When Does Filtering Help You See a Break? Latent-State Diagnostics for Structural Change in Scalar Linear-Gaussian AR(1) State-Space Models at Calibrated False-Alarm Rates

We ask when filtering a latent state helps detect structural change, using a protocol that calibrates every detector to the same false-alarm rate (5% per 500 observations) on matched null data. For scalar linear-Gaussian AR(1) state-space models-the DGP class studied herethe answer is a trichotomy-no, yes, no-across three break types. (i) Level shifts in a persistent state: no-a raw-data CUSUM dominates at every SNR in the main grid, and the innovation CUSUM is provably "fast or never," a boundary condition of persistence (μ∞ sorts detection, Spearman 0.94). (ii) Observation-noise variance changes: yes, but the advantage is prewhitening, not the state estimate-the observable is exactly ARMA(1,1), so ARIMA and Kalman whitening are provably the same filter. (iii) State-innovation (shock) variance changesthe Great-Moderation/crisis-volatility channel: no, a partial null-raw matches or beats whitening on the coarse break, but the subtle break leaves every rung near the false-alarm floor, where whitening fails to recover it. Each leg of this trichotomy holds at the specific operating point studied throughout the main grid-estimated parameters, a two-sided CUSUM, φ = 0.95-and not unconditionally: §4 documents a convention-dependent tie at the flagship level-shift cell; §5 documents a boundary reversal in the observation-noise ordering near the unit root that is not uniform across SNR (raw actually wins at lower SNRs there, contrary to what "prewhitening wins" would predict); and §5 separately checks, at one point so far, that the state-innovation ordering does not yet show the same reversal. None of this is a hedge added after the fact-it is what the φ-sweeps in §4-5 were run to find, and the numbers are there, not here. On real data (industrial production, GDP, unemployment, and Treasury yields-the latter an exploratory fourth series on author-selected events, §9) every alarm attributes to a second-moment feature, but no series clears both the multiple-testing bar and a model-fit check at once: the headline NBER association reaches an uncorrected permutation p = 0.008, yet does not survive a family-wise or FDR correction across the full grid of tests the real-data section actually runs (§9), and real-time vintages confirm COVID while downgrading 2008. Two extensions probe the protocol's edges: offline PELT matches raw CUSUM on level breaks but not variance breaks, and a bounded-memory statistic fixes raw CUSUM's blindness to a second level break, though not a second variance break. Read together, the results are deflationary for the latent layer's detection power: a raw or ARIMAwhitened single-feature benchmark matches or beats the state-aware detector on every break type studied, and what filtering buys instead is breadth and attribution, not power (§10)-with one qualification: feeding the same 11-feature composite ARIMA inputs instead of Kalman ones (§5) shows that, away from the detection ceiling, the composite built on the genuine filtered state decisively beats the same composite built on ARIMA's fitted-value analog (e.g. 0.818 vs. 0.226 at the flagship r-channel subtle-break, SNR 0.1 cell)-the "raw vs. whitened, not the state" reading holds exactly for the single innovation-series statistic (a proven identity) but not for the full composite, where the state does buy real power.