Double-Bracket Projector Refinement

Integrate this at the attention-weight or MoE-router level for a small matrix state, rather than replacing the whole Transformer optimizer. For each token batch, construct a symmetric cost matrix $H$ from existing logits. For attention, let $S=QK^{\mathsf T}/\sqrt d$, symmetrize it as $H=-(S+S^{\mathsf T})/2$, and optionally add a diagonal penalty for undesirable keys. For routing, use a symmetric expert-affinity or load-balancing cost matrix. Maintain a rank-$r$ projector $\Psi$ per head or routing group. Pseudocode: initialize $\Psi_0=U_0U_0^{\mathsf T}$; for $k=0,\ldots,K-1$, compute $C_k=\Psi_kH-H\Psi_k$, compute $D_k=C_k\Psi_k-\Psi_kC_k=[[\Psi_k,H],\Psi_k]$, set $\widetilde\Psi=\Psi_k+\eta D_k$, eigendecompose $\widetilde\Psi=V\operatorname{diag}(\lambda)V^{\mathsf T}$, and set $\Psi_{k+1}=V_{[:,1:r]}V_{[:,1:r]}^{\mathsf T}$. Convert the final projector to nonnegative attention weights with $a_i\propto\operatorname{softplus}((\Psi z)_i)$ followed by normalization, or use $\Psi$ directly as a low-rank mixing operator. The paper supplies the commutator flow; the engineer estimates $H$ from logits and chooses $\eta$, $K$, and rank $r$ empirically. First test on a 2-layer Transformer trained on CIFAR-10 or WikiText-2, comparing standard softmax attention against $K=1,2,4$ projector refinements at equal forward FLOPs. Measure validation loss, attention entropy, gradient-norm variance, projector constraint violation, and whether the constrained version avoids attention collapse or improves validation accuracy at matched parameter count.