Gate-Zero Growth: A Geometric Framework for Function-Preserving Continual Learning

In dynamic environments, large language models need to keep adapting to new tasks, but continual learning often suffers from forgetting, limited transfer, and vulnerability to adversarial perturbations. To address this, we present AdvCL, which repurposes adversarial perturbations as a geometric control signal for stable continual adaptation. AdvCL combines three plug-in modules: Intra-Smooth promotes local smoothness via small adversarial perturbations; Proto-Clip uses similarity clipping to prevent excessive alignment to current task prototype; and Inter-Align applies directional alignment toward previous task prototype to reduce representational gaps. Experiments show consistent gains in both standard performance and robustness, with lower forgetting and stronger transfer. We further analyze key mechanisms by quantifying the sensitivity of Intra-Smooth to perturbation settings and the effect of Inter-Align on task similarity and geometric distance. In summary, the modules provide complementary gains when combined, and each can also be integrated individually into diverse CL paradigms, including replay, regularization, and dynamic architectures, thereby offering a geometric control mechanism for continual learning.

Continual learning commonly relies on post-hoc mechanisms such as replay, elastic regularization, or distillation. This work argues that forgetting should instead be modeled directly as interference between tasks. In the frozen-feature regime, forgetting from learning a new task is exactly the interference energy induced on the old task. In deep networks, the same quantity is recovered through path-averaged curvature with minimal additional forward passes. When task supports are disjoint, forgetting can be eliminated structurally and when task supports overlap in conflicting directions, a non-zero distortion floor is unavoidable. The same geometry optimally merges models through task-aware orthogonalization. From this analysis we derive Interference-Gated Functional Allocation (IGFA), a replay-free, Fisher-free method that shares directions when tasks align and protects them when they conflict. Across benchmarks, IGFA achieves lossless retention when tasks are structurally separable and moves unavoidable cost from irreversible forgetting into deferred but recoverable plasticity when they are not. It matches the strongest replay-free structural baselines on dissimilar-task streams and improves on unconditional projection when similarity makes transfer worth preserving.

Catastrophic forgetting is often viewed as the destruction of previously learned knowledge during sequential learning. Building on the Accessibility Collapse framework, we investigate the geometric structure of recoverability in continual learning. Using Split CIFAR-100 and a sequentially trained ResNet-18, we analyze recoverability, representational drift, and recovery complexity across ten tasks. We introduce Recovery Subspace Dimensionality (k_t), a measure of the minimum number of singular directions required to preserve 90 percent of full probe performance. Contrary to our Recoverability Diffusion hypothesis, recovery dimensionality remains stable throughout training (mean k_t = 8.0) despite substantial representational drift. Principal-angle drift strongly predicts recoverability (r = -0.862), and a simple geometric model explains 82.2 percent of recoverability variance. These findings support the Stable Recovery Manifold hypothesis, suggesting that forgotten knowledge remains compactly decodable despite representational reorganization. The results indicate that catastrophic forgetting is primarily an accessibility and manifold-alignment problem rather than information destruction.