Modular forms for chromatic homotopy: Supersingular congruences

In this note, we confirm a conjecture of Larson that arises in the Adams--Novikov spectral sequence (ANSS) for the stable homotopy groups of spheres and, specifically, in Behrens' program on explicit modular forms detecting v₂--periodic classes in the divided $β$-family. The conjecture predicts the supersingular order of the weight $12t$ form L₂(Δᵗ), when (p-1) 12t, attached to the Γ₀(2) Hecke correspondence. We prove the prediction for all primes p≥5, thereby providing the precise modular input that calibrates the relevant ANSS differentials in the Behrens program and removes the last obstruction to using pure $Δ$-power across the range of indices where Hodge-scaling cancels.