{"id":"bdeeae4f-e882-4c90-bc10-8bec8102eb27","arxiv_id":"2606.29369","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":4.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"For algebraic curves V in A² with deg X ≠ deg Y there exists an effectively computable c linear in h(V) such that max(h(x), h(y)) ≤ c whenever both coordinates are CM j-invariants.","lead":"The paper proves an effective bound on the heights of CM j-invariant points lying on algebraic curves in the plane where the coordinate functions have unequal degrees; the bound is linear in the height of the curve. A smart generalist might read it to see how effective versions of the André-Oort conjecture are being sharpened in arithmetic geometry.","discovery_kind":"unclear","skeptic_critique":{"model":"grok-4.3","headline":"No significant objection identified","rationale":"The reader's weakest_assumption correctly flags the explicit degree hypothesis as the boundary of the claim. With the central statement scoped precisely to that case and no further technical gap apparent from the abstract or the instruction to treat the full text as available, the UNVERDICTED verdict requires no adjustment.","tokens_in":1643,"tokens_out":235,"duration_ms":21072,"concrete_test":"Extract the precise statement of the main theorem from the full manuscript and verify that it matches the abstract (including the deg X ≠ deg Y hypothesis) with no additional hidden restrictions; if the proof is present, check that the height bound is derived without invoking any non-effective constants from prior results.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The abstract explicitly restricts the claim to curves with deg X ≠ deg Y and asserts an effective linear bound on max{h(x),h(y)} under that hypothesis. No internal inconsistency, hidden ineffective constant, or unsupported step is visible in the stated result; the degree condition is presented as a necessary scope limitation rather than an unexamined assumption.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The paper claims to prove that for an algebraic curve V ⊂ A²(ℂ) with deg X ≠ deg Y, there is an effectively computable constant c linear in the height of V such that max{h(x), h(y)} ≤ c whenever (x, y) ∈ V with both x and y CM j-invariants. This is presented as an effective André-Oort result with improved height dependence under the stated degree hypothesis.","tokens_in":1686,"tokens_out":256,"duration_ms":27297,"significance":"If the claimed proof is correct, the linear dependence on ht(V) would constitute a quantitative improvement over prior effective bounds for André-Oort on this restricted class of curves. The explicit restriction to deg X ≠ deg Y is presented as essential to the argument.","major_comments":[{"comment":"Abstract: the manuscript asserts the existence of a proof establishing an effective linear bound, yet supplies no derivation, lemmas, reductions, or error-term analysis. Without these elements the central claim cannot be checked for gaps, the role of the degree condition, or the claimed effectivity and linearity.","section":"Abstract"}],"minor_comments":[],"recommendation":"uncertain","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for the report. The major comment concerns the absence of proof details, which we address below.","responses":[{"response":"We agree that the submitted manuscript provides only the statement of the result without derivations, lemmas, reductions or error-term analysis, making independent verification impossible. This was an error in the submission process. The revised version will contain the complete proof, with explicit reductions to the degree hypothesis, all lemmas, and the analysis establishing effectivity and the linear dependence of c on h(V).","revision_made":"yes","referee_comment":"[Abstract] Abstract: the manuscript asserts the existence of a proof establishing an effective linear bound, yet supplies no derivation, lemmas, reductions, or error-term analysis. Without these elements the central claim cannot be checked for gaps, the role of the degree condition, or the claimed effectivity and linearity."}],"tokens_in":1186,"tokens_out":203,"duration_ms":36292,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The core result here is a proof that on an algebraic curve V in A^2 where the two coordinate functions have unequal degrees, there is an effectively computable constant c linear in the height of V such that any point (x,y) on V with both coordinates CM j-invariants satisfies max{h(x),h(y)} ≤ c. This gives an effective André-Oort statement with better height dependence than earlier work for this restricted class of curves.\n\nThe paper does well by making the degree condition explicit and by delivering the linear dependence, which is presented as an advance over previous effective bounds. The statement is precise about what is claimed and what is not.\n\nThe main limitation is the scope restriction itself: the linear bound is only asserted when deg X ≠ deg Y, so the result does not cover the equal-degree case. Without the full manuscript the derivation cannot be inspected for gaps in the estimates or reductions, though the abstract and stress-test note show no visible circularity or hidden ineffective constants. The claim is stated as a direct existence proof rather than a post-hoc fit.\n\nThis is for arithmetic geometers who track effective versions of the André-Oort conjecture and height bounds on modular curves. A reader already working in that area will see the incremental gain in the linear dependence and the clear statement of the hypothesis.\n\nThe work is coherent on its own terms and the claim is specific enough to merit checking, so it should go to peer review rather than desk rejection.","headline":"Fowler proves a linear effective height bound for CM points on plane curves with unequal coordinate degrees, a clean but scoped improvement over prior effective André-Oort results.","tokens_in":2149,"tokens_out":377,"would_cite":false,"duration_ms":22718,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"On curves with unequal coordinate degrees, CM j-invariants have heights bounded by a constant linear in the curve height.","keywords":["André-Oort conjecture","CM j-invariants","heights","algebraic curves","effective bounds","number fields"],"falsifier":"An explicit curve V with deg X ≠ deg Y together with a point (x, y) on V at which both coordinates are CM j-invariants and max{h(x), h(y)} exceeds any linear function of the height of V.","tokens_in":2534,"feed_emoji":"","tokens_out":626,"duration_ms":35027,"temperature":0.7,"pith_summary":"The paper establishes a bound on the heights of points (x, y) on an algebraic curve V in the affine plane where both x and y are complex multiplication j-invariants. When the degrees of the restricted coordinate functions X and Y on V are unequal, there exists an effectively computable constant c that depends linearly on the height of V and satisfies max{h(x), h(y)} ≤ c. This supplies an effective form of the André-Oort conjecture for these curves. A sympathetic reader cares because the linear dependence improves on the height dependence in earlier effective results, making the bound more useful for explicit calculations.","feed_headline":"Linear height bound holds for CM points on unequal-degree curves","feed_subtitle":"Effective André-Oort result gives max height ≤ c where c is linear in curve height when coordinate degrees differ.","key_machinery":"The algebraic curve V equipped with the degree condition deg X ≠ deg Y, which permits the linear height bound on its CM j-invariant points.","core_discovery":"Let V be an algebraic curve in A²(C) such that deg X ≠ deg Y. Then there exists an effectively computable constant c, depending linearly on the height of V, such that max{h(x), h(y)} ≤ c for every point (x, y) in V at which both x and y are CM j-invariants. This yields an effective version of the André-Oort conjecture for such curves with improved dependence on the height of V compared with previous effective results.","pith_inferences":["When deg X equals deg Y the linear bound is not claimed, so a separate argument would be needed to handle that case.","The effective computability of c could in principle allow exhaustive search for CM points on curves of small height.","The degree-separation condition might extend to give similar linear bounds in higher-dimensional André-Oort settings."],"forward_implications":["The result supplies an effective André-Oort statement for all curves satisfying the degree condition.","The bound on heights is linear in the height of V.","The constant c is effectively computable from the data of V."],"fun_headline_variants":["Linear bound on CM heights for curves with deg X ≠ deg Y","Effective André-Oort for unequal-degree curves with linear c","CM points on V have max height linear in height of V","Weakly bounded CM heights on algebraic curves with unequal degrees"],"cache_read_input_tokens":64,"weakest_assumption_plain":"The assumption that the degrees of the two coordinate functions on V are unequal.","fun_headline_variants_meta":{"raw":{"variants":["Linear bound on CM heights for curves with deg X ≠ deg Y","Effective André-Oort for unequal-degree curves with linear c","CM points on V have max height linear in height of V","Weakly bounded CM heights on algebraic curves with unequal degrees"]},"model":"grok-4.3","cost_usd":0.010421,"raw_usage":{"total_tokens":4579,"prompt_tokens":606,"num_sources_used":0,"completion_tokens":68,"cost_in_usd_ticks":104212000,"prompt_tokens_details":{"text_tokens":606,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":3905,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":606,"tokens_out":68,"duration_ms":48687,"temperature":1.0,"reasoning_tokens":3905,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-30T02:19:23.127155+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"An explicit curve V with deg X ≠ deg Y together with a point (x, y) on V at which both coordinates are CM j-invariants and max{h(x), h(y)} exceeds any linear function of the height of V.","supporting_citations":[],"review_version":1}