Adaptive Domain Models: Bayesian Evolution, Warm Rotation, and Principled Training for Geometric and Neuromorphic AI

We investigate whether independently trained language models converge to geometrically compatible latent representations, and whether this compatibility can be exploited to correct model behavior at inference time without any weight updates. We learn a linear projection matrix that maps activation vectors from a large teacher model into the coordinate system of a smaller student model, then intervene on the student's residual stream during generation by substituting its internal state with the translated teacher representation. Across a fully crossed experimental matrix of 20 heterogeneous teacher-student pairings spanning mixture-of-experts, dense, code-specialized, and synthetically trained architectures, the Ridge projection consistently achieves R^2 = 0.50 on verbal reasoning and R^2 = 0.40 on mathematical reasoning, collapsing to R^2 = -0.22 under permutation control and R^2 = 0.01 under L_1 regularization. Behavioral correction rates range from 14.0% to 50.0% on TruthfulQA (mean 25.2%) and from 8.5% to 43.3% on GSM8K arithmetic reasoning (mean 25.5%), demonstrating that the method generalizes across fundamentally different reasoning domains. We report a near-zero correlation between geometric alignment quality and behavioral correction rate (r = -0.07), revealing a dissociation between representation space fidelity and output space impact. Intervention strength is architecture-specific: student models exhibit characteristic sensitivity profiles that invert across domains, with the most steerable verbal student becoming the least steerable mathematical student. Finally, a double dissociation experiment conducted across all 20 model pairings confirms without exception that projection matrices collapse catastrophically when transferred across reasoning domains (mean R^2 = -3.83 in both transfer directions), establishing domain-specific subspace geometry as a universal property of LMs.

Instruction-tuned models are deployed into environments where domains are heterogeneous and evolve, yet adding new domains or data typically requires costly retraining. We present AutoAdapt, a modular framework that incorporates new domains and data via targeted single-adapter training without modifying other adapters. The framework automatically discovers latent domains, uses them to train per-domain Low-Rank Adaptation (LoRA) adapters independently in parallel and performs parameter-free routing. Across 14 domain-specific benchmarks and GPT-4o pairwise judgements, AutoAdapt achieves parity with a LoRA adapter trained on all domains without requiring full-model retraining. We also find evidence of specialisation effect convergence across independent discovery methods. Overall, training each adapter on its own domain prevents domain interference by construction, thus enabling modular, taxonomy-free domain specialisation without aggregate performance loss or full model retraining.

Adapting LLMs to new domains causes forgetting because standard methods (e.g., full fine-tuning, LoRA) inject new directions into the weight space. We show that forgetting is governed by one algebraic property: whether the update preserves the column span of the pretrained weight matrix (Proposition 1). We propose GAIN, the simplest multiplicative alternative (W_new = S * W), which satisfies this by construction and can be absorbed into existing weights for zero inference cost. Across five models (774M to 70B) adapted sequentially over eight domains, GAIN improves earlier-domain perplexity by 7-13%, while LoRA degrades it by 18-36%. GAIN matches replay-augmented LoRA without storing prior data and dominates EWC on the forgetting-adaptation Pareto front. While LoRA can only reduce forgetting by sacrificing in-domain adaptation, GAIN achieves both with no domain boundaries and no regularization. The principle generalises: (IA)^3, an independent multiplicative method, also improves earlier domains.