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REVIEW 2 major objections 3 minor 129 references

Quantum communication and fault-tolerance

T0 review · 2 major / 3 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read This thesis proves that entanglement-assisted communication survives faulty encoder and decoder gates: below a gate-error threshold the achievable rate is within $f(p)$ of the noiseless capacity, with $f(p)\to0$, and it bounds the rate…

desk verdict Useful thesis with a real gap in the AVP proof: the stated δ does not make Eq. (2.4) vanish, so the explicit capacity bound in Chapter 2 is unsupported as written, though the qualitative recovery likely survives a repair. read the letter →

arxiv 2412.20736 v1 pith:I2RYKAYU submitted 2024-12-30 quant-ph

Paula Belzig This is my paper · ORCID
classification quant-ph MSC 81P4581P68
keywords quantumcommunicationentanglement-assistedcapacityfault-toleranceerrorcorrectionthresholdtheorementanglementdistillationarbitrarilyvaryingperturbationShannontheory
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The thesis's central result is that entanglement-assisted communication over any finite-dimensional quantum channel is robust to noise in the encoding and decoding hardware: when the circuits that implement the encoder and decoder suffer independent gate errors of probability $p$ below a threshold, communication can still be carried out at a rate within $f(p)$ of the ideal entanglement-assisted capacity $C^{\mathrm{ea}}(T)$, with $f(p)\to 0$ as $p\to 0$. The proof compiles a standard noiseless coding scheme into a concatenated quantum error-correcting code, moves qubits between the code space and the raw channel through interface circuits, and accounts for residual interface failures as a weak perturbation of the channel for which a new coding theorem is proved. A fault-tolerant entanglement distillation subroutine restores the maximally entangled resource states that pass through noisy interfaces. The thesis also contains a hardware-oriented study of a four-qubit error-detection code on trapped-ion devices and a separate result bounding how much entanglement assistance can improve rates over unassisted communication for a large class of channels.

What carries the argument

The load-bearing mechanism is the effective-channel reduction. A noiseless entanglement-assisted coding scheme is compiled into fault-tolerant circuits in a concatenated seven-qubit code; interface circuits, whose failure probability is imported as at most $2cp$ under the i.i.d. Pauli model, move qubits between the code space and the physical channel. The effective-interface lemmas show that the noisy compiled circuit is close, in induced 1-to-1 norm, to the ideal circuit acting on an effective channel $T_{p,\mathcal N}=(1-q)(T\otimes \mathrm{Tr}_S)+q\mathcal N$ with $q=2(j_1+j_2)cp$, fed with an arbitrary syndrome state $\sigma_S$. The thesis then proves a coding theorem for entanglement-assisted communication under 'arbitrarily varying perturbation'—an infimum over syndrome states and perturbations $\mathcal N$—giving $C^{\mathrm{ea}}_{\mathrm{AVP}}(p,T)\ge C^{\mathrm{ea}}(T)-g(p)$ with $g(p)=O(p\log p)$. A fault-tolerant entanglement distillation subroutine, based on a one-way distillation protocol and using some channel copies for the classical communication it needs, converts the noisy resource states produced by the interfaces into near-perfect maximally entangled states in the code space.

What would settle it

Simulate the interface circuits of the concatenated seven-qubit code under the i.i.d. Pauli noise model and measure the probability that an encoding or decoding interface is incorrect as a function of $p$; if this probability does not scale as $O(p)$ for small $p$, the bound $\mathrm{Prob}(\text{interface incorrect})\le 2cp$ that carries the effective-channel theorem and the capacity theorem is false. Alternatively, on a channel such as the qubit depolarizing channel, compute the explicit lower bound of the main capacity theorem and search for gate errors $p$ below threshold where the bound is violated by a numerical estimate of the fault-tolerant capacity.

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Extended reading notes

Core claim

On the paper's own terms, the main discovery is a threshold-type coding theorem for fault-tolerant entanglement-assisted communication. For every finite-dimensional quantum channel $T$ and every target accuracy $\eta>0$, there is a gate-error threshold $p_{\mathrm{th}}(\eta,T)>0$ such that for all $0\le p\le p_{\mathrm{th}}$, $C^{\mathrm{ea}}_{F(p)}(T)\ge C^{\mathrm{ea}}(T)-\eta$, where $F(p)$ is the i.i.d. Pauli noise model on the encoder and decoder circuits; equivalently, $\lim_{p\to0}C^{\mathrm{ea}}_{F(p)}(T)=C^{\mathrm{ea}}(T)$. The theorem is proved by exhibiting explicit fault-tolerant encoders and decoders: a noiseless entanglement-assisted scheme is first approximated by circuits, then implemented in a concatenated seven-qubit code whose level grows with the block length, with interface circuits converting between the code space and the physical channel. After the fault-tolerant compilation, the whole setup is close to a noiseless scheme for an effective channel of the form $(1-q)(T\otimes\mathrm{Tr}_S)+q\mathcal N$ with a correlated syndrome input, and the thesis proves a coding theorem showing that such 'arbitrarily varying perturbations' reduce the rate by at most a function $g(q)=O(q\log q)$. Fault-tolerant entanglement distillation, run through a subset of channel uses, supplies the clean entanglement needed by the assisted scheme.

Load-bearing premise

The whole argument borrows an imported guarantee that the interface circuits moving qubits in and out of the error-correcting code fail with probability at most a constant times the gate error $p$; the thesis does not re-prove that guarantee, and if it fails the effective-channel reduction collapses.

Editorial extensions

If this is right

  • For any finite-dimensional quantum channel, fault-tolerant encoder and decoder circuits achieve entanglement-assisted communication at rates arbitrarily close to the noiseless capacity once the local gate error is below a threshold.
  • The communication channel itself does not need to be below the fault-tolerance threshold; only the local gates in the encoder and decoder do, so the result applies to on-chip communication and to links that are noisier than the local hardware.
  • As $p\to0$, the fault-tolerant entanglement-assisted capacity equals the standard entanglement-assisted capacity, recovering the noiseless quantum Shannon-theoretic limit.
  • The same modular construction yields fault-tolerant entanglement distillation: noisy resource states entering the code space through faulty interfaces can be purified at the price of a fraction of channel uses, rather than being unusable.
  • The resource-inequality version of the coding theorem expresses the trade-off as $\langle T\rangle + H(A)(T\otimes\mathrm{id})(\varphi)[qq]\ge_{\mathrm{FT}(p)} I(A':B)(T\otimes\mathrm{id})(\varphi)[c\to c]$ up to $O(p\log p)$ corrections.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The effective-channel reduction is modular in the error-correcting code: any family of concatenated codes whose interface circuits fail with probability $O(p)$ and whose extended rectangles satisfy a threshold theorem would inherit the same capacity-recovery result, so verifying such interface bounds for topological or LDPC constructions is a natural next test.
  • Because the loss bound depends on the ratio $C^{\mathrm{ea}}(T)/C(T)$, the thesis's own conjecture that this ratio is bounded by a dimension-dependent factor would upgrade the threshold result from channel-dependent to uniform in dimension; the thesis explicitly notes this dependence.
  • On real hardware the explicit rate loss is dominated by a $p\log p$ term and is far too large for practical finite-block-length use; the thesis's plots suggest the proof-of-principle nature and point toward tighter continuity bounds or different codes as the route to practical rates.
  • The trapped-ion comparison offers a device-level check of the i.i.d. Pauli assumption: a newer device with lower gate error but no observed encoded gain is consistent with correlated errors that the model excludes, so hardware experiments can directly inform which noise models the capacity theorem should be extended to.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 3 minor

Summary. This PhD thesis studies the limits of quantum communication under noise, with and without entanglement assistance. Chapter 2, reproducing arXiv:2210.02939, introduces the fault-tolerant entanglement-assisted capacity C^ea_F(p)(T) for the i.i.d. Pauli gate-error model and develops the technical machinery: effective interface lemmas, an entanglement-assisted capacity under arbitrarily varying perturbation (AVP), fault-tolerant entanglement distillation, and finally a lower bound C^ea_F(p)(T) >= C^ea(T) - ..., with a threshold theorem recovering C^ea(T) as p tends to zero. Chapter 3 reports on experimental investigations of the [[4,2,2]] error-detection code on trapped-ion hardware and states a threshold theorem for demonstrations of fault-tolerance. Chapter 4, explicitly in preparation, proposes bounding the ratio C^ea(T)/C(T) for a large class of finite-dimensional channels via channel-divergence techniques. The thesis is honest about which parts are published, ongoing, or in preparation.

Significance. If the main result is established, Chapter 2 provides a meaningful fault-tolerant analogue of the entanglement-assisted capacity, extending Christandl-Muller-Hermes' unassisted results to the entanglement-assisted setting and showing that noisy encoder/decoder circuits need not reduce the asymptotic rate. The modular proof structure--separating circuit-level fault-tolerance, the AVP information-theoretic model, and distillation--is a genuine strength, and the explicit interface lemmas and theorem statements make the dependence on prior work clear. The quantitative claims in Theorem 2.4.4 and Theorem 2.6.3 are, however, not proved as written because of the rate-balance error described below; the qualitative threshold theorem appears likely repairable. Chapters 3 and 4 are ongoing/in preparation and therefore carry less weight in the evaluation, though they are clearly labeled as such.

major comments (2)
  1. [§2.4.2, Theorem 2.4.4 and Eq. (2.4)] The proof's rate balance is not valid as written. After applying [27, Lem. IV.10], Eq. (2.4) requires d_B^{(p+\tildeδ)n} e^{-n δ^2/(2 log^2 λ_min)} -> 0. With the stated δ = sqrt(2 log(d_B) p |log λ_min|) and λ_min >= p^2/(d_A d_B), the decay rate is δ^2/(2 log^2 λ_min) ≈ (log d_B) p / (2 log(1/p)) per channel use, while the prefactor contributes at least (p + \tildeδ) n log d_B; even for \tildeδ -> 0 the prefactor outruns the exponential, so Eq. (2.4) does not vanish and the rate condition (2.6) does not imply Eq. (2.5). A choice δ = Θ(√p |log λ_min|) would restore the convergence with a penalty Θ(√p log^2(1/p)), but then the stated bound g(p) = O(p log p) in Theorem 2.4.4 is no longer what is proved. Independently of the balance issue, the displayed g(p) is Θ(√p log^(3/2)(1/p)) rather than O(p log p), so the asymptotic claim in Theorem 2.4.4 and the scaling discussion in §2.8.2 need correction. The qualitative continuity in Theorem 2.4.5 and Corollary 2.4.6 may survive such a repair.
  2. [§2.6, Theorem 2.6.3] The statement and proof of Theorem 2.6.3 disagree on the definition of f1(p): the displayed theorem has f1(p) = (h(4cp) + 4cp log(3)) j2 / (1 - h(4cp) - 4cp log(3)), while the proof derives f1(p) = (h(4cp) + 4cp log(3)) log(2) / (1 - h(4cp) - 4cp log(3)). Since the final lower bound uses 4 f1(p) C^ea(T)/C(T), the stated bound is not the bound actually proved when j2 ≠ 1. Moreover, because the proof invokes Theorem 2.4.4 at perturbation strength 2(j1+j2)cp, the quantitative bound in Theorem 2.6.3 inherits the unresolved AVP gap from the first comment. The threshold formulation in Theorem 2.6.1 is qualitative and may be recoverable, but Theorem 2.6.3 as printed is not established.
minor comments (3)
  1. [Throughout] There are several typographical issues, including 'comparitive', 'occuring', and 'architechtures' (e.g., Chapter 3 and §2.8.3), and expressions like '2j1+j2' in Theorem 2.6.3 should be typeset as 2^{j1+j2} to avoid ambiguity.
  2. [§2.5, Theorem 2.5.4] The proof of Theorem 2.5.4 is only a sketch; since this theorem is a load-bearing subroutine in Section 2.6, the final version should expand the proof or give a precise citation/statement of the underlying threshold argument.
  3. [Chapter 4] The class of channels to which the claimed capacity-ratio bound applies is introduced only through a far/near-replacer dichotomy; the boundary of this class should be stated formally in the main text, not only in the chapter's opening discussion.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the capacity bound is derived from external benchmark capacities and antecedent fault-tolerance theorems, not from a fitted or self-defined input.

full rationale

The central result C_F^ea(p)(T) >= C_ea(T) - f(p) is a lower bound expressed in terms of externally defined, established capacities C_ea(T) and C(T), which are not fitted or constructed from the paper's own target claim. The proof reduces fault-tolerant entanglement-assisted communication to an arbitrarily varying perturbation (AVP) coding problem and then to a standard entanglement-assisted coding theorem for a slightly depolarized effective channel; the lower bound C_AVP^ea(p,T) >= C_ea(T) - g(p) is obtained by an explicit code construction and continuity estimates, not by definition. The imported interface circuits, threshold theorem, postselection-type lemma, and classical fault-tolerant capacity results from [27,80] are stated theorems with their own assumptions (i.i.d. Pauli noise, concatenated 7-qubit Steane code) and are used as building blocks; they are parameter-free with respect to the target result and are not fitted to the capacities being bounded. The fact that several of these citations share authors with the thesis is a normal research continuity, and under the stated rules self-citation is not circular when the cited result is an independent theorem with stated assumptions that do not include the paper's conclusion. The reviewer's separate concern about the delta-rate balance in Eq. (2.4) is a potential correctness gap in the AVP proof, not a circularity: the theorem is not being made true by definition, by renaming a fitted quantity, or by an unverified self-citation chain. No circular step can be exhibited in the derivation chain.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No numerical constants are fitted to data in the portions reviewed. The rate penalties f1(p), f2(p) and the AVP gap g(p) are derived functions of p, dA, dB, and the interface constant c; c is inherited from the Steane code analysis rather than fitted here. The thesis introduces no new physical entities beyond effective channel noise maps and syndrome states, which are mathematical bookkeeping rather than postulates.

assumptions (5)
  • domain assumption Concatenated 7-qubit Steane code has fault-tolerant interfaces with failure probability at most 2cp under i.i.d. Pauli noise for p below threshold.
    Imported from [27, Theorem III.3] and restated as Theorem 1.3.2; the effective channel lemmas in Chapter 2 (Theorem 2.2.3) and the entanglement distillation theorem depend on this bound.
  • domain assumption Circuit noise is i.i.d. Pauli: each gate location independently suffers a Pauli error with probability p.
    Adopted in Section 1.2.1 and used throughout Chapter 2 and Section 3.2. The thesis itself notes the Aria experiments suggest this model may be inadequate, and correlated noise could invalidate the fault-tolerance demonstration.
  • domain assumption Postselection-type domination bound for arbitrarily varying perturbation channels (T_p,N versus T_p) from [27, Lemma IV.10].
    Used in the proof of Theorem 2.4.4 to relate the AVP channel to the noisier depolarized channel; the lower bound on C^ea_AVP depends on this lemma.
  • domain assumption One-way entanglement distillation protocol from [32] converts noisy Bell-diagonal states to maximally entangled states with the stated fidelity and rate.
    Used in Theorem 2.5.4 and Section 2.6; the rate penalty f1(p) in the main capacity bound is computed from this protocol.
  • ad hoc to paper The class of channels in Chapter 4 is partitioned into those far from a replacer channel and those near one, and the boundary of the class is set by this division.
    Section 4.4 divides the proof into far-from-replacer and vicinity-of-replacer cases; the exact class for which the quotient bound holds is defined by this split and could not be fully verified in the truncated text.

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Pith. "Pith review of Quantum communication and fault-tolerance." pith.science (2026). https://pith.science/paper/I2RYKAYU

@misc{pith2026241220736,
  author       = {Pith},
  title        = {Pith review of: Quantum communication and fault-tolerance},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/I2RYKAYU}},
  note         = {Machine review of arXiv:2412.20736}
}
read the original abstract

In this thesis, we are interested in the limits of quantum communication with and without entanglement, and with and without noise assumptions on the communication setup. When a sender and a receiver are connected by a communication line that is governed by noise which is modelled by a quantum channel, they hope to design a coding scheme, i.e. messages and message decoders, in such a way that they are robust to this noise. The amount of message bits per channel use is called the achievable rate of the coding scheme, and the maximal achievable rate for a given quantum channel is called capacity of the quantum channel. Here, we are interested in coding schemes and capacities under various assumptions, in particular in the case where the sender and the receiver share quantum entanglement, which turns out to be the most natural generalization of the classical communication analogue that lies at the basis of many of our modern technologies.

Figures

Figures reproduced from arXiv: 2412.20736 by the authors.

Figure 1.1
Figure 1.1. Basic setup for classical communication over a classical channel. An encoder encodes a message of m bits, x ∈ {1, 2, ..., 2 m} into a codeword an ∈ An which is a bitstring of n bits. The codeword is sent from the sender to the receiver by n uses of the channel T. After receiving the noisy output bn ∈ Bn of n applications of T, the decoder produces an estimate x ′ ∈ {1, 2, ..., 2 m}, which should ideally be identical… view at source ↗
Figure 1.2
Figure 1.2. Basic setup for classical communication over a quantum channel. The encoding map E maps a bit string x of length m to a quantum state in M⊗n dA . The quantum channel T acts on each of the n subsystems, and the decoder D decodes the received quantum state to a bit string x ′ , which should be identical to the input bit string x. Note that classical information transfer is indicated by double lines (input into encoder… view at source ↗
Figure 1.3
Figure 1.3. Basic setup of the superdense coding protocol. Let the sender and the receiver each hold one qubit from a maximally entangled state |ϕ+⟩ = √ 1 2 (|00⟩ + |11⟩). The sender performs an encoding E : C2 ⊗ M2 → M2 of two classical bits a ∈ {0, 1} and b ∈ {0, 1} by performing one of the 4 quantum gates Gab = {12, σx, σy, σz} on their part of the maximally entangled state. Subsequently, they send their part of the entangle… view at source ↗
Figures from the paper (25 more)
Figure 1.4
Figure 1.4. Figure 1.4: Basic setup for entanglement-assisted communication. The encoding map E maps a bit string x of length m and one part of each entangled state ϕ+ to a quantum state in M⊗n dA . The quantum channel T acts on each of the n subsystems, and the decoder D uses the other par…
Figure 1.5
Figure 1.5. Figure 1.5: Pictorial representation of well-behaved exRecs for cases 1,3 and 4 from [PITH_FULL_IMAGE:figures/full_fig_p039_1_5.png]
Figure 1.6
Figure 1.6. Figure 1.6: If every exRec in a circuit is well-behaved, the transformation rules ensure [PITH_FULL_IMAGE:figures/full_fig_p041_1_6.png]
Figure 1.7
Figure 1.7. Figure 1.7: Illustration of the connectivity layout for the 7-qubit Steane code because [PITH_FULL_IMAGE:figures/full_fig_p045_1_7.png]
Figure 1.8
Figure 1.8. Figure 1.8: Sketch of the blockwise application of Dec∗ for the concatenated 7-qubit Steane code in order to mathematically analyze the circuit. Here, we draw the construction for the second level l = 2, where 7 2 = 49 qubits are employed to encode one qubit. For a circuit with …
Figure 2.1
Figure 2.1. Figure 2.1: Sketch of the setup for the effective channel. The fault-tolerantly implemented encoder Γ E Cl takes input in the form of m classical bits x and r physical qubits which are encoded in the code space. The resulting codewords are sent through n copies of the quantum ch…
Figure 2.2
Figure 2.2. Figure 2.2: Basic setup for our coding scheme for fault-tolerant entanglement￾assisted communication, see Definition 2.3.1. The encoding map E (yellow) encodes a bit string of length m into a quantum state that serves as input into n copies of the quantum channel T, and the deco…
Figure 2.3
Figure 2.3. Figure 2.3: Setup for entanglement distillation based on the protocol in [32]. Two parties each have access to one part of k noisy entangled states ϕq. One party performs local operations E Dist and sends one-way classical communication to the other, who performs local operation…
Figure 2.4
Figure 2.4. Figure 2.4: Setup for fault-tolerant entanglement distillation. The local opera￾tions performed in [PITH_FULL_IMAGE:figures/full_fig_p071_2_4.png]
Figure 2.5
Figure 2.5. Figure 2.5: Setup for fault-tolerant entanglement distillation with communica￾tion via a channel T. The local operations performed in [PITH_FULL_IMAGE:figures/full_fig_p072_2_5.png]
Figure 2.6
Figure 2.6. Figure 2.6: An illustration of the building blocks in our analysis. The circuits for encoding and decoding in our fault-tolerant entanglement-assisted communication setup, as illustrated in [PITH_FULL_IMAGE:figures/full_fig_p076_2_6.png]
Figure 2.7
Figure 2.7. Figure 2.7: Plot of the fault-tolerant entanglement-assisted capacity of the qubit [PITH_FULL_IMAGE:figures/full_fig_p086_2_7.png]
Figure 2.8
Figure 2.8. Figure 2.8: Plot of the lower bound on the fault-tolerant entanglement-assisted [PITH_FULL_IMAGE:figures/full_fig_p086_2_8.png]
Figure 2.9
Figure 2.9. Figure 2.9: Plot of the scaling of the different contributions to the function [PITH_FULL_IMAGE:figures/full_fig_p088_2_9.png]
Figure 3.1
Figure 3.1. Figure 3.1: Circuits for preparing |ϕ+⟩ and [PITH_FULL_IMAGE:figures/full_fig_p103_3_1.png]
Figure 3.2
Figure 3.2. Figure 3.2: Preparation circuits for [PITH_FULL_IMAGE:figures/full_fig_p104_3_2.png]
Figure 3.3
Figure 3.3. Figure 3.3: Thresholds pth(T) for demonstrating fault-tolerance for different gate sequence lengths T applied to the preparation circuits listed in [PITH_FULL_IMAGE:figures/full_fig_p106_3_3.png]
Figure 3.4
Figure 3.4. Figure 3.4: Threshold pth(T) for demonstrating fault-tolerance as a function of the gate sequence length T, where locations are counted in terms of the native gate set of IonQ Harmony and Aria. 3.4 Error detection with the [[4,2,2]] code on current hardware The cloud-based open …
Figure 3.5
Figure 3.5. Figure 3.5: Simulation of the circuits for |ϕ+⟩ and increasing lengths of gate se￾quences applied to it, with a depolarizing noise model with depolarizing probability p at each gate, and an amplitude damping noise model with damping parameter γ at each gate. All simulations were…
Figure 3.6
Figure 3.6. Figure 3.6: TV-distance of the probability distributions obtained as an output of [PITH_FULL_IMAGE:figures/full_fig_p111_3_6.png]
Figure 3.7
Figure 3.7. Figure 3.7: TV-distance of the probability distributions obtained as an output of [PITH_FULL_IMAGE:figures/full_fig_p112_3_7.png]
Figure 3.8
Figure 3.8. Figure 3.8: TV-distance of the probability distributions obtained as an output of [PITH_FULL_IMAGE:figures/full_fig_p113_3_8.png]
Figure 4.1
Figure 4.1. Figure 4.1: Numerical results for the qubit amplitude damping channel. [PITH_FULL_IMAGE:figures/full_fig_p122_4_1.png]
Figure 4.2
Figure 4.2. Figure 4.2: Plot of the parameter region for p0 and p3. The light gray region is the region where case 2 applies. The dark gray region is the region where case 1 applies. This can be solved numerically, and the solutions are plotted in [PITH_FULL_IMAGE:figures/full_fig_p138_4_2.png]
Figure 4.3
Figure 4.3. Figure 4.3: For pairs of values p0 and p3, we plot the quotient of C ea(Tp0,p3 ) and CH(Tp0,p3 ). The quotient is maximal on the border between the domains highlighted in [PITH_FULL_IMAGE:figures/full_fig_p140_4_3.png]

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Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.