Exact analytical allowances at the reported parameters

Here $z=2$, and Lemma lem:geometric gives

The packet certificate is

and the survival certificate is $q(1)\le 4/{\pi }^2$.

The script reports floating evaluations:

These are analytical formula evaluations, not residuals obtained from continuum evolution. In particular, $G_{24}$ bounds exact truncation error, not the total error of a floating-point eigendecomposition.

Without relying on numerical evolution, Theorem thm:C8 supplies, for exact preparation and decoding,

This is conservative but is a continuum theorem-level bound independent of floating-point execution.

The exact continuum survival is $J_1(2{)}^2$. For an arithmetic-independent evaluation route, define

The alternating-series estimate gives, for $m\ge 1$,

Both bounds are positive, so their squares enclose $q(1)$. Rational arithmetic can therefore certify survival digits without treating a proxy's printed digits as exact.