• Definitions / Notation used
• The Technical Heart: Decomposition of Tangent Space Along $\iota(X)$
• One Diagram in Words
• Assumptions vs Consequences
• Why This Matters
• Key Takeaway
• Technical Takeaway

What does it mean to observe a field? In GU, it means restricting a 14D object to a 4D submanifold—and interpreting the result. This process isn’t metaphorical. It’s geometrically encoded by the immersion map $\iota: X \to Y$, which embeds our observed 4D world into the ambient space. Without this immersion, there is no observer, no spacetime, no physics.

But $\iota$ doesn’t just localize the arena. It enforces a split between what can be seen and what can’t. Along $\iota(X)$, the tangent bundle $TY$ of the 14D observerse splits cleanly into tangential (observable) and normal (hidden) directions. This is the decomposition:

$$ TY|_X \simeq TX \oplus N_\iota .$$

It’s the workhorse of the whole theory. It defines which parts of a 14D field project onto spacetime and which parts vanish. It’s how fermions reduce to chiral 4D modes. It’s how curvature becomes observable. And it’s the first nontrivial use of the $\mathrm{Spin}(7,7)$ structure.

Let’s build the framework that will power this transport-based formulation of GU.

§Definitions / Notation used

• X = $X^4$: 4D observed spacetime
• Y = $Y^{14}$: 14D ambient space
• $\iota: X \to Y$: immersion / observation map
• $\iota(X) \subset Y$: the image of X inside Y
• $TY|_X$: restriction of TY to $\iota(X)$
• $TY|_X \simeq TX \oplus N_\iota$: decomposition of tangent vectors into observed ($TX$) and hidden ($N_\iota$) directions
• $g_X := \iota^* g_Y$: induced metric on X
• Pullback $\iota^*$: operation sending $Y$-objects to their projections on $X$

§The Technical Heart: Decomposition of Tangent Space Along $\iota(X)$

Given the immersion map $\iota$, every tangent vector in $TY|_X$ splits into two orthogonal components:

• TX: the part tangent to the submanifold $\iota(X)$
• $N_\iota$: the normal bundle—directions in $Y$ that are orthogonal to $TX$ at every point

This is more than a geometric curiosity. It’s the machinery behind observation. Pullback acts like a sieve: it only lets $TX$-aligned components survive.

Formally, for any vector field $V \in \Gamma(TY)$, its restriction to $\iota(X)$ decomposes as:

$$ V|_X = V^\parallel + V^\perp $$

where:

• $V^\parallel \in TX$ is the projection onto the 4D tangent space
• $V^\perp \in N_\iota$ lies in the 10D normal directions

The connection, curvature, and spinors on $Y$ all inherit this decomposition. After pullback, we get:

• $\iota^*\alpha = \alpha^\parallel$ for forms: only components aligned with $TX$ survive
• $\iota^*\Psi(x, n) = \varphi_0(n) \otimes \psi(x)$ for spinors: only base-mode contributions in the normal direction project onto $X$
• $g_X := \iota^* g_Y$: the induced metric uses only the $TX$–$TX$ block

This is the precise operational meaning of “fields are not native to spacetime; they are observed through immersion.”

§One Diagram in Words

Visualize $Y^{14}$ as a vast space, and $X^4$ as a sheet immersed in it. At every point of $X$, $TY$ splits into a 4D plane ($TX$) and a 10D orthogonal space ($N_\iota$). A field defined on $Y$ may have components pointing in all 14 directions. But once you pull it back to $X$, anything pointing even partly in a normal direction is projected out. The pullback enforces observational consistency.

§Assumptions vs Consequences

Assumptions (Definitional):

• There exists a smooth immersion $\iota: X \to Y$
• The ambient manifold $Y$ has a $\mathrm{Spin}(7,7)$ structure and split signature $(7,7)$
• The tangent bundle $TY|_X$ decomposes as $TX \oplus N_\iota$

Consequences (Operational):

• All pullback fields on $X$ are purely tangential
• Normal components vanish under pullback, and cannot be “promoted” into scalar or spinor fields on $X$
• Observation selects only a subset of possible modes—this is how mode selection and chirality arise without inserting them by hand

§Why This Matters

• Fermion sectors: Spinors on $Y$ decompose into tangential and normal modes. After pullback, only certain chiral components remain.
• Gauge structure: Only tangential components of gauge fields (like $A_\mu$) pull back; normal components vanish or decouple.
• Torsion formulation: The torsion-first ansatz we’ll use in this instantiation relies on expressing torsion in normal directions but evaluating its effects on $TX$.
• Action formulation: The action $I_1(\omega)$ integrates over $Y$ using $*_Y$ and depends on observing fields via $\iota$.

§Key Takeaway

Immersion is not optional. It defines how fields, geometry, and spinors on $Y$ reduce to observable content on $X$. Only tangential components survive pullback.

§Technical Takeaway

• Tangent split: $TY|_X \simeq TX \oplus N_\iota$
• Pullback: $\iota^*\alpha = \alpha^\parallel$, normal components vanish
• Observation = restriction + pullback via $\iota$