One-Pass Exact Hessians via Hypercomplex Perturbation: A Vectorized Implementation with XLA Backend and Implicit Layer Applications

Version 4.1 — adds JAX-XLA backend (137–260× faster than JAX v0.3), BackPACK/ViViT comparison in related work, and sharpened IFT correctness proof. The JAX-XLA backend represents the entire coefficient vector as a single JAX array and implements the Hyper algebra via jnp.einsum — one call, XLA-fusable, jax.jit traceable. XLA matches or beats NumPy across all tested dimensions (d=3–64). For implicit layers z*=f(z*,θ), the hypercomplex method recovers the IFT-correct Hessian (gradient error < 10⁻¹³). JAX unrolled AD gives a systematically different result by differentiating through iteration history. hcderiv is 27–403× faster than JAX unrolled for n=3–8. Trust-region demo: exact and FD converge in 16 iterations; diagonal stalls at f=6.2e-3 after 150. 117 tests pass (80 NumPy + 21 JAX + 16 layout + 37 XLA). GitHub: https://github.com/zetta55byte/hypercomplex | PyPI: https://pypi.org/project/hcderiv/ | Software DOI: 10.5281/zenodo.19389522