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Tripartite entanglement of qudits

T0 review · 3 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read The paper proves a decomposition theorem: the four algebraic invariants of any direct-sum tripartite state are exact functions of the invariants of its diagonal-block components, reducing entanglement classification to a compositional…

desk verdict The decomposition theorem is real and proven; the three-tribit completeness claims rest on an unproved 0/1-representative assumption. read the letter →

arxiv 2412.10728 v1 pith:FU5KT4EK submitted 2024-12-14 quant-ph math-phmath.MP

Roman V. Buniy , Thomas W. Kephart This is my paper · ORCID
classification quant-phmath-phmath.MP PACS 03.67.Mn
keywords tripartiteentanglementquditsalgebraicinvariantsclassificationdirectsumdecompositionirreducibleclassesthreetribitsnullity
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

This paper aims to organize the entanglement of three quantum systems of arbitrary dimension (qudits) into a small set of building blocks. Its main result is a decomposition theorem: if a tripartite state is written as a direct sum of component states living on matching diagonal blocks, then the four algebraic invariants that define its entanglement class are determined exactly by the invariants of the components and the block dimensions. The same theorem introduces a natural distinction between reducible classes, which split into smaller pieces, and irreducible classes, which do not. The authors compute the invariants for a catalogue of irreducible classes and use them to give a complete list of all 39 entanglement classes for three tribits, each expressed as a sum of irreducible pieces. A sympathetic reader would care because the result turns the open-ended problem of classifying tripartite entanglement into a finite combinatorial task plus a library of irreducible classes.

What carries the argument

The central objects are the kernel spaces $K_a(v)$, $K_{b,c}(v)$, and $K_{1,2,3}(v)$: spaces of dual vectors whose contraction with the state $v$ vanishes; their dimensions $n_a$, $n_{b,c}$, $n_{1,2,3}$ are the four algebraic invariants that define an entanglement class. The load-bearing mechanism is the simultaneous direct-sum operation $+_{1,2,3}$ on all three parties. When a state is assembled from diagonal blocks, each kernel space of the sum is a union of the corresponding component kernel spaces together with fresh off-diagonal constraints whose dimensions are controlled by the block dimensions; Lemma 4.3 gives exact additivity formulas for this operation, and Theorem 4.1 iterates them over $p$ blocks. The direct-sum calculus, rather than any property of specific coordinates, is what carries the classification.

What would settle it

Take a three-tribit state with a coordinate value outside $\{0,1\}$, for example $[1,1,1]+[2,2,2]+\lambda[3,3,3]$ with $\lambda=2$, compute the four nullities $n_1,n_2,n_3,n_{1,2,3}$ directly from the kernel definitions, and check whether the tuple appears in Table 9. For Theorem 4.1 itself, choose random vectors $v'$, $v''$ on diagonal blocks, form $v=v'+_{1,2,3}v''$, and compare the theorem's three formulas with a direct rank-nullity computation of the kernel spaces.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central claim is Theorem 4.1. For direct sum decompositions $V_a = (\oplus_a)_{q=1}^p V_a^{(q)}$ with $1\le a\le 3$, let $V^{(q)} = V_1^{(q)}\otimes V_2^{(q)}\otimes V_3^{(q)}$ be the diagonal blocks and let $v = (+_{1,2,3})_{q=1}^p v^{(q)}$. Then the invariant $n_a$ is the sum of the component $n_a$'s; $n_{b,c}$ is the sum of the component $n_{b,c}$'s plus the cross-block term $\sum_{q_1\ne q_2} d_b^{(q_1)} d_c^{(q_2)}$; and $n_{1,2,3}$ obeys an analogous exact formula involving the component $n_{1,2,3}$, $n_{b,c}$, and the block dimensions. The theorem holds for any number $p$ of blocks, so the invariants of any state built from components by direct sums are computable without re-deriving kernel spaces in the full space. The paper then defines a class to be reducible when one of its vectors splits across diagonal blocks, irreducible otherwise, and expresses every reducible class as a finite direct sum of irreducible classes. Applying this to three tribits yields the complete class list of Table 9 and the decomposition table of Table 10; Section 8 extends the same calculus to infinite families of higher-dimensional states.

Load-bearing premise

The load-bearing premise is the unproved assertion in Section 6 that for entanglement classification it is enough to consider vectors whose coordinates all lie in $\{0,1\}$; every irreducible class and the three-tribit class list in Table 9 are represented by such vectors, and if some class needs different coordinates, the list would be incomplete.

Editorial extensions

If this is right

  • For any tripartite system, the invariants of a state composed by direct sums can be computed from the invariants of its components and block dimensions alone, without recomputing kernel nullities in the full Hilbert space.
  • Reducible classes decompose into finite lists of irreducible classes, and the same formulas apply at every level of the decomposition, so classification reduces to three tasks: list irreducible classes, list valid block combinations, and apply Theorem 4.1.
  • For three tribits the classification is complete: the 39 classes of Table 9 are exactly the direct sums of the irreducible classes constructed in Section 6, and the invariant values from the decomposition agree with direct computation (Table 8).
  • Annihilation operators generate a reduction graph between classes (Figure 8), showing which classes can be reached by deleting basis directions.
  • The same construction computes $n_{1,2,3}$ for infinitely many classes with dimensions larger than three (Tables 11-18), and exhibits multiple decompositions of the same class into different irreducible components.

Reading between the lines

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

  • A testable extension is to sample three-tribit states whose coordinates include values outside $\{0,1\}$ and check whether any new invariant tuple appears; the paper's completeness claim rests on an unproved assertion that 0/1 representatives suffice.
  • Because the formulas in Theorem 4.1 are purely additive in block dimensions and component invariants, they suggest a recursive algorithm for arbitrary states: repeatedly split a state into diagonal blocks until the blocks are irreducible, then apply the theorem bottom-up; the paper does not spell out such an algorithm.
  • The multiple-decomposition identities in Section 8 imply that the same entanglement class can be factored in different ways, so the irreducible classes do not form a unique factorization basis; this structural point is noted in the paper but not developed further.
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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

3 major / 4 minor

Summary. This paper studies algebraic entanglement invariants for tripartite qudit systems, focusing on the four invariants n1, n2, n3, and n123 associated with kernel nullities. It proves Theorem 3.1, giving a complete characterization of the allowed triples (n1,n2,n3), and states several conjectured bounds for n123. It then introduces tensor-product and direct-sum operations and proves Theorem 4.1, the decomposition theorem, which shows how all four invariants of a direct sum of blocks are computed from the invariants of the blocks. This is used to define reducible and irreducible entanglement classes, to tabulate irreducible classes for dimensions up to three, and to present a complete list of 39 entanglement classes for three tribits with their decompositions into irreducible components. The paper closes with numerous examples constructing higher-dimensional classes from these components.

Significance. The decomposition theorem (Theorem 4.1) is a genuine and useful result: if correct, it reduces the computation of all four algebraic invariants for any direct-sum-composed state to the invariants of its blocks, and it gives a principled organization of entanglement classes by irreducible components. The proofs of Theorem 3.1, Lemmas 4.1-4.3, and Theorem 4.1 are presented in the appendices and appear internally consistent. The explicit invariant computations and the consistency checks in Table 8 are valuable. However, the paper's headline classification claim for three tribits is not fully supported within the manuscript: the completeness of the class list rests on an unproved 0/1-representative assertion and on an undetermined allowed-value set for n123. These are load-bearing gaps rather than mere presentation issues, and they need to be closed before the classification can be accepted as complete.

major comments (3)
  1. [Section 6, paragraph after Figure 3; Tables 4-9] The statement "It turns out that it is sufficient for the purposes of entanglement classification to consider only vectors v with coordinates v_{i,j,k} ∈ {0,1}" is stated without proof, yet every representative in Tables 4-9 is of this form. This assertion is load-bearing for the completeness of the three-tribit classification in Table 9: if some entanglement class requires a representative with non-0/1 coordinates, the list of classes and the decomposition-based enumeration would be incomplete. The authors should provide a proof of this reduction or explicitly cite and summarize a proof from [38] that establishes it.
  2. [Section 3.2 and Table 9] The allowed values of n_{1,2,3} for fixed (n1,n2,n3) are not determined. Section 3.2 gives only an upper bound (Eq. 15) and the conjectured lower bounds Conjectures 3.2-3.4; it does not prove which n123 values occur. For example, Table 9 lists n123 = 10, 8, 7, 6, 5, 4, 3, 2 for (n1,n2,n3)=(0,0,0), but the text does not show that 9 is impossible or that no other values occur for this and the other 15 triples. Since classes are defined by the full quadruple, this gap undermines the claimed exhaustiveness of the classification. The assertion in Section 9 that "We have completely solved" the classification for three tribits is not supported by the derivations in this manuscript.
  3. [Section 6, subsection on small dimensions; Section 7, Tables 9-10] The completeness of the set of irreducible classes and of the decomposition list for three tribits is not self-contained. Section 6 explicitly says that most results are "statements of the results, of which only a few are with derivations," and Section 7 refers to [38] for more information on the three-tribit example. Table 10 decomposes all 39 classes using the irreducible classes of Section 6, but the paper does not prove that this list of irreducibles is exhaustive or that every class in Table 9 is obtained. The authors should either include an appendix proving the completeness of the irreducible-class list and the decomposition table, or state precisely which parts are established in [38] and reproduce those proofs in sufficient detail for the present claims.
minor comments (4)
  1. [Table 10] Several rows of Table 10 use zero-class labels with vanishing dimensions, for example the row for C5, which do not match the notation 0^{d1,d2,d3}_0 introduced in Section 6.1.1; these entries should be corrected or the notation should be clarified.
  2. [Throughout] The text frequently refers to equation numbers such as (2), (4.2), and (4.3) that do not match the displayed equation numbers in the arXiv rendering; the cross-references should be harmonized.
  3. [Section 3.1, Eq. (14)] The approximation \tilde{N}_{d,d,d} is claimed to become exact as d→∞, but the relative errors in Table 1 are not monotonically decreasing and the d=1 row has an error of roughly -47%; a short discussion of the rate of convergence would help avoid overstating the approximation.
  4. [Section 4.1] The inequality for n_{1,2,3}(v'⊗v'') is stated without derivation, unlike the other tensor-product relations; adding a one-sentence justification or a reference would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the decomposition theorem and allowed-value theorem are proved from the definitions; the three-tribit completeness gap is an unproved assertion, not a circular reduction.

full rationale

The central derivation chain is self-contained. Theorem 3.1 is proved in Appendix A by explicitly constructing a vector with prescribed nullities, using only the kernel definitions and Schmidt decompositions; it does not presuppose the allowed-value list. Lemma 4.3 and Theorem 4.1 are proved in Appendix B by direct computation of the kernel spaces for vectors in diagonal-block direct sums, followed by induction; the formulas give the invariants of the sum in terms of invariants of the components, and no step substitutes the claimed output as an input. Section 6's irreducible classes are computed directly from representative vectors via the same kernel definitions. The main unresolved point is the completeness of the three-tribit classification in Table 9, which rests on the unproved statement 'It turns out that it is sufficient for the purposes of entanglement classification to consider only vectors v ∈ V with coordinates v_{i,j,k} ∈ {0, 1}' and on unproved exhaustiveness of the n_{1,2,3} values. This is a genuine correctness/completeness gap, but it is not circular: the table is an enumeration claim, not a derivation whose conclusion is assumed among its premises. Citations [8,9,38] are to the authors' earlier work, but they are used as background and for further references, not as the load-bearing justification of a result that the present paper purports to derive. No fitted parameter is renamed as a prediction, no uniqueness theorem is imported from the authors, and no ansatz is smuggled in via citation. The paper therefore does not exhibit circular reasoning in the sense of the review criteria.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The central theorem is self-contained linear algebra. The main unstated inputs are the choice of the invariant-based classification and the completeness of the small-dimension class lists, which are inherited from the authors' prior work.

assumptions (3)
  • domain assumption The four algebraic invariants n1, n2, n3, n123 classify tripartite entanglement in a physically meaningful way, as proposed in [8,9].
    The paper adopts this classification framework without re-deriving its physical relevance; all classes are defined by these invariants.
  • ad hoc to paper For the purpose of entanglement classification it suffices to consider vectors with coordinates in {0,1}.
    Stated in Section 6 as 'it turns out' without proof; the representative vectors in all tables are 0/1, and completeness relies on this.
  • domain assumption The list of irreducible classes for (3,3,3) in Table 7 is complete.
    The paper states in Section 9 that the first problem is completely solved for three tribits, but the derivation is not given here; Section 6 states most results are without derivations and Section 7 refers to companion [38].

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Pith. "Pith review of Tripartite entanglement of qudits." pith.science (2026). https://pith.science/paper/FU5KT4EK

@misc{pith2026241210728,
  author       = {Pith},
  title        = {Pith review of: Tripartite entanglement of qudits},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FU5KT4EK}},
  note         = {Machine review of arXiv:2412.10728}
}
read the original abstract

We provide an in-depth study of tripartite entanglement of qudits. We start with a short review of tripartite entanglement invariants, prove a theorem about the complete list of all allowed values of three (out of the total of four) such invariants, and give several bounds on the allowed values of the fourth invariant. After introducing several operations on entangled states (that allow us to build new states from old states) and deriving general properties pertaining to their invariants, we arrive at the decomposition theorem as one of our main results. The theorem relates the algebraic invariants of any entanglement class with the invariants of its corresponding components in each of its direct sum decompositions. This naturally leads to the definition of reducible and irreducible entanglement classes. We explicitly compute algebraic invariants for several families of irreducible classes and show how the decomposition theorem allows computations of invariants for compounded classes to be carried out efficiently. This theorem also allows us to compute the invariants for the infinite number of entanglement classes constructed from irreducible components. We proceed with the complete list of the entanglement classes for three tribits with decompositions of each class into irreducible components, and provide a visual guide to interrelations of these decompositions. We conclude with numerous examples of building classes for higher-spin qudits.

Figures

Figures reproduced from arXiv: 2412.10728 by the authors.

Figure 1
Figure 1. This area is 0 for 𝑥3 ≤ 1 and ∫ 𝑑+ 1 2 𝑥3 𝑑+ 1 2 [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 1
Figure 1. The shaded region is the part of the integration region for the variables [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. An illustration of how partitions of 𝑆1, 𝑆2, 𝑆3 lead to diagonal subsets of the set 𝑆 = 𝑆1 × 𝑆2 × 𝑆3. In the first, second and third row, we partition one, two and three sets, respectively. In deriving several inequalities in the following lemmas, we use the following elementary fact about the numbers of constraints 𝑁 associated with arbitrary linear functions {𝐹𝑖(𝑤)}𝑖∈𝐼 that are imposed on a vector 𝑤 ∈ 𝑊 : 𝑁 [PITH… view at source ↗
Figures from the paper (6 more)
Figure 3
Figure 3. Figure 3: A diagram representing 𝑉1 ⊗ 𝑉2 ⊗ 𝑉3 for 𝑑1 = 𝑑2 = 𝑑3 = 3. It will be convenient to use such diagrams to represent vectors in 𝑉 . It turns out that it is sufficient for the purposes of entanglement classification to consider only vectors 𝑣 ∈ 𝑉 with coordinates 𝑣𝑖,𝑗,𝑘 ∈ …
Figure 4
Figure 4. Figure 4: Diagrams of representative elements for the irreducible classes with [PITH_FULL_IMAGE:figures/full_fig_p018_4.png]
Figure 5
Figure 5. Figure 5: Diagrams of representative elements for the irreducible classes with [PITH_FULL_IMAGE:figures/full_fig_p019_5.png]
Figure 6
Figure 6. Figure 6: Diagrams of representative elements for the irreducible classes with [PITH_FULL_IMAGE:figures/full_fig_p019_6.png]
Figure 7
Figure 7. Figure 7: Diagrams of representative elements for the irreducible classes with [PITH_FULL_IMAGE:figures/full_fig_p020_7.png]
Figure 8
Figure 8. Figure 8: All the sets 𝐴 𝑘𝐶𝑖 \ (∪𝑘−1 𝑚=1𝐴 𝑚𝐶𝑖) for all the classes𝐶𝑖 for three tribits. For a row number 0 ≤ 𝑖 ≤ 38 and a column number 0 ≤ 𝑗 ≤ 38, we have 𝐶𝑗 ∈ 𝐴 𝑘𝐶𝑖 \ (∪𝑘−1 𝑚=1𝐴 𝑚𝐶𝑖), where 𝑘 is the number in the position (𝑖, 𝑗) in the table. (These 𝐶𝑗 are the only classes to …

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  32. [40]

    For example, for (B) we have 𝑛1,2,3(𝑣′+1,2𝑣′′) =𝑑1𝑑2𝑑3−𝑁 𝑤1∈𝑉(1),...,𝑤 4∈𝑉(4) : 𝐶1𝐶2(𝑣′⊗𝑤∗ 1)+ 𝐶1𝐶2(𝑣′′⊗𝑤∗

    = 0} and the lemma follows directly by using (4.2). For example, for (B) we have 𝑛1,2,3(𝑣′+1,2𝑣′′) =𝑑1𝑑2𝑑3−𝑁 𝑤1∈𝑉(1),...,𝑤 4∈𝑉(4) : 𝐶1𝐶2(𝑣′⊗𝑤∗ 1)+ 𝐶1𝐶2(𝑣′′⊗𝑤∗

  33. [41]

    = 0 ≥𝑑1𝑑2𝑑3 − min n 𝑑1𝑑2𝑑3,𝑁 𝑤1∈𝑉(1) :𝐶1𝐶2(𝑣′⊗𝑤∗

  34. [42]

    = 0 +𝑁 𝑤4∈𝑉(4) :𝐶1𝐶2(𝑣′′⊗𝑤∗

  35. [43]

    = 0 +𝑁 𝑤2∈𝑉(2) :𝐶1𝐶3(𝑣′⊗𝑤∗

  36. [45]

    = 0 o =𝑑1𝑑2𝑑3− min n 𝑑1𝑑2𝑑3,(𝑑′ 1𝑑′ 2𝑑3−𝑛1,2,3(𝑣′))+( 𝑑′′ 1𝑑′′ 2𝑑3−𝑛1,2,3(𝑣′′)) +𝑁 𝑤2∈𝑉(2) :𝐶1𝐶3(𝑣′⊗𝑤∗

  37. [46]

    = 0 +𝑁 𝑤3∈𝑉(3) :𝐶1𝐶3(𝑣′′⊗𝑤∗

  38. [47]

    = 0 o We now use 𝑁 𝑤2∈𝑉(2) :𝐶1𝐶3(𝑣′⊗𝑤∗

  39. [48]

    = 0 ≤ min 𝑑′ 1𝑑′′ 1+𝑑′ 2𝑑′′ 2,𝑑′′ 2(𝑑′ 1𝑑3−𝑛1,3(𝑣′))+ 𝑑′ 1(𝑑′′ 2𝑑3−𝑛2,3(𝑣′′)) =𝑑′′ 2(𝑑′ 1𝑑3−𝑛1,3(𝑣′))+ 𝑑′ 1(𝑑′′ 2𝑑3−𝑛2,3(𝑣′′)), (46) 𝑁 𝑤3∈𝑉(3) :𝐶1𝐶3(𝑣′′⊗𝑤∗

  40. [49]

    = 0 ≤ min 𝑑′ 1𝑑′′ 1+𝑑′ 2𝑑′′ 2,𝑑′′ 1(𝑑′ 2𝑑3−𝑛2,3(𝑣′))+ 𝑑′ 2(𝑑′′ 1𝑑3−𝑛1,3(𝑣′′)) =𝑑′′ 1(𝑑′ 2𝑑3−𝑛2,3(𝑣′))+ 𝑑′ 2(𝑑′′ 1𝑑3−𝑛1,3(𝑣′′)), (47) which follow from the definitions of𝑛1,3(𝑣′),𝑛2,3(𝑣′′),𝑛2,3(𝑣′) and𝑛1,3(𝑣′′) together with the requirement that the number of constraints cannot...

  41. [50]

    = 0}∪{ 𝑤3∈𝑉(3) :𝐶1𝐶3(𝑣′⊗𝑤∗

  42. [51]

    = 0} ∪{𝑤4∈𝑉(4) :𝐶2𝐶3(𝑣′′⊗𝑤∗

  43. [52]

    = 0}∪{ 𝑤5∈𝑉(5) :𝐶2𝐶3(𝑣′⊗𝑤∗

  44. [53]

    = 0}, ∪{𝑤6∈𝑉(6) :𝐶1𝐶3(𝑣′′⊗𝑤∗

  45. [54]

    = 0}∪{ 𝑤7∈𝑉(7) :𝐶1𝐶2(𝑣′′⊗𝑤∗

  46. [55]

    For example, (B) follows from 𝑛1,2,3(𝑣′+1,2,3𝑣′′) =𝑛1,2,3(𝑣′)+ 𝑛1,2,3(𝑣′′) +𝑑′ 1𝑑′ 2𝑑′′ 3−𝑁 𝑤2∈𝑉(2) :𝐶1𝐶2(𝑣′⊗𝑤∗

    = 0} and the lemma follows directly. For example, (B) follows from 𝑛1,2,3(𝑣′+1,2,3𝑣′′) =𝑛1,2,3(𝑣′)+ 𝑛1,2,3(𝑣′′) +𝑑′ 1𝑑′ 2𝑑′′ 3−𝑁 𝑤2∈𝑉(2) :𝐶1𝐶2(𝑣′⊗𝑤∗

  47. [56]

    = 0 +𝑑′ 1𝑑′′ 2𝑑′ 3−𝑁 𝑤3∈𝑉(3) :𝐶1𝐶3(𝑣′⊗𝑤∗

  48. [57]

    = 0 +𝑑′ 1𝑑′′ 2𝑑′′ 3−𝑁 𝑤4∈𝑉(4) :𝐶2𝐶3(𝑣′′⊗𝑤∗

  49. [58]

    = 0 +𝑑′′ 1𝑑′ 2𝑑′ 3−𝑁 𝑤5∈𝑉(5) :𝐶2𝐶3(𝑣′⊗𝑤∗

  50. [59]

    = 0 +𝑑′′ 1𝑑′ 2𝑑′′ 3−𝑁 𝑤6∈𝑉(6) :𝐶1𝐶3(𝑣′′⊗𝑤∗

  51. [60]

    = 0 +𝑑′′ 1𝑑′′ 2𝑑′ 3−𝑁 𝑤7∈𝑉(7) :𝐶1𝐶2(𝑣′′⊗𝑤∗

  52. [61]

    □ Theorem 4.1

    = 0 =𝑛1,2,3(𝑣′)+ 𝑛1,2,3(𝑣′′)+ 𝑑′′ 3𝑛1,2(𝑣′)+ 𝑑′′ 2𝑛1,3(𝑣′)+ 𝑑′ 1𝑛2,3(𝑣′′) +𝑑′′ 1𝑛2,3(𝑣′)+ 𝑑′ 2𝑛1,3(𝑣′′)+ 𝑑′ 3𝑛1,2(𝑣′′). □ Theorem 4.1. For the direct sum decompositions 𝑉𝑎 =(⊕𝑎)𝑝 𝑞=1𝑉(𝑞) 𝑎 , 1≤ 𝑎≤ 3,𝑝≥ 1, let 𝑉(𝑞) =𝑉(𝑞) 1 ⊗ 𝑉(𝑞) 2 ⊗𝑉(𝑞) 3 , 1≤ 𝑞≤ 𝑝 be the{1, 2, 3} diagonal blo...

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.