Sharp Schatten Certificate for Adapter Fusion

Integrate this at a parallel linear block: a transformer MLP with m residual experts, a collection of LoRA updates applied to one frozen weight matrix, or m independently trained low-rank adapters that will later be fused. Let A_i be the dense branch matrix or the materialized low-rank product B_i C_i; the merged update is Delta=sum_i A_i. Use p=2 for the first MVP because the positive matrices |A_i| can be formed exactly on a small model and the constant simplifies to C_{2,m}=sqrt((1+sqrt(m))/2). At each training step: (1) compute A_i; (2) form S_i=A_i^T A_i+epsilon I; (3) compute H_i=S_i^{1/2} by eigendecomposition; (4) form H=sum_i H_i; (5) add lambda*Tr(H^{p/2}) to the task loss; and (6) optionally rescale Delta by alpha=min(1,tau/(C_{p,m}||H||_p)), where tau is the allowed merged-update norm. For large matrices, estimate traces with Hutchinson probes and approximate square roots with Newton-Schulz iterations. The theorem supplies the certificate and C_{p,m}; epsilon, lambda, tau, probe count, and approximation accuracy are empirical choices. Start with a 6-layer transformer on a small language-modeling corpus, m=2 or 4 LoRA adapters, and compare this penalty against ordinary Frobenius penalties and naive m-times-sum budgets. Record validation perplexity, merged-update norm, gradient spikes, and the output discrepancy between fused and unfused adapters. Success means fewer instability spikes or lower perplexity at equal certificate budget, or safe adapter fusion with a smaller norm margin.