This paper present QI-HRNN, a hybrid recurrent architecture that incorporates mathematical structures from quantum mechanics into classical neural networks for daily foreign exchange forecasting. The framework operates entirely on classical hardware and introduces three components whose designs are motivated—but not physically justified—by quantum formalism: (i) a Quantum-Inspired Activation Function (QIAF) derived from the \(R_y(\theta )\) rotation gate, for which we prove that \(R_y\) uniquely preserves real-valued amplitudes among single-qubit rotations and that the resulting \(\sin ^2(\theta /2)\) nonlinearity admits bounded gradients suitable for stable optimization; (ii) a kurtosis-adaptive normalization scheme with provable coverage guarantees for leptokurtic financial distributions; and (iii) a discrete Lorenz-inspired chaotic oscillator that provides bounded aperiodic modulation as a heuristic functional analog—not a physical simulation—of quantum dynamical properties. We evaluate QI-HRNN on five currency pairs (AUD/JPY, NZD/JPY, EUR/JPY, EUR/USD, GBP/USD) over an 11-year period (2014–2024) using Walk-Forward Validation with strict temporal causality. Benchmarking against ten baselines—including Informer, Autoformer, N-BEATS, and DLinear—across input windows from 7 to 90 days reveals context-dependent tradeoffs: QI-HRNN achieves the lowest MSE ( \(0.72 \times 10^{-4}\) ) and highest Sharpe ratio (2.78) at the short windows ( \(k \le 14\) ) most relevant to daily forex execution, while Transformer architectures become preferable for \(k \ge 30\) . Controlled ablation experiments with randomized and Gaussian-noise lookup tables confirm that the observed gains stem from the specific structure of the chaotic nonlinearity rather than from added parameters or generic stochastic regularization. Under institutional transaction costs (1-pip spread), QI-HRNN attains a net Sharpe ratio of 2.31 and remains profitable up to a break-even spread of 6.2 pips, whereas standard recurrent baselines become unviable above 2.5 pips. During the COVID-19 crash (annualized volatility 28.7%, excess kurtosis 7.8), QI-HRNN maintains a positive Sharpe ratio (0.42) and limits maximum drawdown to \(-18.2\%\) , compared with \(-42.7\%\) for GRU and \(-45.3\%\) for LSTM. These results are achieved with 73% fewer parameters than Informer (184K vs. 672K). All statistical comparisons employ Newey–West corrected standard errors to account for overlapping Walk-Forward folds, and significance is confirmed via block bootstrap.
