A Beckmann boundary form of Talagrand's conjecture on the discrete cube

We introduce the Beckmann boundary of a Boolean function $$\mathsf{B}(f)=\underset{\mathrm{div}V=Lf}{\inf }\mathbb{E}\mathrm{\parallel }V(x){\mathrm{\parallel }}_2\mathrm{.}$$ Here $$L=\sum _i{D}_i,D_{i}f(x)=\frac{f(x)-f(x^{\oplus i})}{2},$$ and $\mathrm{div}V(x)={\sum }_i(V_i(x)-V_i(x^{\oplus i}))$. This nonlocal quantity is no larger than the usual two-sided, one-sided, colored, optimized colored, or optimized fractional colored boundaries. Nevertheless, every nonconstant Boolean $f$ satisfies $$\mathsf{B}(f)\gtrsim \mathrm{Var}(f)\sqrt{\log \text{\hspace{0.17em}⁣}(1+\frac{1}{\sum _i{\mathrm{Inf}}_i(f{)}^2})}\mathrm{.}$$ We also prove strong one-sided fractional spectral estimates. If $A\subset \{-1,1{\}}^n$ and $$h_A(x)=\mathrm{\#}\{i:x\in A,x^{\oplus i}\mathrm{\notin }A\},$$ then, for $0<\alpha <1$, $$\sum _{S\mathrm{\ne }\mathrm{\varnothing }}\mathrm{\mid }S{\mathrm{\mid }}^{\alpha }\widehat{{\mathbf{1}}_A}(S{)}^2{\lesssim }_{\alpha }\mathbb{E}{\omega }_{\alpha }(h_A),$$ where ${\omega }_{\alpha }(m)=\sqrt{m}$ for $\alpha <1\mathrm{/}2$, ${\omega }_{1\mathrm{/}2}(m)=\sqrt{m}\log (e+m)$, and ${\omega }_{\alpha }(m)=m^{\alpha }$ for $\alpha >1\mathrm{/}2$. These profiles are sharp, up to $\alpha$-dependent constants, for majority. We also show that the comparison is genuinely nonreversible: an explicit quotient-cube family makes the optimized fractional, and hence optimized colored, boundary exceed $\mathsf{B}$ by a factor $\gtrsim \sqrt{\log n}$. We further obtain a driftless Bernstein-multiplier inequality.