{"id":"ac16dafd-0d9a-45a3-8360-26499b860ace","arxiv_id":"2507.02885","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"high","formal_verification":"none","parameter_count":1,"one_line_summary":"A new, slightly smaller exponent, 56/85, is claimed for the exceptional set of abc triples with c at most X, improving the previous 33/50.","lead":"This paper claims to improve the best known upper bound on the number of abc triples up to X from X^{0.66} to X^{56/85+epsilon}, about X^{0.6588}. It does so by optimizing the combinatorial case analysis in a recent paper by Browning, Lichtman and Teravainen.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Subcase 1.1's contradiction is an identity: (48) reduces to k ≤ 49/12 − 23/4·θ, equality at θ = 56/85, so Theorem 1.3 is not established.","rationale":"The central claim is the improved exponent 56/85 in Theorem 1.3. The proof splits into several cases, and Subcase 1.1 is not optional: it handles all admissible parameters with s2 ≥ k and b3 ≤ 1 − θ − s1 − s2. The alleged contradiction in (48) is not a contradiction; at the target value θ = 56/85 it is exactly the equality defining k. The only way to read it as a contradiction is to silently drop the δ_a terms, but δ_a is a constrained variable defined by (16) and bounded in (19), not an arbitrary small error. Moreover, the derivation of (46) uses δ_a where the parent inequality (37) has δ_s, and this substitution is not justified. Thus the proof has a genuine gap in a load-bearing case. This is not a difference from consensus; it is an internal failure of the inequality argument. A repaired version might exist, but the current manuscript does not support the claimed theorem.","tokens_in":13483,"tokens_out":4687,"duration_ms":47518,"concrete_test":"Recompute Subcase 1.1 retaining δ_s in (46) and retaining all δ terms through (48); verify whether the chain yields a strict inequality contradicting k = 49/12 − 23/4·θ at θ = 56/85. If the chain terminates at k ≤ 49/12 − 23/4·θ, the claimed contradiction is an identity and the proof fails.","verdict_should_be":"REJECT","load_bearing_attack":"In §4.1.1, the proof of Subcase 1.1 does not close. Equation (37) controls s3 with δ_s, but (46) uses δ_a; no justification is given for the substitution. Even granting that, (47)–(48) produce 11/4·θ + k − 25/12 ≤ 3/2·δ_a ≤ 2 − 3θ, i.e. k ≤ 49/12 − 23/4·θ, which is the definition of k and is an equality at θ = 56/85. The sentence 'Now (48) contradicts our assumption' is valid only if the δ_a terms are dropped, but δ_a is a constrained variable (19), not a negligible error term, and cannot be set to zero. Since Subcase 1.1 is a required branch of Case 1, Theorem 1.3 is left without proof.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript claims to improve the known upper bound for the number of exceptional abc-triples with c ≤ X from X^{33/50} to X^{56/85+ε} for λ < 1+δ(ε). Following Browning–Lichtman–Teräväinen, the author reduces the problem to bounding the number of solutions to a Diophantine equation, imports four analytic bounds, rewrites them as linear inequalities in the exponents a_i,b_i,c_i, and then performs a case split according to whether s2 is at least a threshold k. The main theorem (Theorem 1.3) is the claimed improved bound. The paper's contribution relative to [1] is the optimization and the explicit choice of k and of the exponent 56/85.","tokens_in":13674,"tokens_out":27166,"duration_ms":266930,"significance":"If correct, the result would be a modest but genuine improvement over the BLT exponent 33/50 in a range of λ near 1, and it would confirm that the BLT framework can be pushed to the exponent 56/85. The paper is transparent about its reduction and the algebraic manipulations are mostly clear. However, the manuscript has two load-bearing gaps: the supposed contradiction in Subcase 1.1 is actually an identity, and the case analysis rests on an unjustified ordering assumption on a3,b3,c3. No machine-checked proofs or code are supplied, so the burden is entirely on the written argument. As it stands, the theorem is not established.","major_comments":[{"comment":"The claimed contradiction is not a contradiction. Equation (46) is a legitimate lower bound on a3 obtained from (35), but the subsequent chain (47)–(48) yields exactly k ≤ 49/12 − 23/4θ. Since k is defined in (38) by the equality k = 49/12 − 23/4θ, the final inequality is an identity (equality at θ=56/85), not a contradiction. The sentence 'Now (48) contradicts our assumption since the contributions of δ are omitted' is therefore false: δ_a is a constrained variable from (19), not a negligible error that can be set to zero. Subcase 1.1 is a required branch of Case 1, so Theorem 1.3 is left unproved.","section":"§4.1.1, Eqs. (46)–(48)"},{"comment":"The 'without loss of generality' assumption a3 ≥ b3 ≥ c3 is not valid. The equation a+b=c and the counting condition are symmetric only in a and b; c is distinguished. Swapping a and b can ensure a3 ≥ b3, but it cannot make c3 ≤ b3. Several subsequent inequalities, e.g. (41), (72), and the arguments in Subcases 2.1, 2.2, and 2.6.2, rely on c3 ≤ b3. If c3 is the largest of the three, these steps fail. This gap is independent of the Subcase 1.1 issue and is load-bearing for the case analysis.","section":"§4, paragraph before §4.1"}],"minor_comments":[{"comment":"Section 4 is titled 'Proof of Theorem 1.1' but it proves Theorem 1.3; the header should be corrected.","section":"§4"},{"comment":"The paper uses both ε and ϵ with different meanings (the exponent in the theorem and a small dyadic parameter), and the relationship between them is not specified; a single consistent notation would remove ambiguity.","section":"§1"},{"comment":"Equation (1.2) of [1] is invoked in (9), (14) and (15) but is never stated; the argument is not self-contained without quoting it.","section":"§3"},{"comment":"The quantifier order in Theorem 1.3 ('for every ε there exists δ(ε)') is not reconciled with the internal assumptions 0<δ<10^{-100} and θ=56/85+ε; the proof should explain why it suffices to treat sufficiently small ε.","section":"§1, §4"}],"recommendation":"reject","confidential_remarks":"The two gaps are independent and both concern essential branches of the proof. The error in Subcase 1.1 is not a presentation issue; the final inequality (48) is exactly the definition of k, so the branch remains open. The ordering assumption is also not harmless. I would not advise the editor to seek a minor revision; a substantive new argument would be needed."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Runbo Li's paper tries to shave the BLT exceptional-set exponent from 33/50 to 56/85 by optimizing the same combinatorial framework. The claimed exponent is new, and the write-up is honest about its debt to [1]: no self-citations, no fitted parameters, and a clear statement that this is an optimization inside an existing method. The organization of the case analysis is sound, and the choice of the threshold k is derived explicitly. That part is fine.\n\nThe problem is that the proof does not close. In Subcase 1.1 (Section 4.1.1), inequality (46) replaces the delta_s in (37) with delta_a without any justification. Even if that substitution were allowed, the final inequality (48) reduces to k <= 49/12 - 23/4 theta, which is exactly the definition of k and is an equality at theta = 56/85. Calling this a contradiction is not valid: delta_a is a constrained variable, not a negligible error term, and the inequality is not strict. Since Subcase 1.1 is a required branch of Case 1, Theorem 1.3 is left unproved. The reader's stress-test is right on the money here.\n\nLesser issues: the proof relies heavily on imported BLT lemmas and on many 'similar discussions,' which makes verification slow. But that alone would not be fatal. The core difficulty is the gap in Subcase 1.1.\n\nThe framework is coherent, the target exponent is a real if tiny improvement, and the gap looks like a repairable oversight rather than a fundamental conceptual error. A corrected derivation might well succeed. But as written, the central claim is not established. I would send this to a referee because the result, if fixed, is worth having, and the flaw is specific enough to evaluate. I would not cite it as a theorem until the gap is patched. For a reading group, it is a decent example of parameter optimization inside BLT's method, but I would flag the flaw in advance.","headline":"The claimed exponent is new but the proof of Subcase 1.1 does not close; as written, Theorem 1.3 is not established.","tokens_in":14185,"tokens_out":7953,"would_cite":false,"duration_ms":70929,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11D45","11D75"],"pacs":[],"model":"deepseek-v4-flash","headline":"For every ε>0 there is a δ>0 such that the number of abc triples with c≤X and rad(abc)<c^λ for λ<1+δ is O(X^{56/85+ε}), improving the previous exponent 33/50.","keywords":["abc conjecture","exceptional set","radical","Diophantine equations","combinatorial optimization","power-saving bound","counting function"],"falsifier":"Run Subcase 1.1 with the correct error term kept: derive the lower bound for $a_3$ from (37) without replacing $\\delta_s$ by $\\delta_a$, and check whether the resulting inequality still contradicts (47) at $\\theta=56/85$. If it does not, the proof as written has a gap there; if it does, the substitution is harmless. A symbolic or numerical audit of all subcases at $\\theta=56/85$ would settle the matter.","tokens_in":13251,"feed_emoji":"🔢","tokens_out":11009,"duration_ms":116714,"temperature":0.7,"pith_summary":"The paper is trying to establish a stronger quantitative version of the abc exceptional-set statement: it counts coprime triples $(a,b,c)$ with $a+b=c$ whose radical $\\operatorname{rad}(abc)$ is smaller than $c^{\\lambda}$. The main result, Theorem 1.3, says that for every $\\varepsilon>0$ there is a small positive $\\delta=\\delta(\\varepsilon)$ such that for $0<\\lambda<1+\\delta$, the number $N_{\\lambda}(X)$ of such triples with $c\\leq X$ is $O(X^{56/85+\\varepsilon})$, where $56/85\\approx 0.658824$. This improves the previous power-saving bound $O(X^{33/50})\\approx X^{0.66}$. The point of the improvement is that even for exponents $\\lambda$ slightly above $1$, the exceptional triples form a genuinely thin set measured by a power of $X$, which is the kind of control the abc conjecture asks for but does not yet provide.","feed_headline":"abc exception count drops to exponent 56/85","feed_subtitle":"For any epsilon, abc triples up to X number O(X^{56/85+ε}), beating the prior X^{0.66} bound.","key_machinery":"The load-bearing object is the count $B_d(c,X,Y,Z)$ of integer solutions in dyadic ranges to $c_1\\prod_{j\\leq d}x_j^j + c_2\\prod_{j\\leq d}y_j^j = c_3\\prod_{j\\leq d}z_j^j$ with the relevant gcd equal to 1; Lemma 2.1 shows the original count is controlled by such counts up to a factor $X^{\\varepsilon}$. Writing $X_i=X^{a_i}$, $Y_i=X^{b_i}$, $Z_i=X^{c_i}$ and $s_i=a_i+b_i+c_i$, the paper converts the four imported bounds into explicit linear inequalities on the exponents, with target $\\nu = \\log B_d/\\log X + 2\\varepsilon^2 \\leq \\theta$. The proof then becomes a finite case split, mostly on $s_1$ and $s_2$, with threshold $k=49/12-23\\theta/4\\approx 0.2951$, using chains such as (24)–(37) to force $\\nu\\leq\\theta$. The decisive mechanism is a contradiction in Subcase 1.1: lower and upper bounds on $a_3$ collide at $\\theta=56/85$, and that collision is exactly the inequality that defines $k$, which is why the method cannot pass this exponent.","core_discovery":"On its own terms, the paper establishes Theorem 1.3: for any $\\varepsilon>0$, there exists a constant $\\delta=\\delta(\\varepsilon)>0$ such that for $0<\\lambda<1+\\delta$, $N_{\\lambda}(X)\\ll X^{56/85+\\varepsilon}$. The key claim is that the existing combinatorial framework—reducing the counting problem to a Diophantine count $B_d(c,X,Y,Z)$ and bounding it by Fourier, geometry, determinant, and Thue inequalities—can be optimized to the exponent $56/85$ without introducing any new counting input. The paper also shows that this is the limiting exponent of that framework: the case analysis terminates exactly when the parameter $k=49/12-23\\theta/4$ enters, and the contradiction at (48) is precisely the defining inequality for $k$ at $\\theta=56/85$.","pith_inferences":["One could convert the proof's case analysis into a small linear program in the variables $a_i,b_i,c_i$ and verify the $56/85$ threshold computer-assisted; this would test the internal consistency of the subcase splits and make the optimization reproducible.","The author's stopping point suggests that a genuinely new upper bound on $B_d$, or a better reduction lemma that produces fewer variables, is needed to go below $56/85$; this is an implicit challenge to the method rather than a claim proved here.","A natural testable extension is to ask whether the same technique bounds counts with the radical replaced by the largest squarefree divisor in a subset of primes, where the Diophantine reduction might behave differently."],"forward_implications":["If Theorem 1.3 is correct, then for every $\\varepsilon>0$ the number of abc triples with $c\\leq X$ and exponent below $1+\\delta$ is $O(X^{0.658824+\\varepsilon})$, a power-saving improvement over the previous $O(X^{0.66})$.","The proof identifies the exact limit of the current optimization framework: any further lowering of the exponent requires a new counting bound or a change in the reduction step, not just a sharper case analysis.","Because the theorem allows $\\lambda$ slightly larger than 1, it says the triples that evade the abc conjecture's inequality are sparse even if the inequality is relaxed by a tiny amount.","In combination with the trivial bound, the result gives what is currently the best quantitative control on the exceptional set for exponents just below and just above 1."],"supporting_citations":[{"why":"Supplies the entire combinatorial framework: the reduction to $B_d$, the four counting bounds rewritten as Lemmas 2.2–2.5, the criterion (1.2), and the previous bound $O(X^{33/50})$ that this paper improves.","marker":"[1]"},{"why":"Supplies the proof of the trivial bound stated as Theorem 1.1, which is the baseline the new exponent must beat.","marker":"[2]"}],"fun_headline_variants":["abc exception count bound dips to exponent 56/85","abc exceptional set: count O(X^{56/85+ε}) achieved","Improved abc count: exponent 56/85 beats prior 33/50","abc triples: exceptional set bound now exponent 56/85","New abc bound: O(X^{56/85+ε}) sets better count limit"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof assumes the cited work's four counting bounds and its reduction criterion are exactly correct, and it assumes that in Subcase 1.1 the error term $\\delta_s$ in inequality (37) may be replaced by $\\delta_a$ when deriving (46); if that replacement fails, the contradiction at (48) is not established.","fun_headline_variants_meta":{"raw":{"variants":["abc exception count bound dips to exponent 56/85","abc exceptional set: count O(X^{56/85+ε}) achieved","Improved abc count: exponent 56/85 beats prior 33/50","abc triples: exceptional set bound now exponent 56/85","New abc bound: O(X^{56/85+ε}) sets better count limit"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000822,"raw_usage":{"total_tokens":3546,"prompt_tokens":847,"completion_tokens":2699,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":463,"completion_tokens_details":{"reasoning_tokens":2602}},"tokens_in":463,"tokens_out":2699,"duration_ms":20432,"temperature":1.0,"reasoning_tokens":2602,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T23:48:05.710589+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run Subcase 1.1 with the correct error term kept: derive the lower bound for $a_3$ from (37) without replacing $\\delta_s$ by $\\delta_a$, and check whether the resulting inequality still contradicts (47) at $\\theta=56/85$. If it does not, the proof as written has a gap there; if it does, the substitution is harmless. A symbolic or numerical audit of all subcases at $\\theta=56/85$ would settle the matter.","supporting_citations":[{"cited_title":"The $abc$ conjecture is true almost always","cited_arxiv_id":"2505.13991","evidence_quote":"Supplies the proof of the trivial bound stated as Theorem 1.1, which is the baseline the new exponent must beat."}],"review_version":1}