THERE ARE AT MOST FINITELY MANY SINGULAR MODULI THAT ARE S-UNITS

The paper proves the finiteness of singular moduli j with j − j0 an S-unit via a height-lowering contradiction that combines (i) a global lower bound hW(j−j0) ≥ A log|D| − B, (ii) local bounds −log|j−j0|v ≤ Av log|D| + Bv at all places v (archimedean from Habegger; non-archimedean from a new p-adic dispersion/proximity estimate), and (iii) p-adic dispersion (Theorem B) to show only a small proportion of conjugates can be v-adically close, yielding an upper bound strictly smaller than the global lower bound, hence a contradiction for large |D|. The model’s proof outline critically misidentifies N_{Q(j)/Q}(j − j0) with the two-orbit product J(Δ,Δ0), and relies on an unsubstantiated uniform bound v_p(J(Δ,Δ0)) ≤ C_p h(Δ). Even after correcting the norm identity to N_{Q(j)/Q}(j − j0) = H_Δ(j0), the model’s approach cannot produce a Δ-independent upper bound: Proposition 3.1 implies p-adic contributions may grow like O(log|Δ|) per conjugate, so v_p(N)/h(Δ) = O(log|Δ|), not O(1). Thus the model’s argument does not close, while the paper’s does.