Projection, Memory, and the Validity of Scalar Early-Warning Signals: A Schur--Volterra Theory with an AMOC Application

RESUMO

Early-warning signals are usually computed from one or a few observables, although the underlying dynamics are high dimensional. Projection creates a structural ambiguity: the same scalar record may reflect spectral loss of stability, hidden-state memory, nonnormal amplification, a turning point of a physical balance, or a combination of these. We develop a unified operator theory that makes these alternatives explicit. For a resolved-hidden block linearization we derive exact Volterra reductions, finite and diffusive Markovian liftings, and a Schur-resolvent factorization of the stability problem, distinguishing three notions of memory: realized pathwise, structural Schur return, and finite-horizon projected. A covariance theorem gives sufficient conditions for a scalar to inherit critical slowing down from a real multiplier approaching the unit circle: spectral separation, bounded eigenvector conditioning, observability of the critical mode, stochastic excitation of that mode, and quasi-stationary forcing. We then prove failure mechanisms: a critical mode can be invisible or unexcited; an oscillatory crossing need not raise lag-one correlation; a nonnormal family can produce diverging scalar variance with a fixed stable spectrum; a physics-based turning point need not coincide with a spectral boundary; and finite-horizon data cannot identify long memory uniquely. The theory is illustrated with a 4400-year CESM AMOC hysteresis experiment. EOF1 alone detects altered susceptibility, but EOF1-EOF2 is the smallest resolved space revealing changed Schur return and projected memory after recovery; F_ovS reaches its minimum 26 years before collapse while the modal margin does not decrease toward zero: the two diagnostics probe complementary mechanisms. The application is mechanistic and single-trajectory; no validated early-warning signal is claimed.