In GU’s transport formulation, the combination of affine connections and immersion geometry makes it especially easy to write expressions that look innocent but fail gauge covariance. Two recurring problems show up: (i) connection-like objects transform inhomogeneously, and (ii) $\mathrm{ad}$-valued curvature does not tolerate naive “Ricci trace” instincts. The systematic cure is a pattern: build variables as differences of two equally diseased expressions so the disease cancels. We’ll call it double disease cancellation.

Definitions / Notation used

Double disease cancellation (template): If two objects $K$ and $L$ transform as

$$ K^\prime = h^{-1} K h + \Phi(h), L^\prime = h^{-1} L h + \Phi(h), $$

then $J := K - L$ transforms covariantly:

$$ J^\prime = h^{-1} J h. $$

Also: placement discipline remains strict:

• $A_0$, $B_{\omega}$, $F_B$, $T$ are native to $Y$.
• Only covariant/tensorial objects may be pulled back to $X$ ($\iota^*\delta A$, $\iota^*T$, etc.).
• “Einstein-like contraction” is via Shiab $\bullet_{\varepsilon}$, not naive Ricci traces (we do not develop Shiab here, you'll have to wait a bit longer).

Main technical argument

Lemma (double disease cancellation). Given $K^\prime = h^{-1} K h + \Phi(h)$ and $L^\prime = h^{-1} L h + \Phi(h)$, the difference $J := K - L$ satisfies $J^\prime = h^{-1} J h$.

Proof#

Subtract:

$$ J^\prime = K^\prime - L^\prime = (h^{-1} K h + \Phi) - (h^{-1} L h + \Phi) = h^{-1} (K - L) h. $$

Example (a): the torsion construction

We have

$$ \eta^\prime = h^{-1} \eta h + h^{-1} d_{A_0} h, $$

$$ (\varepsilon^{-1} d_{A_0} \varepsilon)^\prime = h^{-1} (\varepsilon^{-1} d_{A_0} \varepsilon) h + h^{-1} d_{A_0} h. $$

Same disease term, same sign. Therefore

$$ T := \eta - \varepsilon^{-1} d_{A_0} \varepsilon $$

transforms as

$$ T^\prime = h^{-1} T h. $$

Example (b): replacing “connection-as-field” by $\delta A$ or a pulled-back invariant

Choose $A_0$ and use $\delta A := A - A_0 \in \Omega^1(Y, \mathrm{ad}(P_H))$.

Under the $A_0$-aware bookkeeping, the inhomogeneous term cancels in the difference, yielding

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

and then $\iota^*(\delta A)$ is legal on $X$.

Common illegal moves

Illegal move 1: Pull back raw connections#

• Temptation: define $A_{\mu}(x) := (\iota^* A)_{\mu}$ and proceed.
• Fix: pull back $\delta A := A - A_0$ or $T$, i.e. $\iota^*(\delta A)$ or $\iota^*(T)$.

Illegal move 2: Naive Ricci traces on $\mathrm{ad}$-valued curvature#

• Temptation: contract indices on $F_B$ like a Riemann tensor and trace in $\mathrm{ad}$.
• Fix: Einstein-like contraction is replaced by Shiab $\bullet_{\varepsilon}$ (do not use naive Ricci traces here).

Illegal move 3: Mixing $TX$ and $N_{\iota}$ components without projection#

• Temptation: silently identify $Y$-indices with $X$-indices.
• Fix: respect $TY|_X \simeq TX \oplus N_{\iota}$; use explicit projections before contracting or interpreting on $X$.

Assumptions vs Consequences

Assumptions#

• Strict native ($Y$) vs invasive (pulled back to $X$) discipline.
• Use $A_0$ and $A_0$-aware compensators; accept that inhomogeneous terms exist.

Consequences#

• The right variables are compensated differences: $\delta A$, $T$, and later Shiab-contracted curvature objects.
• You get a practical legality test: if a quantity inherits a disease term, pair it with another object carrying the same disease and subtract.

Why this matters

As the theory grows (curvature, torsion, spinors, immersion-induced structure), the main failure mode is not algebra but illegality: writing expressions that are not gauge-covariant or not pullback-safe. Double disease cancellation is the simplest systematic discipline that prevents that. It’s the reason torsion-first transport doesn’t immediately contradict gauge symmetry, and it’s the same discipline that forces Shiab-style contraction rather than naive Ricci traces.

Key takeaway

When two objects transform badly in the same way, subtract them, and you get a good field.

Technical takeaway

The entire torsion-first transport variable $T$ is an instance of the general lemma: shared inhomogeneous terms cancel in the difference, producing adjoint covariance.