Physical adiabatic phase with explicit scale

Assume Countable Choice. Let $d\ge 1$, let $H\in C^2([0,1],\mathrm{Herm}({\mathbb{C}}^d))$ have entries whose derivatives through order two extend continuously to the endpoints, and assign its eigenvalues energy unit joules. Let $\mathrm{\hslash }>0$ have unit joule-seconds. A spectral cluster at $s$ is a subset ${\sigma }_P(s)$ of the finite spectrum of $H(s)$, and $P(s)$ is the orthogonal projection onto the sum of its eigenspaces. Coherently followed means that $s\mapsto P(s)$ is continuous in operator norm (equivalently, it agrees locally with the Riesz continuation of its initial cluster). Assume the distance from ${\sigma }_P(s)$ to the complementary spectrum is at least a uniform $g>0$ in joules. Let $T>0$ be a total duration in seconds, let $s=t\mathrm{/}T$ be dimensionless, and let $U_T(0)=I$ solve $i\mathrm{\hslash }{\mathrm{\partial }}_s{U}_T(s)=TH(s)U_T(s)\mathrm{.}$ Put $K(s)=[P^{\mathrm{\prime }}(s),P(s)]$. There is a $C^1$ Hermitian matrix $X(s)$ solving $[H(s),X(s)]=K(s)$. With $C=2{\sup }_{s\in [0,1]}\mathrm{\parallel }X(s)\mathrm{\parallel }+{\int }_0^1(\mathrm{\parallel }{X}^{\mathrm{\prime }}(s)\mathrm{\parallel }+\mathrm{\parallel }K(s)\mathrm{\parallel }\mathrm{\parallel }X(s)\mathrm{\parallel })ds,$ the leakage satisfies $\mathrm{\parallel }(I-P(s))U_T(s)P(0)\mathrm{\parallel }\le \frac{\mathrm{\hslash }C}{T}(0\le s\le 1)\mathrm{.}$ The supplier provides a $C^1$ Hermitian $X$ solving the displayed commutator equation. For that $X$, $C$ has unit joule-inverse, so $\mathrm{\hslash }C\mathrm{/}T$ is dimensionless. If the cluster has rank one, define $E(s):=\mathrm{tr}(H(s)P(s))$; then $E$ is real and $C^2$ and $H(s)P(s)=E(s)P(s)$. Define $U_A(0)=I$ by $i{\mathrm{\partial }}_s{U}_A(s)=(\frac{T}{\mathrm{\hslash }}H(s)+iK(s))U_A(s)\mathrm{.}$ For Kato transport $W^{\mathrm{\prime }}(s)=K(s)W(s)$, $W(0)=I$, the comparison evolution factors exactly on the initial eigenspace: $U_A(s)P(0)=\exp \text{\hspace{0.17em}⁣}(-\frac{iT}{\mathrm{\hslash }}{\int }_0^{s}E(u)du)W(s)P(0)\mathrm{.}$ The leakage estimate is finite-dimensional and conditional on the stated uniform positive gap.