• 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 \(\iota\)