{"id":"93be6827-a381-4f12-a5b7-ecedcd937558","arxiv_id":"2608.13366","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"Distinguished varieties in Theta_n are shown to admit a determinantal representation, and pure Theta_n-contractions satisfying an assumed fundamental-operator condition are given dilations, functional models, and a von Neumann inequality on distinguished varieties.","lead":"This paper introduces distinguished varieties in the domain Theta_n, proves a determinantal representation for them, and builds dilations and functional models for a class of pure Theta_n-contractions. It also proves a von Neumann inequality on distinguished varieties for certain Theta_n-contractions whose last adjoint is a pure contraction.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The converse of Theorem 2.6 relies on an unproved transfer of [32, Lemma 3.4] to Θ_n; that step is what forces H^2(µ)⊖Ran M_{θ_n} to be finite-dimensional, so without it the determinantal representation has no foundation.","rationale":"Agree with the reader that the most fragile premise is the import of [32, Lemma 3.4]. This is load-bearing because it is the only step producing finite matrices; without it, every distinguished variety need not admit a determinantal representation of the stated form. The paper's title and abstract advertise this as a main theorem. A secondary observation: Lemma 2.3 as stated is not correct (the standard identity is M_f^*k_λ = \\overline{f(λ)}k_λ, not f(λ)k_λ), but replacing it by the conjugate version still yields the intended representation, so this is a fixable sign error rather than the main threat. The abstract also overstates Theorem 4.1 by omitting the assumed coefficient family; however, the verdict is already CONDITIONAL and the missing existence of A_l is a separate issue. Recommended verdict: keep CONDITIONAL; require either a self-contained proof of the transferred lemma or an explicit statement that Theorem 2.6 is conditional on it.","tokens_in":20616,"tokens_out":27267,"duration_ms":258133,"concrete_test":"Take the stated hypotheses of [32, Lemma 3.4] and verify them for an arbitrary distinguished variety in Θ_n. If they do not match, attempt a direct proof of the needed algebraic dependence for a concrete nontrivial case, e.g. n=3, p=2, Ω = θ(ˆΩ) for a distinguished variety ˆΩ in D^3 not contained in a coordinate hyperplane. Compute the quotient H^2(µ)/(θ_n H^2(µ)) for the measure from Lemma 2.2; if it is infinite-dimensional, the converse of Theorem 2.6 is false. Alternatively, run the same computation for the symmetrized polydisc (p=1) where [32, Lemma 3.4] is known, and compare where the proof uses p=1 in an essential way.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section 2, converse half of Theorem 2.6: after defining q_i(z) from the finite fibre over θ'_n, the proof invokes [32, Lemma 3.4] to conclude θ_i^k ∈ span{1,...,θ_i^{k-1}} + θ_n H^2(µ). This inclusion is then used verbatim to get H^2(µ) = span{θ_1^{l_1}...θ_{n-1}^{l_{n-1}}: 0≤l_i<k} + θ_n H^2(µ), hence finite-dimensional H^2(µ)⊖Ran M_{θ_n}. That finite-dimensionality is the entire source of the N×N matrices A_l^{(i)} and of the representation (2.2). No proof of [32, Lemma 3.4] is given, nor are its hypotheses checked for Θ_n; the lemma was proved in Pal's program for a family of domains, and the paper does not show that family includes Θ_n. If the lemma does not transfer, the converse of Theorem 2.6 collapses, and with it Theorem 2.8 and the geometric input to Theorem 4.1. The step is not a routine application: it converts an algebraic statement about a single fiber (q_i vanishes on Ω_{θ'_n}) into a global algebraic dependence modulo θ_n H^2(µ), which is exactly the kind of domain-specific fact that fails for general algebraic curves.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript develops an operator-theoretic and complex-geometric theory for the generalized symmetrized domains Θ_n. Its main structural claim, Theorem 2.6, is a determinantal representation for every distinguished variety in Θ_n: each such variety is contained in an affine algebraic curve that is a set-theoretic complete intersection, and the variety is described by joint spectra of matrix-valued polynomials built from coefficient matrices A_l^{(i)}. On this basis the paper proves polynomial convexity of closures of distinguished varieties (Theorem 2.8), establishes a correspondence between distinguished varieties in Θ_n and in the polydisc (Theorem 2.7), constructs minimal Θ_n-isometric dilations and functional models for pure Θ_n-contractions satisfying equations (3.4) (Theorems 3.4 and 3.6), and proves a matricial von Neumann inequality on distinguished varieties for Θ_n-contractions with pure T_n^* under additional coefficient hypotheses (Theorem 4.1).","tokens_in":20952,"tokens_out":7972,"duration_ms":75283,"significance":"If the structural theorem and the von Neumann inequality are correct, the paper would be a substantial contribution, extending Agler–McCarthy distinguished varieties and Pal's program to the Θ_n family. The dilation and functional-model sections are cleanly written and the intertwining computations in Theorem 3.4 are explicit and verifiable. The potential strength of the paper, however, is conditional: the central theorem rests on a circularly used complete-intersection assumption in Lemma 2.2, on an imported algebraic-dependence lemma from [32] whose hypotheses are not checked for Θ_n, and on an unstated or circularly used polynomial-convexity property in Lemma 2.3. Because these issues affect the foundation of the determinantal representation, the significance of the paper cannot be assessed until they are resolved.","major_comments":[{"comment":"The proof begins by choosing linearly independent polynomials f_1,...,f_{n-1} such that Ω = { (θ_1,...,θ_n) ∈ Θ_n : f_i(θ)=0 }. This assumes that every one-dimensional distinguished algebraic variety in Θ_n is a complete intersection cut out by n−1 polynomials. That assumption is exactly one of the conclusions of Theorem 2.6, whose converse proof uses Lemma 2.3 and Lemma 2.4, which in turn depend on Lemma 2.2. No independent proof of the complete-intersection representation is given before Lemma 2.2. Thus the existence of the measure μ, the space H^2(μ), and the subsequent joint-eigenvalue argument are not established for an arbitrary distinguished variety.","section":"Section 2, Lemma 2.2"},{"comment":"The converse half of Lemma 2.3 argues that the inequality |f(θ')| ≤ sup_Ω |f| implies θ' ∈ Ω by invoking 'the polynomial convexity of Ω.' Polynomial convexity of the closure of a distinguished variety is the statement of Theorem 2.8, whose proof uses the determinantal representation of Theorem 2.6, and Theorem 2.6's converse uses Lemma 2.3. If polynomial convexity is part of the definition of distinguished variety in Θ_n, that definition is nowhere stated; if it is not part of the definition, the argument is circular. The manuscript also never formally defines 'distinguished variety in Θ_n,' so the logical status of Lemma 2.3 and Theorem 2.8 cannot be checked.","section":"Section 2, Lemma 2.3 and Theorem 2.8"},{"comment":"The load-bearing step is the invocation of [32, Lemma 3.4] to conclude θ_i^k ∈ span{1,θ_i,...,θ_i^{k-1}} + θ_n H^2(μ) from the vanishing of q_i on the finite fibre over θ'_n. This inclusion is then used verbatim to obtain H^2(μ) = span{θ_1^{l_1}...θ_{n-1}^{l_{n-1}} : 0≤l_i<k} + θ_n H^2(μ), hence finite-dimensional H^2(μ)⊖Ran M_{θ_n}. That finite-dimensionality is the entire source of the N×N matrices A_l^{(i)} and representation (2.2). No proof of [32, Lemma 3.4] is given, and its hypotheses are not verified for Θ_n; the lemma was proved for a different family of domains in Pal's program. This step is not a routine application, since it converts a single-fibre algebraic statement into a global algebraic dependence modulo θ_n H^2(μ). Without an adapted proof, the converse of Theorem 2.6, Theorem 2.8, and the geometric input to Theorem 4.1 collapse. The earlier use of [32, Lemma 3.5] to conclude dim Ω = 1 from finiteness of fibres is imported with the same lack of verification.","section":"Section 2, Theorem 2.6, converse"},{"comment":"The proof applies condition (2) only for |θ_n|<1 and unit joint eigenvectors, and then asserts: 'By the description of Θ_n and the symmetry condition ... the closure of Ω can meet ∂Θ_n only on bΘ_n.' This is not demonstrated. Condition (2) controls the quadratic forms ⟨A_l^{(i)}v,v⟩ only for interior θ_n, and no limiting argument is given to show that points of the algebraic curve lying over |θ_n|=1 belong to bΘ_n. Since the distinguished-boundary-exit condition is part of the definition of a distinguished variety, the forward direction of Theorem 2.6 is incomplete at exactly this point.","section":"Section 2, Theorem 2.6, forward direction"},{"comment":"The proof of the forward direction asserts that if dim Ω̂ > 1, then 'by the same argument used in the proof of Theorem 2.6, an algebraic set of dimension greater than one cannot exit D^n only through the distinguished boundary T^n.' The argument in Theorem 2.6 uses the specific map F(α) = (α_1+α_{n-1}θ'_n{}^p, ...), the homeomorphism property of F, and [12, Theorem 2.5]; none of these steps is reproduced or adapted to the polydisc. The claimed dimension-one conclusion for Ω̂ is therefore unsupported as written.","section":"Section 2, Theorem 2.7"},{"comment":"The proof begins 'Since dim D_{T_n^*}<∞, put m=dim D_{T_n^*},' but the theorem statement does not assume T_n^* has finite defect. A pure contraction can have infinite defect space, and in that case the operators A_l^{(i)} need not be matrices and the distinguished variety Ω_T constructed from them is not defined. Either the finite-defect assumption must be added to the hypotheses or the theorem must be proved for operator-valued coefficients. Additionally, the hypothesis that the A_l^{(i)} 'determine a distinguished variety Ω_T through the determinantal representation of Theorem 2.6' is conditional and no verifiable criterion is supplied, so the applicability of the theorem is unclear.","section":"Section 4, Theorem 4.1"}],"minor_comments":[{"comment":"Theorem 1.2 is cited as Sz.-Nagy [35], but the bibliography entry [35] is V. Paulsen's book; the Sz.-Nagy–Foiaş book is [26] and should be cited instead.","section":"References"},{"comment":"The distinguished boundary bΘ_n is used without being defined. A definition or a reference for the distinguished boundary of Θ_n should be included.","section":"Section 1, Definition 1.4"},{"comment":"The first sentence of Section 3 is grammatically incomplete: 'Every Θ_n-isometry is the restriction of a Θ_n-unitary to a joint invariant subspace, it follows immediately...' should be rephrased.","section":"Section 3, opening"},{"comment":"The proof speaks of 'linearly independent polynomials' f_1,...,f_{n-1}, but linear independence is not what is needed for a regular sequence; if this is intended to be part of the hypotheses, it should be stated as such.","section":"Section 2, Lemma 2.2"},{"comment":"The convention A_l^{(i)}=0 for l∉{0,...,p} is introduced only at the coefficient-comparison step; it should be stated before equation (2.5).","section":"Section 2, Theorem 2.6, converse"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is part of the same research group's program and depends heavily on several unpublished or barely accessible preprints: [12], [23], [24], and [32]. The published [32] concerns a different family of domains, and the transfer to Θ_n is not shown. The circular structure of Section 2 is the main obstacle: Lemma 2.2 assumes the complete-intersection representation that Theorem 2.6 is meant to prove, and Lemma 2.3 invokes polynomial convexity before it is established. I would recommend requiring the authors to provide a self-contained definition of distinguished variety, an independent proof or precise transfer of [32, Lemma 3.4] to Θ_n, and a proof of the finite-dimensional defect assumption in Theorem 4.1. These are substantial but may be addressable in revision; the current version is not yet ready for publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"You should know two things about this paper: it is a genuine extension of the distinguished variety program to the generalized symmetrized domains Θ_n, and its advertised main theorem is conditional in a way the abstract does not admit. The new notion of distinguished variety in Θ_n is coherent, and the determinantal representation, polynomial convexity, dilation, functional model, and von Neumann inequality form a real package. If the structural theorem holds, this is important.\n\nThe forward part of Theorem 2.6—that the determinantal set defined by commuting matrix polynomials satisfying conditions (1)–(4) is a distinguished variety—looks sound. The dilation and model results in Section 3 are standard Sz.-Nagy–Foias machinery once you accept the existence of the coefficient family A_l^(i). The von Neumann inequality in Theorem 4.1 follows from normal boundary dilations and spectral mapping, a known technique.\n\nNow the soft spots. First, the abstract claims that for every pure Θ_n-contraction there exists a variety on which von Neumann holds. Theorem 4.1 requires an assumed family A_l^(i) satisfying (4.1); existence is not established. That is a gap between abstract and theorem, easy to fix by restating the abstract but not negligible.\n\nSecond, and more serious: the converse half of Theorem 2.6 invokes [32, Lemma 3.4] to get the global algebraic dependence θ_i^k ∈ span{1,...,θ_i^{k-1}} + θ_n H^2(µ). That lemma was proved in Pal's program for a family of domains, and the paper does not verify that Θ_n belongs to that family or supply a proof. This step is load-bearing: it is what makes H^2(µ) ⊖ Ran M_{θ_n} finite-dimensional, hence produces the N×N matrices. If the lemma does not transfer, the determinantal representation collapses. The stress-test note is right that this is not a routine application; it converts a single-fiber algebraic fact into a global statement.\n\nThird, several foundations are imported from same-group preprints ([12], [22], [23], [24]), including the characterization of Θ_n-isometries used in Theorem 3.4. That makes verification harder, though not disqualifying.\n\nMy overall take: the paper deserves a serious referee. It is not ready as is; the authors should prove the transferred lemma or state the results conditionally, and align the abstract with Theorem 4.1. But the framework is promising and the forward directions are solid. Worth reading even now, and worth citing once tightened.","headline":"A real but conditional extension of distinguished variety theory to Θ_n; the converse of the determinantal representation leans on an unproved transfer from Pal's earlier work, and the abstract overstates the von Neumann theorem.","tokens_in":21479,"tokens_out":3897,"would_cite":true,"duration_ms":34419,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14H50","14M10","47A20","47A25","32A60","32M15"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper establishes a determinantal representation for every distinguished variety in the domain Θ_n, and uses it to prove polynomial convexity, dilations, functional models, and a von Neumann inequality for pure Θ_n-contractions.","keywords":["Theta_n-contractions","distinguished varieties","determinantal representation","Taylor joint spectrum","von Neumann inequality","functional model","complete intersections","polynomial convexity"],"falsifier":"Exhibit a distinguished variety Ω in Θ_n for which $H^{2}$(μ) ⊖ Ran M_{θ_n} is infinite-dimensional, or for which some fibre over a point θ_n' ∈ D is infinite; either would refute the complete-intersection and determinantal claims of Theorem 2.6.","tokens_in":20373,"feed_emoji":"📐","tokens_out":9106,"duration_ms":80912,"temperature":0.7,"pith_summary":"The paper introduces distinguished varieties in the domain Θ_n, a family of n-dimensional domains that generalizes the symmetrized polydisc. Its central result is a determinantal representation: every distinguished variety in Θ_n can be written as the set of points where the first n−1 coordinates lie in the Taylor joint spectrum of n−1 commuting matrix polynomials evaluated at the last coordinate. The representation makes each distinguished variety part of an affine algebraic curve that is a set-theoretic complete intersection, and it implies the closure is polynomially convex. The same machinery yields a minimal pure Θ_n-isometric dilation and a functional model for pure Θ_n-contractions whose fundamental operators satisfy a natural adjoint relation. As a consequence, matrix polynomials evaluated at such contractions obey a von Neumann inequality with the supremum taken over the distinguished boundary of the associated distinguished variety.","feed_headline":"Theta_n varieties are determinantal complete intersections","feed_subtitle":"For pure Theta_n-contractions the same matrices give a dilation, model, and boundary inequality.","key_machinery":"The machinery is the determinantal curve Ω = {(θ_1,...,θ_n) ∈ Θ_n : (θ_1,...,θ_{n−1}) ∈ σ_T(Φ_1(θ_n),...,Φ_{n−1}(θ_n))}, with Φ_i(z) = Σ_{l=0}^p $A_l^{{(i)}}$ z^l. This representation converts the geometry of a distinguished variety into the joint spectrum of finite commuting matrices, so polynomial convexity becomes separation by determinants and the von Neumann inequality becomes a normal-boundary dilation estimate. The companion mechanism is the fundamental-operator family $A_l^{{(i)}}$ on the defect space D_{T_n^*}, which satisfies the symmetry relation and the equations D_{T_n^*}T_i^* = Σ $A_l^{{(i)*}}$D_{T_n^*}$T_n^{{*l}}$; these make the model operators V_i = Σ_{l=0}^p M_z^l ⊗ $A_l^{{(i)}}$ and V_n = M_z ⊗ I_{D_{T_n^*}} commute and give the Θ_n-isometry relations V_i = V_{n−i}^* V_n^p. The model space H_{T_n} = ($H^{2}$(D) ⊗ D_{T_n^*}) ⊖ M_{Θ_{T_n}}($H^{2}$(D) ⊗ D_{T_n}) then supports the unitary intertwining with the original contraction.","core_discovery":"On the paper's own terms, the discovery is Theorem 2.6: distinguished varieties in Θ_n are exactly the sets of points (θ_1,...,θ_n) ∈ Θ_n for which (θ_1,...,θ_{n−1}) lies in the Taylor joint spectrum of the commuting matrix polynomials Φ_i(θ_n) = Σ_{l=0}^p $A_l^{{(i)}}$ θ_n^l, where the coefficient matrices satisfy the symmetry $A_l^{{(i)}}$ = A_{p−l}^{(n−i)*}, the commutator condition Σ_{l=0}^k [$A_l^{{(i)}}$, A_{k−l}^{(j)}] = 0, a spectral condition connecting joint eigenvectors to the symmetrized (n−1)-disc, and the conditions that the determinantal polynomials f_i(θ) = det(Φ_i(θ_n) − θ_i I) form a regular sequence and generate an irreducible algebraic set. Conversely, every distinguished variety in Θ_n admits such a representation, is contained in an affine algebraic curve that is a set-theoretic complete intersection, and is the image under the map θ of a distinguished variety in the polydisc D^n. From this structure the paper derives polynomial convexity of the closure, a minimal pure Θ_n-isometric dilation and functional model for pure Θ_n-contractions satisfying D_{T_n^*}T_i^* = Σ $A_l^{{(i)*}}$ D_{T_n^*} $T_n^{{*l}}$, and the matricial von Neumann inequality ||P(T_1,...,T_n)|| ≤ sup_{θ ∈ Ω_T ∩ bΘ_n} ||P(θ)||, with the same bound for the adjoint tuple.","pith_inferences":["If the determinantal representation holds for all distinguished varieties, the distinguished-variety programme developed for the bidisc—interpolation, reproducing kernels, complete spectral sets—can likely be pushed through on Θ_n; the first testable step is whether every point of a distinguished variety is a bounded point evaluation for H^2(μ) with dense evaluation vectors.","The pointwise Γ_{n−1}-contraction hypothesis in Theorems 3.4 and 4.1 may be redundant: if the fundamental equations (1.4) together with the symmetry A_l^{(i)} = A_{p−l}^{(n−i)*} already force the matrices to satisfy that condition on the circle, the dilation and von Neumann inequality would cover all pure Θ_n-contractions whose fundamental operators exist.","When dim D_{T_n^*} is infinite, a finite-rank approximation argument using compressions of the fundamental operators may extend the von Neumann inequality to the full class, with the boundary variety replaced by a limit variety.","The determinantal representation suggests an algorithmic test for whether a given algebraic set is distinguished: check whether its defining ideal is generated by n−1 determinants of commuting matrix polynomials satisfying conditions (1)–(4)."],"forward_implications":["Every distinguished variety in Θ_n has finite fibres over the θ_n-coordinate and lies inside an algebraic curve cut out by n−1 polynomials.","The closure of every distinguished variety is polynomially convex, so the distinguished boundary is a natural spectral set for tuples whose spectrum lies on the variety.","Pure Θ_n-contractions satisfying the fundamental-operator relation admit a minimal pure Θ_n-isometric dilation of the form (Σ M_z^l ⊗ A_l^{(1)},...,M_z ⊗ I), and a functional model on the defect space H_{T_n}.","For such contractions, every matrix polynomial P obeys ||P(T)|| ≤ sup_{θ ∈ Ω_T ∩ bΘ_n} ||P(θ)||, and likewise for P(T*), so the boundary of the associated variety is a complete spectral set for the tuple.","Both T and T* admit normal boundary dilations whose joint spectra lie in the distinguished boundary of Θ_n and, more precisely, in the distinguished varieties Ω_T ∩ bΘ_n and Ω_T^* ∩ bΘ_n."],"supporting_citations":[{"why":"Supplies the definition of distinguished variety, the boundary-measure lemma used to build H^2(μ), and the original bidisc model this paper extends.","marker":"[2]"},{"why":"Introduces the domain Θ_n and the classes of Θ_n-contractions, Θ_n-isometries, and purity used throughout.","marker":"[12]"},{"why":"Introduces the generalized symmetrization map θ and the original description of Θ_n used in the characterization.","marker":"[22]"},{"why":"Provides the structure theorems for Θ_n-isometries and Θ_n-contractions and the dilation criterion invoked in the construction.","marker":"[23]"},{"why":"Supplies the parameterization of points of Θ_n by the symmetrized (n−1)-disc, used in condition (2) and in the boundary argument.","marker":"[24]"},{"why":"Gives the classical spectral-set, dilation, and functional-model framework on which Sections 3 and 4 rely.","marker":"[26]"},{"why":"Provides the defect-space identity WW* + M_{Θ_{T_n}}M_{Θ_{T_n}}* = I that identifies the model space in Theorem 3.6.","marker":"[31]"},{"why":"The source of the algebraic-dependence lemma and finite-fibre lemma that carry the converse half of Theorem 2.6, and of the boundary spectral-set lemma used in Theorem 4.1.","marker":"[32]"}],"fun_headline_variants":["Distinguished varieties in Theta_n are determinantal","Pure Theta_n-contractions get a dilation and inequality","Theta_n: von Neumann inequality on distinguished varieties","Closed distinguished varieties are polynomially convex"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The converse of the determinantal representation rests on a cited algebraic-dependence lemma, proved for a different family of domains, which forces each slice of a distinguished variety to be finite; if that lemma does not transfer to Θ_n, the complete-intersection and determinantal conclusions for every distinguished variety collapse.","fun_headline_variants_meta":{"raw":{"variants":["Distinguished varieties in Theta_n are determinantal","Pure Theta_n-contractions get a dilation and inequality","Theta_n: von Neumann inequality on distinguished varieties","Closed distinguished varieties are polynomially convex"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001365,"raw_usage":{"total_tokens":5586,"prompt_tokens":1043,"completion_tokens":4543,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":659,"completion_tokens_details":{"reasoning_tokens":4482}},"tokens_in":659,"tokens_out":4543,"duration_ms":34315,"temperature":1.0,"reasoning_tokens":4482,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T12:45:50.780301+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Exhibit a distinguished variety Ω in Θ_n for which $H^{2}$(μ) ⊖ Ran M_{θ_n} is infinite-dimensional, or for which some fibre over a point θ_n' ∈ D is infinite; either would refute the complete-intersection and determinantal claims of Theorem 2.6.","supporting_citations":[{"cited_title":"Agler and J","cited_arxiv_id":null,"evidence_quote":"Supplies the definition of distinguished variety, the boundary-measure lemma used to build H^2(μ), and the original bidisc model this paper extends."},{"cited_title":"On the Dilation Theory and Canonical Decomposition of $\\mathbf{\\Theta}_n$-Contractions","cited_arxiv_id":"2608.03574","evidence_quote":"Provides the structure theorems for Θ_n-isometries and Θ_n-contractions and the dilation criterion invoked in the construction."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the parameterization of points of Θ_n by the symmetrized (n−1)-disc, used in condition (2) and in the boundary argument."},{"cited_title":"Sz.-Nagy, C","cited_arxiv_id":null,"evidence_quote":"Gives the classical spectral-set, dilation, and functional-model framework on which Sections 3 and 4 rely."},{"cited_title":"Pal,Dilation, functional model and a complete unitary invariant forC .0-contractions, Infinite Dimensional Analysis, Quantum Probability and Related Topics,26, No","cited_arxiv_id":null,"evidence_quote":"Provides the defect-space identity WW* + M_{Θ_{T_n}}M_{Θ_{T_n}}* = I that identifies the model space in Theorem 3.6."},{"cited_title":"Pal,Distinguished varieties in a family of domains associated with spectral interpolation and operator theory, Ann","cited_arxiv_id":null,"evidence_quote":"The source of the algebraic-dependence lemma and finite-fibre lemma that carry the converse half of Theorem 2.6, and of the boundary spectral-set lemma used in Theorem 4.1."}],"review_version":1}