Theorem: Structural and predictive complete-output bridges

For the identified Hawk02-B endpoint,

where

The packet approximation ${\hat{W}}_N$ can be represented by a suitably fitted Hawk02-B endpoint. That fit is structural.

Separately, for the finite autonomous chain isometry $W^{[N]}(t)$ embedded with vacant unused modes,

The uncapped $S_N(T)$ bounds the isometry operator-norm difference, hence also component-vector differences. Either partial trace satisfies the channel bound.

For two exact finite chains of lengths $L>N$, the same uncapped $S_N(T)$ bounds their embedded isometry-norm difference.

If $M=N+1\ge 2e(2cT)$, then

Proof. The source and field branches and label channels are orthogonal, so

Apply Theorem thm:C1.

For a structural fit with $p_N>0$, choose

Set

When ${q_{j-1}}^{\mathrm{B}}>0$, take

Once a preceding weight is zero, all later weights are zero and subsequent angles may be zero. Fixed cell and source phases supply the desired phases. Zero-success endpoints use the modes chosen in Theorem thm:C4. The fitted endpoint is ${\hat{W}}_N$. It uses the calculated solution, so it is not a predictive approximation theorem.

For the chain comparison, decouple the link between nodes $N$ and $N+1$, retaining the compressed infinite tail. Call the resulting operator ${\hat{K}}_N$. Both norms are at most $2c$, and

A path cannot reach the removed link sooner. Expanding the exponentials gives

The common ${\omega }_0$ phase cancels. Orthogonal logical copies turn this scalar-vector bound into the code-isometry bound. Apply Theorem thm:C1.

The same power-agreement argument works for two finite chains $L>N$, each padded into a common space.

Finally, with $z=2cT$ and $M\ge 2ez$, successive tail ratios are at most $1/2$ and $M!\ge (M/e{)}^M$, giving