Cuneiform recorded typologically unrelated languages for more than three millennia, raising a natural question: which of its statistical regularities are properties of the shared writing system, and which belong to individual languages? We answer this with two probe pairs targeting the same script-versus-language decomposition. The first compares two unsupervised, dictionary-free segmentation methods: transitional probability (TP), grounded in Saffran et al. (1996), and Morfessor (Creutz and Lagus, 2007). To our knowledge, this is the first systematic evaluation of dictionary-free segmentation on cuneiform; the only prior published baseline (Homburg and Chiarcos, 2016) relies on a hand-built dictionary (MaxMatch, F1 = 0.74). Both our methods reach F1 > 0.97 within language (Morfessor reaches F1 > 0.99 on Akkadian and Sumerian), exceeding MaxMatch by more than 23 F1 points, but diverge sharply in cross-language transfer: TP holds F1 > 0.95 across all six language pairs, while Morfessor collapses to F1 = 0.27 from the 13K-token Elamite source. A learning curve and a Hittite stress test (34.7% foreign-script logograms, where TP drops 35 points and Morfessor only 8) further isolate the two underlying mechanisms. The second probe pair extends the same decomposition to representations: concatenating Unicode and Latin features improves classification in 11 of 12 morphosyntactic tasks (p < 0.01). We release cunei-tools and a three-language benchmark.