Moki & Julio: Ramanujan's septic theta identity at every odd prime - explicit analogues, a Galois rule, the abelian cases at every odd level, and Ramanujan's units

On page 206 of the lost notebook Ramanujan recorded an incomplete septic identity for phi(q); Rebak (2023) supplied the missing terms. For every odd prime p the p-dissection of phi at q = e-pi/sqrt p gives pieces u_k; we study b_k = u_k^p at the CM point and their relatives at other levels. Results: explicit identities at p = 11 and 13 and certified minimal polynomials of b_1 at every prime 5 <= p <= 97 (ball arithmetic); the field F_p of the pieces, an explicit Galois rule, integrality and the 2-adic valuation law, units exactly for p = 7 mod 8, with an index formula in Bernoulli numbers; the cyclotomic cases (p in {3, 5, 7, 13, 37}; odd n in {3, 5, 7, 9, 13, 25, 37}); one abelian-over-Q law at every odd level in terms of Euler's idoneal numbers (Theorem G), covering Berndt-Rebak's cubic and quintic pieces and the Rogers-Ramanujan continued fraction; even-level laws for the Goellnitz-Gordon (m = 3, 7, 15) and cubic (n = 2, 8) continued fractions; and a Stark-unit dictionary for the pieces. Proofs were checked by independent AI reviewers, not yet by a human expert; status words and condition tags are given for every statement; nothing rests on GRH. Code, data, sealed predictions and a clean-room record are included. Text CC BY 4.0; code MIT.