A Walsh-Quotient Obstruction for Fourier Frames on Odd Reciprocal-Power Bernoulli Convolutions

We introduce a Walsh-quotient obstruction to study Fourier-frame existence for symmetric two-branch Bernoulli convolutions. Suppose that 0 < rho < 1/2 and rho^{-m} = B for some integer m >= 1 and odd integer B >= 3. We prove that L^2(mu_{rho,d}) admits no Fourier frame. For m = 1, our argument independently proves the non-frame theorem for odd-integer-base Cantor measures and hence resolves Strichartz's long-standing open problem for the middle-third Cantor measure. A contemporaneous independent proof of the case m = 1 was obtained by Pont, Liehr and Taylor. For m > 1, our theorem treats the non-integer reciprocal-power contraction ratios rho = B^{-1/m}, which fall outside the classical integer-base Cantor-measure setting. Our proof is self-contained. It uses finite-coordinate Walsh packets to transform the frame inequalities into incompatible tangent-quotient estimates, while the identity rho^{-m} = B supplies the exact m-step scale relation leading to the contradiction.