Toric folds and the n^(-3/2) rate in the Boyd-Lawton limit

Let P(x,y) be a Laurent polynomial whose roots in y, for every x on the unit circle, are symmetric under y ↦ 1/ȳ, and suppose that the extreme coefficients in y have no zeros on the unit circle. We show that if the double roots of P on the torus are finitely many simple folds, at which two roots in y coalesce on the unit circle and separate transversally, then m(P(x,xn)) − m(P) = κP(n) n−3/2 + o(n−3/2), where κP(n) is an almost periodic sequence formed from one value of Li3/2 on the unit circle for each fold, with explicit coefficients. For double roots of higher contact order k the exponent becomes 1 + k/2 when such a point contributes, and we determine which points do. For Q = 1 + x + x−1 + y + y−1, whose Mahler measure is L′(E15, 0) by a theorem of Rogers and Zudilin, n3/2(m(Q(x,xn)) − m(Q)) converges along each residue class n ≡ r (mod 6) to 31/4 π−1/2 Re(e−3πi/4 Li3/2(−eπir/3)). This explains the exponent 3/2 and the period six observed numerically by Condon, a question raised again by Brunault, Guilloux, Mehrabdollahei and Pengo. The six limits are combinations of ζ(3/2) and L(χ−3, 3/2), and none of them vanishes. We also give an example in which the folds lie over irrational angles and the normalized difference has no limit along any arithmetic progression. MSC 2020: Primary 11R06; Secondary 41A60, 42A16, 11M35. The ZIP archive contains the complete LaTeX source with the data of the figures and tables, programs and data for the numerical checks, and a Lean 4 file formalising auxiliary lemmas of the proof. Assisted by AI.