Forced coarse-graining of non-normal generators: the eigenvector arc and the time-reversal selector

A coarse-graining is "forced" when the dynamics, rather than the modeller, decide which microstates merge into a mesostate. For a Markov generator, the standard test is the lumpability defect: a non-negative quantity measuring how far a candidate partition is from commuting with the dynamics, with the forced grain being the partition that minimises it. When the generator — whitened by its stationary covariance — is normal, the picture is classical: the best partition is the slow eigenspace, where the defect equals a variance of eigenvalues and vanishes exactly on eigenvectors. Driven, genuinely out-of-equilibrium generators are non-normal: their eigenvectors are not orthogonal, equivalently the symmetric and antisymmetric parts of the whitened generator fail to commute. This is exactly the non-normality that Nagayama, Kolchinsky and Ito isolate as a distinct, non-negative part of the steady-state entropy production. We ask whether the lumpability defect still identifies the coarse-graining here, and prove a sharp dichotomy controlled by one quantity: the condition number of the slow eigenvalue (the reciprocal of the overlap between its right and left eigenvectors). The defect pins down a unique coarse-graining if and only if the slow mode is normal — when this condition number equals one. Otherwise it cannot separate a whole family of partitions, the "right–left eigenvector arc," whose width is fixed by the condition number; and the blindness is exact, not approximate, because the defect assigns the same value to a generator and to its time-reversal. Choosing the right partition within the arc therefore requires a time-asymmetric quantity. The minimal one is a self-commutator form, which we identify as the stationary arrow of time (the operation sending the generator to its adjoint); it shares its zero with the non-normal part of the entropy production, vanishing exactly when the symmetric and antisymmetric parts commute. Selecting the coarse-graining thus amounts to reading the arrow of time off the dissipation. The identified set is stable under perturbation, with the spectral gap replaced by a pseudospectral separation (a smallest-singular-value quantity); the classical Davis–Kahan theory is the normal special case. A single number — the norm of the slow spectral projector, equal to the condition number — casts two shadows: a spatial one, the arc, and a temporal one, a resolvable-lifetime band given by the logarithm of that projector norm, measuring the extra time a distinction persists beyond its spectral decay rate. On the contractive cone — the physical, stationary-diffusion regime, where the symmetric part is negative definite — the best partition stays robust well past the certificate's bare threshold (shown numerically); off the cone it can be destabilised, which bounds the result's reach. A multi-mode cluster version (covering several slow modes at once) is given in an appendix.