Six-month Nino 3.4 cooling imbalance exceeded six fitted seasonal linear model ranges in 1950-2025

We tested whether seasonal linear memory and monthly noise distributions reproduce a selected six-month cooling imbalance in NOAA ERSSTv5 Nino 3.4 for 1950-2025. The observed cubic change statistic was -0.5066. We fit calendar-month autoregressions with two or six months of memory. Each used Gaussian noise, monthly residual draws, or sign-symmetric residual draws. The observed statistic was below the central 95% range in all six checks with 4,999 simulations each. Lower-tail fractions were 0.0024 to 0.0052. The empirical-noise two-month range was -0.3320 to 0.3265. Six-month memory better matched annual correlation and retained the departure. A symmetry argument also shows why seasonal time direction need not produce a nonzero ensemble cubic change in a centered Gaussian model. These are fixed-parameter model checks on the same record used for pattern selection and fitting. They omit parameter and source uncertainty and do not establish a physical cause or reject all seasonal linear models. The package gives original code, aggregate simulation evidence, source hashes, a figure and an independent numerical audit. External monthly data remain outside the bundle.

ENSO · Nino 3.4 · seasonal autoregression · time reversal · non-Gaussian noise · conditional model check