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Artificial systems increasingly operate in environments where task distributions, input statistics, and system conditions evolve over time. This creates two concurrent challenges: avoiding catastrophic forgetting when learning new tasks, and recovering from parameter corruption caused by noise, adversarial perturbation, or storage faults—a failure mode largely orthogonal to forgetting and unaddressed by existing continual learning methods. We present Progressive Self-Healing Neural Networks (PSHNN), a modular architecture that couples a progressive column network with an autonomous healing controller. PSHNN maintains a shared encoder, allocates dedicated task columns with lateral transfer connections, and stores an episodic memory bank per task. A threshold-triggered controller continuously monitors per-task accuracy and Population Stability Index (PSI) drift; upon detecting degradation, it reinitialises the affected column and retrains it via replay—without disturbing any other column. We evaluate PSHNN on Split-MNIST (2 tasks), Split-CIFAR-10 (5 tasks), and Split-CIFAR-100 (20 tasks). Healing recovers mean accuracy from 0.5576 to 0.9353 on Split-MNIST (+0.378), from 0.6653 to 0.7740 on Split-CIFAR-10 (+0.109), and from 0.4807 to 0.5799 on Split-CIFAR-100 (+0.099)—in every case meeting or exceeding the pre-corruption baseline. A novel positive transfer effect is identified: replay-driven encoder updates during healing improve accuracy on unaffected tasks by a mean of +0.042 on CIFAR-10 and +0.073 on CIFAR-100. These results establish PSHNN as an effective framework for fault-tolerant continual learning.