immersion $\iota$, $TY|_X \simeq TX \oplus N_\iota$, indices $\mu,\nu$ / $a,b$ / $M,N$, $g_X = \iota^* g_Y$, $\sigma$-split, $\ast_X$ vs $\ast_Y$.
• $H$, $N = \Omega^1(Y,\mathrm{ad})$, $G = H \ltimes N$, $\omega = (\varepsilon, \eta)$, $A_0$, $B_\omega$, $F_B$, augmented torsion $T$.
• $E$ is the $A_0$-parallel adjoint projector selecting the gravitational block: $D_{A_0}E=0$.
• $\Theta_E$ is fixed here as an $\mathrm{ad}$-valued 11-form, covariantly constant: $D\Theta_E = 0$.

§What $\Theta_E$ is

Critical specifics for this instantiation:

• $\Theta_E$ is $E$ tensored into an 11-form that saturates the 10 normal directions and leaves one $X$-slot.
• $\Theta_E$ is covariantly constant: $D\Theta_E = 0$.
• The combination $\ast_Y(\Theta_E \wedge (\cdot))$ selects an X-trace of curvature and induces a characteristic $\sigma$-scaling (details of cosmological dynamics deferred).

Concretely, think of $\Theta_E$ as living in $\Omega^{11}(Y,\mathrm{ad})$, with the $\mathrm{ad}$-factor equal to $E$ and the differential-form factor having the schematic type $(10\ \text{normals}) \wedge (1\ \text{tangent})$. You do not need coordinates to say this: you are specifying its degree and its type relative to the splitting $TY|_X \simeq TX \oplus N_\iota$.

§The lemma-level heart: $\Theta_E$ implements “normal saturation → $X$-visible contraction”

§Lemma (Normal saturation projector)

Let $\alpha$ be an $\mathrm{ad}$-valued $k$-form on $Y$ restricted to $\iota(X)$. Write its decomposition by the number of normal legs: $\alpha = \sum_{r=0}^{\min(k,10)} \alpha^{(r)}$, where $\alpha^{(r)}$ has exactly $r$ normal components (and $k-r$ tangent components). Let $\Theta_E$ be an $\mathrm{ad}$-valued 11-form whose form part contains all 10 normal directions exactly once (a normal volume-type factor) and one tangent slot. Then:

1. $\Theta_E \wedge \alpha$ annihilates every component $\alpha^{(r)}$ with $r \geq 1$ (it “kills” anything carrying normal legs), because the normal directions are already saturated.
2. $\Theta_E \wedge \alpha$ depends only on $\alpha^{(0)}$ (the purely tangent part).
3. Applying the ambient Hodge star $\ast_Y$ to $\Theta_E \wedge \alpha$ produces a form that is canonically identifiable with an $X$-form (up to the $\sigma$-scaling inherited from the $\sigma$-split metric).

In short: $\ast_Y(\Theta_E \wedge \alpha) = (\text{X-visible contraction of }\alpha^{(0)}) \times (\sigma\text{-scaling})$.

§“One diagram in words” (how the saturation works)

Picture the tangent space at a point of $\iota(X)$ as a direct sum: 4 tangent directions + 10 normal directions.

Now run the following mental diagram:

(1) Start with $F_B$, a 2-form on $Y$ with values in $\mathrm{ad}$. It contains three types of pieces:

• tangent–tangent: $F_{\mu\nu}$
• tangent–normal: $F_{\mu a}$
• normal–normal: $F_{ab}$

(2) Wedge with $\Theta_E$. The form part of $\Theta_E$ already uses “all ten normal directions” once. That means:

• If you try to wedge in a component of $F_B$ that already has a normal leg ($\mu a$ or $ab$), you would be attempting to wedge in an extra normal direction beyond saturation. That term vanishes.
• Only the purely tangent component $F_{\mu\nu}$ survives the wedge.

(3) Apply $\ast_Y$. Because the normal directions are saturated inside $\Theta_E$, the Hodge star effectively collapses the ambient dualization down to the tangent (X) sector. What comes out is an object that behaves like an $X$-form built from the tangent curvature data—i.e., the $X$-visible “trace-like” contraction you wanted, but obtained without ever performing an illegal index trace on $Y$.

This is the allowed replacement for “take the Ricci trace” in a gauge-covariant ambient formulation.

§Why $D\Theta_E = 0$ is non-negotiable here

We are not merely choosing a convenient 11-form; we are choosing a calibrator that must be stable under transport. $D\Theta_E=0$ means $\Theta_E$ is parallel with respect to the combined covariant derivative (on the $\mathrm{ad}$-factor and the form factor). In this instantiation the $\mathrm{ad}$-factor is $E$, already $A_0$-parallel. The remaining content is the 11-form that saturates normals and leaves one $X$-slot, chosen to be covariantly constant as part of the background structure.

§Operational consequences of $D\Theta_E=0$

• The map $\alpha \mapsto \ast_Y(\Theta_E \wedge \alpha)$ is stable under background transport; it does not drift along $Y$.
• When we vary actions later, derivatives do not produce “boundary leakage” terms from $\Theta_E$ itself; $\Theta_E$ behaves like a fixed calibrating tensor, not a dynamical field.

§How $\ast_Y(\Theta_E \wedge (\cdot))$ induces $\sigma$-scaling

The $\sigma$-split metric assigns a factor $\sigma^2(x)$ to each normal direction. As a result:

• A normal 10-volume element scales like $\sigma^{10}$.
• When you build $\Theta_E$ with a normal-saturating factor, you are inserting something with that normal-volume scaling.
• The ambient Hodge star $\ast_Y$ on a form that already contains the saturated normal volume contributes the inverse scaling, schematically $\sigma^{-10}$, because $\ast_Y$ uses the inverse metric to dualize and the normal part of the metric carries $\sigma^2$.

Therefore the composite operator $\ast_Y(\Theta_E \wedge (\cdot))$ carries a characteristic $\sigma^{-10}$ weight in the effective $X$-sector. This is exactly why, when the dust settles in the GR(+$\Lambda$) corner, the induced cosmological term acquires a $\sigma$-dependent prefactor. The detailed cosmology (stabilization, evolution, and effective $\Lambda$ running) will be addressed later; here we only need the logic: normal saturation + ambient duality imports a $\sigma$-scaling into whatever $X$-visible contraction you build.

§Assumptions vs Consequences

§Assumptions

• $\Theta_E$ is fixed as an $\mathrm{ad}$-valued 11-form: $\Theta_E = (E) \otimes (\text{normal-saturating 10-form} \wedge \text{one }X\text{-slot})$.
• $\Theta_E$ is covariantly constant: $D\Theta_E = 0$.
• The metric on $Y$ is split as $g_Y \simeq g_X \oplus \sigma^2 \delta_{ab} \hat{n}^a \hat{n}^b$, giving distinct $\ast_X$ and $\ast_Y$.

§Consequences

• The operator $\ast_Y(\Theta_E \wedge (\cdot))$ annihilates curvature components with normal legs and keeps only the $X$-tangent curvature content in the gravitational block.
• The resulting contraction is gauge-covariant and transport-stable (because $D\Theta_E=0$ and $D_{A_0}E=0$).
• A $\sigma$-dependent weight (qualitatively $\sigma^{-10}$) is inherited by the $X$-visible sector, setting up the cosmological scaling mechanism used later.

§Why this matters

Without $\Theta_E$, you can talk about “gravity” as an adjoint block ($E$) but you cannot actually make ambient curvature produce an $X$-level Einsteinian contraction without smuggling in coordinates. $\Theta_E$ is the legitimate replacement for forbidden traces: it is the geometric operation that (i) kills the normal contamination, (ii) keeps the tangent curvature that an $X$-observer can measure, and (iii) does so in a way compatible with $\mathrm{Spin}(7,7)$ split signature and background transport.