Parity Decision Trees of Depth O(4ᵈ log\| f\|₁) for Boolean Functions f of F₂-Degree d

Let f F₂ⁿ→\-1,+1\ have F₂-degree d≥2. We prove that its deterministic parity decision tree depth satisfies \[ D⊕ (f)= O(4^d log_2\| f\|_1) = O(4^d log_2 s(f)), \] where \| f\|₁=∑_ξ| f(ξ)| is the Fourier spectral norm and $s(f)$ is the Fourier sparsity. The bound is linear in the logarithm of the spectral norm for every fixed degree, with singly exponential dependence on d. This improves the previously best known bound of O(2d²/2 logᵈ⁻² \| f\|₁) due to Tsang, Wong, Xie and Zhang (FOCS 2013). Our result establishes the log-rank conjecture for XOR functions of F₂-degree O(loglog s(f)), extending the previously accessible range $d=O(1)$. The proof combines field lifting, multiplicative derivatives, a multiplicity-based generic-rank comparison, a Schur-complement vanishing argument, and dimension-preserving descent to F₂ to lower the degree on every coset of a common subspace.