Where Quantum Fourier Sampling Stops Short: A Three-Gate Audit Protocol for Delay-PUF Security Models

Quantum Fourier sampling (QFS) appears to offer a direct route to auditing the spectral learnability of delay-based physical unclonable functions (PUFs). We ask whether that promise survives access matching, a strong classical comparator, and oracle synthesis, and report where it stops short. Three gates structure the evaluation. \textbf{Structure:} low degree is not small support at reachable challenge lengths; for 4-XOR at $n=14$, degree $\le d_f(0.1)$ admits $91\%$ of all $2^n$ characters and the median $90\%$-mass set spans a third of the spectrum. \textbf{Algorithmics:} constructing the phase oracle logically implies classical membership access, making Kushilevitz--Mansour the correct baseline; across 45 tasks it exhausts each finite domain, and no 4-XOR ideal-sampling case reaches $90\%$ mass within $2^n$ calls. A quantum-kernel diagnostic appears more favorable, with geometric difference rising to $2.151$ at $N=512$ challenges, but it correlates $0.991$ with $1/\sqrt{\lambda_{\min}(K_C)}$ for the classical Gram matrix $K_C$, and the 4-XOR label-complexity ratio does not exceed a balance-preserving permutation null ($p=0.930$). Trace-normalized geometric difference can therefore grow through classical ill-conditioning alone, without task-label alignment. \textbf{Implementation:} a simulator-validated fixed-point phase oracle based on the quantum Fourier transform (QFT) admits an $18.9\%$ routed-depth reduction, yet the least certified precisions have estimated durations of $1.18$--$1.55\times$ the median dephasing time $T_2$ of the mapped qubits on a static backend snapshot, without hardware execution. We find no end-to-end advantage in the evaluated regime, although ideal sampling does use fewer coherent calls on the thresholded task. The contribution is the Three-Gate Quantum Audit Protocol: a reproducible procedure separating an ideal query advantage from a realizable security benefit. This is not a claim about deployed silicon and not an impossibility result.