How Much Does the Quantum Part Do? Auditing Quantum Implicit Neural Representations

Quantum implicit neural representations place a parametrised circuit inside a coordinate network and report gains over classical baselines. We rebuild two of them under a common evaluation harness at matched parameter budgets, recover the reported performance level for both, and then ask what produces it. For Quantum Visual Fields the answer is almost entirely classical: changing one activation in the classical embedder is worth $5.52$ dB, while training the $80$-parameter ansatz, $0.17\%$ of the model, is worth $-0.05$ dB at width $128$ and $+1.48$ dB at width $256$, the largest effect we observe and the only multi-seed configuration whose paired $95\%$ confidence interval excludes zero, at $40$–$75\times$ the training cost. A proposition explains the ceiling: the circuit's output is a quadratic form in the embedder's own output; each channel's form lies in a $136$-dimensional symmetric-matrix space, while the jointly reachable family has intrinsic dimension at most $80$. A budget-matched classical quadratic readout with the same $80$ trainable parameters matches the circuit while training roughly three orders of magnitude faster; an unconstrained relaxation performs $0.78$ dB better. The same proposition predicts why the architecture is flat in classical width and why its one effective scaling direction costs $2^n$. At matched parameter budgets SIREN leads in fitting quality, while QIREN requires three to four orders of magnitude more compute. We also document that the released artifact does not run as shipped and omits the variant producing its published numbers.