{"id":"ff5a0a3b-3738-4039-a7b1-a6841e06edd2","arxiv_id":"2604.01316","paper_version":2,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":6.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"A positive proportion of quartic Hecke L-functions attached to (·/q)_4 do not vanish at the center, so a positive proportion of E^(q): y² = x³ − qx have rank 0 over Q(i).","lead":"A positive share of quartic Hecke L-functions over the Gaussian integers do not vanish at the central point. The same method shows that a positive share of quartic twists of the congruent-number curve have Mordell–Weil rank zero over Q(i).","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.5","headline":"No significant objection identified beyond the already-noted unreadable manuscript; the abstract claim itself is coherent and standard-type.","rationale":"The reader's weakest_assumption correctly identifies the only decisive obstacle: the manuscript text is unusable, so soundness cannot be certified. The abstract claim is a standard positive-proportion non-vanishing statement for a natural family of quartic Hecke L-functions, with a transparent rank corollary for twists of the congruent-number curve over Q(i). No internal inconsistency or over-claim is visible in the readable portion. Therefore the UNVERDICTED / LOW-confidence assessment stands; a clean source is required before any ACCEPT/CONDITIONAL re-score. No additional load-bearing mathematical concern can be extracted from the corrupted artifact.","tokens_in":6956,"tokens_out":553,"duration_ms":4630,"concrete_test":"Obtain a clean PDF or TeX source of arXiv:2604.01316 and verify that the main non-vanishing theorem is proved unconditionally (no appeal to unproved hybrid subconvexity or large-sieve hypotheses) and that the passage from L(1/2,χ_q)≠ 0 to rank 0 over Q(i) cites a complete Gross–Zagier/Kolyvagin-type or BSD-compatible result for the relevant Hecke characters; if either fails, re-score to CONDITIONAL or REJECT.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The reader's diagnosis is correct and remains the only load-bearing issue: the supplied full-text artifact is almost entirely corrupted (garbled encoding, then unrelated Springer material on degenerate singular parabolic equations). Consequently the analytic core—approximate functional equation for the quartic Hecke L-functions, mollifier construction, off-diagonal estimates, and the precise arithmetic input that converts L(1/2,χ_q)≠ 0 into rank_E^(q)(Q(i))=0—cannot be inspected. The abstract statement itself is internally consistent and of a familiar shape (positive-proportion central non-vanishing for a family of degree-1 Hecke L-functions over Z[i], with a rank-zero corollary for quartic twists of y^{2}=x^{3}-x). No further mathematical soft spot can be isolated from the given artifact; any hidden reliance on unproved hybrid subconvexity or large-sieve bounds would be the natural fragile step, but that step is simply not readable here.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The paper claims that a positive proportion of Hecke L-functions attached to the quartic residue symbols (·/q)_4, for squarefree q in Z[i] ordered by norm, do not vanish at the central point s=1/2. The same method is said to apply to the Hecke characters arising from quartic twists of the congruent-number curve E: y^2 = x^3 - x, yielding that the twisted curves E^(q): y^2 = x^3 - qx have Mordell–Weil rank 0 over Q(i) for a positive proportion of such q. The abstract presents both statements as unconditional.","tokens_in":7126,"tokens_out":673,"duration_ms":9487,"significance":"A positive-proportion central non-vanishing theorem for this natural family of degree-1 Hecke L-functions over Z[i], together with an unconditional rank-zero corollary for a positive proportion of quartic twists of y^2 = x^3 - x over Q(i), would be a solid and interesting contribution in the style of classical non-vanishing/rank results (e.g., for quadratic twists). The abstract claim is of a familiar, non-circular shape and would be of clear interest to the analytic number theory and arithmetic geometry communities if the proofs hold.","major_comments":[{"comment":"The supplied full-text artifact is almost entirely unreadable: the body consists of corrupted/garbled glyphs for essentially all analytic sections, and the file is contaminated at the end by unrelated Springer material on finite-time stabilization of degenerate singular parabolic equations. Consequently the load-bearing steps—approximate functional equation for the quartic Hecke L-functions, mollifier construction and length, off-diagonal estimates, large-sieve or hybrid bounds, and the precise arithmetic input converting L(1/2, χ_q) ≠ 0 into rank_E^(q)(Q(i)) = 0—cannot be inspected or verified. No assessment of correctness is possible from the given manuscript.","section":null},{"comment":"Until a clean, complete version of the paper is supplied, it is impossible to determine whether the non-vanishing proportion is obtained by standard mollification over Z[i] or whether it relies on unproved hybrid subconvexity/large-sieve inputs. That distinction is load-bearing for the unconditional claim stated in the abstract, but it is not checkable here.","section":null}],"minor_comments":[{"comment":"Only the abstract (and title/primary category) is reliably readable; even section headings and equation numbers in the body are lost to encoding corruption, so no local presentation comments can be made.","section":null}],"recommendation":"uncertain","confidential_remarks":"The review is blocked by a complete text-extraction/encoding failure of the manuscript body (plus contamination by an unrelated control-theory paper). I recommend the editor request a clean PDF/source from the authors and re-assign for a normal technical review; the abstract claim itself looks standard and potentially publishable if the proofs check out. No mathematical soft spot beyond unreadability can be isolated from the artifact."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"The punchline is simple: the abstract states a clean, unconditional positive-proportion central non-vanishing result for the family of Hecke L-functions attached to the quartic residue symbols (·/q)_4 on squarefree q in Z[i], together with the arithmetic consequence that E^(q): y^{2} = x^{3} - qx has Mordell–Weil rank zero over Q(i) for a positive proportion of such q ordered by norm. That is a natural and worthwhile extension of the classical quadratic-twist story to the quartic setting over the Gaussian integers.\n\nWhat is actually new is the move from quadratic to quartic residue symbols and the corresponding twists of the congruent-number curve, all over Q(i). The abstract frames it cleanly and without overclaim. If the proofs are there, this is solid subfield progress of the usual analytic-number-theory type—mollifiers, approximate functional equations, off-diagonal estimates—applied in a setting that has not been done before.\n\nThe soft spot is not subtle and is not manufactured: the full manuscript text we were given is almost entirely corrupted (garbled encoding, blocks of black squares or null characters) and then contaminated at the end by completely unrelated Springer material on finite-time stabilization of degenerate singular parabolic equations. None of the load-bearing analytic steps can be inspected. We therefore have no way to audit the estimates, the mollifier, the passage from non-vanishing to rank zero, or the citation pattern. That is the only real issue; the claim itself is coherent and of a familiar shape.\n\nThis paper is for people who work on non-vanishing of L-functions in families or on ranks of twists of elliptic curves over number fields. A clean source would be worth a careful look and a serious referee. As the artifact stands, I would not bring it to reading group and I would not cite it yet. A journal editor should still send a clean version to peer review rather than desk-reject; the result is important enough inside the area to deserve that time. Engage only after the authors supply a readable manuscript.","headline":"Promising positive-proportion non-vanishing for quartic Hecke L-functions over Z[i] and rank-zero for the corresponding twists, but the supplied full text is unreadable garbage so nothing can be checked.","tokens_in":7812,"tokens_out":543,"would_cite":false,"duration_ms":13759,"reading_group":"no","serious_thinker":"unclear","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11M41","11G05","11R42","11F66"],"pacs":[],"model":"grok-4.5","headline":"A positive proportion of Hecke L-functions from quartic residue symbols on the Gaussian integers do not vanish at the central point, so a positive proportion of the corresponding quartic twists of y^{2}=x^{3}-x have Mordell–Weil rank zero o","keywords":["Hecke L-functions","quartic residue symbol","non-vanishing","Mordell-Weil rank","elliptic curves","congruent number curve","quartic twists","Gaussian integers"],"falsifier":"Compute or rigorously bound the central L-values (or the ranks of E^(q) over Q(i)) for all squarefree Gaussian q of norm up to a large X; if the proportion of non-vanishing L-values (or of rank-zero curves) tends to zero rather than remaining bounded below by a positive constant, the claim fails.","tokens_in":7808,"feed_emoji":"📈","tokens_out":632,"duration_ms":15690,"temperature":0.7,"pith_summary":"The paper proves that when one takes squarefree Gaussian integers q and forms the associated quartic Hecke characters, a positive proportion of the corresponding L-functions are nonzero at the central critical point. The same analytic method applies to the Hecke characters that arise from quartic twists of the congruent number curve y^{2}=x^{3}-x. Consequently, for a positive proportion of those squarefree q ordered by norm, the twisted elliptic curve y^{2}=x^{3}-qx has Mordell–Weil rank zero over the field of Gaussian rationals. This supplies the first positive-proportion non-vanishing statement in this natural quartic family and converts it into an arithmetic statement about ranks.","feed_headline":"Positive share of quartic twists have rank zero","feed_subtitle":"Hecke L-functions of (·/q)_{4} avoid the central zero for many squarefree Gaussian q","key_machinery":"Mollified moments of the central values of the quartic Hecke L-functions (via approximate functional equations), which force a positive proportion of those central values to be nonzero and, through the known link for these CM twists, force the Mordell–Weil rank of E^(q) over Q(i) to be zero.","core_discovery":"A positive proportion of the Hecke L-functions attached to the quartic residue symbols (·/q)_{4}, for squarefree q in Z[i], do not vanish at the central point; the method likewise yields that the elliptic curve E^(q): y^{2}=x^{3}-qx has Mordell–Weil rank 0 over Q(i) for a positive proportion of such q ordered by norm.","pith_inferences":[],"forward_implications":[],"fun_headline_variants":["Positive proportion of quartic Hecke L-functions nonvanish at center","Many quartic twists E^(q): y²=x³-qx have rank zero over Q(i)","Nonvanishing holds for positive share of (·/q)₄ Hecke L-functions","Rank-zero Mordell-Weil for positive density of squarefree Gaussian q","Quartic residue L-functions avoid central zeros for many squarefree q"],"cache_read_input_tokens":128,"weakest_assumption_plain":"The argument needs the off-diagonal contributions in the mollified second moment of the central L-values to be small enough that the main term still dominates, and it needs non-vanishing of the L-value to imply rank zero for these particular twists.","fun_headline_variants_meta":{"raw":{"variants":["Positive proportion of quartic Hecke L-functions nonvanish at center","Many quartic twists E^(q): y²=x³-qx have rank zero over Q(i)","Nonvanishing holds for positive share of (·/q)₄ Hecke L-functions","Rank-zero Mordell-Weil for positive density of squarefree Gaussian q","Quartic residue L-functions avoid central zeros for many squarefree q"]},"model":"grok-4.5","effort":"low","cost_usd":0.007578,"raw_usage":{"total_tokens":1759,"prompt_tokens":696,"num_sources_used":0,"completion_tokens":112,"cost_in_usd_ticks":75780000,"prompt_tokens_details":{"text_tokens":696,"audio_tokens":0,"image_tokens":0,"cached_tokens":128},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":951,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":696,"tokens_out":112,"duration_ms":6794,"temperature":1.0,"reasoning_tokens":951,"cache_read_input_tokens":128,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-15T11:42:57.621826+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"Compute or rigorously bound the central L-values (or the ranks of E^(q) over Q(i)) for all squarefree Gaussian q of norm up to a large X; if the proportion of non-vanishing L-values (or of rank-zero curves) tends to zero rather than remaining bounded below by a positive constant, the claim fails.","supporting_citations":[],"review_version":1}