Joint-tuple counting moves the window-13 excess3 maximum into developed chaos

Version 2. Version 1 placed the window-13 excess3 maximum at r = 3.5675. That location came from nested-recd ≤ 0.2.2, which flattened joint tuples inside Syn. This version counts joint tuples (nested-recd 0.2.3, 10.5281/zenodo.22970079). The six-seed window-13 maximum is at r = 3.95, excess3 = 2.897 ± 0.006. Surprise still peaks in the accumulation band on about 4.11 joint tuples, and the full-block residual there remains about 0.0003. From window length 78 the excess3 argmax returns to the accumulation band. Version 1 remains 10.5281/zenodo.22945800. This is not a new version of the methods deposit 10.5281/zenodo.21385937. On four diffusively coupled logistic maps, the hybrid proxy excess3 = 0.6·Syn + 0.4·Surp reaches its maximum near the period-doubling accumulation and is lower in developed chaos. A six-seed grid places the mean maximum at r = 3.5675, where excess3 = 2.378 ± 0.002 in windows of 13 symbols. Surprise against independence carries that elevation. The synergy term is 0.0065 ± 0.0004 there. The Kullback–Leibler residual beyond pairwise margins is 0.0003 ± 0.0001 on the full symbol block at the maximum and 0.192 ± 0.002 at r = 3.85. On this generator the proxy and the irreducible residual mark different parts of the cascade. This preprint is a separate record. It is not a new version of the excess3 methods deposit 10.5281/zenodo.21385937. Source and the 24 September 2026 decomposition: https://github.com/johelpadilla/excess3, folder accumulation/.