Finite-budget Newton iteration for the matrix sign

Input a stored complex square matrix $A\in {\mathbb{C}}^{n\times n}$, $n\ge 1$, with the admissibility hypothesis $\sigma (A)\cap i\mathbb{R}=\mathrm{\varnothing }$, a finite iteration budget $N\ge 0$, and finite Householder-QR controls including a nonnegative integer candidate-operation budget $K_H$. The procedure does not itself certify the spectral hypothesis. Set $X_0=A$. For $k=0,\dots ,N-1$:

1. Call Finite Householder QR solve for a Padé denominator with explicit statuses on $X_{k}Y=I$, passing $K_H$ to its instrumented kernel dispatcher. If it does not return verified-solution, return its failure status with the last candidate.
2. Compute $X_{k+1}=\mathrm{fl}((X_k+Y)\mathrm{/}2)$ componentwise in the operationwise model of Operationwise real and complex floating point arithmetic with rounded square roots and fused multiply-add. A range failure returns range-failure with the last candidate.

After $N$ completed steps, return $X_N$ with status budget-exhausted and the number of completed steps. The exact Newton sequence is analyzed separately; the procedure makes no forward-error or floating-point convergence claim.