Exact Scalar Threshold and Extended Precision Bounds

We study the fixed CG recurrence and adversarial scalar relative-error
envelope of the problem. The scalar threshold is determined exactly by a
degree-five rational inequality. In general dimension, set
$H=\min\{K,64/\varepsilon^2\}$ and $D=1024H^2$, and let $m$ be the minimum of
the dimension, an explicit Chebyshev horizon of order
$\sqrt K\log(K/\varepsilon)$, and a backward-error horizon of order
$\varepsilon^{-2/3}$. We prove the sufficient precision
$\lceil\log_2(1024(n+1)KD^{12m+20}/\varepsilon^2)\rceil$. The cap in $H$
follows from the fact that an excessively long step would already have
small backward error. The early-horizon arguments transfer polynomial
approximation to the actual rounded recurrence, rather than assuming that
computed iterates are exact Krylov projections. A prefix-wise matvec
estimate and a new breakdown family also give
$P_{\mathrm{CG}}(n,K,\varepsilon)=\log_2(nK)+O_\varepsilon(1)$ for each fixed
tolerance, uniformly in $n\ge2$ and $K\ge1$. The additive constants are
universal when $3/8\le\varepsilon<1/2$. There is also a sharp order
uniformly for $1\le K\le2$.