Deterministic Seismic Inter-Story Drift and Structural Health Monitoring via Idempotent Spatio-Temporal Projections: Causal Streaming and Post-Event Screening

Structural health monitoring (SHM), rapid seismic screening, and post-earthquake safety assessment of civil structures require estimating dynamic inter-story drift ratios θi(t) = Δui(t)/hi. Because direct displacement sensing via optical or satellite instruments is constrained during strong ground shaking by line-of-sight loss and multipath errors, monitoring systems rely on floor accelerometers. However, double numerical integration üi(t) dt2 is ill-posed: instrumental DC offsets, co-seismic rotational slab tilts (g·sin θtilt), and suspension hysteresis integrate into quadratic baseline drift (1/2 ε0 t2), producing fictitious displacements on the order of tens of meters. Conventional acausal polynomial detrending requires the earthquake to have terminated, while causal high-pass filtering induces phase distortion and eliminates permanent residual drift. Here, we present IdemSHM, a deterministic framework formulated as an algebra of idempotent metric projection operators (Π2 = Π) across spatial and temporal dimensions. IdemSHM introduces a dual-mode operational architecture: (1) Mode 1 (Online Causal Streaming): a moving-horizon polynomial drift estimator using precomputed orthogonal Legendre boundary state extraction kernels over sliding ring buffers, bounding baseline divergence with per-sample latencies of 6.43–7.75 μs on a 1 kHz DAQ clock for shock detection and event triggering; and (2) Mode 2 (Post-Event Screening): a global orthogonal Legendre projector combined with mass-weighted modal dynamic equilibrium (Πshear) and symmetric polyhedral capacity clamping (Πcapacity) for automated structural safety tagging under ASCE 41-23. The implementation incorporates compile-time static memory arenas (0.0 B dynamic heap allocation) for deterministic interrupt execution, coprime round-robin bus polling schedules to eliminate packet collisions on multiplexed fieldbuses (CAN-FD, RS-485), and stiffness tracking (K(T)) under non-uniform vertical thermal gradients. Benchmarked on 5-story steel and 10-story RC frames under 1940 El Centro and 1995 Kobe records, Mode 2 IdemSHM recovers dynamic displacements with 0.26 mm RMSE (R = 0.9996) and 6.71 mm RMSE (R = 0.9991), providing certified baseline correction and automated damage classification without empirical tuning.