Cubic-regularized Newton methods have the optimal $\mathcal{O}(\varepsilon^{-3/2})$ global rate and an automatic saddle-escape mechanism, but their subproblem is most often solved by a full eigendecomposition, limiting feasible model size. We introduce a blockwise optimizer that partitions parameters by tensor, minimizes an independent cubic model with an adaptive cubic constant $M_b$ per block, and accepts or rejects each block step against a monotone guard on the full loss. The subproblem solver is chosen by block size: small blocks use lazy exact cubic steps from explicitly formed per-block Hessians; arbitrarily large tensors use a matrix-free Chebyshev-bounded Krylov subspace built by the Lanczos process. The cubic shift bounds the required polynomial degree whenever the gradient-driven shift dominates negative curvature, renders the shifted operator positive semidefinite before any polynomial is applied, and preserves the $\mathcal{O}(\varepsilon^{-3/2})$ rate under inexact subproblem solves. We prove these claims, and the blockwise scheme carries a monotone per-block descent guarantee. Experiments cover FINER INRs (about 199k parameters) and a 91.4M-parameter ViSIR INR, where the blockwise cubic step remains exact in the cubic-model sense on every block, including the 88.5M-parameter decoder tensor (97\% of the model). Run to full convergence on FINER, the ARC-$\varphi_1$ optimizer reaches 133.5 dB PSNR while tuned Adam plateaus at 78.2 dB at the same extended budget; in the roughly 70 minutes Adam takes to reach its peak, ARC-$\varphi_1$ reaches 95.6 dB. A companion report isolates the loss-landscape features responsible for Adam's behavior.

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