• Definitions / Notation used
• Main technical argument
  • Argument sketch
• Concrete example: what becomes well-defined only after fixing $A_0$
• Assumptions vs Consequences
  • Assumptions
  • Consequences
• Why this matters
• Key takeaway
• Technical takeaway

The space of gauge connections is not linear — it’s affine. If we want to talk about differences of connections, we must choose an origin. In this torsion-first, transport-based instantiation, that origin is the distinguished background connection $A_0$. Choosing $A_0$ is not new physics; it is coordinate choice in field space. It’s what makes “transport displacement” and “augmented torsion” into legitimate, pullback-safe variables.

§Definitions / Notation used

• $X = X^4$, $Y = Y^{14}$, immersion $\iota: X \to Y$, pullback $\iota^*$.
• Along $\iota(X): TY|_X \simeq TX \oplus N_{\iota},$ with indices $\mu,\nu$ on $TX$; $a,b$ on $N_{\iota}$; $M,N$ on $Y$.
• $Y$ has split signature $(7,7)$; Spin$(7,7)$ is structural.
• Principal $H$-bundle $P_H \to Y$; $\mathrm{ad}(P_H)$ its adjoint bundle.
• $N := \Omega^1(Y, \mathrm{ad}(P_H))$ (translation space).
• $G := H \ltimes N$, with $\omega = (\varepsilon, $)$, $\varepsilon \in H$, $$ \in N$.
• Distinguished background connection $A_0 \in \mathrm{Conn}(P_H)$; covariant exterior derivative $d_{A_0}$.
• Rotated connection $B_{\omega} := A_0 \cdot \varepsilon$; curvature $F_B := dB_{\omega} + B_{\omega} \wedge B_{\omega}$.
• “Connections aren’t tensors”: pull back $\delta A$ or $T$, not $A$ itself.

Native vs invasive reminder: Fields are native to $Y$; $X$ only receives invasive data via pullback $\iota^*$ of covariant/tensorial objects.

§Main technical argument

Lemma ( $A_0$ as affine origin makes a covariant displacement coordinate). Fix $A_0 \in \mathrm{Conn}(P_H)$. Then every connection $A$ is uniquely expressible as

$$ A = A_0 + \delta A, \delta A \in \Omega^1(Y, \mathrm{ad}(P_H)) = N. $$

Moreover, under the $A_0$-preserving (tilted) gauge/transport bookkeeping, $\delta A$ transforms homogeneously (adjointly), hence $\iota^*(\delta A)$ is well-defined on $X$.

§Argument sketch

$\mathrm{Conn}(P_H)$ is an affine space modeled on $N$. Without an origin, “the connection field” is not a vector variable, and any attempt to treat $A$ as a tensor causes inhomogeneous transformation terms to show up.

Once $A_0$ is fixed, $\delta A := A - A_0$ is a bona fide $\mathrm{ad}(P_H)$-valued 1-form. The point of the transport group $G = H \ltimes N$ is that it packages the inhomogeneous “connection disease term” into a compensator. Restricting to the tilted embedding of $H$ inside $G$ (the $A_0$-preserving subgroup), the inhomogeneous pieces cancel, leaving $\delta A$ transforming as

$$ \delta A \mapsto h^{-1} (\delta A) h. $$

That makes $\delta A$ pullback-safe:

$$ \iota^*(\delta A) \in \Omega^1(X, \mathrm{ad}(\iota^* P_H)). $$

§Concrete example: what becomes well-defined only after fixing $A_0$

• Illegal (not covariant): $\iota^*(A)$.
• Correct: $\iota^*(A - A_0) = \iota^*(\delta A)$.

The cancellation is exactly “subtract the same disease term twice.”

§Assumptions vs Consequences

§Assumptions

• Split signature on $Y$ is fixed (Spin$(7,7)$ context).
• $A_0$ is chosen once and held fixed.
• Use $G = H \ltimes N$ so translation and rotation bookkeeping is explicit.

§Consequences

• Displacements $\delta A = A - A_0$ are legitimate fields in $N$.
• Pullback to $X$ is legal for $\delta A$ (and later $T$), not for raw $A$.
• The rotated connection $B_{\omega} = A_0 \cdot \varepsilon$ becomes operational in a way that is consistent with the transport bookkeeping.

§Why this matters

A torsion-first instantiation lives or dies on whether you can write down variables that are both gauge-covariant and pullback-safe. Fixing $A_0$ is the minimal move that makes “difference-from-background” a meaningful field. It’s not adding structure to $Y$; it’s selecting an affine origin so transport variables exist. Without $A_0$, the torsion coordinate you actually want cannot even be stated cleanly.

§Key takeaway

Choosing $A_0$ is choosing an affine origin, not adding physics.

§Technical takeaway

The only pullback-safe “connection-like” data are differences from $A_0$ ($\delta A$) or compensated combinations ($T$). Raw $A$ is not tensorial.