A quadratic term gave no clear gain in six-month Nino 3.4 forecasts for 1981-2025

We tested a quadratic temperature term in six-month Nino 3.4 forecasts from a seasonal linear state model. We used a fixed NOAA ERSSTv5 snapshot and 540 target months in 1981-2025. Each fit used targets strictly before its forecast origin. Mean squared error fell by 0.000514 C squared, or 0.106%. The paired five-year block interval was -0.024430 to 0.021321 C squared. Three- and seven-year blocks also left the sign uncertain. Gains varied by season and period. This test gave no clear gain from this one added term. It does not show that ENSO is linear or identify a physical cause. Revised data, a modern anomaly reference, 45 test years, conditional intervals and prior pattern selection on the same record limit inference. Code, aggregate errors, draw means, provenance and numerical checks are included. External monthly data stay outside the bundle.

ENSO · Nino 3.4 · forecast · quadratic regression · seasonality · null result · ERSSTv5