• Definitions / Notation used
• The Technical Heart: Why $(7,7)$ Is a Structure, Not a Preference
• One Diagram in Words
• Assumptions vs Consequences
• Why This Matters
• Key Takeaway
• Technical Takeaway

What if the dimensions you don’t see are the reason you see anything at all?

This instantiation of Geometric Unity has a rigid starting point: $Y$ has split signature $(7,7)$ and carries a $\mathrm{Spin}(7,7)$ structure. This isn’t a speculative flourish. It’s what makes the whole setup possible—from chiral fermions and torsion-first transport, to gravitational dynamics and normal-mode quantization. Without this split signature, the entire construction collapses.

It is equally important to realize that this signature and structure are a choice. Geometric Unity does not prescribe this. But it turns out it’s a choice you are required to make. It could be a different choice, but you need to decide so you can actually instantiate it. I will not lead you down the many paths I took to discover that other signatures didn’t quite work out for me, but you are invited to try them out yourself.

Let’s unpack why this 14D stage is indispensable, and how the $\mathrm{Spin}(7,7)$ structure anchors every move downstream.

§Definitions / Notation used

• $X = X^4$ (observed spacetime), $Y = Y^{14}$ (ambient/observerse space); immersion $\iota: X \to Y$; pullback $\iota^*$.
• Tangent/normal split along $\iota(X)$: $TY|_X \simeq TX \oplus N_\iota,$ with $\mathrm{rank}(N_\iota)=10;$ indices $\mu,\nu$ for $TX;$ $a,b$ for $N_\iota;$ $M,N$ for $Y.$
• $\mathrm{Spin}(7,7)$ structure group on $Y$; induces split signature $(7,7).$
• Metric: $g_Y$ ambient metric on $Y$, induced metric on $X$ via pullback $g_X := \iota^* g_Y;$ metric split ansatz $g_Y \simeq g_X \oplus \sigma(x)^2 \delta_{ab} n^a \otimes n^b;$ Hodge stars $*_X$ and $*_Y.$
• $\sigma(x)$ scalar breathing mode in the metric split.

§The Technical Heart: Why $(7,7)$ Is a Structure, Not a Preference

Let’s start with a basic question: what kind of space can host the fields we need—fermions, gauge bosons, curvature, and torsion—in a unified framework where pullback defines observation?

We demand that:

1. Spinor bundles exist on $Y$ and decompose under the immersion.
2. Tangent and normal directions split cleanly, with each having meaningful representations.
3. Chirality in 4D emerges naturally from the structure of $Y$.

These constraints lead us to a split-signature space, with equal numbers of spacelike and timelike directions. Why?

• The spin group $\mathrm{Spin}(7,7)$ has a 64-dimensional real Majorana–Weyl representation.
• Upon immersion $\iota: X \hookrightarrow Y$, the spinors on $Y$ decompose into tensor products involving $\mathrm{Spin}(1,3)$ (our observed Lorentz symmetry) and internal components in the normal bundle.
• The split signature guarantees the existence of chirality: it supports Majorana–Weyl spinors in 14D and allows their decomposition into chiral 4D spinors after pullback.

§One Diagram in Words

Imagine $Y^{14}$ as a vast arena with 7 forward-time axes and 7 space axes. At each point of spacetime $X$, the immersion $\iota$ carves out a 4D plane tangent to $X$ and a 10D orthogonal slice (the normal bundle $N_\iota$). The $\mathrm{Spin}(7,7)$ structure controls how spinors behave across both regions—tangential and normal—and enables the coherent pullback of fields from $Y$ to $X$.

§Assumptions vs Consequences

Assumptions (Definitional):

• $Y$ is a 14-dimensional smooth manifold.
• $Y$ is equipped with a split-signature metric $(7,7)$.
• The structure group on $Y$ is fixed to $\mathrm{Spin}(7,7)$.

Consequences (Structural/Operational):

• The tangent bundle $TY$ supports spinors with definite chirality (Majorana–Weyl).
• The immersion $\iota: X \to Y$ induces a spinor decomposition aligned with $TX$ and $N_\iota$.
• The pullback $\iota^*$ selects a real slice of spinor modes on $X$, one chirality surviving generically.
• Later: This structure allows axial torsion to break left/right symmetry via geometry, not by hand.

§Why This Matters

• Chiral fermions in the Standard Model emerge as pulled-back modes from a $\mathrm{Spin}(7,7)$ structure, not as fundamental insertions.
• Axial torsion lives naturally in the normal directions of $Y$, consistent with the split signature.
• The Shiab operator and its gravitational contraction are defined with respect to $*Y$—not $*X$—and depend on having the right spin group.
• The $(7,7)$ signature allows for a real, covariant treatment of spinor normal modes, setting up the derivation of Yukawa-like interactions in later sections.

§Key Takeaway

The geometry of the observerse $Y$ is not a blank slate. It’s fixed to split signature $(7,7)$ and $\mathrm{Spin}(7,7)$ structure, which directly enables the spinor, torsion, and chirality content of GU. This is the baseboard—not a variable.

§Technical Takeaway

• $\mathrm{Spin}(7,7)$ on $Y$
• Signature: $(7,7)$
• Pullback of spinors: $\Psi(x, n) = \varphi_k(n) \otimes \psi(x)$, with only one chirality surviving generically on $X$