• Definitions / Notation used
• The Technical Heart: Pullback is a Sieve
• One Diagram in Words
• Assumptions vs Consequences
• Two Examples of “Illegal Moves”
• Why This Matters
• Key Takeaway
• Technical Takeaway

In GU, what you see is not what you put in—it’s what survives projection. Most field theories start by declaring fields on spacetime. Not here. In the torsion-first instantiation of Geometric Unity, the only truly native field on spacetime $X$ is the immersion map $\iota: X \hookrightarrow Y$. Everything else—metric, gauge fields, spinors, torsion—is induced from the ambient 14D geometry of $Y$.

This is a hard structural rule. And it has sharp consequences. A field on $Y$ can only affect physics on $X$ if it has no legs in the normal direction—that is, if it survives pullback via $\iota^*$. Connections, being affine, require special handling. So do torsion and curvature. If you get the pullback wrong, you’re no longer working in GU.

In this post, we make the pullback rules explicit. We show what’s legal, what’s not, and how to avoid cheating.

§Definitions / Notation used

• $X$ = $X^4$: observed spacetime
• $Y$ = $Y^{14}$: ambient space
• $\iota: X \to Y$: immersion map
• $\iota^*$: pullback operation ($Y \to X$)
• $TX \subset TY|_X$: tangent bundle of $X$
• $N_\iota \subset TY|_X$: normal bundle
• $\alpha \in \Omega^p(Y)$: $p$-form on $Y$
• $A \in \mathrm{Conn}(P_H)$: gauge connection on $Y$
• $A_0$: fixed background connection
• $\delta A := A - A_0$: tensorial connection difference
• $T := \xi - \varepsilon^{-1} d_{A_0} \varepsilon$: augmented torsion (pullback-legal)
• $F \in \Omega^2(Y, \mathfrak{ad})$: curvature 2-form
• $*_X, *_Y$: Hodge stars on $X$ and $Y$ respectively

§The Technical Heart: Pullback is a Sieve

When you pull back a $p$-form $\alpha$ from $Y$ to $X$, what survives?

Only the components that are fully tangential.

Let’s say:

$$ \alpha = \alpha^\parallel + \alpha^\perp $$

where:

• $\alpha^\parallel$: components with only $TX$ legs
• $\alpha^\perp$: any component involving at least one $N_\iota$ leg

Then:

$$ \iota^*\alpha = \alpha^\parallel $$

This applies to:

• 1-forms: only tangential components survive
• 2-forms: mixed terms vanish
• Spinor-valued forms: same story—normal-indexed components drop

Moreover, connections are not tensors. So you can’t pull back $A$ directly. Instead, you use differences like:

$$ \delta A := A - A_0 \in \Omega^1(Y, \mathfrak{ad}) $$

which is a bona fide 1-form and can be pulled back safely.

Same for torsion:

$$ T := \xi - \varepsilon^{-1} d_{A_0} \varepsilon \in \Omega^1(Y, \mathfrak{ad}) $$

This is the augmented torsion, designed to be a pullbackable, gauge-covariant object.

§One Diagram in Words

Imagine a net stretched across $Y$ along $\iota(X)$. Only those field components that align entirely along the mesh ($TX$) make it through. Anything pointing even partly into the bulk ($N_\iota$) is caught and discarded. The pullback acts like a sieve—not a copying machine.

§Assumptions vs Consequences

Assumptions (Definitional):

• Fields on $X$ are induced via pullback $\iota^*$
• Pullback kills normal components: $\iota^*(\alpha_\perp) = 0$
• Connections are affine and not directly pullbackable
• Only tensorial objects (like $\delta A$, $T$) can be pulled back

Consequences (Operational):

• No normal components survive on $X$
• Naively assigning a “scalar” on $X$ from a normal vector on $Y$ is forbidden
• Action terms must use properly pulled-back objects
• Gauge contractions must use Shiab ($\bullet_\varepsilon$), not naive traces

§Two Examples of “Illegal Moves”

1. Promoting Normal Components to Scalars

What you might try:

“I’ll take the normal components $A_a$ of the gauge field and treat them as scalars on $X$.”

Why this fails:

$A_a n^a$ is not tangent. Pullback kills it. You can’t assign $A_a(x)$ as a scalar field—it never existed on $X$ to begin with.

Correct move:

Treat $A$ as a full ambient connection. Compute $F_B$, then apply Shiab operator $\bullet_\varepsilon$ and only afterward pull back components aligned with $TX$.

2. Tracing Curvature like a Ricci Tensor

What you might try:

“Let’s contract $F_{MN}$ using $g_Y$ to get a Ricci-like trace: $R \sim \mathrm{Tr}(g^{MP} F_{PN})$”

Why this fails:

$F \in \Omega^2(Y, \mathfrak{ad})$ is not a tensor—it’s $\mathfrak{ad}$-valued. Contracting it with $g_Y$ makes no gauge-theoretic sense. Worse: you’re implicitly summing over normal indices, which vanish under pullback.

Correct move:

Use Shiab contraction:

$$ \bullet_\varepsilon(F) := *_Y [\Theta_E \wedge \varepsilon^{-1} F \varepsilon] $$

This defines a gauge-covariant, pullback-compatible “trace” using the fixed gravitational block (E) and proper dualization.

§Why This Matters

• Field content on $X$ is strictly derived—nothing is added arbitrarily
• Gauge covariance is preserved by using only pullback-legal objects
• Torsion and Yukawa couplings emerge from overlap geometry, not hand-waved scalar insertions
• Lagrangians built from incorrect contractions break both covariance and geometry

§Key Takeaway

GU enforces a strict rule: only tangential, covariant objects survive pullback. If you try to sneak in new fields by promoting internal components or misusing curvature, the geometry pushes back.

§Technical Takeaway

• $\iota^*(\alpha_\perp) = 0$
• $\delta A := A - A_0$ pulls back; $A$ does not
• $T := \xi - \varepsilon^{-1} d_{A_0} \varepsilon \in \Omega^1(Y, \mathfrak{ad})$, pullback-legal
• Ricci-like traces → replaced by Shiab $\bullet_\varepsilon$