{"id":"f1460284-2ca6-49ff-8e87-a3b24f6ed642","arxiv_id":"2507.01015","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Partial flag cluster algebras reproduce a large part of the two-loop five- and six-point massless scattering symbol alphabets, with the momentum twistor embedding stronger at six points after permutation completion.","lead":"This paper checks whether cluster algebras attached to certain partial flag varieties reproduce the known singularity patterns of five- and six-particle massless scattering. It finds partial success, with the momentum twistor version capturing more six-point singularity classes than the spinor helicity version, and some observed patterns matching cluster mutation rules.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The six-point evidence hinges on an unproven tropical truncation; the missing-class assertions would be tested by computing the full Gr(4,8) cluster support of the 244 rational letters.","rationale":"The reader's weakest_assumption identified the tropical truncation as unproven, and I agree that is the central worry. I mark this partial rather than full agreement because I would also flag the absence of a statistical baseline for the recovery counts; however, the truncation is indeed the single most load-bearing assumption because the negative evidence (missing classes, failure of spinor-helicity embedding to find new classes) is only meaningful if the positive set is the true set of flag-expressible letters. The paper is honest about its status as evidence: Section VII explicitly lists failures and open tasks, and the F(2,4;5) and F(2,3;5) five-point analyses use finite cluster algebras that are fully computed, giving the method independent support there. The six-point algebraic-letter recovery via the single limit ray and the permutation completion to all five square roots is striking and provides real, non-circular evidence. The proposed concrete test directly targets the saturation claim: by expanding the search beyond the tropical list and checking whether missing classes become expressible, it would settle whether the truncation is a true invariant of the flag subalgebra or an artifact. I would not raise the verdict to REJECT because the paper frames itself as an evidence report, not a proof, and the claimed recoveries are concrete computations that can be checked from the ancillary data. I would keep CONDITIONAL to require either the saturation test above or an explicit statistical baseline before the failure pattern is presented as the load-bearing evidence.","tokens_in":20846,"tokens_out":1972,"duration_ms":124653,"concrete_test":"Compute, for the F(2,4;6) subalgebra of Gr(4,8), the saturation of the 77-variable tropical list by a direct cluster-algebraic search beyond the tropical truncation: collect all A-coordinates of Gr(4,8) whose g-vectors lie within, say, two mutation layers beyond the 272-variable tropical list, identify the subset expressible in F(2,4;6) variables, and test whether each currently missing permutation class (S6, S7, S9, S10, S11, S12, S15, S17) becomes expressible as a multiplicative combination. If none of the missing classes appear, the negative evidence is robust; if even one class appears, the paper's saturation claim fails and the conditional verdict should be strengthened to require reframing or a revised truncation.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The paper's six-point claims are evidence-level, not theorem-level, and the key load-bearing assumption is explicitly stated in Section V: a tropical truncation of the infinite Gr(4,8) cluster algebra—272 active coordinates, of which 71 plus six frozen coordinates realize F(2,4;6)—is declared sufficient. The saturation statement, \"generating more A-coordinates by performing many mutations does not yield anything further,\" is a computational observation, not a proof. This matters because the meaningful signature of the central claim is asymmetric: the momentum-twistor embedding recovers eleven of nineteen permutation classes (S1-S5, S8, S13-S14, S16, S18-S19) and fails on S6-S7, S9-S12, S15, S17. If the truncation is incomplete, or if the completion-by-permutation procedure misses flag orientations, these misses could be artifacts rather than genuine cluster-algebraic exclusions. The paper's own \"Comment on unused flag variables\" exposes the fragility: p1378 is unused while its cyclic image p2478 is used, merely because a partner coordinate is absent from the truncated list. This shows the recovery set depends on accidental availability of cofactors, so the negative statements inherit that dependence. A second-order concern is that no baseline is provided for how many of the 244 rational letters would be recovered by a random or permutation-generous multiplicative basis of comparable size; without that, the 11/19 class rate is suggestive but not calibrated. The central claim—that the cluster algebra \"contains information relevant\" to these alphabets—can survive even if some classes are unrecovered, but the specific failure pattern cited as evidence depends on the truncation being genuinely complete for the flag subalgebra.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes that cluster algebras associated to partial flag varieties encode information about the symbol alphabets of massless scattering amplitudes without dual conformal symmetry. Two embeddings are studied: a momentum-twistor embedding into F(2,4;n), interpreting four-index Plücker coordinates as momentum twistors and two-index Plückers as angle brackets, and a spinor-helicity embedding into F(2,n-2;n), interpreting (n-2)-index Plückers as square brackets. For five points, both embeddings reproduce the known two-loop pentagon alphabet (up to permutation completion), and the momentum-twistor F(2,4;5) algebra is used to interpret certain triples of quadratic letters in existing two-loop data. For six points, the paper uses a tropical truncation of the Gr(4,8) cluster algebra to obtain 77 flag-compatible A-coordinates and reports that the momentum-twistor embedding captures 54 rational letters of the planar hexagon alphabet, rising to eleven of the nineteen permutation classes after permutation completion, while the spinor-helicity embedding captures a subset of the same classes. The paper also matches the five square-root letters and all algebraic letters to origin-cluster/limit-ray constructions in F(2,4;6), and analyzes adjacency relations in a two-loop Wilson-loop correlator. The central claim is presented as evidence, not a theorem.","tokens_in":21152,"tokens_out":5183,"duration_ms":61894,"significance":"If the evidence is sound, the paper extends cluster-algebra technology beyond dual-conformal planar MSYM to broader classes of massless observables, which is a potentially important step for symbol-bootstrap calculations in QCD-like settings. The strength of the paper is that it provides explicit formulas for the recovered letters, includes ancillary files documenting each identification and permutation, and uses no fitted numerical parameters; the partial flag cluster algebras are independently grounded in the geometry of momentum twistors and spinor helicity variables. The main limitations are that the six-point conclusions rely on an unproven tropical truncation, and the matching procedure is retrospective and searches over a very large space of permutations and multiplicative combinations of cluster variables. These issues do not invalidate the positive observations, but they do mean the negative statements about missing classes of letters cannot yet be read as genuine structural exclusions.","major_comments":[{"comment":"The saturation statement at the start of Section V ('generating more A-coordinates by performing many mutations does not yield anything further') is a computational observation, not a proof, and it is load-bearing for the paper's negative conclusions about the missing permutation classes S6, S7, S9, S10, S11, S12, S15, and S17. The paper's own 'Comment on unused flag variables' makes the dependence concrete: the variable p1378 is unused while its cyclic image p2478 is used, solely because a partner coordinate is absent from the truncated list. This shows that the recovery set depends on accidental availability of cofactors in the chosen truncation, so the failures to recover those classes are not yet established as genuine cluster-algebraic exclusions. Please either provide a systematic characterization of the relevant finite set of flag-compatible cluster variables (for example, by specifying the finite set of clusters and mutations explored and making the verification code available), or reformulate the six-point claims as 'not found within this truncation' rather than as failures of the cluster structure.","section":"Section V, tropical truncation and negative claims"},{"comment":"The central six-point comparison is retrospective over a large search space: 54 rational letters are identified as multiplicative combinations of variables from a 77-element coordinate set, and all particle-label permutations are then used to complete the result to eleven of nineteen permutation classes. No baseline is given for how many of the 244 rational letters would be recovered by a similarly generous multiplicative closure over a random set of 77 coordinates, or over another subalgebra of comparable size. Without such a calibration, the 11/19 class rate and the 54/244 pre-permutation count are suggestive but do not by themselves establish that the cluster structure, rather than the size and flexibility of the coordinate set, is responsible for the matches. I would ask for a null-model estimate, for example the expected recovery rate under random choices of 77 coordinates with the same multiplicative and permutation closure, or at least a report of the number of distinct multiplicative combinations tested and the maximum degree of the products involved.","section":"Section V.A, recovery counts and permutation completion"},{"comment":"The abstract states that the cluster structures 'correctly predict the appearance of certain triples of symbol letters related to cluster mutation relations,' but the triples in Eqs. (34)-(42) and in Section V.C are extracted from already computed two-loop data and then matched to a mutation pair in the flag algebra, with an additional permutation on particle labels allowed in the matching step (for example, the permutation {2,4,3,1,5,6} before the W23-W10 triple). This is a retrospective consistency check, not a prediction. I recommend softening the language to 'are consistent with' or 'are organized by,' and stating explicitly that the permutation step was not fixed in advance. This does not reduce the value of the observation, but it removes the appearance that the cluster algebra was used to anticipate the data.","section":"Section IV.B and Section V.C, retrospective triple matching"}],"minor_comments":[{"comment":"The caption of Table II contains 'presented in []'; the citation [30] should be inserted.","section":"Table II caption"},{"comment":"The text refers to 'F(2, n-4; n) with the spinor helicity embedding,' but the embedding was defined earlier as F(2, n-2; n) in Section III.B; this appears to be a typo and should be corrected.","section":"Section VII, Conclusions"},{"comment":"In Eq. (10), the listed active coordinates are written as '{a1, a2, a4, a4}'; this should presumably be '{a1, a2, a3, a4}'.","section":"Section II.C, Example F(2,4;5)"},{"comment":"The word 'Grasmmannian' in the third paragraph is a typo for 'Grassmannian'.","section":"Section I, Introduction"}],"recommendation":"major_revision","confidential_remarks":"The paper overlaps with the concurrent work cited as [3], but it acknowledges this and its emphasis on explicit alphabet matching and partial flag embeddings is sufficiently distinct. The main editorial question is whether the authors can make the truncation and calibration issues precise enough for the negative six-point claims to be interpretable; if they instead soften those claims, the paper would be suitable as an evidence-level contribution."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Punchline: the new content here is the systematic six-point comparison of the two flag embeddings and the observation that certain triples of symbol letters are explained by cluster mutation relations. The structural setup — spinor-helicity and momentum twistor cluster structures on F(2,4;n) and F(2,n−2;n) — was already in Bossinger–Li, Pokraka et al., and Bossinger–Drummond–Glew; the authors are honest about that. What they add is concrete: explicit five-point recoveries, the 19 permutation classes of the 244 rational six-point letters, recovery of 11 classes via the momentum-twistor embedding after permutation completion, the algebraic letters from F(2,4;6) origin clusters matching the square-root letters, and a form-factor test. The ancillary files with per-letter expressions are a real plus.\n\nCredit where due: the cluster algebra is independently grounded in geometry; there are no fitted parameters. The five-point formulas are explicit and checkable, and the A4 adjacency analysis — the 339 vs 370 word count — is a genuine quantitative observation. The paper is well organized and careful about what is evidence versus what is theorem.\n\nSoft spots, in order of severity:\n\n1. The six-point negative statements rest on the saturation claim for the tropically truncated list of 272 Gr(4,8) coordinates. That claim is a computational observation, not a proof. The paper's own 'Comment on unused flag variables' shows the recovery set depends on accidental presence of cofactors: p1378 is unused while its cyclic image p2478 is used, because the partner variable is missing from the truncated list. Worse, the form-factor section shows that in a closely related setup, going beyond the truncated list does recover a missing letter class. So the truncation is not generally sufficient, and the six-point 'misses' may be artifacts. The authors acknowledge this in the conclusions, but it means the headline evidence is weaker than the presentation suggests.\n\n2. The matching procedure is a search over multiplicative combinations of cluster coordinates, with arbitrary permutation completion. There is no baseline for how much matching would occur by chance. The 11/19 class rate is suggestive but uncalibrated. A random or permutation-generous basis of comparable size might do surprisingly well.\n\n3. Everything is postdiction: the alphabets were already computed at two loops. There is no not-yet-computed observable predicted. The authors frame the paper as evidence, which is fair, but the reframing matters for how much weight the reader should give.\n\nIs the central argument sound? As evidence, mostly yes; as a proof of relevance, no. The framework is plausible and the five-point part is solid. The six-point comparison is a useful data point but needs either a larger/complete truncation test, a chance baseline, or a prediction for a new observable.\n\nWho this is for: amplitudes people working on symbol alphabets and cluster bootstrap, especially those interested in non-dual-conformal observables. It deserves a serious referee; the question is important and the claims are concrete and checkable. My recommendation: send to peer review, with a request that the authors either prove or carefully bound the truncation, provide a chance baseline, and avoid overclaiming the negatives.","headline":"A careful, honest evidence report; the six-point story is postdictive and rests on an unproven tropical truncation, but the framework, five-point checks, and ancillary files make it worth refereeing.","tokens_in":21697,"tokens_out":2823,"would_cite":true,"duration_ms":31338,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"Cluster algebras of partial flag varieties encode much of the singularity alphabet of massless five- and six-point scattering, with the momentum-twistor embedding doing better at six points.","keywords":["cluster algebras","partial flag varieties","scattering amplitudes","symbol alphabets","momentum twistors","spinor helicity","cluster adjacency","tropical geometry"],"falsifier":"Compute the untruncated flag cluster algebra of F(2,4;6) (or its Gr(4,8) embedding) far enough to search for a multiplicative combination of A-coordinates equal to a letter in a missing permutation class such as S6 or S9; the paper reports no such combination within the truncation, so an explicit counterexample would disprove the saturation claim. A second check is whether a letter like W138, which the paper compares to the five-point letter V31, appears in any finite two-loop observable; if it does, the dimensional-regularisation explanation for some missing classes would be weakened.","tokens_in":2670,"feed_emoji":"📐","tokens_out":5963,"duration_ms":154258,"temperature":0.7,"pith_summary":"Massless scattering amplitudes and related observables have a set of possible singularities, captured in a symbol alphabet, and for planar maximally supersymmetric Yang-Mills theory that alphabet is known to come from Grassmannian cluster algebras. This paper asks whether the same is true for observables that lack dual conformal symmetry, and argues that partial flag varieties supply the right cluster algebras: F(2,4;n) for momentum-twistor kinematics and F(2,n-2;n) for spinor-helicity kinematics. At five points both embeddings reproduce the known two-loop pentagon alphabet after permutation completion, and the finite A4 flag explains observed adjacency triples of quadratic letters. At six points the momentum-twistor embedding of F(2,4;6) recovers a larger subset of the 289-letter planar hexagon alphabet than the spinor-helicity embedding, and the flag's limit rays reproduce the square-root letters. A sympathetic reader would care because this suggests a systematic way to predict singularity alphabets for observables without dual conformal symmetry, potentially extending symbol-bootstrap methods beyond the special amplitudes where they already work.","feed_headline":"Partial flag cluster algebras capture scattering alphabets","feed_subtitle":"At six points the momentum-twistor flag reproduces more hexagon letters and predicts letter triples.","key_machinery":"The central objects are the cluster algebras of partial flag varieties $F(2,4;n)$ and $F(2,n-2;n)$, where Plücker coordinates serve as $A$-coordinates and are interpreted either as momentum twistors ($p_{ijkl}=\\langle ijkl\\rangle$, $p_{ij}=\\langle ij\\rangle$) or spinor-helicity variables ($p_{ij}=\\langle ij\\rangle$, $p_{\\widehat{ij}}=\\pm[ij]$). For six points the flag algebra $F(2,4;6)$ is embedded in $Gr(4,8)$ by requiring twistor labels 7 and 8 to appear together, and the infinite list of $A$-coordinates is truncated using a tropical construction to 77 flag-compatible variables. The square-root letters come from origin clusters whose repeated mutations obey $z_{n+2}z_n = b f_z F^n + z_{n+1}^2$, with discriminant $\\Delta = P_z^2 - 4F_z$ defining the algebraic letters $\\varphi_z$ and $\\varphi_w$. The argument also relies on the cluster mutation relation, which makes the $X$-coordinate between mutation-related letters the unique rational function appearing between them in integrable symbol words.","core_discovery":"The central claim is that the cluster algebras of partial flag varieties contain the singularity data of non-dual-conformal massless scattering. For five points, both the spinor-helicity flag F(2,3;5) and the momentum-twistor flag F(2,4;5) reproduce the two-loop pentagon alphabet once all permutations are used; the momentum-twistor flag of type A4 also predicts that quadratic letters whose factors form a mutation pair appear one slot apart with a unique X-coordinate between them, matching pentagonal Wilson loop data. For six points both embeddings use the same flag F(2,4;6), realised as a subalgebra of Gr(4,8), and a tropical truncation to 77 flag-compatible cluster variables. The momentum-twistor embedding identifies 54 rational letters of the planar hexagon alphabet, eleven of nineteen permutation classes after permutation completion, whereas the spinor-helicity embedding recovers fewer letters and loses the permutation class S5. The algebraic letters are produced from sixteen origin clusters of F(2,4;6), which share one limit ray and yield the square roots r1,...,r5 after permutation completion. The paper also shows that specific observed triples in the hexagonal Wilson loop, such as W23 ⊗ (W1W7)/(W2W28) ⊗ W10, are realised as mutation pairs in the momentum-twistor embedding.","pith_inferences":["The failure of both embeddings to recover classes $S_6$, $S_7$, $S_9$, $S_{10}$, $S_{11}$, $S_{12}$, $S_{15}$, and $S_{17}$ suggests those letters may require a different cluster algebra or embedding; the exchange-type relation $W_{26}W_{22} = W_1 W_{31} - W_{61}$ noted for one missing triple is a hint that such a structure exists.","A testable extension is that the $D_4$-type prediction for five points, namely that quadratic pairs of Type 3 appear only when separated by two slots, should become visible at weight six or higher, exactly in the unique weight-two function the paper identifies from $F(2,3;5)$.","The observation that the four-particle form factor needs $Gr(4,8)$ variables beyond the 272-variable truncation suggests the tropical truncation is observable-dependent; one could test this by checking whether any higher-loop or non-planar observable requires letters outside the truncated list.","If the limit-ray dictionary is correct, the unrecovered three-loop pentagon letters mentioned in the paper might be identifiable as limit rays of $F(2,4;5)$ or $F(2,4;6)$, extending the flag-cluster prescription to the next loop order."],"forward_implications":["At five points, both $F(2,3;5)$ and $F(2,4;5)$ yield the full two-loop pentagon symbol alphabet after permutation completion, and $F(2,4;5)$ explains the observed one-slot separation of quadratic letters in planar and non-planar data.","At six points, the momentum-twistor embedding of $F(2,4;6)$ reproduces 54 rational letters of the planar hexagon alphabet and, after permutation completion, eleven of nineteen permutation classes; the spinor-helicity embedding reproduces a strictly smaller subset and loses class $S_5$.","All five square roots of the six-point alphabet arise from the single limit ray of $F(2,4;6)$ after permutation completion, and the algebraic letters built from the sixteen origin clusters span the observed algebraic letters involving $r_1$ in one embedding or $r_2$ in the other.","Cluster adjacency in the flag algebras constrains symbol words: quadratic letters that are cluster-incompatible and not mutation partners cannot appear adjacent or one slot apart, and the unique letters between them are $X$-coordinates of mutations.","The $A_4$ cluster-adjacent weight-four polylogarithm basis built from $F(2,4;5)$ is sufficient to express the leading-singularity symbols of the pentagonal Wilson loop with Lagrangian insertion."],"supporting_citations":[{"why":"Supplies the cluster structures on spinor-helicity and momentum-twistor varieties that define the two embeddings of kinematics into flag variables.","marker":"[2]"},{"why":"Provides the infinite mutation sequences, origin clusters, and limit-ray construction that the paper uses to obtain square-root letters at six points.","marker":"[17]"},{"why":"Establishes cluster adjacency for scattering amplitudes, the constraint the paper uses to interpret triples and separated quadratic letters.","marker":"[22]"},{"why":"Gives the five-point flag analysis and the D4/A4 cluster facts, including the identification F(2,3;5) ≅ Gr(3,6), used for the pentagon alphabet and adjacency.","marker":"[24]"},{"why":"Supplies the two-loop pentagonal Wilson loop with Lagrangian insertion data whose letter triples and leading-singularity symbols are tested against the A4 flag algebra.","marker":"[28]"},{"why":"Provides the 289-letter planar two-loop six-point alphabet and its permutation classes, the dataset the paper tries to reproduce from flag cluster coordinates.","marker":"[30]"},{"why":"Gives the tropically truncated list of 272 Gr(4,8) A-coordinates that the six-point flag analysis uses after restricting to flag-compatible variables.","marker":"[31]"},{"why":"Supplies the hexagonal Wilson loop with Lagrangian insertion data in which the six-point adjacency observations and the W23 ⊗ X ⊗ W10 triple appear.","marker":"[32]"},{"why":"Provides the three-loop four-particle form factor alphabet whose rational letters the paper recovers from Gr(4,8) cluster variables beyond the truncated list.","marker":"[33]"}],"fun_headline_variants":["Cluster algebras of partial flags encode scattering alphabets","Momentum-twistor flags reproduce more hexagon letters","Flag cluster structures predict letter triples in scattering","Partial flag algebras link massless scattering singularities","At five and six points, flag cluster algebras capture alphabets"],"cache_read_input_tokens":23808,"weakest_assumption_plain":"The six-point comparison assumes that the tropically truncated list of 77 flag-compatible cluster variables is exhaustive: the saturation is a computational observation, not a proof, so a missing cluster variable could change which permutation classes appear recoverable.","fun_headline_variants_meta":{"raw":{"variants":["Cluster algebras of partial flags encode scattering alphabets","Momentum-twistor flags reproduce more hexagon letters","Flag cluster structures predict letter triples in scattering","Partial flag algebras link massless scattering singularities","At five and six points, flag cluster algebras capture alphabets"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000699,"raw_usage":{"total_tokens":3156,"prompt_tokens":941,"completion_tokens":2215,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":557,"completion_tokens_details":{"reasoning_tokens":2139}},"tokens_in":557,"tokens_out":2215,"duration_ms":18812,"temperature":1.0,"reasoning_tokens":2139,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T21:00:26.914955+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the untruncated flag cluster algebra of F(2,4;6) (or its Gr(4,8) embedding) far enough to search for a multiplicative combination of A-coordinates equal to a letter in a missing permutation class such as S6 or S9; the paper reports no such combination within the truncation, so an explicit counterexample would disprove the saturation claim. A second check is whether a letter like W138, which the paper compares to the five-point letter V31, appears in any finite two-loop observable; if it does, the dimensional-regularisation explanation for some missing classes would be weakened.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the cluster structures on spinor-helicity and momentum-twistor varieties that define the two embeddings of kinematics into flag variables."},{"cited_title":"Express these in terms of the letters of the planar pentagon al- phabet","cited_arxiv_id":null,"evidence_quote":"Provides the infinite mutation sequences, origin clusters, and limit-ray construction that the paper uses to obtain square-root letters at six points."},{"cited_title":"Cluster structures on spinor helicity and momentum twistor varieties","cited_arxiv_id":null,"evidence_quote":"Establishes cluster adjacency for scattering amplitudes, the constraint the paper uses to interpret triples and separated quadratic letters."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the 289-letter planar two-loop six-point alphabet and its permutation classes, the dataset the paper tries to reproduce from flag cluster coordinates."},{"cited_title":"Tropical grassmannians, clus- ter algebras and scattering amplitudes","cited_arxiv_id":null,"evidence_quote":"Gives the tropically truncated list of 272 Gr(4,8) A-coordinates that the six-point flag analysis uses after restricting to flag-compatible variables."},{"cited_title":"Henn, and Georgios Pap- athanasiou","cited_arxiv_id":null,"evidence_quote":"Supplies the hexagonal Wilson loop with Lagrangian insertion data in which the six-point adjacency observations and the W23 ⊗ X ⊗ W10 triple appear."},{"cited_title":"Dixon, ¨Omer G¨ urdo˘ gan, Yu-Ting Liu, An- drew J","cited_arxiv_id":null,"evidence_quote":"Provides the three-loop four-particle form factor alphabet whose rational letters the paper recovers from Gr(4,8) cluster variables beyond the truncated list."}],"review_version":1}