{"id":"11cb7ca7-a028-4bf8-a988-8e3a659d7677","arxiv_id":"2610.28196","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A new anticyclotomic Euler system over imaginary biquadratic fields is constructed from diagonal cycles and applied to Bloch-Kato ranks 0, 1, and 2 and to the Iwasawa-Greenberg main conjecture.","lead":"This paper constructs a new Euler system, a family of cohomology classes with norm relations, for a self-dual Hecke character over an imaginary biquadratic field, and uses it to prove cases of the Bloch-Kato conjecture and the Iwasawa main conjecture. If correct, it extends results previously known for imaginary quadratic fields and CM elliptic curves to certain CM abelian surfaces over biquadratic fields.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek/deepseek-v4-flash","headline":"Theorem A's construction is not the weakest spot; the applications hinge on the unpublished JNS framework, and the paper never verifies its hypotheses for the classes (6.16)/(6.17), so a failure there would invalidate Theorems B–D.","rationale":"The central Euler-system construction in Section 3 appears coherent: the tame norm relations are derived from published sources [LLZ15, LZ19, Rub00], and the general-weight step follows the published framework of [Do26] and [CD25]. I found no evidence that Theorem 3.3.3 is circular or internally inconsistent. However, the advertised arithmetic consequences do not follow from Theorem A alone; they require an external theorem from an unpublished preprint. The step in Section 6.4 is an assertion, not a verification: the text says the classes form an Euler system in the sense of [JNS] and lists only simple Galois-theoretic properties, without checking the JNS Selmer-structure axioms. Because [JNS] is not publicly available, this is a genuine correctness risk for Theorems B, C, and D. This is the same load-bearing point the reader identified, so the conditional verdict should stand unchanged.","tokens_in":41200,"tokens_out":32519,"duration_ms":406493,"concrete_test":"Obtain the current [JNS] preprint and transcribe its definition of a split anticyclotomic Euler system into a numbered hypothesis list; then check each hypothesis against the classes c_z_{mu3,chi} of Theorem 3.3.3 and the Selmer conditions of Proposition 6.3.1. In particular, verify that the local condition at P3/P4 induced by (6.13) equals the JNS local condition H^1_F for the one-dimensional coefficient O_chi, and that \\bar{chi} satisfies the JNS residual non-degeneracy axiom. If all hypotheses check, Theorems B–D are supported; if any fails, the applications collapse even though Theorem A may stand.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The load-bearing step is not the Euler-system construction itself but the identification of (6.16)/(6.17) with a Jetchev–Nekovar–Skinner split anticyclotomic Euler system. In Theorems 6.4.1 and 6.4.2 the proof asserts this identification and then checks only three properties: absolute irreducibility of a one-dimensional residual character, existence of sigma with a rank-one quotient, and existence of gamma with a nontrivial scalar action. It does not verify the JNS hypotheses on the Selmer structure: that the local condition (6.13) at the four primes above p coincides with the JNS ordinary condition for the one-dimensional character O_chi, that the Euler factor P_{L4} equals the JNS Euler factor, that the restricted index set N'_chi satisfies the JNS conductor/ramification hypotheses, and that the residual character \\bar{chi} satisfies the JNS non-degeneracy axiom. Since [JNS] is unpublished, a referee cannot check these conditions from the text. If, for example, JNS requires a rank-two Galois representation or a different local filtration than (∅,0,∅,0), then the Selmer rank-one conclusion and the divisibility of Theorem 6.4.2 do not follow, and Theorems B, C, and D are unsupported. No internal inconsistency in the core norm-relation construction of Theorem 3.3.3 was found.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek/deepseek-v4-flash","summary":"The paper constructs an anticyclotomic Euler system over an imaginary biquadratic field K0 for a self-dual Hecke character χ. The construction starts from diagonal cycles in the triple product of modular curves, uses the norm-map formalism of Loeffler–Zerbes, and specializes CM Hida families to obtain Iwasawa cohomology classes z_{μ3,χ} indexed by squarefree ideals of the real quadratic subfield. The main theorem (Theorem 3.3.3) asserts the Euler-system norm relation with Euler factor P_{L4}(Frob_{L4}) = det(1 - Frob_{L4} X | O_{χ^{-1}}(1)). The paper then identifies these classes with the 'split' anticyclotomic Euler systems of the unpublished Jetchev–Nekovář–Skinner framework, uses a reciprocity law for triple product p-adic L-functions, and derives applications: a rank-zero Bloch–Kato statement (Theorem 6.5.1), a full Iwasawa–Greenberg main conjecture (Theorem 6.7.1), and rank-one/rank-two Bloch–Kato consequences (Theorems 6.8.1 and 6.8.5).","tokens_in":41702,"tokens_out":8714,"duration_ms":57067,"significance":"If the construction is correct, this is a substantial contribution: it is a new Euler system in a biquadratic setting with one-dimensional self-dual coefficients, generalizing Rubin's elliptic-unit approach and complementing Hida and Hsieh's anticyclotomic results. The norm-relation computations in Section 3 are explicit and use no fitted parameters, and the local-condition computation in Section 6.3 is a useful independent contribution. However, the arithmetic applications are conditional on unpublished material: the rank-one, divisibility, and nonvanishing steps rely on the preprints [JNS] and [BHTY26], and the paper does not verify the hypotheses of [JNS] for the specific classes (6.16) and (6.17). The central Euler-system construction appears plausible, but the advertised theorems do not yet follow from the text as written.","major_comments":[{"comment":"The proof that the collections (6.16) and (6.17) form Euler systems in the sense of Jetchev–Nekovář–Skinner consists of an assertion followed by checks (i)–(iii), but these checks are not the hypotheses of [JNS]. The paper does not verify that the local condition (6.13) at the four primes above p agrees with the JNS ordinary condition for the one-dimensional coefficient O_χ, that P_{L4} is the Euler factor required by [JNS], that the index set N'_χ satisfies the JNS conductor and ramification hypotheses, or that the residual character \\barχ satisfies the JNS non-degeneracy axiom. Since [JNS] is listed as an unpublished preprint, these points cannot be checked from the text. This gap is load-bearing: Theorem 6.4.1 supplies the rank-one Selmer conclusion and Theorem 6.4.2 supplies the divisibility used in each of Theorems 6.5.1, 6.7.1, and 6.8.5.","section":"§6.4, Theorems 6.4.1 and 6.4.2","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The proof obtains only a divisibility in the anticyclotomic algebra Λ^-_{K0}, and then asserts that 'following the proof of [Hsi14a, Thm. 8.17], replacing Hida's solution to the anticyclotomic main conjecture in [Hid06] by our divisibility, we obtain the equality over the full Λ_{K0}'. This is not a formality: one must show that the hypotheses of [Hsi14a] encoded in Assumption 6.3 apply to the specific pair (X_{0,∅,∅,0}(K_{0,∞},O_{χ^{-1}}(1)), L^{K0}_{P1,P4,χ}) and that a divisibility in the anticyclotomic quotient upgrades to equality of characteristic ideals over the full Iwasawa algebra. Without that argument, Theorem C is not established beyond a divisibility.","section":"§6.7, proof of Theorem 6.7.1","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The implication L(χ,0) ≠ 0 ⇒ L(V†,0) ≠ 0 in the proof of Theorem 6.5.1, and the non-torsion of the base class in the proof of Theorem 6.7.1, both depend on a nonvanishing theorem from the unpublished preprint [BHTY26]. The paper does not state the exact form of the theorem used, and after the twist (ψ1,ψ1') ↦ (ψ1φ,ψ1'φ) it does not verify that the twisted triple still satisfies the running hypotheses, including condition (6.7) and the minimality condition (6.19). Since Theorems B, C, and D rely on this step, the dependence on [BHTY26] should either be removed, made completely explicit, or clearly declared as a standing assumption.","section":"§6.5 and §6.7, use of [BHTY26]","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The factorization of the triple product p-adic L-function is a key step, but its proof is highly compressed. In particular, the derivation of (6.24)–(6.26) combines Hida's adjoint L-value formula, Dirichlet's class number formula, and the relation (6.25) between the Katz p-adic L-function and the chosen generator η_g of the congruence ideal, all up to p-adic units. The period identity for the biquadratic Katz L-function, obtained via the CM abelian surface E×E, is also only asserted. An error in any one of these unit identities would break the factorization. The author should state each period and unit identity as a precise lemma with its hypotheses and either prove it or give a complete reference.","section":"§6.6.3, Proposition 6.6.6","authors_acknowledge":null,"authors_acknowledge_p":null}],"minor_comments":[{"comment":"The statement of Theorem A says the norm relation holds 'for every prime λ3 ∈ N', but the theorem in Section 3.3 is stated for λ3 ∈ L'_χ; the notation should be made consistent.","section":"Introduction, Theorem A","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"In the table for Sel_BK(L,V_ψ), the first two rows are identical: both the case k'_1 ≥ k_1+k_2 and the case k_1 ≥ k'_1+k_2 are assigned Sel_{∅,0,0,∅}. If this is intentional, a remark would help; if it is a typo, the second row should be corrected.","section":"§4.2, Lemma 4.2.1","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The condition written as 'Q_{p|N} ε_p = 1' is not defined; the intended meaning appears to be a product over primes p|N of local root numbers, but the notation should be made explicit.","section":"§6.5, Theorem 6.5.1","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The text says the Iwasawa cohomology decomposition follows as in equation (3.5), but (3.5) was written for ordinary cohomology; the Shapiro-lemma and twist identifications needed for the Iwasawa version should be spelled out.","section":"§3.3, equation (3.8)","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"There are several typographical issues that should be fixed: the title has 'BIQUADRA TIC', the phrase 'Here, we enlarging Q_p to F' is ungrammatical, and the reference to the 'left superscript notation' in the Introduction does not display the superscripts being discussed.","section":"Throughout","authors_acknowledge":null,"authors_acknowledge_p":null}],"recommendation":"major_revision","confidential_remarks":"The core Euler-system construction seems potentially sound and is not the main weakness; the decisive issue is that Theorems B, C, and D rest on the unpublished [JNS] framework and on [BHTY26], with the JNS hypotheses never verified for the specific classes. I would advise the editor that acceptance should require either a verification appendix for the JNS hypotheses or a reformulation of the theorems as conditional on clearly stated assumptions, together with a completed proof of the final equality step in Theorem 6.7.1. The paper's contribution to the Euler-system construction itself is significant enough to warrant revision rather than rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek/deepseek-v4-flash","letter":"The real news here is Theorem A: an anticyclotomic Euler system for a self-dual Hecke character over an imaginary biquadratic field, with the norm relations written out. That construction is a genuine extension of the author's earlier biquadratic work [Do26]. The tame relation is quoted from there, but the projection to the self-dual component, the wild relations through Hida families, and the Euler-factor massaging over the restricted index set N'_chi are done here, and nothing in the core looks circular or fitted. The local-condition computation in Section 6.3 is explicit, the factorization of the triple-product p-adic L-function (Proposition 6.6.6) is a substantial interpolation computation, and the concrete LMFDB examples help.\n\nThe soft spots are all in the application layer, and the reader's flag lands. Section 6.4 asserts that the classes (6.16)/(6.17) form a Jetchev-Nekovar-Skinner split Euler system and verifies three properties, all near-trivial for a one-dimensional representation (sigma = 1, plus a gamma outside ker psi). The substantive hypotheses — that (6.13) at the four primes above p is the JNS ordinary filtration, that P_{L4} is the JNS Euler factor, that N'_chi meets the conductor and ramification axioms, that the residual character is non-degenerate for JNS — are asserted, not checked, and [JNS] is unpublished, so a referee cannot check them from the text. I don't think this is fatal; the framework exists for exactly this one-dimensional conjugate self-dual setting, and the paper points to [Do22, Sec. 4.3] and [ACR25b, Sec. 8] for the analogous matching in the imaginary quadratic case. But the burden is real: as written, Theorems B, C, and D rest on an unverified identification, and the fix is the author's.\n\nTwo smaller points. Theorem C reduces the hard direction of the full Iwasawa main conjecture to [Hsi14a, Thm 8.17] in one sentence, which is a large black box, and the nonvanishing that makes the base class non-torsion comes from [BHTY26], another unpublished preprint. Theorem D is an implication conditional on non-torsion; the abstract's \"rank 2\" wording oversells that a little, though the proof of Theorem 6.7.1 does supply non-torsion under its hypotheses. The manuscript is also rough in places — missing words, a repeated theorem number — minor.\n\nThis paper is for people working in anticyclotomic Iwasawa theory for CM fields. It deserves a serious referee: the conditional verdict is right, and the referee should push hard on the JNS verification before the applications are accepted.","headline":"The Euler system construction is real and the norm relations are written out; the applications hinge on an unpublished JNS framework whose hypotheses are asserted, not verified.","tokens_in":42135,"tokens_out":10443,"would_cite":true,"duration_ms":64617,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"pith_extraction":{"msc":["11G05","11G40","11R23"],"pacs":[],"model":"deepseek/deepseek-v4-flash","headline":"The paper builds a new Euler system for self-dual Hecke characters over imaginary biquadratic fields and derives Bloch–Kato rank-0, 1, and 2 results plus the Iwasawa main conjecture.","keywords":["Euler systems","Iwasawa theory","Hecke characters","biquadratic fields","diagonal cycles","Bloch–Kato conjecture","Iwasawa main conjecture","CM abelian surfaces"],"falsifier":"For the paper's explicit example — $\\chi$ of infinity type $(1,0,0,1)$ and conductor $(42+42i)$ over $K_0 = \\mathbb{Q}(\\zeta_{12})$ — the tame norm relation at weights $(2,2,2)$ can be tested directly: choose a small split prime $\\ell \\equiv 1 \\pmod p$ whose Frobenius is conjugate to $\\tau_\\chi$, compute the Hecke character values $\\psi'_1(\\lambda_1)$, $\\psi_1(\\lambda_1)$, $\\psi_2(\\bar{\\lambda}_2)$ at the four primes above $\\ell$, and check that the displayed norm-relation factor equals $P_{L_4}(\\mathrm{Frob}_{L_4})$; a single prime where the two sides disagree would falsify the construction, while a numerical sweep over small primes would confirm the weight-$(2,2,2)$ core on which all applications rest.","tokens_in":42681,"feed_emoji":"🔢","tokens_out":21700,"duration_ms":119804,"temperature":0.7,"pith_summary":"An Euler system is a compatible family of Galois cohomology classes whose norm relations reproduce the Euler factors of an $L$-function; when one exists it is the standard tool for bounding Selmer groups and proving Iwasawa main conjectures. This paper constructs a new one for a self-dual Hecke character $\\chi$ of an imaginary biquadratic field $K_0$, starting from diagonal cycles in the triple product of modular curves whose three $\\theta$ series come from Hecke characters of two distinct imaginary quadratic subfields. The classes are indexed by ideals of the real quadratic subfield $K_3$, and the norm relation is governed by the local Euler factor at one of the four primes above each auxiliary prime $\\ell$. Feeding this system into the Jetchev–Nekovář–Skinner machinery, the paper derives the Bloch–Kato conjecture in analytic ranks 0, 1, and 2 (the last for certain CM abelian surfaces) and the full Iwasawa–Greenberg main conjecture for $\\chi$, generalizing Rubin's elliptic-units results from imaginary quadratic fields to biquadratic fields.","feed_headline":"New Euler system proves Bloch–Kato cases over biquadratic fields","feed_subtitle":"It extends Rubin's elliptic-units proof to biquadratic fields and reaches CM abelian surfaces.","key_machinery":"The load-bearing object is the diagonal cycle class in the triple product of modular curves, pushed through the norm-compatible isomorphisms $\\nu_n: \\mathcal{O}[H^{(p)}_n] \\otimes_{\\mathbb{T}'(N),\\phi_n} H^1_{\\mathrm{et}}(Y_1(N)_{\\overline{\\mathbb{Q}}}, \\mathbb{Z}_p(1)) \\cong \\mathrm{Ind}^{\\mathbb{Q}}_{K(n)} \\mathcal{O}(\\psi_P^{-1})$ of Theorem 2.3.2. These isomorphisms convert the cycle into classes over ring class fields of $K_0$, and Shapiro's lemma then splits the resulting representation into two one-dimensional self-dual pieces $\\psi$ and $\\chi$ (equation (3.5)). The norm relation is extracted by a congruence at primes $\\ell \\equiv 1 \\pmod p$ split in $K_0$: the four-way prime decomposition makes the displayed combination of Hecke character values collapse modulo $(\\ell-1)$ to $(1 - \\psi^{-1}(L_4)\\mathrm{Frob}_{L_4})(1 - \\chi^{-1}(L_4)\\mathrm{Frob}_{L_4})$, and the mod $p$ distinction between $\\psi$ and $\\chi$ at inertia (Assumption 3.1) lets the unwanted $\\chi$ factor be inverted away following the cited Loeffler–Zerbes result. Extending from weights $(2,2,2)$ to arbitrary weights is carried out inside Hida families, which produces the wild norm relations over the anticyclotomic $\\mathbb{Z}_p^2$-extension.","core_discovery":"The central claim is Theorem 3.3.3: there is a collection of Iwasawa cohomology classes $c_{z_{\\mu_3,\\chi}} \\in H^1_{\\mathrm{Iw}}(K_0[\\mu_3 p^\\infty], O_\\chi)$, indexed by squarefree ideals $\\mu_3$ of $\\mathcal{O}_{K_3}$ whose rational norms are squarefree and split in $K_0$, such that for every auxiliary prime $\\lambda_3$ of norm $\\ell$ the corestriction from level $\\mu_3\\lambda_3$ to level $\\mu_3$ sends $c_{z_{\\mu_3\\lambda_3,\\chi}}$ to $P_{L_4}(\\mathrm{Frob}_{L_4})(c_{z_{\\mu_3,\\chi}})$, where $P_{L_4}(X) = \\det(1 - X\\,\\mathrm{Frob}_{L_4} \\mid O_{\\chi^{-1}}(1))$ is the local Euler factor at the prime $L_4$ above $\\ell$. The construction proceeds in two stages: a tame norm relation is proven in weight $(2,2,2)$ by a congruence computation with explicit Hecke character values, and the general weights follow by varying the diagonal cycles inside Hida families, giving the wild norm relations over the anticyclotomic $\\mathbb{Z}_p^2$-extension. The paper then shows the $\\chi$-classes land in the Greenberg-type Selmer group $\\mathrm{Sel}^{\\emptyset,0,\\emptyset,0}$ (Proposition 6.3.1), and that a non-torsion base class forces the associated Selmer groups to have rank one and yields the Iwasawa divisibility of Theorem 6.4.2. The explicit reciprocity law of Theorem 6.1.1 links the base class to the $g$-unbalanced triple product $p$-adic $L$-function, and the factorization of Proposition 6.6.6 splits that $p$-adic $L$-function into the $\\psi$ and $\\chi$ anticyclotomic Katz-type factors, converting the structural statements into the Bloch–Kato and Iwasawa main conjecture theorems.","pith_inferences":["The same diagonal-cycle template suggests a route to Euler systems over composita of three or more imaginary quadratic fields, or over CM fields with several imaginary quadratic subfields, whenever the base-changed representation splits into distinct self-dual one-dimensional characters; the mod $p$ separation condition of Assumption 3.1 would be the natural obstruction to watch there.","The inversion step removing the factor $(1 - \\chi^{-1}(L'_4)\\mathrm{Fr}_{L'_4})$ is only licit while $\\psi$ and $\\chi$ remain distinct modulo $p$, which predicts a sharp boundary at primes $p \\leq k'_1$ where the construction should visibly fail; this is a concrete place to test the method's limits in small examples.","If the Jetchev–Nekovář–Skinner hypotheses were verified directly over $\\mathbb{Z}_p^2$-extensions of biquadratic fields, the divisibility of Theorem 6.4.2 would hold unconditionally for every constructed class, rather than only for classes whose base class is provably non-torsion via the reciprocity law."],"forward_implications":["If $L(\\chi,0) \\neq 0$, then $\\mathrm{Sel}^{\\mathrm{BK}}(K_0, V_\\chi) = 0$, so the Bloch–Kato conjecture holds in analytic rank 0 for $\\chi$ (Theorem 6.5.1).","In the balanced range, a non-torsion base class forces $\\dim_F \\mathrm{Sel}^{\\mathrm{BK}}(K_0, V_\\chi) \\geq 1$, with equality when the class remains non-torsion after projection to the finite level (Theorem 6.8.1 and Remark 6.8.2).","Under the sign and ramification hypotheses of Assumption 6.3, the characteristic ideal of the anticyclotomic Iwasawa Selmer group equals the ideal generated by the anticyclotomic $p$-adic $L$-function $L_{K_0}^{P_1,P_4,\\chi}$, i.e. the full Iwasawa–Greenberg main conjecture (Theorem 6.7.1).","For $k_2 = 1$, even $k'_1$, and $k_1 = k'_1 + 1$, a non-torsion base class for a CM-abelian-surface character with $L(\\chi,0) = 0$ and sign $+1$ forces $\\dim_F \\mathrm{Sel}^{\\mathrm{BK}}(K_0, V_\\chi) = 2$ (Theorem 6.8.5)."],"supporting_citations":[{"why":"Supplies the 'split' anticyclotomic Euler system machinery that turns the constructed classes into rank-one Selmer statements, Iwasawa divisibility, and the rank-two result.","marker":"[JNS]"},{"why":"Provides the balanced diagonal classes and the explicit reciprocity law (Theorem 6.1.1) relating them to the triple product $p$-adic $L$-function.","marker":"[BSV22]"},{"why":"Provides the norm-compatible isomorphisms and commutative diagrams (Theorem 2.3.2) that convert the diagonal cycle into classes over ring class fields and produce the norm relation.","marker":"[LLZ15]"},{"why":"The author's prior work supplies the projection to ring class fields of $K_0$ and the tame norm relation (Propositions 3.2.1 and 3.2.2).","marker":"[Do26]"},{"why":"Supplies the test-vector framework and the device for quotienting out diamond operators to strip the $(\\ell-1)$ factor from the tame norm relation.","marker":"[CD25]"},{"why":"The cited Theorem 5.3.3 justifies inverting the unwanted character factor in the norm relation once the mod $p$ distinction of Assumption 3.1 is granted.","marker":"[LZ19]"},{"why":"Rubin's Euler system lemmas (9.6.1 and 9.6.3) and Theorem 6.4.1 justify the character elimination and the passage to Iwasawa cohomology classes.","marker":"[Rub00]"},{"why":"Constructs the $g$-unbalanced triple product $p$-adic $L$-function and its interpolation formula (Theorem 5.3.1) used in the reciprocity law.","marker":"[Hsi21]"},{"why":"Main theorem 2 supplies the ramification conditions (Assumption 6.3) and the argument that upgrades the Iwasawa divisibility to an equality over the full Iwasawa algebra.","marker":"[Hsi14a]"}],"fun_headline_variants":["Euler system over biquadratic fields proves Bloch–Kato ranks 0–2","Iwasawa–Greenberg main conjecture for biquadratic fields via Euler systems","Rubin's elliptic-unit method extended to biquadratic fields","Euler systems over biquadratic fields apply to CM abelian surfaces"],"cache_read_input_tokens":0,"weakest_assumption_plain":"The theorems rest on the applicability of the unpublished Jetchev–Nekovář–Skinner Euler-system machinery to the anticyclotomic tower over the biquadratic field with one-dimensional coefficients $\\psi$ and $\\chi$, a property that Section 6.4 asserts but does not verify hypothesis by hypothesis.","fun_headline_variants_meta":{"raw":{"variants":["Euler system over biquadratic fields proves Bloch–Kato ranks 0–2","Iwasawa–Greenberg main conjecture for biquadratic fields via Euler systems","Rubin's elliptic-unit method extended to biquadratic fields","Euler systems over biquadratic fields apply to CM abelian surfaces"]},"model":"deepseek/deepseek-v4-flash","effort":"low","cost_usd":0.000182,"raw_usage":{"cost":0.0001821456,"is_byok":false,"cost_details":{"upstream_inference_cost":0.0001821456,"upstream_inference_prompt_cost":3.69096e-05,"upstream_inference_completions_cost":0.000145236},"total_tokens":2034,"prompt_tokens":1236,"completion_tokens":798,"prompt_tokens_details":{"audio_tokens":0,"video_tokens":0,"cached_tokens":1038,"cache_write_tokens":0},"completion_tokens_details":{"audio_tokens":0,"image_tokens":0,"reasoning_tokens":722}},"tokens_in":1236,"tokens_out":798,"cost_source":"openrouter_usage","duration_ms":7184,"image_pages":[],"temperature":1.0,"billed_attempts":1,"reasoning_tokens":722,"transport_provider":"openrouter","cache_read_input_tokens":0,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-10-21T08:07:48.038283+00:00","model_set":{"reader":"deepseek/deepseek-v4-flash"},"falsifier":"For the paper's explicit example — $\\chi$ of infinity type $(1,0,0,1)$ and conductor $(42+42i)$ over $K_0 = \\mathbb{Q}(\\zeta_{12})$ — the tame norm relation at weights $(2,2,2)$ can be tested directly: choose a small split prime $\\ell \\equiv 1 \\pmod p$ whose Frobenius is conjugate to $\\tau_\\chi$, compute the Hecke character values $\\psi'_1(\\lambda_1)$, $\\psi_1(\\lambda_1)$, $\\psi_2(\\bar{\\lambda}_2)$ at the four primes above $\\ell$, and check that the displayed norm-relation factor equals $P_{L_4}(\\mathrm{Frob}_{L_4})$; a single prime where the two sides disagree would falsify the construction, while a numerical sweep over small primes would confirm the weight-$(2,2,2)$ core on which all applications rest.","supporting_citations":[],"review_version":1}