Dyadic rigidity in p-adic L-functions: symmetry-forced vanishing and locking congruences

We show that the (+)-involution on the Katz 2-adic L-function of a CM motive forces a double zero at T = 0, and that the leading coefficient is a 𝔓-adic unit. The vanishing c₁ = 0 is of a different nature from both exceptional zeros in the sense of Mazur–Tate–Teitelbaum and trivial zeros arising from Γ-factors; it is intrinsic to the signed projection. The sharpening to ordT=0 L_p^+(T) = 2 with c₂ ∈ ℤ_p× follows from the p-adic Colmez formula, since the balanced CM type forces the Katz period ratio to be a 𝔓-adic unit. We then convert this analytic double zero into arithmetic congruences on Hecke units of unbounded depth: for ray class families ℛ_e of positive Chebotarev density, every q ∈ ℛ_e satisfies v_𝔓(u(𝔮) − 1) ≥ 2e. The genus barrier for the Fermat octic is completely classified, with σ̃_q(8) = 0 only for q ∈ {17, 41, 113}. final edited version