Protected Unbiased Sparsification: A Sharp Loewner Curve

Many learning systems replace a probability vector $p\in {\Delta }_k$—token attention, data weights, expert gates—by a random vector $Q$ supported on at most $m\ll k$ coordinates. Such a replacement is trustworthy only if it is unbiased, preserves prescribed linear statistics $AQ=Ap$ on every realization (class proportions, group masses, control variates), and remains reliable for downstream responses that are not known when $Q$ is drawn. We give a complete answer to this problem in Loewner order. Writing $d=\mathrm{rank}(A{C}_p{A}^{\top })\le k-2$ for the effective number of protected directions, $C_p=\mathrm{diag}(p)-p{p}^{\top }$, and $C_{p\mid A}$ for the covariance of a categorical draw that remains after regressing on the protected statistics, we construct, for every budget $d+1\le m\le k$, an unbiased exactly protected $m$-sparse $Q$ with $\mathrm{Cov}(Q)⪯\frac{k-m}{(m-d)(k-d-1)}{C}_{p\mid A},$ and we prove that the coefficient cannot be improved uniformly over instances at any budget. The result lifts a concurrent unprotected coefficient to arbitrary pathwise protection through a new residual-covariance bridge: the protected problem on $k$ coordinates with budget $m$ behaves exactly like the unprotected problem on $k-d$ coordinates with budget $m-d$. For rational inputs, an exact-rational Las Vegas sampler draws $Q$ in expected polynomial time and attains the curve within a factor $1+\epsilon$. Finally, the universal guarantee is essentially the strongest constructible one: optimizing a law for a single revealed quadratic query $W$ already encodes Minimum Uncut exactly, so no polynomial-time approximation scheme outputs an explicitly represented optimal law unless $\mathsf{P}=\mathsf{NP}$. A small numerical study confirms that the constructed law tracks the curve and that the minimax step is necessary.