No Free Qubit: A Falsification-Driven Investigation of Quantum Kernel Advantage Under Causal Data Structure

A growing body of work has questioned whether quantum machine learning (QML) offers any genuine advantage over classical methods on realistic data, with recent results showing that many apparent quantum kernel advantages are reproducible by classical surrogates once sample size, hyperparameters, and estimator variance are properly controlled. We conduct a pre-registered, falsification-first investigation into one specific, previously unexamined variant of this question: does the causal sparsity of a data-generating process as opposed to generic statistical structure modulate quantum kernel advantage or variational quantum circuit (VQC) trainability? Using a controllable structural-causal-model (SCM) data generator, a moment-matched dense-null control, and a battery of eight experiments spanning tabular regression, PDE solving, and gradient-landscape diagnostics, we find: (i) an apparent quantum-kernel advantage at small sample sizes N>=200 that consistently vanishes and reverses sign by N(approx):300, uniformly across all tested sparsity levels; (ii) no causal link between data sparsity and this advantage, once tested against a properly matched non-causal control (factorial ANOVA interaction $p=0.194$); (iii) no genuinely-quantum causal contribution to predictive performance distinguishable from estimator noise at any tested condition, including the resolution of one apparent positive result as a single-seed statistical artifact; (iv) a dissociation between classical-shadow kernel-matrix reconstruction (which fails) and task-level dequantization (which succeeds), indicating that matrix fidelity is an unreliable proxy for classical simulability; (v) a previously undocumented, reproducible failure mode in the PHASE domain-decomposition PINN architecture's boundary-continuity constraint, whose removal yields a solver that is simultaneously faster and more accurate across the full tested regime; and (vi) a robust, reproducible, but ultimately unexplained resonance phenomenon in VQC gradient variance at specific integer causal-edge counts, for which we construct, pre-register, and subsequently falsify a candidate mechanistic explanation via out-of-sample prediction. We argue that the recurring pattern of apparent effects collapsing under targeted multi-seed and out-of-sample scrutiny is itself a methodologically important finding, and we recommend such validation be treated as a mandatory component of QML advantage claims rather than an optional check applied only to surprising results.