{"id":"6b8e1d2f-a024-4cfc-a698-33c5249ed904","arxiv_id":"2412.15202","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"An orthosymplectic extension of the Decay and Fission algorithm is proposed, validated on class S and 6d orbi-instanton quivers, and used to predict new Higgs branch RG flows.","lead":"This paper proposes a rule-based algorithm for computing Coulomb branch Hasse diagrams of orthosymplectic 3d N=4 quiver gauge theories. The authors test the algorithm against known class S and 6d orbi-instanton examples and use it to predict new Higgs branch flows of higher-dimensional SCFTs.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The algorithm's admissibility criterion is calibrated only for U Sp(2) neighbours of SO(6); higher-rank symplectic neighbours and b=−1 special-orthogonal nodes are admitted without the required vacuum analysis, so 'all descendant theories' is not established for general orthosymplectic quivers.","rationale":"The reader's weakest_assumption is the redundancy conjecture used to convert the tentative diagram of Figure 2.1a into the Hasse diagram of Figure 2.1b. That concern is real and well identified: without the conjecture, the algorithm does not uniquely determine the stratification. However, I judge a different gap to be more load-bearing: the completeness of the goodness criterion that decides which quivers are admissible at all. The paper's entire output is a set of quivers generated by decay and fission steps, each step requiring a goodness check. If that check is wrong, the algorithm produces invalid strata or misses valid ones, so the central claim 'systematically predicts all descendant theories' fails irrespective of how redundancies are merged. The paper itself flags the missing vacuum analysis for b = −1 special-orthogonal nodes in Section 4.2, and Section 4.1 treats only U Sp(2) neighbours of SO(6), even though the examples of Section 5.1 use higher-rank symplectic neighbours. The redundancy conjecture is at least explicitly stated, motivated by the Hilbert series equality in equation (2.9), and used carefully; the goodness gap is unacknowledged. The concrete test I propose settles whether Rule 1 is complete: a negative result would show that the calibration does not extend to generic orthosymplectic quivers, undermining the Abstract's claim. The verdict remains CONDITIONAL, as the paper has substantial validation for the calibrated class and the new 6d prediction is interesting, but the condition should include either a general derivation of the balance corrections or a systematic test on quivers with higher-rank symplectic neighbours.","tokens_in":31191,"tokens_out":13168,"duration_ms":115112,"concrete_test":"Compute, via the monopole formula of [11], the R-charge of the minimal half-integer-lattice monopole for a quiver consisting of one SO(6) node adjacent to n USp(4) gauge nodes, for n = 3, 4, 5, 6, with all other nodes balanced. Conventional balance (2.2) gives b = n − 5, so n ≥ 5 is nominally good; for U Sp(2) neighbours the analogous computation in §4.1 shows badness for n < 8. If any minimally charged half-integer monopole has Δ < 1 for n ≥ 5, Rule 1 is incomplete and the algorithm misclassifies a whole family of orthosymplectic quivers as good, directly falsifying the claim that it applies to all orthosymplectic quivers. The test distinguishes whether the 5/8 and 5/4 fractions are an artefact of the U Sp(2) analysis or a genuine general principle.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim requires that every quiver produced by the algorithm is a valid stratum, i.e., its Coulomb branch is a single symplectic singularity. The paper's proxy for this is the balance (2.2) together with Rule 1, which renormalises only U Sp(2) contributions to SO(6) nodes (5/8 and 5/4). Section 4.1 derives these fractions by matching two class-S families and checks half-integer lattice monopoles only for an SO(2K) node with n U Sp(2) neighbours. No analogous computation is given for SO(6) attached to USp(4) or larger symplectic nodes, where the bifundamental weight sum is larger and the half-integer monopole R-charge can be lower; conventional balance can then be non-negative while the theory is bad in the paper's own 'single singular point' sense. Section 4.2 explicitly states that an analysis for even special-orthogonal nodes with balance −1 is lacking, yet Rule 2 and Rule 3b admit b = −1 nodes of any kind and prescribe their decays. The higher-rank examples of Section 5.1 (e.g., the magnetic quiver (5.8) for the D-type orbi-instanton) contain SO(6) nodes adjacent to USp(4) and larger symplectic nodes, precisely the regime where the calibration is silent. Thus the admissibility decisions that generate the tentative diagram, and hence the final Hasse diagram, rest on an unproven and possibly incomplete goodness criterion. This is more load-bearing than the redundancy conjecture: if the goodness criterion is wrong, the algorithm produces spurious or missing descendants regardless of how redundancies are identified.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes an algorithm, termed Orthosymplectic Decay and Fission, for computing Coulomb branch Hasse diagrams of 3d N=4 orthosymplectic quiver gauge theories. The algorithm extends the known unitary Decay and Fission procedure: Rule 1 modifies the balance of SO(6) nodes by assigning effective fractional flavour contributions to neighbouring USp(2) nodes (5/8 or 5/4); Rules 2 and 3 specify when decays are admissible and which decays are allowed; Rules 4a and 4b introduce fission and unitarisation into products with unitary quivers. The authors validate the algorithm on su(4) ~ so(6) class S theories, where orthosymplectic mirrors can be compared with unitary mirrors, and then apply it to 6d D-type orbi-instanton theories and higher-rank so(2N) class S theories, obtaining new predictions for Higgs branch RG flows. The paper contains a detailed worked example, an extended class S comparison, and an appendix application to T^sigma_rho(SO(16)).","tokens_in":31652,"tokens_out":5038,"duration_ms":40560,"significance":"If the proposed algorithm is correct, it is a significant technical advance: it extends a powerful method for extracting symplectic stratifications from unitary quivers to orthosymplectic quivers, with immediate applications to the Higgs branches of higher-dimensional SCFTs. The paper's concrete strengths are its systematic treatment of the su(4)-type class S examples with enhanced flavour symmetry, the cross-checks using Hilbert series and known geometries (for instance the d4 x d4 identification in eq. (2.9)), and the D-type orbi-instanton fission prediction in eqs. (5.9) and (5.10), which goes beyond previously studied g = su(K) cases. The presentation is generally clear and the examples are worked in detail. However, the central admissibility criterion is calibrated rather than derived, and the announced scope -- 'all descendant theories' for all simply-laced orthosymplectic quivers with edge multiplicity one -- is not yet established by the evidence presented.","major_comments":[{"comment":"The effective fractional flavour contributions in Rule 1 are fitted to the su(4) ~ so(6) class S examples in (4.1), and the only half-integer-lattice monopole computation given is for an SO(2K) node with n USp(2) neighbours, eq. (3.20). The paper then applies the algorithm to quivers containing SO(6) nodes adjacent to USp(4) or larger symplectic nodes, e.g. the D-type orbi-instanton magnetic quiver (5.8) and the class S quiver (5.17). In this regime the calibrated 5/8 and 5/4 contributions have no demonstrated validity, so the admissibility of the generated quivers -- and hence the claimed Hasse diagrams -- is not established by the manuscript. A concrete way to close this gap would be to generalise the monopole R-charge computation of eq. (3.20) to USp(2r) neighbours and to test whether Rule 1 reproduces those thresholds.","section":"§4.1 and eq. (3.20), Rule 1"},{"comment":"Rule 2 and Rule 3b admit gauge nodes with balance b = -1 of any orthosymplectic kind, but §4.2 explicitly states that an analysis for even special-orthogonal nodes with b = -1 is lacking and is left for future work. This is load-bearing because Rule 3b prescribes decays of such b = -1 nodes, and the higher-rank examples in §5.1 and §5.2 rely on those admissibility decisions. Since the paper's own criterion for applicability is that the Coulomb branch has exactly one singular point (footnote 2 and §4.2), the absence of this analysis leaves open the possibility that some admitted b = -1 special-orthogonal nodes are bad in a stronger sense that changes the set of allowed decays.","section":"§4.2, Rules 2 and 3b"},{"comment":"The paper introduces an additional admissibility criterion when discussing eq. (3.18): a bad 4d class S theory is taken to imply that its 3d mirror quiver is bad and should be excluded from the Decay and Fission products. This criterion is used to eliminate the transition in eq. (3.18), and it is not derived from the 3d quiver data alone. Its validity outside the su(4) ~ so(6) class S setting is not assessed, yet it functions as an input to the algorithm whenever a candidate quiver is discarded on these grounds. The paper should either derive this criterion from the 3d perspective or explicitly state it as an additional conjecture with a precise domain of applicability.","section":"§3, discussion of eq. (3.18)"},{"comment":"The algorithm does not decide which of its output quivers are redundant; converting the tentative diagram in Figure 2.1a into the Hasse diagram in Figure 2.1b uses the conjecture, stated after eq. (2.9), that two theories with identical Coulomb branch Hasse diagrams are different presentations of the same theory. The paper provides one Hilbert series check, eq. (2.9), and one Hasse-diagram comparison for the pair in eq. (3.12), but these do not establish the conjecture in general. Since the central claim is to 'systematically predict all descendant theories' and thereby determine the stratification, this conjecture is part of the algorithm's operating assumptions and should be formulated as a separate, explicitly flagged axiom.","section":"§2, Figure 2.1 and eq. (2.9)"}],"minor_comments":[{"comment":"The sentence 'an alternative one for even special-orthogonal nodes is lacking' is grammatically unclear and should be reworded to something like 'an analogous analysis for even special-orthogonal nodes is lacking'.","section":"§4.2"},{"comment":"In the passage after eq. (3.19), 'only for n = 7, the theory we get is, in fact, free' is awkwardly phrased; consider rephrasing to clarify that for n = 7 the resulting theory is a free theory.","section":"§3"},{"comment":"The balance calculation b = 4 * 5/4 + 5/8 - 5 for the central SO(6) node in eq. (2.4) relies on the distinction between the two USp(2) contribution cases in Rule 1; the text should state explicitly which USp(2) nodes are counted with 5/4 and which with 5/8, since the example is discussed before Rule 1 is formally introduced.","section":"§2, eq. (2.4)"},{"comment":"The abstract and Section 6 say the algorithm applies to 'all orthosymplectic quivers', while the body restricts to simply-laced quivers with edge multiplicity one and excludes cases whose Coulomb branch symmetry involves g2 or f4; the scope statement should be consistent from the outset.","section":"§2 and §6"},{"comment":"The claim that the D-type orbi-instanton fission is 'the first time that an explicit example of this phenomenon was carried out beyond the g = su(K) case' would benefit from a brief comparison with the results of Ref. [37], which is cited but not discussed in relation to this claim.","section":"§5.1"}],"recommendation":"major_revision","confidential_remarks":"The paper is a solid and useful contribution to the magnetic quiver literature, and its class S comparisons are convincing as far as they go. My main reservation is that the central admissibility criterion is calibrated on su(4)-type examples and then applied to higher-rank orthosymplectic quivers without the analogous monopole computation; the authors themselves flag the missing even special-orthogonal b = -1 analysis. This is fixable within the manuscript's scope by extending the computation in eq. (3.20) and by explicitly listing the axioms/conjectures on which the algorithm relies. I would support publication after such a revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know about this paper. It genuinely extends Decay and Fission from unitary to orthosymplectic quivers, and that is not just bookkeeping: new processes (unitarisation, mixed orthosymplectic/unitary fission) and a modified balance rule for SO(6) nodes are needed. And the validation is real but partly fitted: the 5/8 and 5/4 balance contributions in Rule 1 are chosen to reproduce the known goodness thresholds of su(4)~so(6) class S, so matching that class S data is partly a restatement of the input rather than an independent check. The paper is upfront about this, and then goes beyond the calibration to make new predictions for D-type orbi-instantons and higher-rank so(2N) class S, which is where the value lies.\n\nWhat the paper does well: it checks the algorithm against all enhanced-symmetry su(4)~so(6) class S mirrors, and gets the unitary Hasse diagrams right. The 6d orbi-instanton fission into the orthosymplectic quiver times an E-string quiver is a concrete, novel result that matches F-theory expectations. The appendix on T^σ_ρ(SO(16)) is also a nice sanity check against nilpotent orbit dominance ordering. The authors are also unusually honest about limitations: they state the redundancy conjecture explicitly, and in Section 4.2 they say the b=-1 special-orthogonal vacuum analysis is lacking.\n\nThe main soft spot is more serious than the redundancy conjecture. The admissibility criterion that decides which quivers the algorithm outputs is calibrated only for USp(2) neighbours of SO(6). Section 4.1 derives the effective fractional flavour contributions for exactly that configuration and checks half-integer monopoles for an SO(2K) node with n USp(2) neighbours. The algorithm then freely admits b=-1 nodes of any kind (Rule 2 and Rule 3b), and is applied to quivers with USp(4) and larger symplectic neighbours - the D-type orbi-instanton quiver (5.8) has exactly those adjacencies. No vacuum analysis is given for those regimes. So the central claim that the algorithm predicts all descendant theories is not established for general orthosymplectic quivers. If the goodness criterion is wrong in those regimes, the algorithm will generate spurious or missing descendants regardless of how redundancies are identified. This is a genuine gap, not a manufactured one, and the paper's own text points at it.\n\nWho it is for: people working on magnetic quivers, symplectic singularity stratifications, and 5d/6d SCFT Higgs branch RG flows. It deserves a serious referee, because the algorithm is useful, the predictions are concrete and checkable, and the gaps are explicitly flagged. I would push the authors to extend the monopole analysis to higher-rank symplectic neighbours and to either prove or substantially broaden the validation of the balance rule. I would cite it, and I would bring it to reading group.","headline":"Real extension of Decay and Fission to orthosymplectic quivers with genuine new predictions, but the goodness criterion is calibrated only for USp(2) neighbours of SO(6) and the b=-1 vacuum analysis is missing, so 'all descendants' is not yet established.","tokens_in":32187,"tokens_out":3279,"would_cite":true,"duration_ms":21871,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"An algorithm—Orthosymplectic Decay and Fission—derives the full Coulomb branch Hasse diagram of any simply-laced orthosymplectic 3d N=4 quiver, predicting every descendant theory from Coulomb branch Higgsing.","keywords":["orthosymplectic quivers","Decay and Fission","Coulomb branch","Hasse diagram","symplectic singularity","Higgs branch RG flow","magnetic quiver","class S theories"],"falsifier":"Compute an independent invariant—the Coulomb branch Hilbert series where the quiver is good, or the Hall–Littlewood index or the class $\\mathcal{S}$ spectrum where it is not—for two quivers that the algorithm identifies as equivalent, such as the pair in equation (3.12); if the invariants differ, the identification conjecture fails. A single orthosymplectic quiver whose algorithmically predicted Hasse diagram disagrees with the diagram of its independently known unitary dual, for any of the $\\mathfrak{so}(2N)$ class $\\mathcal{S}$ flows checked against the existing D-type classification, would similarly falsify the rules.","tokens_in":30921,"feed_emoji":"🧲","tokens_out":12112,"duration_ms":94976,"temperature":0.7,"pith_summary":"This paper proposes an algorithm that takes any 3d $\\mathcal{N}=4$ quiver gauge theory built from orthogonal and symplectic gauge nodes and returns the full Hasse diagram of its Coulomb branch: the ordered collection of singular strata, each connected to the next by a transverse slice, which is the moduli space of the residual theory after Higgsing. Physically, each stratum is a distinct way of giving vacuum expectation values to monopole operators, so the diagram is a complete account of Coulomb branch Higgsing. The rules are validated through the Lie algebra isomorphism $\\mathfrak{su}(4)\\cong\\mathfrak{so}(6)$, where the same moduli space admits both a unitary and an orthosymplectic quiver description, allowing direct comparison with the established unitary Decay and Fission algorithm. The paper then uses the algorithm to predict new Higgs branch renormalization group flows of 6d $\\mathcal{N}=(1,0)$ D-type orbi-instanton theories and of class $\\mathcal{S}$ theories of type $\\mathfrak{so}(2N)$, including flows that split a theory into a product of interacting fixed points. A stated conjecture—that quiver theories with identical Coulomb branch Hasse diagrams are different presentations of the same theory—is what lets the algorithm remove redundant descendant quivers and produce a genuine Hasse diagram.","feed_headline":"New rules predict all Higgsing paths of orthosymplectic quiver theories","feed_subtitle":"Validated on su(4)-so(6) class S, the rules reveal Higgs flows of 6d orbi-instantons.","key_machinery":"The load-bearing object is the Orthosymplectic Decay and Fission algorithm of Section 2. It starts from a per-node balance, the local measure of whether monopole operators stay above the unitarity bound, together with a modified balance rule for $\\mathrm{SO}(6)$ nodes: a neighbouring $\\mathrm{Usp}(2)$ node counts as $5/8$ of a flavour, or $5/4$ when it sits between an $\\mathrm{SO}(6)$ and an $\\mathrm{SO}(2)$ node. This modification makes the goodness test sensitive to monopole operators valued in the half-integer charge lattice, which the ordinary balance misses. The algorithm then applies two operations: decay, which subtracts the maximal allowed labels from balanced nodes according to a table of transverse slices ($d_n$, $b_n$, $a_7$, $a_5$, $a_4$, $a_2$, $a_1$, $e_7$, $e_8$), and fission, which uses the Levi decompositions $\\mathfrak{so}(2N)\\to\\mathfrak{so}(2N-2K)\\oplus\\mathfrak{u}(K)$ and $\\mathfrak{usp}(2N)\\to\\mathfrak{usp}(2N-2K)\\oplus\\mathfrak{u}(K)$ to split the quiver into an orthosymplectic factor and a unitary factor; unitarisation is the case $K=N$, where only the unitary factor remains. The unitary factor is then processed further by the standard unitary Decay and Fission rules.","core_discovery":"The central claim is that Coulomb branch Higgsing of simply-laced orthosymplectic quivers is governed by two elementary moves, decay and fission, with unitarisation as the limiting case of fission. Decay lowers the ranks of balanced gauge nodes according to a small table of allowed transverse slices; fission splits an $\\mathfrak{so}(2N)$ node into $\\mathfrak{so}(2N-2K)\\oplus\\mathfrak{u}(K)$ or a $\\mathfrak{usp}(2N)$ node into $\\mathfrak{usp}(2N-2K)\\oplus\\mathfrak{u}(K)$, producing a product of an orthosymplectic quiver and a unitary quiver, and unitarisation is the case where the orthosymplectic factor disappears entirely. Through the $\\mathfrak{su}(4)\\cong\\mathfrak{so}(6)$ class $\\mathcal{S}$ mirror pairs, the paper shows that these rules reproduce the full Coulomb branch Hasse diagrams already known from unitary Decay and Fission, including all cases with enhanced flavour symmetry. Beyond that benchmark, the algorithm predicts that the magnetic quiver of the rank-$N$ D-type orbi-instanton fissions, for each $0<\\ell\\leq N$, into the rank-$(N-\\ell)$ orbi-instanton quiver times the rank-$\\ell$ E-string quiver (the 6d theory of M5-branes without an orbifold), matching the picture of separating stacks of M5-branes, and that class $\\mathcal{S}$ theories of type $\\mathfrak{so}(12)$ flow through the Levi splitting $\\mathfrak{so}(12)\\to\\mathfrak{so}(8)\\oplus\\mathfrak{su}(2)$ into a product of class $\\mathcal{S}$ theories of type $\\mathfrak{so}(8)$ and $\\mathfrak{su}(2)$.","pith_inferences":["If the identical-Hasse-diagram conjecture is correct, the same criterion could serve as a practical equivalence test for symplectic singularities, deciding when two different-looking 3d $\\mathcal{N}=4$ quivers describe the same space even when a Hilbert series comparison is unavailable.","The fractional-balance prescription for $\\mathrm{SO}(6)$ suggests that half-integer lattice monopole effects might be localisable as fractional flavour contributions more generally; testing whether an analogous readjustment exists for other $\\mathrm{SO}(2K)$ nodes would sharpen or generalise Rule 1.","Applying the algorithm to the additional unitary/orthosymplectic dual pairs catalogued in the quotient-quiver-subtraction literature would provide further independent checks, as the paper itself suggests.","Because the 6d flows are mirrored by complex-structure deformations in F-theory, the algorithm could be run in reverse: given a predicted Hasse diagram, one could search for the corresponding Calabi–Yau deformation or tensor-branch curve configuration that engineers the same Higgsing pattern."],"forward_implications":["For any simply-laced orthosymplectic quiver with edge multiplicity one, the Coulomb branch stratification—all symplectic leaves and their transverse slices—becomes algorithmically accessible, not only for star-shaped or class $\\mathcal{S}$ examples.","Class $\\mathcal{S}$ theories of type $\\mathfrak{so}(2n)$ obtain a complete Higgs branch Hasse diagram from their 3d mirror, including Higgsings beyond partial puncture closure that were previously inaccessible from the mirror perspective.","The D-type orbi-instanton magnetic quiver fissions, for each $0<\\ell\\leq N$, into the product of the rank-$(N-\\ell)$ orbi-instanton quiver and the rank-$\\ell$ E-string quiver, realising the expected separation of M5-branes; the paper states this is the first explicit demonstration beyond the $\\mathfrak{su}(K)$ case.","The algorithm predicts new class $\\mathcal{S}$ Higgs branch RG flows from Levi decompositions, such as $\\mathfrak{so}(12)\\to\\mathfrak{so}(8)\\oplus\\mathfrak{su}(2)$, including two distinct flow paths between $S_{\\mathfrak{so}(12)}$ and $S_{\\mathfrak{so}(6)}$, which supports the simultaneous-deletion proposal for higher-rank class $\\mathcal{S}$.","Orthosymplectic quivers can fission into mixed products with unitary quivers or unitarise entirely to a unitary quiver, after which the standard unitary Decay and Fission algorithm takes over; redundant equivalent quivers are identified by the paper's identical-Hasse-diagram conjecture."],"supporting_citations":[{"why":"Defines the Tρ(G) theories and the balance/goodness criterion that the algorithm adapts to orthosymplectic nodes.","marker":"[17]"},{"why":"Provides the 3d mirrors of class S theories used to build unitary and orthosymplectic realisations of the same moduli space.","marker":"[31]"},{"why":"Introduces unitary Decay and Fission, the baseline algorithm that the orthosymplectic version extends.","marker":"[21]"},{"why":"Develops unitary Decay and Fission further and is the reference algorithm applied after unitarisation.","marker":"[22]"},{"why":"Gives orthosymplectic magnetic quivers for Higgs branches of 6d theories, the family of quivers the algorithm is designed to treat.","marker":"[20]"},{"why":"Establishes balanced B- and D-type orthosymplectic quivers as magnetic quivers for product theories, motivating fission into products.","marker":"[35]"},{"why":"Provides the atomic Higgsing results for 6D SCFTs that the algorithm recovers and generalises.","marker":"[37]"},{"why":"Supplies the independent class S analysis of D-type theories used to verify the higher-rank so(2N) flows.","marker":"[83]"}],"fun_headline_variants":["Decay and fission decode orthosymplectic quiver moduli","New moves chart Higgsing of orthosymplectic quivers","Quiver decay rules map all Coulomb branch slices","From orbi-instantons to E-strings via quiver fission","Orthosymplectic quivers: complete Higgsing algorithm"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The algorithm does not itself decide which of the descendant quivers it produces are redundant; it relies on the paper's conjecture that two quiver theories with identical Coulomb branch diagrams are different presentations of the same theory, and without that identification the tentative diagram cannot be reduced to a unique final answer.","fun_headline_variants_meta":{"raw":{"variants":["Decay and fission decode orthosymplectic quiver moduli","New moves chart Higgsing of orthosymplectic quivers","Quiver decay rules map all Coulomb branch slices","From orbi-instantons to E-strings via quiver fission","Orthosymplectic quivers: complete Higgsing algorithm"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000288,"raw_usage":{"total_tokens":1801,"prompt_tokens":1172,"completion_tokens":629,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":788,"completion_tokens_details":{"reasoning_tokens":541}},"tokens_in":788,"tokens_out":629,"duration_ms":5264,"temperature":1.0,"reasoning_tokens":541,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T11:32:43.940926+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute an independent invariant—the Coulomb branch Hilbert series where the quiver is good, or the Hall–Littlewood index or the class $\\mathcal{S}$ spectrum where it is not—for two quivers that the algorithm identifies as equivalent, such as the pair in equation (3.12); if the invariants differ, the identification conjecture fails. A single orthosymplectic quiver whose algorithmically predicted Hasse diagram disagrees with the diagram of its independently known unitary dual, for any of the $\\mathfrak{so}(2N)$ class $\\mathcal{S}$ flows checked against the existing D-type classification, would similarly falsify the rules.","supporting_citations":[],"review_version":1}