Characterization of natural completion via coefficient perturbations

Prove that a partial Boolean function f admits the natural completion (sign of an approximating polynomial) without superpolynomial degree blowup if and only if there exists a perturbation Δ of the polynomial’s coefficients such that the perturbed polynomial p_Δ(x) is a degree poly(d) approximator for some completion F extending f, where d is the degree of the original approximator.

Background

The paper studies completing partial functions by extending an approximating polynomial beyond the promise, proposing a 'natural' completion via the sign of the approximator.

They conjecture an exact equivalence between the feasibility of such a natural completion and the existence of bounded-size coefficient perturbations that maintain good approximation everywhere, thereby formalizing when completion without large degree blowup is possible.

References

We conjecture that this relaxation on $\Delta$ exactly characterizes when the natural completion of $f$ is possible. Conjecture A partial function $f: {f} \rightarrow {-1,1}$ with an approximating polynomial $p(x)$ of degree $d$ admits the natural completion $F$ (for approximate degree) if and only if there exists a perturbation $\Delta$ such that $p_{\Delta}(x)$ is a degree $(d)$ approximating polynomial for $F$.

— From Promises to Totality: A Framework for Ruling Out Quantum Speedups  (2603.29256 - Huffstutler et al., 31 Mar 2026) in Subsubsection: Perturbing approximate polynomials (Towards general partial function completions)