{"id":"89ba68c9-96f6-4475-8c6b-bbf122930a19","arxiv_id":"2505.09121","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Under large image and a localized Iwasawa main conjecture, the Selmer group of the central critical twist of a modular form is explicitly determined by Kurihara numbers, giving exact rank and module structure.","lead":"This paper proves a refined Birch and Swinnerton-Dyer type formula: the exact structure of Bloch-Kato Selmer groups attached to modular forms is pinned down by discrete Kolyvagin derivatives of L-values, known as Kurihara numbers. Under a localized Iwasawa main conjecture, and for many ordinary cases unconditionally, the result gives exact Selmer ranks and module structure across weights and local behaviors.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 3.18(2)/(3) asserts an equivalence between non-trivial specialization of the Λ-adic Kolyvagin system and the localized Iwasawa main conjecture; this is false as stated because the IMC equality can hold with positive P-adic order while the specialization vanishes.","rationale":"The reader correctly flagged that 'one claimed equivalence goes beyond the cited literature,' but the issue is stronger than a missing reference: Theorem 3.18(2)–(3) is false as stated under the paper's own definition of Λ-primitive and of the localized IMC. This is not a peripheral technicality. The equivalence is the mechanism by which Theorem 1.2 converts the Iwasawa main conjecture localized at the augmentation ideal into the non-vanishing of the Kurihara numbers, and it is the mechanism by which the unconditional good-ordinary case (Corollary 1.5) is derived from Skinner–Urban and Wan. If the equivalence fails, those unconditional conclusions are unsupported, and the theorem reduces to a conditional statement whose hypothesis (non-vanishing of e^{min,†}) is exactly the kind of input the paper claims to supply via the IMC. I do not dispute that the conditional structure theorem may be salvageable, and the numerical appendix provides useful evidence for the non-vanishing phenomenon. But as submitted, the central package advertised in the abstract — a formula 'only under the Iwasawa main conjecture localized at the augmentation ideal' — rests on an invalid implication. A major revision that either proves the missing converse under additional hypotheses or explicitly repositions Theorem 1.2 as conditional on the non-trivial specialization of Kato's Kolyvagin system would be needed before the main claims can be accepted.","tokens_in":44052,"tokens_out":36018,"duration_ms":398018,"concrete_test":"Analytical check: write κ∞ = f·κ⁰ with KS(T⊗Λ) free of rank one over Λ generated by κ⁰. Take a height-one prime P with ord_P(f) = ord_P(char Sel∨) > 0; for instance, in the model Λ = Z_p[[T]], take Sel∨ with characteristic ideal (T) and κ∞ = T·κ⁰. Verify that this configuration is not excluded by the large-image hypotheses. If it is admitted, Theorem 3.18(2) fails because κ∞ mod P = 0 while the localized IMC holds. Concretely, re-derive Theorem 3.18 from [MR04, Thm. 5.3.10] and locate the exact step proving the converse; that step must show f∉P from the IMC equality, which is impossible without an additional primitivity assumption. If no such step exists, the equivalence in Theorem 1.2 should be replaced by the explicit hypothesis that Kato's Kolyvagin system specializes non-trivially at the augmentation ideal.","verdict_should_be":"REJECT","load_bearing_attack":"The load-bearing bridge is Theorem 3.18(2)–(3) in §3.5.3. The paper claims that, for each height-one prime P of Λ, non-triviality of the specialization κKato,∞ mod P is equivalent to the localized Iwasawa main conjecture (equality of ord_P of the two characteristic ideals), and that Λ-primitivity is equivalent to equality of the full characteristic ideals. This is inconsistent with the paper's own definition of Λ-primitive. Since KS(Tf(k−r)⊗Λ) is free of rank one over Λ (Theorem 3.11 and [MR04, Thm. 5.2.10, 5.3.3]), write κ∞ = f·κ⁰. The reduction κ∞ mod P is zero exactly when f∈P. The localized IMC is the equality ord_P(f) = ord_P(char Sel∨). This equality is compatible with ord_P(f) > 0: then the IMC holds at P while the specialization is zero. Likewise, if char Sel∨ is non-trivial, equality of characteristic ideals can hold while κ∞ is not Λ-primitive, because every prime divisor of the common ideal is a prime at which the specialization vanishes. The proof note that '[MR04] claims one direction... the proof implies the equivalence' cannot supply the missing converse: that converse would require ord_P(char Sel∨) = 0, which is not part of the IMC. Theorem 1.2 uses exactly this equivalence to pass from (IMC at 1), and hence from Skinner–Urban/Wan in the ordinary case, to non-vanishing of the Kurihara numbers e^{min,†}. Without it, the unconditional Corollary 1.5 and the 'conversely' direction of Theorem 1.2 are not established; the non-vanishing hypothesis of Theorem 1.1 must simply be assumed.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a refined Tamagawa-number-type formula for the Bloch–Kato Selmer group Sel(Q,W†_f) of the central critical twist of a cuspidal newform f, expressed in terms of \"Kurihara numbers\" e^{min,†}_n built from Kolyvagin derivatives of modular symbols. Theorem 1.1 states that if the collection e^{min,†} does not vanish, then Kato's Kolyvagin system is non-trivial and Sel(Q,W†_f) is isomorphic to (F/O)^{⊕r} ⊕ ⊕(O/π^{e_i}O)^{⊕2}, with explicit BSD-type formulas for corank and length. Theorem 1.2 asserts that non-vanishing of e^{min,†} is equivalent to the Iwasawa main conjecture localized at the augmentation ideal, and uses this to derive unconditional results in the good ordinary case via Skinner–Urban and Wan. The paper also contains a rank-one p-converse, a p-parity statement, a computational upper bound, and a numerical appendix with Pollack.","tokens_in":44535,"tokens_out":12432,"duration_ms":134831,"significance":"If the main results were correct, the paper would be a significant advance: it would give a structural BSD-type formula for Selmer groups that is insensitive to weight and local behavior at p, and it would upgrade known Iwasawa main conjecture results to exact structure theorems. The idea of using Kolyvagin derivatives of modular symbols rather than p-adic L-function derivatives is original and appealing, and the numerical appendix with Pollack is a useful, reproducible computational contribution. However, the central bridge between the Iwasawa main conjecture and non-vanishing of the Kurihara numbers, Theorem 3.18(2)–(3), is false as stated. This invalidates the unconditional applications in Theorem 1.2, Corollary 1.5, and Theorem 6.2, so the advertised significance is not achieved in the present version.","major_comments":[{"comment":"The asserted equivalences are false as stated. By Theorem 3.11, KS(Tf(k−r)⊗Λ) is free of rank one over Λ; write κ∞ = f·κ⁰. The reduction κ∞ mod P is zero exactly when f∈P. The localized IMC in Conjecture 3.17 is the equality ord_P(f) = ord_P(char Sel0(Q∞,Wf(r))∨). This equality is compatible with ord_P(f) > 0, in which case the specialization is zero while the IMC holds at P. Likewise, equality of full characteristic ideals (f) = (char Sel∨) can hold with f a non-unit, so κ∞ need not be Λ-primitive. The proof line in Theorem 3.18 that \"the proof implies the equivalence\" cannot supply the missing converse, because that converse would require ord_P(char Sel∨) = 0, which is not part of the IMC. Consequently, the implication '(IMC at 1) ⇒ e^{min,†} ≠ 0' in Theorem 1.2 is not established, and the unconditional consequences in Corollary 1.5 and in the proof of Theorem 6.2, which rely on that implication, do not follow. The converse direction (e^{min,†} ≠ 0 ⇒ IMC at 1) remains valid, since it follows from the one-sided divisibility in Theorem 3.18(1) once κ is known to be non-trivial.","section":"§3.5.3, Theorem 3.18(2)–(3)"},{"comment":"The integrality of Kato's zeta morphism over finite abelian extensions L is a load-bearing input: it is used to construct Kato's Kolyvagin system that underlies Theorems 1.1 and 1.2. The proof of Theorem 3.1 delegates a key step to the author's submitted paper [Kim]. Since [Kim] is not part of the published record, the referee cannot verify this step from the manuscript. Please either include a complete proof of the required integrality statement or state precisely which result of [Kim] is being used and its status. This dependency is especially concerning because the paper otherwise advertises independence from integral p-adic Hodge theory.","section":"§3.1.1, Theorem 3.1"},{"comment":"The numerical invariants of the Kurihara numbers are used in §5 to compute ordπ(loc_s^p κKato) only up to a non-zero uniform constant C±_per and a fixed Euler factor. The manuscript asserts that this constant is cancelled in the differences that appear in Theorem 1.1. This cancellation is plausible for differences such as ordπ(e_n) − ordπ(e_{nℓ}), and the structure formula indeed uses only such differences. However, the paper should state this cancellation explicitly at the point where exact valuations of loc_s^p κKato are first identified with valuations of e^{min,†}, because Proposition 4.2 as written only gives non-vanishing, not equality of numerical invariants. Without an explicit statement, the transition from the Selmer-theoretic Kolyvagin system invariants to the modular-symbol invariants in §5.3 and §5.4 is incomplete.","section":"§4, Proposition 4.2 and §5"}],"minor_comments":[{"comment":"The functional equation for Kurihara numbers is stated without proof; a brief derivation or a precise reference to the modular-symbol functional equation would help the reader.","section":"§2.3.4, equation (2.3)"},{"comment":"The notation KS and KS is introduced with a comment that they are not distinguished, but the two symbols are visually nearly identical; please use distinct typography or explicitly identify one with the other.","section":"§3.4.1, after Theorem 3.11"},{"comment":"The exception 'except πΛ' in the statement of Wan's theorem is potentially confusing: please clarify whether this exception is the prime ideal generated by a uniformizer of the coefficient ring, and confirm explicitly that the augmentation ideal is not excluded.","section":"§3.5.3, Theorem 3.19"},{"comment":"The reference [Kur24] contains a typo: 'anayltic' should be 'analytic'.","section":"References"}],"recommendation":"reject","confidential_remarks":"For the editor: the false equivalence in Theorem 3.18 is not a minor gap; it is the hinge connecting known Iwasawa main conjecture results to the paper's headline unconditional theorems. A conditional structure theorem under the explicit non-vanishing hypothesis may be salvageable, but the advertised applications in their current form are unsupported. The numerical appendix is valuable, but it does not compensate for the invalid bridge."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know upfront. First, this is a genuinely original and technically rich paper with a plausible structure theorem. Second, the claimed bridge between the Iwasawa main conjecture and the non-vanishing of Kurihara numbers, Theorem 3.18(2)/(3), is false as stated, and that break takes the unconditional applications (Corollary 1.5 and the '(ord)' part of Theorem 1.2) with it.\n\nThe good parts are real. The 'relative' Kolyvagin system argument in Section 5 extracts not just coranks but the full O-module structure of Bloch–Kato Selmer groups from Kolyvagin derivatives of L-values. The connection between Kato's Euler system and Kurihara numbers for higher weight, via explicit reciprocity with torsion coefficients, is new and useful. The numerical appendix with Pollack gives concrete Selmer structures, several of which are out of reach of the BSD conjecture. The main theorem is cleanly stated and, conditional on its non-vanishing hypothesis, is independent of weight, local behavior, and analytic rank.\n\nThe soft spots, in order of severity. The equivalence in Theorem 3.18 is load-bearing. Since the Λ-adic Kolyvagin system module is free of rank one, write κ∞ = f·κ⁰. The specialization mod P vanishes exactly when f ∈ P. The localized IMC is ord_P(f) = ord_P(char Sel∨); this equality is perfectly compatible with ord_P(f) > 0. So the IMC can hold at P while the specialization is zero. The same logic invalidates the Λ-primitivity claim. The proof's note that '[MR04] claims one direction... the proof implies the equivalence' cannot work, because the missing converse would require ord_P(char Sel∨) = 0, which is not part of the IMC. Proposition 4.2 transfers this to the Kurihara numbers, so the claimed equivalence in Theorem 1.2 between e^{min,†} ≠ 0 and (IMC at 1) is not established. The conditional Theorem 1.1 may still be fine, but the unconditional ordinary case is currently over-sold.\n\nMinor issues: the explicit reciprocity law is proved only up to a non-zero uniform constant; the paper acknowledges this and the constant cancels in the differences that matter, so that is acceptable. The proof also leans on the author's submitted paper [Kim] for integrality of Kato's zeta morphism, a dependency to watch.\n\nWho is this for? Iwasawa theorists and anyone working on Selmer group structures, p-converses, or BSD-type formulas. It deserves serious referee time, but it needs a major revision: fix or replace Theorem 3.18 and then re-derive the applications. I'd send it to a specialist, not desk-reject.","headline":"Original and technically rich, but the claimed equivalence between the localized IMC and non-vanishing of the Kolyvagin system / Kurihara numbers is false as stated, so the unconditional applications need repair.","tokens_in":44983,"tokens_out":4919,"would_cite":false,"duration_ms":49551,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"pith_extraction":{"msc":["11F67","11G40","11R23"],"pacs":[],"model":"deepseek-v4-flash","headline":"Kurihara numbers determine the exact Selmer group structure for modular forms with large image, yielding corank, elementary divisors, and the p-part of a BSD-type formula.","keywords":["Tamagawa number conjecture","Kato's Kolyvagin systems","Kurihara numbers","Bloch-Kato Selmer groups","modular symbols","Iwasawa main conjecture","p-converse","central critical twist"],"falsifier":"Take an elliptic curve of analytic rank two in the appendix's class, at a good ordinary prime $p \\geq 5$ with p-distinguished large-image representation, and compute the Kurihara numbers $e^{\\min,\\dagger}_n$ for all Kolyvagin-squarefree n up to a growing bound. The paper predicts a non-vanishing term at some n with $\\nu(n) = 2$; if every computed $e^{\\min,\\dagger}_n$ vanished, that would contradict the unconditional case of Theorem 1.2, since the known Iwasawa main conjecture at the augmentation ideal would force non-vanishing. Such a computation is feasible because the appendix already exhibits many non-vanishing values.","tokens_in":2421,"feed_emoji":"🧮","tokens_out":7642,"duration_ms":128740,"temperature":0.7,"pith_summary":"The paper claims that for a modular form whose attached Galois representation has large image, the full module structure of the central-critical Bloch–Kato Selmer group is determined not by the L-function and its analytic derivatives but by a discrete family of modular-symbol numbers, the Kurihara numbers. These numbers are built from special L-values at auxiliary primes, normalized by minimal integral periods, and the paper treats them as Kolyvagin derivatives of special L-values. If this family does not vanish, the paper proves an explicit isomorphism describing the Selmer group, giving its corank, its elementary divisors, and the p-part of a Birch–Swinnerton-Dyer type formula. The non-vanishing is shown to be equivalent to the Iwasawa main conjecture localized at the augmentation ideal, and for good ordinary primes with p-distinguished large-image representations the formula is unconditional. Consequences include a discrete analogue of the Beilinson–Bloch–Kato conjecture, the p-parity conjecture, and a higher-weight p-converse to the theorem of Gross–Zagier and Kolyvagin.","feed_headline":"Kurihara numbers determine exact Selmer group structure","feed_subtitle":"For modular forms with large Galois image, corank, elementary divisors, and the BSD p-part come from modular-symbol L-values.","key_machinery":"The central objects are the Kurihara numbers $e^{\\min,r} = \\{ e^{\\min,r}_n \\in \\mathcal{O}/I_n \\mathcal{O} \\}_{n \\in \\mathcal{N}_1}$, collections of modular-symbol values at $s = r$, normalized by minimal integral periods and reduced modulo the ideal $I_n$ generated by $\\ell-1$ and $1 - a_\\ell(f) + \\psi^{-1}(\\ell)\\ell^{k-1}$ for primes $\\ell$ dividing $n$. Their numerical invariants $\\mathrm{ord}$ and $\\partial^{(i)}$ mirror the order of vanishing and Taylor coefficients of $L(f,s)$ at $s = r$, which is why the paper calls them Kolyvagin derivatives of special L-values. The proof rests on three load-bearing mechanisms: an explicit reciprocity law with torsion coefficients that equates the image of Kato's Kolyvagin system with the Kurihara numbers up to a nonzero uniform constant; Mazur–Rubin's Kolyvagin system machinery, including the rigidity and rank-one property of Kato's system; and a relative Kolyvagin system argument that compares canonical and Bloch–Kato Selmer structures using global Poitou–Tate duality, Flach's generalized Cassels–Tate pairing, the self-duality of $T^\\dagger_f$, and the functional equation of modular symbols, which supplies the parity constraint in the filtration.","core_discovery":"Let $f$ be a cuspidal newform and $p \\geq 3$ a prime such that $\\rho_f$ has large image. Theorem 1.1 states: if the collection of Kurihara numbers $e^{\\min,\\dagger}$ for $f$ at the central critical point $s = k/2$ does not vanish, then Kato's Kolyvagin system $\\kappa_{\\mathrm{Kato},\\dagger}$ for $T^\\dagger_f$ is non-trivial, and there is an isomorphism of co-finitely generated $\\mathcal{O}$-modules $\\mathrm{Sel}(\\mathbb{Q}, W^\\dagger_f) \\cong (F/\\mathcal{O})^{\\oplus \\,\\mathrm{ord}(e^{\\min,\\dagger})} \\oplus \\bigoplus_{i\\geq 0} (\\mathcal{O}/\\pi^{e_i}\\mathcal{O})^{\\oplus 2}$, where $e_i = \\tfrac{1}{2}\\big(\\partial^{(\\mathrm{ord}(e^{\\min,\\dagger})+2i)}(e^{\\min,\\dagger}) - \\partial^{(\\mathrm{ord}(e^{\\min,\\dagger})+2i+2)}(e^{\\min,\\dagger})\\big)$. Consequently the Selmer corank equals $\\mathrm{ord}(e^{\\min,\\dagger})$ and the length of the quotient by the maximal divisible submodule equals $\\partial^{(\\mathrm{ord})}(e^{\\min,\\dagger}) - \\partial^{(\\infty)}(e^{\\min,\\dagger})$. Theorem 1.2 proves that the non-vanishing of $e^{\\min,\\dagger}$ is equivalent to the Iwasawa main conjecture localized at the augmentation ideal, which makes the structure theorem unconditional in the good ordinary, p-distinguished, large-image case. The discovery is that discrete Kolyvagin derivatives of L-values, rather than complex analytic or p-adic derivatives, carry the exact bound of the Selmer group, so the result is insensitive to weight, to the local behaviour of $f$ at $p$, and to analytic rank.","pith_inferences":["The equivalence between e^{min,†} != 0 and the localized Iwasawa main conjecture suggests a computational route to numerical verification of that conjecture: a single non-vanishing Kurihara number at a suitable Kolyvagin prime would certify the localized main conjecture, and the appendix tables can be read as such numerical evidence.","Because the argument upgrades upper-bound-only Euler-system information to an exact structure formula, other Selmer-group results obtained from Euler systems likely admit the same discrete variation, turning one-sided divisibility into equalities where the corresponding localized Iwasawa main conjecture is known.","The analytic fudge factor ∂^{(∞)} is deliberately left uncontrolled, and the paper conjectures it is the product of local Tamagawa factors away from p; a natural testable extension is to compute both sides numerically for higher-weight modular forms with canonical periods, where the conjectural equality becomes a concrete, checkable congruence.","The relative Kolyvagin-system comparison between canonical and Bloch–Kato Selmer structures appears transplantable to other settings with an Euler system and a self-dual twist, such as anticyclotomic families or higher-rank groups, although such extensions would need new global duality input."],"forward_implications":["If the Kurihara numbers do not vanish, Kato's Kolyvagin system is non-trivial; this is the cyclotomic analogue of Kolyvagin's conjecture and is proved unconditionally for good ordinary p-distinguished forms of large image at p >= 5.","The Selmer corank, all higher Fitting data, and the p-part of a Birch–Swinnerton-Dyer formula follow from the single family e^{min,†}, with no condition on analytic rank, weight, or the local automorphic representation at p.","For analytic rank zero, the length of the Selmer group is the normalized special L-value minus the analytic fudge factor ∂^{(∞)}(e^{min,†}), and this extends to all critical points r != k/2 without the Iwasawa main conjecture.","In the good ordinary, p-distinguished, large-image setting, the paper obtains the p-parity conjecture, the rank-one p-converse to Nekovář and Zhang, and computational upper bounds on Selmer ranks.","The appendix computes Kurihara numbers for elliptic curves of rank up to four and for higher weight modular forms, determining Selmer structures that go beyond what the Birch–Swinnerton-Dyer conjecture alone predicts."],"supporting_citations":[{"why":"Supplies the Kolyvagin system framework, the structure theorem for p-strict Selmer groups, and the core-rank-one rigidity used throughout the proof.","marker":"[MR04]"},{"why":"Provides Kato's zeta elements, the explicit reciprocity law, and the one-sided divisibility of the Iwasawa main conjecture that the paper refines.","marker":"[Kat04]"},{"why":"Proves the Iwasawa main conjecture for GL2 at ordinary primes, making the structure theorem unconditional in the good ordinary p-distinguished case.","marker":"[SU14]"},{"why":"Proves the Iwasawa main conjecture for Hilbert modular forms, used in Theorem 3.19 for the p-distinguished ordinary case.","marker":"[Wan15]"},{"why":"Introduces the Kurihara-number approach to Selmer structure for elliptic curves, which the paper generalizes to higher weight modular forms.","marker":"[Kur14a]"},{"why":"Develops the modular-symbol computations and functional equations for Kurihara numbers that underpin the discrete L-value invariants.","marker":"[Kur14b]"},{"why":"Settles Perrin-Riou's conjecture on Kato's zeta elements, used for the analytic rank-one case in Theorem 1.2.","marker":"[BDV22]"},{"why":"Supplies the generalized Cassels–Tate pairing needed in the relative Kolyvagin argument to track the parity-constrained filtration.","marker":"[Fla90]"},{"why":"Provides the Gross–Zagier formula, used together with Kolyvagin's theorem for the rank-one structural results for weight-two forms.","marker":"[GZ86]"},{"why":"Provides Kolyvagin's Euler-system arguments for elliptic curves, the basis for the rank-one p-converse and for Corollary 1.10.","marker":"[Kol90]"}],"fun_headline_variants":["Kurihara numbers fix Selmer rank and module structure","Exact Selmer structure from modular-symbol L-values","Kolyvagin L-values determine Selmer shape exactly","No analytic rank needed: Kurihara numbers set Selmer","Refined BSD: Selmer structure from Kurihara numbers"],"cache_read_input_tokens":46976,"weakest_assumption_plain":"Everything hangs on the non-vanishing of the collection of Kurihara numbers at the central critical point; the paper proves this is equivalent to the Iwasawa main conjecture localized at the augmentation ideal, so for forms outside the proven ordinary case the structure formula is conditional on that conjecture.","fun_headline_variants_meta":{"raw":{"variants":["Kurihara numbers fix Selmer rank and module structure","Exact Selmer structure from modular-symbol L-values","Kolyvagin L-values determine Selmer shape exactly","No analytic rank needed: Kurihara numbers set Selmer","Refined BSD: Selmer structure from Kurihara numbers"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000378,"raw_usage":{"total_tokens":2170,"prompt_tokens":1262,"completion_tokens":908,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":878,"completion_tokens_details":{"reasoning_tokens":827}},"tokens_in":878,"tokens_out":908,"duration_ms":8157,"temperature":1.0,"reasoning_tokens":827,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T21:39:51.689145+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take an elliptic curve of analytic rank two in the appendix's class, at a good ordinary prime $p \\geq 5$ with p-distinguished large-image representation, and compute the Kurihara numbers $e^{\\min,\\dagger}_n$ for all Kolyvagin-squarefree n up to a growing bound. The paper predicts a non-vanishing term at some n with $\\nu(n) = 2$; if every computed $e^{\\min,\\dagger}_n$ vanished, that would contradict the unconditional case of Theorem 1.2, since the known Iwasawa main conjecture at the augmentation ideal would force non-vanishing. Such a computation is feasible because the appendix already exhibits many non-vanishing values.","supporting_citations":[],"review_version":1}