Coarsening-Aware Global-Consensus Scheduler

1. Integration point: partition a small MLP, CNN, or transformer into blocks such as layers, attention heads, convolutional channels, or parameter shards. Every $K$ optimizer steps, collect the recent update vector $\Delta\theta_b$ for each block and normalize it to $u_b$. Define the block adjacency graph $E_b$ using neighboring layers, adjacent channels, or attention heads in the same layer. 2. Pseudocode: initialize learning rate $\eta$ and target global coherence $C_0$. At each diagnostic interval, compute pairwise cosine similarities on adjacent update vectors to obtain $Q_t$; compute the covariance $G_t$ of block-level directions and its leading-eigenvalue ratio $C_t$ using a low-rank eigensolver; if $Q_t>q_0$ and $C_t<C_0$, set $\eta\leftarrow\eta\exp[-\kappa(1-Q_t)(C_0-C_t)]$ and optionally increase the number of steps before the next decay; otherwise apply the baseline cosine or exponential schedule. 3. Computed versus estimated: $Q_t$, $C_t$, and $G_t$ are empirical estimates, while the paper supplies the mechanism and the prediction that global ordering requires a coarsening time scaling as a power of system size, with exponent at least approximately two in its measured regime. Estimate an effective exponent $\hat z$ by repeating runs with different block counts $B$ and fitting the time to reach $C_0$ as $T_{\mathrm{coal}}(B)\propto B^{\hat z}$. 4. First cheap experiment: train a 3-layer MLP and a 6-layer small transformer on MNIST or CIFAR-10 using SGD or Adam, comparing the baseline schedule against this controller at equal update count. Log training loss, validation accuracy, $Q_t$, $C_t$, and gradient/update disagreement. The falsifiable signature is a plateau with high $Q_t$ but low $C_t$ under the baseline, followed by a measurable increase in $C_t$ after the controller slows learning. Across block counts, the predicted consensus time should follow a power law with $\hat z$ near or above $2$; if no such scaling or delayed-coherence regime appears, reject the transfer.