Two-state generator extraction: property currents and a two-layer arrow of time in pre- and post-selected quantum dynamics

Conditioning on both past and future assigns intermediate-time properties a causal observer does not; these time-symmetric assignments obey exact symmetry theorems and are measurable from trajectories. We use two-state generator extended dynamic mode decomposition (gEDMD): because weak values obey $dA_w/dt=i\langle[H,A]\rangle_w$ exactly, generator extraction, with an exact-derivative baseline, applies unchanged to them. First, a reflection involution on the pre-/post-selected ensemble splits every window-fitted friction uniquely as $\gamma_{fwd}=\gamma_A+\gamma_S$: $\gamma_A$, antisymmetric about the midpoint, carries the modes' boundary-condition physics; $\gamma_S$, symmetric, comes from the differencing scheme; both follow from the same data as $(\gamma_{fwd}\pm\gamma_{bwd})/2$. At a fixed inference resolution the arrow of time has two layers: the coherent-mode arrow reverses at the midpoint, the fluctuation-level one does not, $\gamma_S$ dominating $\gamma_A$ at every size and class. The difference is one of degree: $\gamma_S$ is 34 times larger there than at the mode layer, and with the exact derivative both layers reverse: immunity belongs to the inference, not the ensemble. Second, in a lattice interferometer conditioned only at its ports, the quantum Cheshire-cat structure emerges unimposed: particle and polarization obey separate continuity equations, and a local field in the polarization-carrying arm rotates that phase alone, at exactly twice the field strength, entering the generator as a rigid imaginary shift, while the particle's weak density stays invariant to machine precision. We verify the sample-level identity and the two layers from $2^8$ to $2^{20}$ dimensions: $|\gamma_A|/\gamma_S=0.09$ to $0.27$ across five classes; self-averaging makes it insensitive to class among those sharing a boundary modulation, removing the $2^{-N/2}$ overlap obstruction for $N$ qubits.