Confidence-Calibrated Contractive Fixed-Point Block

1. Integration point: use a small recurrent or equilibrium state update, not the whole transformer. Given hidden state $u_k\in\mathbb{R}^d$, let $F_\theta(u;\xi)$ be the block output on stochastic sample $\xi$, and update only the state with $u_{k+1}=(1-\widehat\beta_k)u_k+\widehat\beta_kF_\theta(u_k;\xi_{k+1})$. Apply the same construction to a target-network Polyak update if an equilibrium block is unavailable. 2. Enforce contraction: choose $\|\cdot\|_c=\|\cdot\|_\infty$ initially; spectral-normalize every linear map in $F_\theta$, bound activation Lipschitz constants, and multiply the residual branch by $\rho/L$ so the estimated global Lipschitz constant is at most $\rho=0.9$. 3. Estimate noise: maintain two independent minibatch evaluations $F_\theta(u;\xi)$ and $F_\theta(u;\xi')$, set $\widehat s_k=\|F_\theta(u;\xi)-F_\theta(u;\xi')\|_\infty/\sqrt{2}$, and fit robustly over a sliding window $\widehat s_k\leq\widehat a+\widehat b\|u_k-\operatorname{EMA}(u_k)\|_\infty$ using quantile regression. Clip $\widehat a,\widehat b$ at the 95th percentile. 4. Set $\widehat\beta_k=\beta_0\log(1/\delta)/(k+h)$, with $h$ chosen so the initial rate is below $0.1$, and clip it to $[10^{-4},0.2]$. Log maximum state norm, divergence frequency, and fixed-point residual. 5. First experiment: compare a four-iteration equilibrium MLP with and without contraction control and confidence calibration on CIFAR-10 or a small language-model recurrent cell, using identical FLOPs. Test $\delta\in\{0.5,0.1,0.01\}$. Success means fewer exploding-state runs, lower worst-window loss, and equal or better validation accuracy; a secondary signal is smaller variance of the final fixed-point residual at the same iteration count.