Markovian PAGE-Halpern Equilibrium Solver

(1) Integration point: build a small deep-equilibrium or recurrent block $T_\theta(x;z)$ whose input is the current hidden state $x$ and the current Markov-correlated sample $z_t$, such as an RL transition, streaming sensor frame, or sequential token chunk. Apply spectral normalization to every linear operator in the state path and use residual scaling so an empirical estimate of the induced Lipschitz constant is near or below one. Run the solver during training or inference to obtain the equilibrium state; initially use truncated or implicit differentiation rather than backpropagating through every solver iteration. (2) Pseudocode: initialize $x=x_0$, $x_{\mathrm{old}}=x$, and $v=0$. For $n=1,\ldots,N$, draw the next contiguous Markov block. With probability $p$, compute $v=B^{-1}\sum_i[x-T_\theta(x;z_i)]$. Otherwise compute $c=b^{-1}\sum_i( [x-T_\theta(x;z_i)]-[x_{\mathrm{old}}-T_\theta(x_{\mathrm{old}};z_i)] )$ and set $v\leftarrow v+c$. Then set $x_{\mathrm{old}}\leftarrow x$ and update $x\leftarrow (n+1)^{-1}x_0+n(n+1)^{-1}(x-\eta v)$. (3) Compute the Halpern coefficients, refresh schedule, and PAGE correction exactly. Estimate non-expansiveness using random pairs through $\|T(x)-T(y)\|/\|x-y\|$, and estimate Markov dependence from residual autocorrelations at several lags. Tune $B$, $b$, and $p$ so refreshes are infrequent while correction variance stays controlled. (4) First cheap experiment: use a 2-4 layer spectral-normalized DEQ on sequential MNIST or a small RL policy-evaluation task driven by an AR(1) or replay-free trajectory stream. Compare ordinary stochastic fixed-point iteration, direct Halpern minibatching, and PAGE-Halpern at equal environment samples and equal wall-clock FLOPs. Measure final fixed-point residual, residual variance, full refresh count, and downstream validation accuracy or return. Success means lower residual at equal samples, or roughly two times fewer expensive refresh/oracle evaluations at the same residual and downstream score.