Finite adiabatic braid compilation input

A finite adiabatic braid input consists of a finite-dimensional complex Hilbert space $V$, a $C^2$ Hermitian path $H(s)$ on $0\le s\le 1$, a norm-continuous rank-$m$ orthogonal spectral projection $P(s)$, with $m\ge 1$, onto a coherently followed band with uniform positive full gap, a slow-time schedule $T>0$, and an explicit initial isometry $A_0:{\mathbb{C}}^m\to \mathrm{ran}P(0)$. These are the path and propagator hypotheses of Finite adiabatic path model; the output propagator $U_T$ solves $i{U}_T^{\mathrm{\prime }}(s)=TH(s)U_T(s)$. Also supply an endpoint isometry $A_1:{\mathbb{C}}^m\to \mathrm{ran}P(1)$ and a specified Fibonacci braid word $b$ with categorical action $\rho (b)$ on a specified $m$-dimensional fusion space $V_{\mathrm{c}\mathrm{a}\mathrm{t}}$, in the fixed convention of Fibonacci fusion data and chiral convention, and a unitary target ${\rho }_{\mathrm{l}\mathrm{o}\mathrm{g}}(b)\in U({\mathbb{C}}^m)$ in the declared logical input/output frames. Supply unitary coordinate maps $Q_j:{\mathbb{C}}^m\to V_{\mathrm{c}\mathrm{a}\mathrm{t}}$ for $j=0,1$ and set ${\rho }_{\mathrm{l}\mathrm{o}\mathrm{g}}(b)=Q_1^{\ast }\rho (b)Q_0$. The band identifications are then ${\iota }_j=A_j{Q}_j^{\ast }$, as in Physical braid implementation interface. The target is the selected categorical $F\mathrm{/}R$ action written in those frames, including the phase convention recorded by Braid phase reference and correction ledger; specifying it does not assert that the physical path realizes it. The in-band transport condition is the separate hypothesis $\mathrm{\parallel }P(1)U_T(1)A_0-A_1{\rho }_{\mathrm{l}\mathrm{o}\mathrm{g}}(b){\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}\le {ϵ}_{\mathrm{t}\mathrm{r}}\mathrm{.}$ This condition is not a consequence of a projector-leakage estimate. The model also specifies a finite output measurement and, if needed, its own implementation-variation bound. The JRS expression and leakage consequence are handled as a separate result.