Architecture Shape Governs QNN Trainability: Jacobian Null Space Growth and Parameter Efficiency

Variational quantum circuits with angle encoding implement truncated Fourier series, and architectures arranging N qubits with L encoding layers each -- sharing encoding budget E = NL -- generate identical frequency spectra, identical frequency redundancy, and require the same minimum parameter count for coefficient control. Despite this equivalence, trainability varies substantially with architecture shape (N,L) at fixed E. We identify structural rank deficiency of the coefficient matching Jacobian J as the mechanism responsible. For serial single-qubit architectures, we prove rank(J) ≤ 2L+1 regardless of parameter count P, with dim(ker J) ≥ P-(2L+1) growing without bound -- a phenomenon we term structural gradient starvation: a growing fraction of parameters become structurally decoupled from the loss as P increases at fixed L. Parallel architectures avoid this via independent phase trajectories, ensuring σmin(J(par)) > 0 generically for P ≤ 2E+1, so no parameter lies in ker J. For practitioners, we further show that the two natural routes to increasing parameter count have fundamentally different effects: adding feature map (FM) layers monotonically strengthens the Jacobian QFIM eigenvalue spectrum and achieves R2 ≥ 0.95 with 1.6--2.2× fewer parameters than adding trainable blocks across all tested architectures, while trainable blocks improve training only through the classical interpolation mechanism with no quantum-specific benefit.