A proof of the Berkovich-Dhar conjecture modulo five

Abstract

We prove the sign-pattern and limiting-transition assertions of the Berkovich--Dhar conjecture modulo five, covering both the square and the cube of the finite Borwein product. In each power, the coefficients in residue class zero are strictly positive, while the sequences in residue classes three and four have exactly one positive-to-negative sign change after zero terms are omitted. We determine all four transition constants, correct the preliminary numerical estimates in the conjecture, and give their linear corrections with explicit bounded errors. A common four-arc expansion governs both powers. Near each vanishing amplitude, strict negativity of a weighted adjacent difference determines the transition. The cubic case also has two identically vanishing components in the infinite product; a shifted saddle expansion resolves the resulting finite-product boundary terms. Positive series identities, uniform remainder estimates, and an exact integer recurrence complete the proof for every positive integer.