• Definitions / Notation used
• Main technical argument
  • Proof
• Example (a): the torsion construction
• Example (b): replacing “connection-as-field” by $\delta A$ or a pulled-back invariant
• Common illegal moves
  • Illegal move 1: Pull back raw connections
  • Illegal move 2: Naive Ricci traces on $\mathrm{ad}$-valued curvature
  • Illegal move 3: Mixing $TX$ and $N_{\iota}$ components without projection
• Assumptions vs Consequences
  • Assumptions
  • Consequences
• Why this matters
• Key takeaway
• Technical takeaway

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.