Strict Hausdorff-dimension drop for Quartic Salem Bernoulli convolutions

For every quartic Salem number $\beta$, we prove that the equal-weight Bernoulli convolution ${\nu }_{{\beta }^{-1}}$ has Hausdorff dimension strictly less than one. The main step converts large Fourier coefficients at geometrically separated frequencies into a deficit in the Shannon entropy of a finite convolution. A positive trigonometric product gives the entropy estimate, while the algebraic norm controls the number of distinct atoms in each spatial cell. For the two quartic Salem numbers in $(1,2)$, reciprocal differences at suitable return times produce the required Fourier coefficients. A finite block cover then gives the dimension bound. This proof is independent of numerical computation. Separate exact calculations give $\text{dimH}{\nu }_{{\beta }_1^{-1}}<1-1.4\cdot {10}^{-23}$ and $\text{dimH}{\nu }_{{\beta }_2^{-1}}<1-7.8\cdot {10}^{-22}$, where ${\beta }_1<{\beta }_2$ are the two parameters in $(1,2)$. The accompanying verification programs use rational interval arithmetic and the Python standard library.