arXiv:2608.00466v3 Announce Type: replace Abstract: Many engineering systems contain states that cannot be measured directly, yet these hidden states act on the measured outputs. We present SPIRAL-PO (Symbolic Physics-Informed Residual Augmentation Loop - Partially Observed), a framework for identifying an interpretable observable-space closure: a model, written in the measured coordinates, of how the unmeasured states act on the measured dynamics. A physics seed is augmented by candidate terms from a mechanism-labeled library; terms are selected by statistical tests designed for numerically differentiated, serially correlated data, using a weak (integral) formulation, and are checked on independent records. For an idealized regression we give sufficient conditions for recovering the closure coefficients, a rank condition under which no estimator can recover them, and a per-term detection bound; these results apply only to systems that admit an instantaneous observable closure. On a simulated four-degree-of-freedom vertical rotor with cubic bearing stiffness, in which only the lateral displacements are measured, the method recovers the gyroscopic and cubic-stiffness signatures of the hidden tilt dynamics in every noise realization at 2% noise; with a mechanism removed it never admits the cubic term and admits the gyroscopic term in 15-25% of realizations, above the nominal level, because the closure is only partly instantaneous. A residual network and a quasi-random projection stage add no measurable benefit, and pointwise tests that ignore the serial correlation of differentiated data admit absent mechanisms more often. On a laboratory rotor with simultaneous x,y measurement and an order-tracked speed, the method detects only a speed-dependent stiffening; gyroscopic and cubic terms obtained with an assumed constant-deceleration speed are consistent with the speed error, which produces such terms in simulation.