{"id":"18e88c36-0bad-4acb-b492-03ba9399216d","arxiv_id":"2505.14109","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Conditionally on sparse 'strange' newforms, the largest prime order of rational torsion on elliptic curves over degree-d fields is at most 3d+1 for large even d and o(d) for odd d.","lead":"The paper proves that, assuming two conjectures about how rare certain modular forms are, the largest possible prime order of a torsion point on an elliptic curve over a degree-d number field grows roughly like 3d when d is even and much more slowly when d is odd. The interest is in the asymptotic shape of torsion over number fields, a question previously settled only for small degrees.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 9.1 rests on two unproved sparsity conjectures, 5.3(1) and 8.4(1); if either fails in the tail, the d0(m) threshold and the S(d) asymptotics do not follow.","rationale":"I read the paper in good faith and checked the derivation from Conjectures 5.3(1) and 8.4(1) to Theorem 9.1. The proof chain is internally coherent: Corollary 7.5 reduces large p to the exceptional cases; Theorem 7.7 uses Conjecture 5.3(1) to make condition (7.1) hold; Theorem 8.5 uses Conjecture 8.4(1) to force the parity of mu; and the final union over divisors in Corollary 9.2 is handled by taking d large relative to m. I found no algebraic or logical mistake in the central derivation. The single most load-bearing point is therefore the unproved sparsity of strange newforms and of positive-even-rank newforms. The weak forms of the conjectures are much weaker than the strong forms and are empirically plausible, but they are asymptotic statements with no effective rate and no released computational artifacts. If either weak conjecture is false along a sparse subsequence, the threshold d0(m) need not exist, and the claimed asymptotics for S(d) are unsupported. This is exactly the concern the reader identified, so I agree with the conditional verdict rather than proposing a rejection or an accept.","tokens_in":25373,"tokens_out":25227,"duration_ms":244757,"concrete_test":"Release the Magma/Modular Symbols code and data behind Proposition 5.2 and Table 3, and independently recompute strdim(p) and perdim(p) for all primes p < 10^6 (and, if feasible, p < 10^7). A single prime with strdim(p) >= log p / log log p or perdim(p) >= log p / log log p would falsify the relevant weak conjecture and show that the proof of Theorem 9.1 cannot produce d0(m). If no such prime occurs, the conditional theorem stands, though the conjecture remains open.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim (Theorem 9.1, Corollary 9.2) is logically sound only if Conjectures 5.3(1) and 8.4(1) hold. Conjecture 5.3(1), strdim(p)/log p -> 0, is used in the proof of Theorem 7.7 and Corollary 7.5 to make condition (7.1) hold for every sufficiently large prime p: it guarantees p > sqrt(C ((2*sqrt(2)+3)^strdim(p)) d + 1), so that in case (d) of Corollary 7.5 one obtains p = 2d/mu + 1 with mu <= 2m. Conjecture 8.4(1), perdim(p)/log p -> 0, enters through Theorem 8.5: for each fixed odd degree e < 2m it gives a p0(e) beyond which X0(p) has no non-cuspidal degree-e points, forcing mu to be even in the proof of Theorem 9.1. If either conjecture fails on a set of primes with strdim(p) >= c log p or perdim(p) >= c log p, the corresponding exponent (2*sqrt(2)+3)^strdim(p) (respectively the failure of the p0(e) bound) destroys the dichotomy in Theorem 9.1, and the asymptotic bounds in Corollary 9.2 need not hold. The numerical support stops at 10^5 for strangeness and 2*10^6 for perdim, and no scripts or data files are provided, so the empirical case for the conjectures cannot be independently checked. This is not an internal error but a genuine load-bearing assumption: the existence of d0(m) is exactly equivalent to these growth statements.","agreement_with_reader":"agree"},"referee_report":null,"author_rebuttal":null,"desk_editor":null,"rs_alignment":null,"lean_confirmation":null,"pith_extraction":null,"created_at":"2026-08-07T15:39:52.138933+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":null,"supporting_citations":[],"review_version":1}