Sparse Fault Paths Preserve Correctable Code Deviation

Write $V_0=I$ and $V_r=V_{r-1}^{\otimes m}V$, with the corresponding typed tensor products for distinct port types. Put ${\mathcal{E}}_{1;s}={\mathcal{E}}_s$ and ${\mathcal{E}}_{r;s}={\mathcal{H}}_s({\mathcal{E}}_{r-1;b})$ for $r\ge 2$, using joint per-block tensor spans. Define $D_1$ to be the declared decoded recovery and $D_r$ by applying the child decoders and then $D_1$ to their logical outputs. Decoder syndrome registers remain attached to their original child edges in the comparison; no decoder is inserted into the faulty circuit.

First justify scalar recovery on the whole error span. In an orthonormal basis of each nonzero physical factor, the identity, diagonal sign flips, pairwise swaps, and pairwise Hermitian swaps with off-diagonal entries $i,-i$ are unitaries spanning its matrix algebra: subtracting the identity outside the selected pair gives the symmetric and skew matrix units, and diagonal sign flips give the diagonal units. Tensor these unitaries on at most $d$ coordinates, with identities elsewhere, and choose a basis from this finite spanning family. If its size is $M$, its elements divided by $\sqrt{M}$ are Kraus operators of a full-space channel corrected by $D_1$. For every recovery Kraus map $R_l$ and basis element $E_a$, [F4] gives $R_l{E}_{a}V=c_{la}I$. Linearity gives this for every error term. Inductively a factor in ${\mathcal{E}}_{r-1;b}$ becomes scalar identity on its decoded logical coordinate, with its syndrome coefficient retained, whereas an unrestricted factor becomes an arbitrary operator on that one coordinate. Expanding a tensor in ${\mathcal{E}}_{r;s}$ leaves at most $s\le d$ outer errors, corrected by $D_1$. Thus $D_r$ corrects the full span. Pairwise Kraus expansion gives the joint and memory-valued version, and full trace preservation makes the total recovered coefficient one. Zero-dimensional input spaces have no density instance.

Prove simultaneously, by level, the computation ${\mathcal{E}}_{r;b}\to {\mathcal{E}}_{r;2b}$ and correction ${\mathcal{E}}_{r;d}\to {\mathcal{E}}_{r;b}$ identities and arbitrary-input attraction for correction and designated refresh. Include every stronger zero-marked clause named by the finite hypothesis. At level one these are the stated one-level contracts and the additional attraction hypotheses. At an induction step all lower intermediate sectors are correctable by step 2.1. For an error basis on such an edge, trace preservation of recovery gives $V^{\ast }{E}_a^{\ast }{E}_{c}V=({\sum }_l\overline{c_{la}}{c}_{lc})I$. Diagonalize this finite positive Gram matrix, discard its zero directions, and normalize the others. The resulting encoded maps $F_{j}V$ satisfy $(F_{j}V{)}^{\ast }{F}_{k}V={\delta }_{jk}I$, so $J(\psi \otimes e_j)=F_{j}V\psi$ identifies that range with logical register times a private syndrome. The lower termwise sector identities give exactly the admitted logical/syndrome maps in the finite experiment.

A bad lower child remains an arbitrary CPTP map on its physical ports; its output is not assumed admitted. Apply the proved finite hypothesis to the lower-port maps from step 3.1, in its specified acyclic order. Its first-use refresh and support checks justify absorption only into that marked map's own ports and include every still-inadmissible output in the same residual support. It yields precisely ${\mathcal{H}}_{2b}({\mathcal{E}}_{r-1;b})$ for computation and ${\mathcal{H}}_b({\mathcal{E}}_{r-1;b})$ for correction, together with arbitrary-input attraction and the retained zero-marked clauses. These are the required level-$r$ assertions. Logical outcome ports and private syndromes keep their actual branch maps throughout.

Initially the encoded boundary has budget zero. At a fixed level, chronological induction now alternates the proved computation bound $2b\le d$ and correction bound $b$, starting from that boundary and the protected preparations. Composition retains complete positivity, full trace preservation on the admitted domain, and every public logical outcome. Step 2.1 decodes the resulting joint hierarchical residual instrument to the ideal logical instrument. Together with the independent level induction in steps 3.1–4.1, this proves the assertion at every level, with the external reference unchanged.