{"id":"3927cb50-c369-4eab-a9ef-43a4c8892df3","arxiv_id":"1908.04938","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"A claimed constructive proof that, for each of the four rational 2-torsion structures, infinitely many good Frey curves exist.","lead":"The paper gives an explicit recursive construction intended to prove Masser's theorem: infinitely many elliptic curves built from ABC triples are good and have each of the four possible 2-torsion structures. A generalist should care because good elliptic curves are the elliptic-curve analogue of high-quality ABC triples, and explicit infinite families are rare.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 2.1(6) is false for T=C2 x C2 and C2 x C4: f_T(t)<0 for t>1 even though theta_T is listed as 1, so Lemma 3.2 and Proposition 3.4 do not establish that the recursion stays in its domain.","rationale":"The reader's weakest assumption identifies exactly the same load-bearing failure: Lemma 2.1(6) is false for C2 x C2 and C2 x C4, and Lemma 3.2 depends on it to propagate the condition b/a>theta_T. I independently checked the Table 6 formulas at t=2 and at t=1; the listed theta_T=1 is not a root of f_T, and f_T is negative throughout the claimed domain t>1. The C2 x C2 example in Section 5 produces a next ratio b_1/a_1 below theta_T, confirming internally that the recursive construction leaves its stated domain. This invalidates Proposition 3.4 and therefore Theorem 6.3 as written. I do not claim Masser's theorem is false; the concern is that this proof's central induction is unsound. Because the reader already rejected on this basis, my verdict does not change. No independent machine-checked proof or reproducible code is supplied to offset the elementary counterexample.","tokens_in":9944,"tokens_out":6625,"duration_ms":63671,"concrete_test":"Evaluate f_{C2 x C2}(2) and f_{C2 x C4}(2) using the formulas in Table 6: both are negative despite 2>theta_T=1. Equivalently, for T=C2 x C2 take a0=32, b0=49 and compute one application of (A_T,B_T,C_T) exactly or in a CAS; confirm that B_T/A_T is about 0.00194, so the next ratio b_1/a_1 lies below theta_T, contradicting Proposition 3.4.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central induction in Theorem 6.3 depends on Proposition 3.4, whose proof uses Lemma 3.2; Lemma 3.2 uses Lemma 2.1(6), which asserts f_T(t)>0 for t>theta_T. That assertion is false for T=C2 x C2 and T=C2 x C4 using the paper's own Table 6. For T=C2 x C2, f_T(t)=((1-t)^4)/(8t(1+t^2))-t, so f_T(2)=1/80-2<0. For T=C2 x C4, f_T(t)=((1-t^2)^2)/((2t)^2)-t, so f_T(2)=9/16-2<0. In both cases t=2>theta_T=1, directly contradicting Lemma 2.1(6); indeed f_T(1)=-1, so theta_T=1 is not even a root. Consequently Lemma 3.2's inference that f_T(b/a)>0 and hence B_T/A_T>b/a>theta_T is unsupported. The paper's own C2 x C2 example illustrates the failure: starting from P0=(2^5,7^2,3^4)=(32,49,81), one step gives b_1/a_1=(a_0-b_0)^4/(8a_0b_0(a_0^2+b_0^2)) approximately 0.00194, which is far below theta_T=1 and contradicts Proposition 3.4. Since Theorem 6.3 requires the inequality b_j/a_j>theta_T for every j, the proof of the main theorem is not sound for two of the four torsion subgroups. The flaw is internal to the manuscript's definitions, not a disagreement with external consensus.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript presents a constructive proof of Masser's theorem that there are infinitely many good Frey curves, and it aims to refine the statement to each of the four possible torsion subgroups C2 x C2N with N = 1, 2, 3, 4. The proof defines rational maps given by the polynomials in Table 6, uses them to construct recursive ABC triples (Proposition 3.4), and then attaches Frey curves whose torsion and modified Szpiro ratio are controlled (Theorem 6.3). The paper also provides numerical examples and tables of the resulting qualities and Szpiro ratios.","tokens_in":10311,"tokens_out":3532,"duration_ms":32640,"significance":"If the construction were sound, the paper would give explicit infinite families of good elliptic curves for each of the four torsion subgroups allowed by Mazur's theorem, which is a concrete strengthening of Masser's existence result and a useful source of test cases for the abc/Szpiro conjectures. The paper also clearly separates the algebraic identities from the arithmetic inequalities and includes explicit starting triples. However, the main induction rests on a positivity claim that is false for two of the four torsion subgroups, so the central theorem is not established.","major_comments":[{"comment":"Lemma 2.1(6) asserts that f_T(t), g_T(t), A_T(1,t), B_T(1,t), C_T(1,t), D_T(1,t) are all positive for t > theta_T. This is false for T = C2 x C2 and T = C2 x C4. From Table 6, for T = C2 x C2 we have f_T(t) = (1-t)^4/(8t(1+t^2)) - t, so f_T(2) = 1/80 - 2 = -159/80 < 0 even though theta_T = 1. Similarly, for T = C2 x C4, f_T(t) = (1-t^2)^2/(4t^2) - t, so f_T(2) = 9/16 - 2 = -23/16 < 0. In fact f_T(1) = -1 for both cases, so theta_T = 1 is not even a root of f_T. Since Lemma 3.2 and all subsequent results use this positivity, the claimed positivity is load-bearing and false.","section":"Lemma 2.1(6)"},{"comment":"Lemma 3.2 relies on Lemma 2.1(6) to infer that f_T(b/a) > 0 and hence B_T/A_T > b/a > theta_T. Because Lemma 2.1(6) is false for T = C2 x C2 and C2 x C4, the proof does not establish the key inequality B_T/A_T > theta_T that Proposition 3.4 needs to keep the recursion in its domain. The paper's own Example 5.2 for T = C2 x C2 starting from P0 = (32, 49, 81) gives b1/a1 = (a-b)^4/(8ab(a^2+b^2)) = 83521/42963200, approximately 0.00194, which is far below theta_T = 1 and contradicts the conclusion of Proposition 3.4 for j = 1. Thus the inductive step fails on the very example the paper presents.","section":"Lemma 3.2, Proposition 3.4"},{"comment":"Theorem 6.3 depends on Proposition 3.4 to ensure that every triple P^T_j satisfies b_j/a_j > theta_T, a_j even, b_j congruent to 1 mod 4, and the extra condition for T = C2 x C6. Since Proposition 3.4 is not valid for T = C2 x C2 and T = C2 x C4, the proof of Theorem 6.3 does not go through for those two torsion subgroups. The numerical examples in Table 4 likewise cannot be produced by the stated recursion without leaving the hypothesized domain, since the first step already violates b_1/a_1 > theta_T for T = C2 x C2.","section":"Theorem 6.3 and Lemma 6.2"}],"minor_comments":[{"comment":"The numerical values in Table 2 for T = C2 x C2 appear inconsistent with the definitions in Table 6: using a0 = 32 and b0 = 49, the recursion gives a1 = 8*32*49*(32^2+49^2) = 42963200 and b1 = (32-49)^4 = 83521, whereas the table lists a1 = 2511214657 and b1 = 38134. Please clarify whether the table entries are scaled or whether a different recursion was used.","section":"Example 5.2 / Table 2"},{"comment":"The statement of Lemma 2.1(7) is vacuously limited to T = C2 x C2N for N = 1,2, but the text labels it as part of Lemma 2.1; it may be clearer to present this as a separate remark or to justify why N = 3,4 are excluded.","section":"Lemma 2.1(7)"},{"comment":"The paper cites [BTW10] for prior proofs of two cases, but that reference appears to be an unpublished MSRI Undergraduate Program report; the author does provide independent proofs here, yet the citation should be clearly marked as unpublished or replaced with a peer-reviewed source if one exists.","section":"References"}],"recommendation":"reject","confidential_remarks":"The core issue is a genuine mathematical error in a lemma that the main theorem directly relies on, and the paper's own examples illustrate the failure. The error appears fixable only by reworking the positivity analysis and the recursive domain for at least two torsion subgroups, which would likely change the statement of the main theorem and the supporting examples. The self-citation [BTW10] also warrants editorial attention, though it does not affect my recommendation."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nThe paper is a genuine attempt to make Masser's theorem constructive, and its uniform treatment of the four possible torsion subgroups is a nice idea. The author builds explicit polynomial families AT, BT, CT from modular curve parameterizations, then defines a recursion on good ABC triples. If the technical inequalities worked, the recursion would produce infinitely many good Frey curves with prescribed torsion. That is a real contribution, and the paper is honest: it cites Masser, the modular curve sources, and its own earlier work for two cases.\n\nThe problem is load-bearing. Lemma 2.1(6) claims f_T(t), g_T(t), and the four polynomials are positive for t > theta_T. That is false. For T = C2 x C2, f_T(t) = ((1-t)^4)/(8t(1+t^2)) - t, and with theta_T listed as 1, f_T(2) = 1/80 - 2 < 0. For T = C2 x C4, f_T(2) = 9/16 - 2 < 0. So the lemma is simply wrong for two of the four cases. This is not a minor misprint: Lemma 3.2 uses the positivity of f_T to infer that b1/a1 > theta_T, and Proposition 3.4, Lemma 6.2, and Theorem 6.3 all depend on that step. The author's own example in Example 5.2 shows the failure: starting from P0 = (32,49,81), the recursion for C2 x C2 gives b1/a1 about 0.002, which is far below theta_T = 1. So the recursion leaves the domain on the very first step.\n\nThe rest of the paper is well-organized and the examples are clearly computed. There are no fitted parameters or invented entities, and the citations look appropriate. But the central induction does not hold as written. The constructive idea may be salvageable if the sign in the positivity statement is reversed or the domain of t is restricted correctly, but that would require rechecking the tables and likely changing the starting triples. As it stands, the proof of the main theorem is unsound.\n\nI would not cite this in its current form, and it would not make my reading group's shortlist as a positive result. It could serve as a cautionary tale, though, and it deserves a serious referee who can check whether the lemma can be repaired. My recommendation: send it to review, and tell the author to fix the positivity claims and re-verify the induction.\n\nBest,\n[You]","headline":"A genuine constructive framework that fails on a false positivity lemma; the recursion leaves its domain for two of the four torsion subgroups.","tokens_in":10871,"tokens_out":6376,"would_cite":false,"duration_ms":58124,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11G05"],"pacs":[],"model":"deepseek-v4-flash","headline":"Four torsion types admit infinitely many good Frey curves, constructed from a recursive abc-triple process.","keywords":["abc conjecture","modified Szpiro conjecture","good elliptic curves","Frey curves","torsion subgroups","recursive abc triples","modular curves","constructive proof"],"falsifier":"Evaluate $f_T(t)$ at $t = 3/2$ for $T = C_2 \\times C_2$: with $f_T(t) = \\frac{(1-t)^4}{8t(1+t^2)} - t$, the value is $\\frac{1}{624} - \\frac{3}{2} < 0$, while the table gives $\\theta_T = 1$; this directly contradicts the positivity claim the recursion relies on. Recompute the first recursive triple for the paper's $C_2 \\times C_2$ seed $P_0 = (25,72,3^4)$ and check whether $b_1/a_1$ remains above $\\theta_T$; if it drops below, Proposition 3.4's conclusion fails.","tokens_in":9691,"feed_emoji":"","tokens_out":8918,"duration_ms":82618,"temperature":0.7,"pith_summary":"The paper gives a constructive proof of the classical theorem that infinitely many good elliptic curves exist, and it does so for each of the four rational torsion subgroups $C_2 \\times C_{2N}$ with $N = 1,2,3,4$. A good elliptic curve here is one whose conductor is small compared with its $c_4$ and $c_6$ invariants, the objects controlled by the modified Szpiro conjecture, which is equivalent to the $abc$ conjecture. The construction starts from a single good $abc$ triple satisfying explicit congruences and a ratio threshold, then applies a polynomial recursion to produce an infinite sequence of good triples. Each resulting Frey curve is shown to be good and to have the prescribed torsion subgroup. If the construction is sound, it turns an existence theorem into explicit infinite families of examples.","feed_headline":"Four torsion types admit infinitely many good Frey curves","feed_subtitle":"A recursive abc-triple construction explicitly builds elliptic curves beating the Szpiro bound.","key_machinery":"The central machinery is a table of four polynomial transformations $(A_T, B_T, C_T)$ together with the identities $A_T + B_T = C_T$ and a Bezout-type identity $U_T B_T + V_T C_T = W_T$ that controls coprimality. Each transformation is accompanied by a ratio function $f_T(t) = B_T(1,t)/A_T(1,t) - t$, which records how the ratio $b/a$ changes under one recursion step, and a positivity function $g_T(t)$ used to keep the resulting triple good. The torsion statement is carried by explicit universal elliptic curve models $X_t(T)$; an admissible change of variables identifies the Frey curve built from the transformed triple with $X_{t_T}(T)$, forcing the rational torsion subgroup to be exactly $T$. The congruences preserved by the recursion, such as $a_j \\equiv 0 \\pmod{16}$ and $b_j \\equiv 1 \\pmod 4$, make the resulting Frey curves semistable.","core_discovery":"The paper's central claim is that for each admissible torsion type $T = C_2 \\times C_{2N}$, a recursive map on $abc$ triples produces an infinite family of good Frey curves. Starting from a good triple $P_0 = (a_0,b_0,c_0)$ with $a_0$ even, $b_0 \\equiv 1 \\pmod 4$, $b_0/a_0 > \\theta_T$, and additionally $a_0 \\equiv 0 \\pmod 3$ when $T = C_2 \\times C_6$, the recursion $P_j = (A_T(a_{j-1},b_{j-1}), B_T(a_{j-1},b_{j-1}), C_T(a_{j-1},b_{j-1}))$ yields, for every $j$, a good $abc$ triple whose induced Frey curve $y^2 = x(x-a_j)(x+b_j)$ is semistable, has torsion subgroup exactly $T$, and satisfies the good-curve inequality $N_E^6 < \\max\\{|c_4|^3, c_6^2\\}$. This establishes Theorem 6.3, the constructive form of the infinitude theorem, uniformly for all four torsion types.","pith_inferences":["The recursion can be viewed as a dynamical system on ratios $r_j = b_j/a_j$ governed by $r_{j+1} = B_T(1,r_j)/A_T(1,r_j)$; if the positivity threshold issue were repaired, one could study growth rates, heights, and whether the ratios converge to $\\theta_T$, yielding quantitative control on the quality decay visible in the computed examples.","The same template may extend to prescribed torsion structures over number fields, since the universal-curve step is what fixes the torsion; the rational classification restricts $T$, but the polynomial recursion itself does not obviously depend on that restriction.","The first few computed modified Szpiro ratios decay toward 6, which suggests that these families do not force a uniform gap above 6; one could test whether any uniform lower bound above 6 can be achieved infinitely often by similar recursive constructions."],"forward_implications":["For each of the four torsion types, there are explicit infinite sequences of good $abc$ triples; for example, $T = C_2 \\times C_2$ and $T = C_2 \\times C_4$ both start from the seed $(25, 72, 3^4)$.","Every curve in these families is semistable and has modified Szpiro ratio strictly greater than 6, so the families are concrete witnesses to the infinitude that the modified Szpiro conjecture would forbid in the limit $\\epsilon \\to 0$.","The recursion preserves the congruence invariants needed for semistability at every step, so the construction is uniform across all four torsion types rather than requiring a separate argument for each.","Because the $abc$ conjecture and the modified Szpiro conjecture are equivalent, each infinite family of good Frey curves corresponds to infinitely many $abc$ triples with $\\operatorname{rad}(abc) < c$, extending the phenomenon of the classical $(1, 9^k - 1, 9^k)$ family."],"supporting_citations":[{"why":"Supplies the theorem being made constructive: infinitely many good Frey curves exist.","marker":"[Mas90]"},{"why":"Provides the equivalence between the $abc$ conjecture and the modified Szpiro conjecture, which motivates the definition of a good elliptic curve.","marker":"[Oes88]"},{"why":"Gives the rational torsion classification that restricts admissible torsion subgroups to $C_2 \\times C_{2N}$ with $N = 1,2,3,4$.","marker":"[Maz77]"},{"why":"Supplies the universal elliptic curve parameterizations for $X_1(2,2N)$ used for $T = C_2 \\times C_6$ and $T = C_2 \\times C_8$.","marker":"[HLP00]"},{"why":"Provides the earlier implicit universal curve expressions that the parameterizations for $N = 3,4$ expand.","marker":"[Kub76]"},{"why":"Gives the model for curves with $C_4 \\times C_4$ over $\\mathbb{Q}(i)$, used to pin down the torsion when $T = C_2 \\times C_4$.","marker":"[Sil97]"},{"why":"Supplies the torsion theorem over $\\mathbb{Q}(i)$ used to exclude $C_2 \\times C_8$ in the $C_2 \\times C_4$ case.","marker":"[Kam92]"},{"why":"Provides the criterion used to show that $8ab(a^2+b^2)$ is not a square, forcing the torsion of the $C_2 \\times C_2$ family to be exactly $C_2 \\times C_2$.","marker":"[Ono96]"},{"why":"Supplies standard facts about semistable Frey curves, minimal discriminants, and conductor-radical identities used to prove goodness.","marker":"[Sil09]"}],"fun_headline_variants":["Constructive proof: infinite good Frey curves for each torsion type","Recursive abc triples build infinite good elliptic curves","Each torsion type yields infinite good Frey curves","Constructive families of good elliptic curves for every torsion"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The recursion stays inside its domain because, for every ratio above the stated threshold $\\theta_T$, certain positivity inequalities (Lemma 2.1(6)) hold; for $T = C_2 \\times C_2$ and $T = C_2 \\times C_4$ that positivity assertion is false, so the step from $P_0$ to $P_1$ can land below the threshold.","fun_headline_variants_meta":{"raw":{"variants":["Constructive proof: infinite good Frey curves for each torsion type","Recursive abc triples build infinite good elliptic curves","Each torsion type yields infinite good Frey curves","Constructive families of good elliptic curves for every torsion"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00072,"raw_usage":{"total_tokens":3234,"prompt_tokens":952,"completion_tokens":2282,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":568,"completion_tokens_details":{"reasoning_tokens":2219}},"tokens_in":568,"tokens_out":2282,"duration_ms":13483,"temperature":1.0,"reasoning_tokens":2219,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T13:29:37.827238+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Evaluate $f_T(t)$ at $t = 3/2$ for $T = C_2 \\times C_2$: with $f_T(t) = \\frac{(1-t)^4}{8t(1+t^2)} - t$, the value is $\\frac{1}{624} - \\frac{3}{2} < 0$, while the table gives $\\theta_T = 1$; this directly contradicts the positivity claim the recursion relies on. Recompute the first recursive triple for the paper's $C_2 \\times C_2$ seed $P_0 = (25,72,3^4)$ and check whether $b_1/a_1$ remains above $\\theta_T$; if it drops below, Proposition 3.4's conclusion fails.","supporting_citations":[],"review_version":1}