Open questions and potential solutions

Direct codeword calculations give $V^{†}{J}_{z}V=0$ and $V^{†}{J}_z^{2}V=vI$, with $v=10$ or $8$. Explicit downward coefficients give $V^{†}{L}_{\delta }^{†}{L}_{r}V=\gamma a_{\delta }{\delta }_{\delta r}I$, where the weights are $(1\mathrm{/}3,1\mathrm{/}3,1\mathrm{/}3)$ or $(5\mathrm{/}14,2\mathrm{/}7,5\mathrm{/}14)$ in order $-1,0,+1$. These identities construct mutually orthogonal syndrome isometries and a trace-preserving decoder. Its action gives $\mathcal{R}\mathcal{V}=\mathrm{i}\mathrm{d}$ and $\mathcal{R}{\mathcal{L}}_{{\mathrm{\Delta }}_0}\mathcal{V}=0$ on every logical matrix. Block diagonality removes the rotor term exactly. The integral Taylor remainder and channel diamond norm one give a one-cycle error at most ${\tau }^2\mathrm{\parallel }{\tilde{\mathcal{L}}}_{{\mathrm{\Delta }}_0}{\mathrm{\parallel }}_{\diamond }^2\mathrm{/}2$; telescoping gives the stated accumulated bound. The numbered argument supplies the coefficient calculations, complete channel and uniform constant.

Direct construction and contraction estimate; proof_strategy: direct. Given: the explicit finite code and stipulated GKSL matrices, with ideal syndrome operations and reencoding. Goal: construct a decoder independent of the lift, prove cancellation of the complete first derivative, and bound the exact repeated channel. The article supplies the question, code data and physical model attribution; its proposed correction or convergence claims are not used as proofs.

1. 1.1 Let $V:{\mathbb{C}}^2\to \mathcal{H}$ encode the selected displayed codewords and put $P=V{V}^{†}$. Their disjoint supports and squared amplitudes show that $V$ is an isometry. For the Æ code the first moments are $(-10+10)\mathrm{/}7=0$ and the second moments are $(20+50)\mathrm{/}7=10$. For the signed spin-four code they are $(-4+4)\mathrm{/}3=0$ and $(16+8)\mathrm{/}3=8$. Off-diagonal matrix elements of both diagonal operators vanish because the supports are disjoint. Therefore $V^{†}{J}_{z}V=0$ and $V^{†}{J}_z^{2}V=v{I}_2$, with $v=10$ or $8$. Consequently $B=J_{z}V\mathrm{/}\sqrt{v}$ is an isometry orthogonal to $V$. [F2, F3, F4, algebra] [F2, F3, F4]
2. 1.2 In compatible real coupling phases, write $L_{\delta }\mathrm{\mid }J,m⟩=\sqrt{\gamma }{b}_{\delta }(J,m)\mathrm{\mid }J-1,m+\delta ⟩$, with $b_0=-\sqrt{(J^2-m^2)\mathrm{/}(J(2J-1))}$ and $b_{\pm 1}=\sqrt{(J\mp m)(J\mp m-1)\mathrm{/}(2J(2J-1))}$; out-of-range outputs are zero. These coefficient formulas can be checked without importing the article's rate identity. Define $u_M=\sqrt{(2J-1)\mathrm{/}(2J+1)}{\sum }_{\delta =-1}^1{b}_{\delta }(J,M-\delta )\mathrm{\mid }1,\delta ⟩\otimes \mathrm{\mid }J,M-\delta ⟩$. Substitution gives norm one and total magnetic weight $M$, and the explicit ladder matrices give $J_{+}^{\mathrm{t}\mathrm{o}\mathrm{t}}{u}_M=\sqrt{(J-1-M)(J+M)}{u}_{M+1}$, including zero at $M=J-1$. Thus these are normalized spin-$(J-1)$ coupled vectors in the stated ordered convention. A different allowed common coupled phase does not change the following products. Directly summing the three displayed squared coefficients at fixed $m$ gives one. Since each fixed-$\delta$ map sends different input basis vectors to different output basis vectors, ${\sum }_{\delta }{L}_{\delta }^{†}{L}_{\delta }=\gamma I$ on every excited block. [F1, F4, algebra] [F1, F4]
3. 1.3 For $\delta =r$, the squared coefficients in step 1.2 are quadratic polynomials in $m$. The moments in step 1.1 give $V^{†}{L}_{\delta }^{†}{L}_{\delta }V=\gamma a_{\delta }{I}_2$, with $a=(1\mathrm{/}3,1\mathrm{/}3,1\mathrm{/}3)$ for $J_0=5$ and $a=(5\mathrm{/}14,2\mathrm{/}7,5\mathrm{/}14)$ for $J_0=4$, ordered $-1,0,+1$. Cross-logical diagonal products vanish by disjoint support. For $\delta \mathrm{\ne }r$, any overlap requires two occupied magnetic labels to differ by $\mathrm{\mid }\delta -r\mathrm{\mid }\le 2$. The Æ support gaps are at least three. For the spin-four code only the two-label differences between $-4,-2$ and between $2,4$ can contribute. In the potentially nonzero cross-logical product $⟨L_{+1}\bar{0},L_{-1}\bar{1}⟩\mathrm{/}\gamma$, the output labels $-3$ and $3$ contribute respectively $1\mathrm{/}\sqrt{126}$ and $-1\mathrm{/}\sqrt{126}$; they cancel. Its adjoint cancels as well, and all remaining mixed products have disjoint output supports. Hence $V^{†}{L}_{\delta }^{†}{L}_{r}V=\gamma a_{\delta }{\delta }_{\delta r}{I}_2$, with all $a_{\delta }>0$ and ${\sum }_{\delta }{a}_{\delta }=1$. [step 1.1, step 1.2, algebra] [step:1, step:2]
4. 2.1 Define $A_{\delta }=L_{\delta }V\mathrm{/}\sqrt{\gamma a_{\delta }}$. Step 1.3 makes the three $A_{\delta }$ orthogonal isometries. Their ranges lie in $V_{J_0-1}$, whereas $V$ and $B$ lie in $V_{J_0}$, so all five syndrome ranges are mutually orthogonal. Thus $Q=I-P-B{B}^{†}-{\sum }_{\delta }{A}_{\delta }{A}_{\delta }^{†}$ is an orthogonal projector. Fix ${\sigma }_{\ast }=\mathrm{\mid }0⟩⟨0\mathrm{\mid }$ on the logical qubit and set $\mathcal{R}(Y)=V^{†}YV+B^{†}YB+{\sum }_{\delta }{A}_{\delta }^{†}Y{A}_{\delta }+\mathrm{tr}(QY){\sigma }_{\ast }$. This is a complete channel: the first five terms have their displayed Kraus operators, and an orthonormal basis $(q_j)$ of $\mathrm{im}Q$ supplies completion Kraus operators $\mathrm{\mid }0⟩⟨q_j\mathrm{\mid }$. The adjoint-square sum is $I$. Each amplified Kraus term preserves positivity by conjugation, and the adjoint-square sum proves trace preservation for every input matrix. Operationally this refines the complementary-manifold outcome by a single-decay recovery measurement and completes every remaining outcome. [F5, step 1.1, step 1.3, construct, algebra] [F5, step:1, step:3]
5. 2.2 Orthogonality in step 2.1 gives $\mathcal{R}\mathcal{V}=\mathrm{i}\mathrm{d}$ and annihilates cross operators between distinct syndrome ranges. On the host block, $HV=h_{0}V+{\mathrm{\Delta }}_0{J}_0\sqrt{v}B$, where $h_0={\omega }_0{J}_0(J_0+1)$. It follows that $\mathcal{R}(-i[H,VX{V}^{†}])=0$ for every logical matrix $X$. Also $\mathcal{R}({\sum }_{\delta }{L}_{\delta }VX{V}^{†}{L}_{\delta }^{†})=\gamma {\sum }_{\delta }{a}_{\delta }X=\gamma X$. Step 1.2 makes the decoded anticommutator term $-\gamma X$. The jump gain and loss therefore cancel, proving $\mathcal{R}{\mathcal{L}}_{{\mathrm{\Delta }}_0}\mathcal{V}=0$. These are linear-map identities on all matrices and hence remain true after tensoring with every identity ancilla. [F1, step 1.1, step 1.2, step 1.3, step 2.1, algebra] [F1, step:1, step:2, step:3, step:4]
6. 2.3 Let $\mathcal{B}={⨁}_{J=0}^{J_0}\mathcal{L}(V_J)$ denote the block-diagonal operator subspace. Encoding takes values in $\mathcal{B}$; diagonal Hamiltonians, the anticommutator terms and all jump gains preserve it. The rotor commutator vanishes on $\mathcal{B}$ because its Hamiltonian is scalar on each block. Consequently ${\mathcal{L}}_{{\mathrm{\Delta }}_0}$ and ${\tilde{\mathcal{L}}}_{{\mathrm{\Delta }}_0}$ agree on this invariant subspace, so their exponential series agree on encoded inputs for every nonnegative time. In particular $S_{\tau }=\mathcal{R}{e}^{\tau {\tilde{\mathcal{L}}}_{{\mathrm{\Delta }}_0}}\mathcal{V}$ and $\mathcal{R}{\tilde{\mathcal{L}}}_{{\mathrm{\Delta }}_0}\mathcal{V}=0$. This removes the rotor exactly rather than requiring a small rotor phase. [F1, step 2.1, step 2.2, algebra] [F1, step:4, step:5]
7. 3.1 Write $G={\tilde{\mathcal{L}}}_{{\mathrm{\Delta }}_0}$ and $M=\mathrm{\parallel }G{\mathrm{\parallel }}_{\diamond }$. The finite GKSL theorem makes $e^{sG}$ a channel for every $s\ge 0$, so its diamond norm is one. Differentiating the finite-dimensional exponential and integrating twice gives the exact identity $e^{\tau G}-I-\tau G={\int }_0^{\tau }(\tau -s)e^{sG}{G}^{2}ds$. Triangle inequality and composition submultiplicativity yield $\mathrm{\parallel }{e}^{\tau G}-I-\tau G{\mathrm{\parallel }}_{\diamond }\le {\tau }^2{M}^2\mathrm{/}2$. Composing with the channels $\mathcal{R}$ and $\mathcal{V}$, and using step 2.2, gives $\mathrm{\parallel }{S}_{\tau }-\mathrm{i}\mathrm{d}{\mathrm{\parallel }}_{\diamond }\le {\tau }^2{M}^2\mathrm{/}2$. This contraction argument improves the exponential-series majorant in the seed. [F6, F7, F8, step 2.2, step 2.3, algebra] [F6, F7, F8, step:5, step:6]
8. 3.2 Each $S_{\tau }$ is a channel. The exact telescoping identity $S_{\tau }^N-\mathrm{i}\mathrm{d}={\sum }_{k=0}^{N-1}{S}_{\tau }^k(S_{\tau }-\mathrm{i}\mathrm{d})$ and channel diamond norm one give $\mathrm{\parallel }{S}_{\tau }^N-\mathrm{i}\mathrm{d}{\mathrm{\parallel }}_{\diamond }\le N{\tau }^2{M}^2\mathrm{/}2=T\tau M^2\mathrm{/}2$. The distance is also at most two by triangle inequality for two channels. Because this uses the exact exponential and the complete decoder, no trajectory order or measurement outcome was discarded. [F8, step 2.1, step 3.1, algebra] [F8, step:4, step:7]
9. 3.3 For completeness, a singular-value decomposition $X={\sum }_j{s}_j{u}_j{v}_j^{†}$ gives $\mathrm{\parallel }AXB{\mathrm{\parallel }}_1\le {\sum }_j{s}_j\mathrm{\parallel }A{u}_j\mathrm{\parallel }\mathrm{\parallel }{B}^{†}{v}_j\mathrm{\parallel }\le \mathrm{\parallel }A{\mathrm{\parallel }}_{\mathrm{\infty }}\mathrm{\parallel }B{\mathrm{\parallel }}_{\mathrm{\infty }}\mathrm{\parallel }X{\mathrm{\parallel }}_1$, using the directly computed rank-one trace norm. The same inequality holds on every ancilla, and tensoring a fixed matrix with an identity preserves its operator norm. Step 1.2 gives $\mathrm{\parallel }{L}_{\delta }{\mathrm{\parallel }}_{\mathrm{\infty }}\le \sqrt{\gamma }$; therefore each of the three jump dissipators has diamond norm at most $2\gamma$. The rotor-free Hamiltonian has operator norm at most $D{J}_0^2$, so its commutator has diamond norm at most $2D{J}_0^2$. Hence $M\le 6\gamma +2D{J}_0^2$ uniformly for $\mathrm{\mid }{\mathrm{\Delta }}_0\mathrm{\mid }\le D$. Substitution in step 3.2 proves the announced uniform estimate and fixed-$T$ convergence. The finite matrix argument uses no infinite choice construction or microscopic bath theorem. [F9, step 1.2, step 2.3, step 3.2, algebra] ∎ [F9, step:2, step:6, step:8]

• F1: Physical model hypothesis: on the finite orthonormal basis |J,m>, use the stipulated equal-rate downward jump matrices and H=ω0 J(J+1)+Δ0 Jm, with γ>0 and fixed |Δ0|≤D. The sector J≤J0 is invariant because all jumps lower J and H is diagonal. No microscopic validity after lifting is inferred. Source: post-qec-distinguishability-general-lift-model; Postulate and Facts & Assumptions; specialize the diagonal lift and restrict to the downward reachable sector.. Reading limits: Read the complete frozen item and original PDF pp. 14–16 text and images. Only the matrix-model postulate is used; inherited microscopic derivations are not premises of this finite calculation.
• F2: Æ code data: J0=5, with logical vectors sqrt(5/7)|5,-2>+sqrt(2/7)|5,5> and sqrt(2/7)|5,-5>+sqrt(5/7)|5,2>. Only the displayed vectors are imported; their correction properties are computed locally. Source: def-qec-distinguishability-ae-code; Definition, first paragraph.. Reading limits: Read the complete frozen item and checked the codewords against original PDF p. 6 image. The article's correction claim is not used as proof.
• F3: Signed Aydin–Barg code data: J0=4, with logical vectors sqrt(1/3)|4,-4>+sqrt(2/3)|4,2> and sqrt(2/3)|4,-2>-sqrt(1/3)|4,4>. The relative minus sign is essential. Source: def-qec-distinguishability-aydin-barg-code; Definition, first paragraph.. Reading limits: Read the complete frozen item and checked the vectors and sign against original PDF p. 6 image. Its general tensor-distance and correction conclusions are not invoked.
• F4: Finite spin convention: VJ has orthonormal magnetic basis, Jz|J,m$\ge$m|J,m>, and J±|J,m$\ge$sqrt((J∓m)(J±m+1))|J,m±1>. Ordered Clebsch–Gordan coefficients are coordinates of normalized coupled ladder vectors, with a common phase for each irreducible summand. Source: def-qec-prereq-spin-tensor-cg-conventions; Definition, first three paragraphs.. Reading limits: Read the complete frozen definition and justification. The downward coefficient formulas used here are explicitly checked in step 1.2 rather than taken from an unread table.
• F5: Operational assumption: instantaneous exact projective syndrome extraction, the conditional inverse isometries constructed in step 2.1, preparation on all remaining outcomes, and exact reencoding are available. Between checks the same stipulated Markov channel applies. All outcomes are retained in the unconditional channel. Source: self-contained; Protocol hypothesis and explicit decoder in step 2.1.. Reading limits: This is an expressly assumed ideal operation model, not a sourced experimental capability. The channel construction is checked locally; physical implementation is not established.
• F6: For finite-dimensional matrices, a Hermitian Hamiltonian and a finite jump family define a GKSL generator whose exponential is completely positive and trace preserving for every nonnegative time. Source: physics:thm-qsm-dt-gksl-sufficient; Statement and Proof, steps 1.1–4.1.. Reading limits: Read the complete serving item. Its actual metadata declares domain: mathematics despite its physics storage namespace. Its finite-dimensional hypotheses apply to both generators here; its cited external texts were not freshly read.
• F7: A bounded operator on a real or complex Banach space has an operator-norm convergent exponential, differentiable with derivative Ae^(tA)=e^(tA)A and with the exponential composition law. Finite matrix and superoperator spaces satisfy these hypotheses. Source: math:lem-exponential-series-of-a-bounded-operator; Statement; Proof, steps 2.1–2.3 and 3.3.. Reading limits: Read the complete serving item. The integral remainder is derived here from its differentiation statement; no reading of the item's cited monographs is claimed.
• F8: On finite complex Euclidean spaces, the diamond norm is the induced trace norm after adjoining an identity on the input space. Channels have diamond norm one, and compatible compositions satisfy ||ΨΦ||diamond≤||Ψ||diamond||Φ||diamond. Source: https://cs.uwaterloo.ca/~watrous/TQI/TQI.pdf; §3.3.2, Definition 3.43, Proposition 3.44(2), and Proposition 3.48(1); printed pp. 170–171 and 174.. Reading limits: Freshly read the extracted definition, channel-norm statement, composition inequality and its proof passage. No reading of the entire book or its earlier norm theory is claimed.
• F9: Finite-matrix norm conventions: the trace norm sums singular values, the operator norm is the largest singular value, and the diamond norm obeys triangle inequality and homogeneity. The sandwich estimate and its ancilla version used for the explicit constant follow from a singular-value decomposition as shown in step 3.3. Source: self-contained; Elementary local norm calculation in step 3.3.. Reading limits: The required sandwich estimate is supplied locally. No additional literature theorem or numerical computation is used as its proof.

Decision: accept

Checked all nine numbered steps. The code moments and jump-pair compressions give mutually orthogonal syndrome isometries and a complete trace-preserving decoder independent of the unknown lift. Its decoded generator vanishes on every logical matrix. Block invariance removes the rotor term exactly; the integral remainder and channel contraction yield the claimed bound for every positive integer N. This is a sound conditional partial answer for the specified completion. Retaining proposed status and the broader physical qualifications is conservative.

Dependency review: F1 is used as a stipulated finite matrix noise model, not as a deduction from microscopic physics. F2 and F3 supply explicit codeword data; their correction claims are recomputed rather than assumed. F4 supplies the spin and coupling conventions. Exact radical-arithmetic checks verified the coupled-vector normalization, raising recurrence and rate identity for $J=1$ through 5, and every logical jump-pair matrix entry for both codes, including the signed spin-four cancellation. F5 explicitly assumes the ideal operations; the displayed Kraus completion independently establishes their channel mathematics. F6 has actual domain mathematics despite its physics namespace, and its finite-dimensional hypotheses apply. F7 supplies the applicable exponential differentiability. F8's channel norm and composition properties were checked against Watrous. F9's sandwich estimate and its amplified version follow from the displayed finite-matrix calculation. The frozen declared dependency closure inspected contained no cycle or open-problem premise. The open-problem item supplies context only. No substantive circular use of the article's correction or convergence claims occurs in this argument.

Review reading limits: Read the complete assigned direction and the individually listed frozen item records. Inspected frozen PDF page images 6 and 14–16 and relevant extracted text, verifying the codewords, model and authors' unresolved-performance passage. Read both listed serving mathematical suppliers, without freshly checking their cited monographs or auditing their full dependency closures. Read the relevant Watrous definition, propositions and composition-proof passage, and Dominy et al.'s abstract, introductory scope and Theorem 1 assumptions. Opened the article's arXiv abstract page and performed two narrow status searches; these were inconclusive about subsequent work. No full-article review, simulation replay or physical implementation check was performed. Earlier assessors' broader reading inventories were not adopted.

Solve the code and decay blocks directly: surviving populations are $e^{-{\gamma }_i\tau }{\rho }_{ii}$, decayed populations are $(1-e^{-{\gamma }_i\tau }){\rho }_{ii}$, surviving coherence is $e^{-s\tau }{\rho }_{01}$, and decayed coherence is $g(1-e^{-s\tau }){\rho }_{01}\mathrm{/}s$. Recovery adds the corresponding blocks. Hence ${\lambda }_{\tau }=1-a\tau +O({\tau }^2)$, which gives the stated fixed-time limit. Complete positivity bounds the useful off-diagonal matrix element of any branch-restricted decoder by one; its useful diagonal elements are also at most one. Applying these bounds to the entanglement-fidelity expression proves the one-cycle optimum without importing the article's equal-rate optimality argument. The effective logical generator is $\mathcal{D}[D]$ with $D=\mathrm{diag}(\sqrt{{\gamma }_0},\sqrt{{\gamma }_1})$, so amplitude imbalance produces first-order dissipative logical dephasing.

Direct finite-dimensional block evolution, followed by exact decoding and a scalar product limit. A separate complete-positivity argument proves one-cycle optimality. The altered dynamics and checking protocol are explicit physical assumptions; all conclusions are conditional deductions. The article supplies question attribution and comparison only.

1. 1.1 Use the article only to identify the question and the recovery being varied. Prescribe the unequal-rate model in A1 independently of the article's claims. Define $W_0\mathrm{\mid }i⟩=\mathrm{\mid }i⟩$ and $W_1\mathrm{\mid }i⟩=\mathrm{\mid }i+2⟩$ for $i=0,1$. The logical decoder is $\mathcal{R}(Y)=W_0^{†}Y{W}_0+W_1^{†}Y{W}_1$. Because $W_0{W}_0^{†}+W_1{W}_1^{†}=I_4$, conjugation positivity and the finite trace identity show that this map is CPTP. Its physical code-supported version is exactly the stated ${\mathcal{R}}_4$. [C1, A1, A2, D1] [C1, A1, A2, D1]
2. 1.2 Start with encoded logical density matrix $\rho$. Let $x_i(t)$ and $y_i(t)$ be its surviving and decayed populations, and let $c(t)$ and $d(t)$ be the corresponding $01$ and $23$ coherences. Direct substitution in A1 gives $x_i^{\mathrm{\prime }}=-{\gamma }_i{x}_i$, $y_i^{\mathrm{\prime }}={\gamma }_i{x}_i$, $c^{\mathrm{\prime }}=-sc$ and $d^{\mathrm{\prime }}=gc$, with $x_i(0)={\rho }_{ii}$, $y_i(0)=d(0)=0$ and $c(0)={\rho }_{01}$. Cross-sector blocks remain zero. The common code energy creates no relative phase within either occupied block. Consequently $x_i=e^{-{\gamma }_{i}t}{\rho }_{ii}$, $y_i=(1-e^{-{\gamma }_{i}t}){\rho }_{ii}$, $c=e^{-st}{\rho }_{01}$ and $d=g(1-e^{-st}){\rho }_{01}\mathrm{/}s$. These equations are solved locally, rather than inferred by substituting unequal rates into an article formula. [A1, D1, F1] [A1, D1, F1]
3. 1.3 The encoded-input evolution has the explicit representation ${\mathcal{N}}_t(W_0\rho W_0^{†})=W_0{A}_t\rho A_t^{†}{W}_0^{†}+{\int }_0^t{W}_1{K}_u\rho K_u^{†}{W}_1^{†}du$, where $A_t=\mathrm{diag}(e^{-{\gamma }_{0}t\mathrm{/}2},e^{-{\gamma }_{1}t\mathrm{/}2})$ and $K_u=\mathrm{diag}(\sqrt{{\gamma }_0}{e}^{-{\gamma }_{0}u\mathrm{/}2},\sqrt{{\gamma }_1}{e}^{-{\gamma }_{1}u\mathrm{/}2})$. Every amplified output is positive for positive input, since its quadratic forms are sums and integrals of nonnegative quadratic forms. Also $A_t^{†}{A}_t+{\int }_0^t{K}_u^{†}{K}_{u}du=I_2$. Thus this explicit restricted evolution is CPTP, without requiring an imported existence theorem for the calculation. Decoding adds the two blocks from step 2: populations return exactly and coherence is multiplied by ${\lambda }_t=e^{-st}+(g\mathrm{/}s)(1-e^{-st})$. Since $0<g\le s$, one has $0<{\lambda }_t\le 1$. [A2, D1, F1, step 1.2] [A2, D1, F1, step:2]
4. 2.1 Under A3, composition multiplies the coherence factors, so $N$ cycles have factor ${\lambda }_{\tau }^N$. Expanding the four matrix units in the maximally entangled projector in D2 gives $F_e=(1+{\lambda }_{\tau }^N)\mathrm{/}2$. For a unit logical vector $\alpha \mathrm{\mid }0⟩+\beta \mathrm{\mid }1⟩$, the state fidelity is $1-2\mathrm{\mid }\alpha {\mathrm{\mid }}^2\mathrm{\mid }\beta {\mathrm{\mid }}^2(1-{\lambda }_{\tau }^N)$; hence its worst-case value is also $(1+{\lambda }_{\tau }^N)\mathrm{/}2$, attained at equal populations. [A3, D2, step 1.3] [A3, D2, step:3]
5. 2.2 At fixed rates, ${\lambda }_{\tau }=1-a\tau +as{\tau }^2\mathrm{/}2+O({\tau }^3)$, so $\log {\lambda }_{\tau }=-a\tau +ag{\tau }^2\mathrm{/}2+O({\tau }^3)$. Therefore, for $\tau =T\mathrm{/}N$, ${\lambda }_{\tau }^N=\exp [-aT+agT\tau \mathrm{/}2+O({\tau }^2)]\to e^{-aT}$, with the remainder understood at fixed $T$ and fixed rates. If the rates coincide then $a=0$ and every cycle is exactly the identity. If they differ and $T>0$, the limiting infidelity is $(1-e^{-aT})\mathrm{/}2>0$. [F1, step 1.3, step 2.1] [F1, step:3, step:4]
6. 3.1 Establish a decoder bound independent of the chosen recovery. For any CPTP logical decoder $\mathcal{Q}:M_4\to M_2$, its restrictions ${\mathcal{Q}}_j(X)=\mathcal{Q}(W_{j}X{W}_j^{†})$ are CPTP. Write $E_{ik}=\mathrm{\mid }i⟩⟨k\mathrm{\mid }$, $x_j=⟨0\mathrm{\mid }{\mathcal{Q}}_j(E_{00})\mathrm{\mid }0⟩$, $y_j=⟨1\mathrm{\mid }{\mathcal{Q}}_j(E_{11})\mathrm{\mid }1⟩$ and $z_j=⟨0\mathrm{\mid }{\mathcal{Q}}_j(E_{01})\mathrm{\mid }1⟩$. Positivity and trace preservation give $0\le x_j,y_j\le 1$. Apply ${\mathrm{i}\mathrm{d}}_2\otimes {\mathcal{Q}}_j$ to the positive operator formed from $\mathrm{\mid }0⟩\otimes \mathrm{\mid }0⟩+\mathrm{\mid }1⟩\otimes \mathrm{\mid }1⟩$. Its principal submatrix on the matching output vectors is $\left(\begin{array}{cc}{x}_j& z_j\\ {\bar{z}}_j& y_j\end{array}\right)\ge 0$, so $\mathrm{\mid }{z}_j{\mathrm{\mid }}^2\le x_j{y}_j\le 1$. This follows from the nonnegative quadratic form of that two-dimensional compression. [D1] [D1]
7. 3.2 Set $p_{0i}=e^{-{\gamma }_i\tau }$, $p_{1i}=1-e^{-{\gamma }_i\tau }$, $b_0=e^{-s\tau }$ and $b_1=g(1-e^{-s\tau })\mathrm{/}s$. Both $b_j$ are nonnegative. Expanding D2 using the block evolution gives $F_e(\mathcal{Q}{\mathcal{N}}_{\tau }\mathcal{V})=\frac{1}{4}{\sum }_{j=0}^1[p_{j0}{x}_j+p_{j1}{y}_j+2b_j\mathrm{Re}{z}_j]$, where $\mathcal{V}(X)=W_{0}X{W}_0^{†}$. Step 6 bounds this by $\frac{1}{4}{\sum }_j(p_{j0}+p_{j1}+2b_j)=(1+{\lambda }_{\tau })\mathrm{/}2$. The decoder in step 1 restricts to the identity on each identified sector, so $x_j=y_j=z_j=1$ and it attains the bound. Thus the stated one-cycle optimum holds over all deterministic system-only CPTP decoders, not only sector-preserving ones. [D2, step 1.1, step 1.2, step 1.3, step 3.1] [D2, step:1, step:2, step:3, step:6]
8. 4.1 With $D=\mathrm{diag}(\sqrt{{\gamma }_0},\sqrt{{\gamma }_1})$, the logical map has expansion ${\mathcal{E}}_{\tau }=\mathrm{i}\mathrm{d}+\tau \mathcal{D}[D]+O({\tau }^2)$, where $\mathcal{D}[D](X)=DXD-\{D^2,X\}\mathrm{/}2$. Its diagonal action vanishes and its off-diagonal action is $-a{X}_{01}$, agreeing with the exact limit. For ${\gamma }_0=\gamma (1+\epsilon {)}^2$, ${\gamma }_1=\gamma (1-\epsilon {)}^2$, $\gamma >0$ and $\mathrm{\mid }\epsilon \mathrm{\mid }<1$, this gives $a=2\gamma {\epsilon }^2$. Already the syndrome probability $1-e^{-{\gamma }_i\tau }$ depends on the logical basis input $i$ when the rates differ. This identifies amplitude-dependent record information without assuming a microscopic bath dilation. The calculation separates that mechanism from the jump-time phase mechanism discussed in C2; it leaves the physical rate-versus-lift relation and broader control problem unresolved. [A1, C2, step 1.2, step 2.2, step 3.2] ∎ [A1, C2, step:2, step:5, step:7]

• D1: Use finite complex matrices in specified orthonormal bases, with adjoint given by conjugate transpose and outer products by matrix multiplication. A density matrix is positive semidefinite with trace one. A CPTP map is complex-linear, trace-preserving and positive after tensoring with every finite identity map. Positivity passes to compressions; positive trace-one matrices have diagonal entries in [0,1]. Source: self-contained; Definitions in this direction; positivity arguments in steps 1, 3 and 6.. Reading limits: These conventions and elementary consequences are stated explicitly and used locally. No library acceptance or independent prerequisite audit is asserted.
• A1: Physical model assumption: on $C^4$ with the specified basis, fixed rates gamma_0,gamma_1>0, real omega_0, H=omega_0 P and L=sqrt(gamma_0)|2><0|+sqrt(gamma_1)|3><1|, the density matrix obeys rho'=-i[H,rho]+L rho L†-{L†L,rho}/2. Inputs are supported on span{|0>,|1>}; the lower sector is absorbing. The energy lift is zero. Source: self-contained; potential_solution.solution.statement; direct component equations in step 2.. Reading limits: New prescribed unequal-amplitude model. Neither microscopic derivation nor experimental verification is claimed. The original article is not evidence for this altered model.
• A2: The encoding is $W_0$|i$\ge$|i>, and $W_1$|i$\ge$|i+2> identifies the decay sector. Recovery measures the two sectors and maps each back to the same logical basis, discarding the outcome: R(Y)=$W_0$†$Y{W}_0$+$W_1$†$Y{W}_1$. The allowed one-cycle comparison class consists of all deterministic system-only CPTP decoders $M_4$ to $M_2$, with no environmental input or postselection. Source: self-contained; potential_solution.solution.statement and proof step 1.. Reading limits: The operation is explicitly defined and its channel property proved locally. The complete frozen alg-qec-distinguishability-r4 record and original PDF p. 2 were inspected for correspondence with the article's procedure.
• A3: Fix total time $T>0$ and a positive integer N, set $tau=T$/N, and assume identical noise intervals followed by ideal instantaneous recovery. After each recovery the state is re-encoded into the same code, and the decoded cycles compose as the same memoryless channel. Rates remain fixed as N tends to infinity. Source: self-contained; potential_solution.solution.statement and proof steps 4–5.. Reading limits: Explicit protocol assumptions, not consequences of a microscopic reservoir calculation. Finite-duration measurement, control errors and bath memory are outside this result.
• D2: For a logical qubit channel E, entanglement fidelity at the maximally mixed input is <Phi|(id tensor E)(|Phi><Phi|)|Phi>, with |Phi$\ge$(|00>+|11>)/sqrt(2). Source: def-qec-prereq-entanglement-fidelity; Frozen context, complete item source, Definition and Justification.. Reading limits: The complete frozen item was read. Its defining functional is used; the underlying Nielsen paper and transitive library suppliers were not freshly read. The fidelity formulas needed here are evaluated directly.
• F1: Scalar linear differential equations are solved by exponentials and integration. At fixed finite parameters, the convergent exponential and logarithm series give e^(-s tau)=1-s tau+$s^2$ $ta{u}^2$/2+O($ta{u}^3$) and log(1+u)=u-$u^2$/2+O($u^3$) near zero. Source: self-contained; Explicit solutions in step 2 and scalar expansions in step 5.. Reading limits: Elementary local calculus input, with applications displayed. Numerical spot checks of the scalar product limit were supplementary only and are not a proof or simulation of an emitter.
• C1: Context and attribution only: the article specifies an equal-amplitude four-level coherent jump and natural sector recovery, and proposes future models in which degeneracy lifting changes jump operators. Its equal-rate recovery and optimality conclusions are not premises of the unequal-rate derivation. Source: https://arxiv.org/abs/2609.26946; Frozen PDF p. 2, Four-level system and Eqs. (2)–(3); p. 5, final Discussion paragraph; pp. 8–9, Supplemental Material S1.. Reading limits: Freshly read the extracted text and inspected the rendered images of these four pages. Complete relevant article-origin records are individually listed in items_read. No full-article or 113-record reading is claimed for this direction.
• C2: Context only: Michael et al. discuss recovery-stage error, compensable no-jump unitary evolution under self-Kerr nonlinearity, and additional dephasing from unknown photon-jump times. These passages motivate distinguishing timing phases from the amplitude mechanism calculated here. Source: https://journals.aps.org/prx/pdf/10.1103/PhysRevX.6.031006; Printed p. 10, end of Section V before Section VI, recovery-stage and self-Kerr paragraphs.. Reading limits: Fresh targeted reading of the searchable PDF passages, including the transition into Section VI. No full-paper reading, source-proof reconstruction, experimental verification or theorem application is claimed.

Decision: accept

Accepted unchanged in category solution. Independently checked all eight argument steps: the restricted evolution is CPTP, decoding gives the stated coherence multiplier, memoryless composition gives the stated fixed-time limit, and the positivity bound proves optimality over all deterministic system-only one-cycle CPTP decoders. The result is a conditional physics-theorem for prescribed dynamics, not an observation or a solution of the authors’ general modified-jump question.

Dependency review: Checked every immediate dependency D1, A1–A3, D2, F1 and C1–C2. D1 supplies explicit finite-matrix conventions; its positivity consequences follow directly from quadratic forms. A1 is a physical model postulate, A2 specifies the recovery and decoder class, and A3 independently assumes memoryless composition. None is presented as experimental evidence. D2 is a mathematical definition: its complete frozen item was inspected, and the defining overlap agrees with Nielsen’s paper, p. 1 and p. 2 before Eq. (11). The required fidelity expressions were checked directly by matrix-unit expansion. F1’s scalar solutions and Taylor expansions give the stated logarithmic limit. C1 correctly attributes the equal-amplitude model, recovery and future question to frozen article pp. 2, 5 and S1 pp. 8–9; those claims are not proof premises for the altered model. C2’s limited contextual description agrees with Michael et al., printed p. 10. In particular, the unequal-rate proof does not import the equal-rate fixed-weight branch argument. No open conclusion or circular article citation supplies a mathematical premise.

Review reading limits: Read the complete assigned direction, all eight proof steps and eight dependency records, and the seven item records individually listed in items_read. Read frozen PDF text and inspected rendered pages 2, 5, 8 and 9. Read Michael et al.’s printed-page-10 recovery and self-Kerr passages through the opening of Section VI, and Nielsen’s cited definition passages. Did not read all 113 original article items, audit transitive library suppliers, reproduce simulations, verify experiments, derive microscopic dynamics or establish global novelty. Earlier assessor reading claims retained in the draft’s source inventory are not additional reading by this reviewer.