Mahler Cliffs and the U_p-Eigenvalue Criterion for Modular Forms

We study the p-adic rigidity of Mahler coefficients and its applicationto Fourier coefficients of modular forms. For a p-pure sequence a, weprove A_k = kU(k) with U(k) in Z_p, hence v_p(A_k) ≥ v_p(k). This extendsto generalized p-pure sequences satisfying a(pn) = λa(n) withv_p(λ) ≥ 1. Atkin-Lehner theory supplies this structure for newforms ofeven weight k ≥ 2, level divisible by p, and v_p(λ_f) ≥ 1, givingunconditional Mahler rigidity for these newforms. A Sturm verificationprinciple reduces prime-index congruence towers to finite computation.We implement this completely for the weight 4, level 8 newformη(2τ)^4η(4τ)^4, proving an optimal tower with depths (8,10,9,10). Acomplete trace formula shows the local density is governed by theKronecker symbol within tower precision. This yields the second floorbound v_2(a(q)) ≥ min(v_2(q+1),11) unconditionally for v_2(q+1) ≤ 10,and the same bound is verified computationally for all primes q < 2^21with v_2(q+1) ≥ 11. Computational verification and cross-weightphenomena support the framework.