A spatial thermal benchmark needs a 1.90-fold late-count allowance in log-weighted indices on 2,075 paired bird routes

We tested how much late undercount can hide a spatial thermal benchmark in paired bird surveys. We used 15,667 visits on 2,075 routes in 1997-2001 and 2021-2024. Species thermal values were fixed. The log-weighted community temperature index rose by 0.107 C. Its spatial benchmark was 0.338 C. We derived sharp bounds when each positive late count can increase by up to Gamma times, with early counts fixed. The benchmark first became possible at about Gamma=1.90 for log weights and 1.33 for raw weights. Route bootstrap ranges were 1.81-1.99 and 1.30-1.36. These allowances correspond to maximum relative detection losses of 47% and 25%. They are sensitivity thresholds, not measured losses. Corrections can vary by species and visit. Zero-count species cannot enter. Unknown early detection, count noise, habitat and shared year changes remain outside the model. Independent linear programs checked the algorithm. The package includes a proof, code, source hashes, aggregate results and a retrieval recipe. Record-level inputs stay external.