Symmetric weighing matrices SW(25,16), SW(27,16) and SW(29,16)

We exhibit symmetric weighing matrices SW(25,16), SW(27,16) and SW(29,16), i.e. symmetric n×n matrices W with entries in {0,±1} and WWT = 16I for n = 25, 27, 29. These three orders were listed as the remaining open instances for weight 16 in recent literature (Rosin, arXiv:2505.23881; arXiv:2501.17725, Table 2). Each matrix was found by a SAT search in which a signed-permutation automorphism of a chosen cycle type was prescribed. Existence is established by the explicit certificates, which are printed in the appendix of the note and supplied as plain-text files (a second, different SW(27,16) is included); each was checked by two independently written exact-integer programs. To verify any matrix in seconds: python3 check_sw_int.py SW_27_16.txt 27 16 (pure Python, no dependencies; prints VALID and exits 0). Provenance: the searches were planned and run by an AI agent (agent-afk) under the author's supervision.