Robust Polynomial Freiman-Ruzsa from Corrupted Set Observations via a Sharp Persistent-Subset BSG Compiler

We study structural inference from an exact, adversarially corrupted set observation over $\mathbb F_2^n$. A hidden nonempty set $A$ satisfies $|A+A|\leq K|A|$, while the algorithm receives deterministic membership and exact uniform-sampling access only to $B$, where $|A\triangle B|\leq\eta|A|$. Since corruption can destroy the doubling of $B$, sharp existential BSG and clean-input Algorithmic PFR do not directly compose in this model. Our main result is a promise-free sharp persistent-subset BSG compiler. Given sample-and-query access to $S$ and $\alpha$, it returns either $\mathsf{FAIL}$ or a descriptor defining one fixed subset $Y\subseteq S$ with $|Y|\geq c\sqrt{\alpha}\,|S|$ and $|Y+Y|\leq C\alpha^{-4}|Y|$. Without an energy promise, every nonfailure output is valid except with the prescribed soundness probability; high energy guarantees success with high probability. The descriptor gives persistent membership under adaptive queries, and a finite-horizon bridge gives conditionally exact product samples. Thus sharp retained mass, promise-free validity, persistence, and exact finite sampling form one composable interface. Combined with certified size-oblivious Algorithmic PFR and deterministic lifting, the compiler yields a randomized FPT-form algorithm for $\eta=O(K^{-1/2})$, outputting $V$ with $|V|\leq|A|$ and $\mathcal N_V(A)\leq K^{O(1)}$. For every supplied $\eta<1$, an iterated residual algorithm outputs a common list serving every compatible hidden set with polynomial covering budget; samples are polynomial, while membership-query and running-time complexity are XP. Finally, every nonempty compatibility class admits one common subspace nonconstructively, whereas an exact two-subspace construction forces common covering cost $\Theta((1-\eta)^{-1/2})$.