A reference-invariant quadratic term gave no clear twelve-month ENSO forecast gain at 1981-2024 origins

We tested whether a quadratic temperature term improves twelve-month monthly Nino 3.4 forecasts beyond a seasonal linear model with recent change. We used a fixed NOAA CPC ERSSTv5 snapshot and 528 origins in 1981-2024. Training used earlier targets only. The quadratic term cut mean squared error by 6.50%. Its conditional 95% five-year block interval was -2.34% to 14.56%, so this test gives no clear gain. A zero-threshold warm-cold model gave a 9.58% reduction, with limits of 0.86% to 17.89%. Fixed threshold changes gave different point gains. We prove that the quadratic and linear model spaces are invariant under uniform temperature reference shifts. Numerical checks agreed within 6.44e-15 C. This invariance did not ensure a clear forecast gain. Revised data, few independent events, model choice and conditional intervals limit inference. The package gives original code, aggregate evidence, source hashes, a figure and a separate numerical audit. External source records and individual predictions are regenerated outside the signed draft.

ENSO · reference sensitivity · quadratic regression · twelve-month forecast · null result · Nino 3.4