{"id":"27ebb23c-26c8-4306-9d64-6d83644865f3","arxiv_id":"2507.17485","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"A strictly k-fold degeneracy point, when perturbed, can produce at most k^2(k^2-1)/12 complex Weyl points, a number independent of the perturbation.","lead":"This paper proves an upper bound on how many Weyl points, the generic two-level crossings of quantum energy bands, are created when a multi-level degeneracy is split by a perturbation. For a k-fold degeneracy the bound is k^2(k^2-1)/12, and it is independent of the perturbation chosen.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The upper bound ♯WP ≤ k^2(k^2−1)/12 rests on the complex-isolation hypothesis f^{-1}(Σ)={0}; for the crystalline example this is only numerically checked for random parameters and left as a conjecture, so the physical claim is not fully established.","rationale":"The reader's weakest_assumption correctly identifies the complex-isolation condition as the load-bearing premise. The paper's mathematical core is a precise reduction of the count of complex Weyl points to the dimension of O_4/J (Theorem 3.1.1), and the numerical value k^2(k^2−1)/12 is obtained when that algebra is finite-dimensional and f is linear (Corollary 3.2.8). Without f^{-1}(Σ)={0}, the quotient can be infinite-dimensional and the perturbation count can be infinite or perturbation-dependent, so the upper bound would not apply. The paper proves the condition for the spin Hamiltonian (Proposition 4.5.2) but leaves it as a numerical conjecture for the crystalline example (Section 4.6). This is exactly the gap that should be closed before the physical upper bound is asserted for that system. The proposed symbolic computation over the parameter field would settle the conjecture for the whole parameter space, not just random samples. The secondary point about topological genericity of transverse perturbations (Remark A.3.2) is acknowledged by the authors and does not affect the conditional mathematical theorem, but it does affect the physical phrasing 'generic perturbation.' Since the reader's conditional verdict already accounts for these gaps, my assessment leaves the verdict unchanged.","tokens_in":64029,"tokens_out":5131,"duration_ms":51464,"concrete_test":"For the crystalline family H^{(α)} in Eq. (4.6.1), form J_α ⊂ C[x,y,z,λ] from the 3×3 minors of H^{(α)}(x,y,z)−λ·1_4 and compute a Gröbner basis over the rational function field C(α0,α1,α2) to determine dim_{C(α)} C[x,y,z,λ]/J_α. If the dimension is 20 for generic α and remains finite at special points such as α2^2=α0^2+α1^2, the conjecture is supported; if it is infinite or varies, the bound ♯WP≤20 is not justified. An equivalent numerical-algebraic test is to evaluate the dimension at many α including the exceptional surfaces and check stability.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central result (Corollary 4.4.7, via Theorem 3.2.1) requires that the complexified linear map germ f:(C^3,0)→(C^{k×k},0) be isolated with respect to the geometric degeneracy variety Σ, i.e. f^{-1}(Σ)={0} (Section 3.1 and Remark 4.4.6). This hypothesis is what makes the quotient algebra O_4/J finite-dimensional; without it the count ♯cWP is not defined as a finite number and the upper bound may simply fail. The hypothesis is not automatic for linear Hamiltonians: one can write down linear maps whose complex degeneracy set has positive dimension. For the spin Hamiltonian the authors prove it (Proposition 4.5.2), but for the crystalline Hamiltonian (4.6.1) they only verify dim O_4/J=20 for random α and conjecture the isolation for all non-exceptional α (Section 4.6). Thus the headline bound for that material class—and more generally for any linear H without a proof of f^{-1}(Σ)={0}—is conditional on an unverified premise. A secondary gap is that 'generic perturbation' is defined as transverse and topological genericity is not proved (Remark A.3.2), but the isolation condition is the primary load-bearing assumption.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper derives an upper bound on the number of Weyl points produced by perturbing a multifold degeneracy point of a parameter-dependent Hamiltonian. After introducing the geometric degeneracy variety Sigma of complex matrices having an eigenvalue of geometric multiplicity at least two, its lift eSigma, and the determinantal variety Sigma', the authors prove that for a holomorphic map germ f:(C^3,0)->(C^{n x n},A0) with f^{-1}(Sigma,A0;lambda0)={0}, the number of preimages of a transverse perturbation equals dim O4/J, where J is generated by the (n-1)x(n-1) minors of f(x)-(lambda+lambda0)1 (Theorem 3.1.1). For a strictly k-fold degenerate eigenvalue this number is evaluated as k^2(k^2-1)/12 (Theorem 3.2.1), giving the bound WP <= cWP = k^2(k^2-1)/12 for linear Hamiltonians whose complexification is isolated with respect to Sigma (Corollary 4.4.7). The paper also computes the analogous multiplicities for complex symmetric and diagonal families, proves that Sigma is not Cohen-Macaulay, gives Chern-number lower bounds, and discusses spin and crystalline examples.","tokens_in":64301,"tokens_out":19011,"duration_ms":204373,"significance":"If the main theorem is fully established, the formula k^2(k^2-1)/12 is a striking parameter-free prediction: the number of complex Weyl points born from a k-fold degeneracy depends only on k and not on the perturbation. The paper supports this with detailed proofs, including an explicit Gulliksen-Negaard free resolution and Hilbert-series computation, and it gives concrete perturbations for the spin-1 case attaining both the lower bound 4 and the upper bound 6. The non-Cohen-Macaulay result for Sigma is a valuable caution that the vanishing ideal of Sigma cannot be used directly for these counts. The authors are also transparent about what is proven versus conjectured, which is a strength.","major_comments":[{"comment":"The proof passes from f_t^{-1}(Sigma,A0;lambda0) to e f_t^{-1}(eSigma) by invoking the fact that the projection eSigma->Sigma is generically one-to-one (Corollary 2.3.33). However, transversality to the branch (Sigma,A0;lambda0) at non-singular points does not exclude the possibility that f_t(x) has a second geometric degenerate eigenvalue near another eigenvalue mu0 of A0. At such a point the same x appears twice in e f_t^{-1}(eSigma), once for each degenerate eigenvalue, so the equality #f_t^{-1}(Sigma,A0;lambda0)=#e f_t^{-1}(eSigma) can fail under the stated definition of a generic perturbation. The proof needs a stronger genericity condition (for example, transverse and avoiding the singular locus of Sigma) or a transversality argument showing that such bad intersections can be avoided by a one-parameter perturbation. This issue is load-bearing because Theorem 3.1.1 and all subsequent corollaries depend on this equality.","section":"Section 3.1, Theorem 3.1.1 and its proof (Eqs. (3.1.6)-(3.1.9))"},{"comment":"For the crystalline Hamiltonian (4.6.1), the complex-isolation hypothesis f^{-1}(Sigma)={0} is verified only numerically for random values of alpha and is left as a conjecture. Consequently the statement near the end of Section 4.6 that cWP=20 is an upper bound in all regions of alpha is not a theorem: Corollary 4.4.7 and Eq. (1.4.3) apply only once the isolation conjecture is proved. The paper should either provide a proof (for example, by showing symbolically that dim O4/J=20 for all non-exceptional alpha) or explicitly label the crystalline application as conjectural in the abstract and in the summary of results in Section 1.4.","section":"Section 4.6 and Remark 4.4.6"},{"comment":"The paper defines 'generic perturbation' as transverse in the sense of Section A.3, but it does not prove that such perturbations form an open dense set or even that they exist for every Hamiltonian germ. The physical statements in the abstract and in Section 4.4 use 'generic' in the usual sense of 'almost all' perturbations. The authors should either prove or cite a stratified transversality theorem that supplies such perturbations, or consistently state all results for 'every transverse perturbation with respect to Sigma' rather than for 'generic perturbations'. This is not merely a terminology issue, because the proof of Theorem 3.1.1 currently relies on an implicit genericity assumption to avoid the overcounting described above.","section":"Remark A.3.2 and Appendix A.3"}],"minor_comments":[{"comment":"The word 'subvariaty' should be 'subvariety'.","section":"Section 2.1"},{"comment":"The word 'strictrly' should be 'strictly'.","section":"Remark 4.2.2"},{"comment":"The word 'Therorem' in the introduction to Section 3.2 should be 'Theorem'.","section":"Section 3.2"},{"comment":"The word 'neighborhhod' should be 'neighborhood'.","section":"Lemma 2.3.30 proof"},{"comment":"The word 'coomplete' should be 'complete'.","section":"Appendix A.3, proof of Proposition A.3.3"},{"comment":"The phrase 'can be apllied' should be 'can be applied'.","section":"Section 4 (introductory paragraph)"}],"recommendation":"major_revision","confidential_remarks":"This is a mathematically ambitious manuscript with substantial correct-looking algebraic work. The main issue is that Theorem 3.1.1, as stated and proved, does not fully justify the transition from counts of Sigma-intersections to counts of eSigma-intersections for all perturbations satisfying the paper's own definition of generic; this needs either a strengthened definition or an additional transversality argument. The crystalline example is explicitly conjectural and should be presented as such in the abstract. With those fixes the paper would be a strong contribution to the math-physics interface."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Read the full paper. Bottom line: the central formula is right and the main theorem is proved carefully. The clean statement is that for a strictly k-fold degeneracy of a linear Hamiltonian, a generic perturbation produces exactly k^2(k^2−1)/12 complex Weyl points, so the real Weyl point count is bounded above by that number, independent of the perturbation. The proof goes through the geometric degeneracy variety, lifts it to the Cohen–Macaulay determinantal variety, and computes the multiplicity with the Gulliksen–Negaard free resolution. That is a substantial generalization of the two-fold result in the authors' earlier work, and the direct computation is a real contribution even though the determinantal multiplicity itself was known. The complex Schrieffer–Wolff chart and the proof that the geometric degeneracy variety is not Cohen–Macaulay are also genuinely new and make the paper more than a repackaged formula.\n\nThe exposition is long but self-contained, with real examples and honest statements about what is and is not proved. The citation pattern is fine: the self-citations to [50] and [51] are for published tools and base cases, and the central calculation does not reduce to them.\n\nThe soft spots are real but mostly proportionate. The load-bearing hypothesis is complex isolation: f^{-1}(Σ)={0}. The theorem is stated conditionally on that hypothesis, so the math is not flawed, but the condition is not automatic. For the spin Hamiltonian it is proved; for the crystalline Hamiltonian of Section 4.6 it is only checked numerically for random alpha and then conjectured for whole regions. That means the physical upper bound for that material class is not fully established. The authors should either prove the conjecture or present the bound for that example as conditional. Minor but worth saying: they define 'generic perturbation' as transverse and explicitly note in Remark A.3.2 that topological genericity is not proved. That is a gap between the physical language and the technical definition, though it does not damage the main theorem.\n\nWho is this for? Mathematical physicists and singularity theorists, plus condensed-matter readers who want a bound on Weyl point counts and are willing to work through local algebra. It deserves a serious referee: the central argument holds up, the new tools are useful, and the open points are clearly localizable rather than fatal. Send it to review.","headline":"For a k-fold degeneracy, the upper bound ♯WP ≤ k^2(k^2−1)/12 is proved with care; the physics examples inherit a genuine, explicitly flagged caveat about complex isolation.","tokens_in":64927,"tokens_out":2397,"would_cite":true,"duration_ms":30826,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14B05","14M12","32S05","15A18","81Q70"],"pacs":["02.40.-k","03.65.Vf","71.18.+y"],"model":"deepseek-v4-flash","headline":"For a Hamiltonian whose k-fold degeneracy is isolated in the complex sense, every generic perturbation produces exactly k^2(k^2-1)/12 complex Weyl points, an upper bound on the number of real Weyl points that can be born.","keywords":["multifold degeneracy","Weyl points","complex Weyl points","degeneracy variety","determinantal variety","multiplicity","Schrieffer-Wolff transformation","singularity theory"],"falsifier":"Take an explicit linear Hamiltonian $H:(\\mathbb{R}^3,0)\\to(\\mathrm{Herm}(k),0)$ with $f^{-1}(\\Sigma)=\\{0\\}$ and compute $\\dim_{\\mathbb{C}}\\mathcal{O}_4/J$ for its minors ideal: any value different from $k^2(k^2-1)/12$, or two generic perturbations with different numbers of complex solutions, would refute the central formula. For the crystalline example of Section 4.6, finding a parameter value $\\alpha$ where the degeneracy is isolated for real $k$ but $\\dim_{\\mathbb{C}}(\\mathcal{O}_4^{\\mathbb{C}}/J^{\\mathbb{C}})$ is infinite would falsify the conjecture that real isolation implies complex isolation, while a proof of finiteness at the exceptional values $\\alpha_2=\\pm\\sqrt{\\alpha_0^2+\\alpha_1^2}$ would extend the bound into the regime currently left open.","tokens_in":63782,"feed_emoji":"⚛️","tokens_out":21824,"duration_ms":171976,"temperature":0.7,"pith_summary":"When the energy levels of a parameter-dependent quantum Hamiltonian coincide at a point of parameter space, that multifold degeneracy is dissolved by a generic perturbation into several ordinary two-fold degeneracies, the Weyl points. This paper proves that the number of Weyl points thus born is controlled by a number that depends only on the degeneracy order: for a linear Hamiltonian with a $k$-fold degeneracy that is isolated in the complex sense, every generic perturbation produces exactly $k^2(k^2-1)/12$ complex Weyl points, and hence at most that many real Weyl points. The count is an algebraic intersection number: the complexified Hamiltonian is a holomorphic map from three complex parameters into the space of complex matrices, and the born Weyl points are its transverse intersections with the geometric degeneracy variety, the set of matrices having some eigenvalue with at least two eigenvectors. The authors show this number is the same for every generic perturbation and does not depend on the size $n$ of the Hamiltonian matrix. The payoff is that a question about quantum band structure becomes a singularity-theory calculation with a clean, parameter-free answer.","feed_headline":"Isolated k-fold crossings split into k²(k²-1)/12 complex Weyl points","feed_subtitle":"For a k-fold crossing, the number of born complex Weyl points depends only on k and caps the real Weyl points.","key_machinery":"The load-bearing objects are three complex matrix varieties: the geometric degeneracy variety $\\Sigma\\subset\\mathbb{C}^{n\\times n}$ of matrices with some eigenvalue of geometric multiplicity at least two; the lifted variety $\\widetilde{\\Sigma}=\\{(A,\\lambda): A-\\lambda 1\\in\\Sigma'\\}$; and the determinantal variety $\\Sigma'$ of matrices of rank at most $n-2$, whose vanishing ideal is generated by the $(n-1)\\times(n-1)$ minors. The argument runs through four mechanisms. (1) The counting formula (Theorem 3.1.1): for a holomorphic map germ $f:(\\mathbb{C}^3,0)\\to(\\mathbb{C}^{n\\times n},A_0)$ isolated against $(\\Sigma,A_0;\\lambda_0)$, the number of complex Weyl points of a generic perturbation equals $\\dim_{\\mathbb{C}}\\mathcal{O}_4/J$, where $J$ is generated by the minors $M_{ij}(f(x)-(\\lambda+\\lambda_0)1)$; finiteness of this dimension is equivalent to isolation in the complex sense (Remark 4.4.6). (2) The reduction: since $\\widetilde{\\Sigma}\\cong\\Sigma'\\times\\mathbb{C}$ is Cohen–Macaulay while $\\Sigma$ itself is not, the count is computed through the lifted variety and the determinantal ideal, where the key perturbation-invariance principle (Proposition A.3.3) applies. (3) Local triviality by the complex Schrieffer–Wolff chart: a local biholomorphism $g(S,C,A_{\\mathrm{eff}})$ puts any strictly $k$-fold degenerate matrix $A_0$ in the same normal form as the origin in $\\mathbb{C}^{k\\times k}$, yielding $(\\Sigma^{(n)},A_0;\\lambda_0)\\cong(\\Sigma^{(k)},0)\\times\\mathbb{C}^{n^2-k^2}$ and transferring all multiplicities to the origin (Corollary 2.3.15). (4) The evaluation $\\mathrm{mult}(\\Sigma'^{(k)},0)=k^2(k^2-1)/12$ via the graded Hilbert series of $\\mathcal{O}_4/I_{k-1}(f)$ computed from the Gulliksen–Negård free resolution of the ideal of $(k-1)\\times(k-1)$ minors of a generic linear $f$ (Section 3.5).","core_discovery":"The paper's central claim is that a strictly $k$-fold degeneracy of a linear Hamiltonian $H:(\\mathbb{R}^3,0)\\to(\\mathrm{Herm}(k),0)$, $H(0)=0$, whose complexification $f$ is isolated against the geometric degeneracy variety $\\Sigma\\subset\\mathbb{C}^{k\\times k}$, dissolves under a generic perturbation into exactly $k^2(k^2-1)/12$ complex Weyl points, so that $\\sharp\\mathrm{WP}\\le\\sharp\\mathrm{cWP}=k^2(k^2-1)/12$ holds for the number of real Weyl points (Eq. (1.4.3), Corollary 4.4.7). In the more general form (Corollary 3.2.6), any map germ with a strictly $k$-fold degenerate eigenvalue that is isolated in the complex sense satisfies $\\sharp f_t^{-1}(\\Sigma,A_0;\\lambda_0)=k^2(k^2-1)/12$, independent of the perturbation and of the ambient matrix size $n$; the authors propose to call such degeneracies $k$-fold Weyl points. The supporting multiplicity identities (Theorem 3.2.1) state that $\\mathrm{mult}(\\Sigma^{(n)},A_0;\\lambda_0)=\\mathrm{mult}(\\widetilde{\\Sigma}^{(n)},(A_0,\\lambda_0))=\\mathrm{mult}(\\Sigma'^{(n)},B_0)=k^2(k^2-1)/12$, with analogues $k(k^2-1)/6$ for complex-symmetric two-parameter families and $k(k-1)/2$ for diagonal one-parameter families. A byproduct is that $\\Sigma$ is not Cohen–Macaulay for $k\\ge 3$ (Theorem 3.7.11), which forces the counting argument to pass through the determinantal variety $\\Sigma'$ and shows that pulling back the vanishing ideal of $\\Sigma$ itself would give the wrong answer.","pith_inferences":["The paper leaves open whether the upper bound is generically sharp for real Weyl points: for spin-1 it exhibits one perturbation reaching the full 6 real points and notes that constant (translation) perturbations of that Hamiltonian cannot reach 6. A direct numerical test would be to search the crystalline fourfold example for a perturbation producing all 20 real Weyl points.","Remark 4.4.9 raises the open question of which Chern-number patterns a $k$-fold Weyl point can carry; if those patterns were classified, the integers reachable between the Chern lower bound and the upper bound $k^2(k^2-1)/12$ would likely follow, yielding a sharper prediction for each symmetry class.","Since the count is a purely holomorphic intersection number, the same formulas should transfer to other matrix ensembles with the same local geometry, such as real-symmetric or antisymmetric families, and to degeneracies at boundaries of parameter regions, where proving the paper's conjectured equivalence between real and complex isolation would make the bound rigorous."],"forward_implications":["The number of complex Weyl points born from a strictly $k$-fold degeneracy depends only on $k$, never on the perturbation, the matrix size $n$, or the physical details of the Hamiltonian, so the bound $\\sharp\\mathrm{WP}\\le k^2(k^2-1)/12$ is universal for linear Hamiltonians with an isolated complex degeneracy.","For a spin-$s$ particle in a magnetic field ($k=2s+1$), the two-sided bound is $k(k^2-1)/6\\le\\sharp\\mathrm{WP}\\le k^2(k^2-1)/12$, and the spin-1 case attains both ends: one perturbation produces 4 real Weyl points, another produces all 6.","For the crystalline fourfold degeneracies of Section 4.6, the formula predicts exactly $4^2(4^2-1)/12=20$ complex Weyl points, so a symmetry-breaking perturbation of such a fourfold crossing creates at most 20 Weyl points when the complex-isolation conjecture holds.","For ordinary two-fold degeneracies, the same algebra recovers the classical description: the local algebra $\\mathcal{O}_4/J$ is isomorphic to the local algebra of the effective map germ, so the degeneracy type is read off directly from the Hamiltonian without performing a Schrieffer–Wolff transformation (Section 4.7).","Because $\\Sigma$ is not Cohen–Macaulay, any counting method that pulls back the vanishing ideal of $\\Sigma$ itself will generically miscount; the route through the lifted variety and the determinantal ideal is essential rather than optional (Theorem 3.7.12)."],"supporting_citations":[{"why":"Supplies the determinantal-variety facts: the ideal of (n-1)-minors is prime, Sigma' is Cohen-Macaulay, and Sigma' is the reduced vanishing locus of that ideal.","marker":"[11]"},{"why":"Proves that the geometric degeneracy variety Sigma is the complexification of the hermitian degeneracy variety Sigma_herm and partially describes its ideal, fixing the central object of the paper.","marker":"[16]"},{"why":"Provides the Gulliksen-Negaard free resolution used in Section 3.5 to compute mult(Sigma', 0) directly from a Hilbert series.","marker":"[34]"},{"why":"Gives the known multiplicity formulas for determinantal and symmetric determinantal varieties that the paper's independent computation reproduces.","marker":"[35]"},{"why":"Supplies the general multiplicity results for minors ideals that serve as the prior baseline for mult(Sigma', 0).","marker":"[37]"},{"why":"Provides the multiplicity theory of holomorphic map germs with respect to analytic set germs, including the key perturbation-invariance principle (Proposition A.3.3).","marker":"[45]"},{"why":"Earlier work by the authors establishing the hermitian Schrieffer-Wolff chart and the geometry of Sigma_herm that the complex SW chart generalizes.","marker":"[50]"},{"why":"The authors' earlier upper bound on Weyl points born from nongeneric two-fold degeneracies, the special case generalized here to multifold degeneracies.","marker":"[51]"},{"why":"Source of the crystalline 4x4 Hamiltonian family used in Section 4.6 to test the prediction sharp-cWP = 20 for fourfold degeneracies.","marker":"[1]"},{"why":"Provides the Jozefiak free resolution used in the symmetric case to prove the linear independence of the pulled-back minors.","marker":"[41]"}],"fun_headline_variants":["k-fold degeneracy yields k²(k²−1)/12 complex Weyl points","Exact count: k²(k²−1)/12 Weyl points from a k-fold point","Multifold crossing fixes Weyl birth count: k²(k²−1)/12","k-fold singularity splits into k²(k²−1)/12 complex Weyl points"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The bounds count complex solutions and apply only when the degeneracy point is isolated in the complex sense: no point of the complexified parameter space arbitrarily close to zero, other than zero itself, may make some eigenvalue geometrically degenerate. For the crystalline band-structure example of Section 4.6 this condition is verified numerically for random parameters and otherwise left as a conjecture.","fun_headline_variants_meta":{"raw":{"variants":["k-fold degeneracy yields k²(k²−1)/12 complex Weyl points","Exact count: k²(k²−1)/12 Weyl points from a k-fold point","Multifold crossing fixes Weyl birth count: k²(k²−1)/12","k-fold singularity splits into k²(k²−1)/12 complex Weyl points"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000781,"raw_usage":{"total_tokens":3571,"prompt_tokens":1189,"completion_tokens":2382,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":805,"completion_tokens_details":{"reasoning_tokens":2287}},"tokens_in":805,"tokens_out":2382,"duration_ms":17567,"temperature":1.0,"reasoning_tokens":2287,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T14:48:17.461823+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take an explicit linear Hamiltonian $H:(\\mathbb{R}^3,0)\\to(\\mathrm{Herm}(k),0)$ with $f^{-1}(\\Sigma)=\\{0\\}$ and compute $\\dim_{\\mathbb{C}}\\mathcal{O}_4/J$ for its minors ideal: any value different from $k^2(k^2-1)/12$, or two generic perturbations with different numbers of complex solutions, would refute the central formula. For the crystalline example of Section 4.6, finding a parameter value $\\alpha$ where the degeneracy is isolated for real $k$ but $\\dim_{\\mathbb{C}}(\\mathcal{O}_4^{\\mathbb{C}}/J^{\\mathbb{C}})$ is infinite would falsify the conjecture that real isolation implies complex isolation, while a proof of finiteness at the exceptional values $\\alpha_2=\\pm\\sqrt{\\alpha_0^2+\\alpha_1^2}$ would extend the bound into the regime currently left open.","supporting_citations":[{"cited_title":"Berry phase, topology, and degeneracies in quantum nanomagnets","cited_arxiv_id":null,"evidence_quote":"Supplies the determinantal-variety facts: the ideal of (n-1)-minors is prime, Sigma' is Cohen-Macaulay, and Sigma' is the reduced vanishing locus of that ideal."},{"cited_title":"Hermitian matrices with a bounded number of eigenvalues","cited_arxiv_id":null,"evidence_quote":"Proves that the geometric degeneracy variety Sigma is the complexification of the hermitian degeneracy variety Sigma_herm and partially describes its ideal, fixing the central object of the paper."},{"cited_title":"Un complexe resolvant pour certain id´ eaux d´ et` erminentiels","cited_arxiv_id":null,"evidence_quote":"Provides the Gulliksen-Negaard free resolution used in Section 3.5 to compute mult(Sigma', 0) directly from a Hilbert series."},{"cited_title":"Weyl points in ball-and-spring mechanical systems","cited_arxiv_id":null,"evidence_quote":"Gives the known multiplicity formulas for determinantal and symmetric determinantal varieties that the paper's independent computation reproduces."},{"cited_title":"On minimality of determinantal varieties","cited_arxiv_id":null,"evidence_quote":"Provides the multiplicity theory of holomorphic map germs with respect to analytic set germs, including the key perturbation-invariance principle (Proposition A.3.3)."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Earlier work by the authors establishing the hermitian Schrieffer-Wolff chart and the geometry of Sigma_herm that the complex SW chart generalizes."},{"cited_title":"Fundamental laws of chiral band crossings: Local constraints, global con- straints, and topological phase diagrams","cited_arxiv_id":null,"evidence_quote":"Source of the crystalline 4x4 Hamiltonian family used in Section 4.6 to test the prediction sharp-cWP = 20 for fourfold degeneracies."},{"cited_title":"A Weyl Fermion semimetal with surface Fermi arcs in the transition metal monopnictide TaAs class","cited_arxiv_id":null,"evidence_quote":"Provides the Jozefiak free resolution used in the symmetric case to prove the linear independence of the pulled-back minors."}],"review_version":1}