REVIEW 2 major objections 4 minor 2 references
On the exceptional set in the $abc$ conjecture
T0 review · 2 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash
Pith's one-line read 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.
desk verdict The claimed exponent is new but the proof of Subcase 1.1 does not close; as written, Theorem 1.3 is not established. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
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.
What would settle it
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.
Extended reading notes
Core claim
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 $\theta=56/85$.
Load-bearing premise
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.
Editorial extensions
If this is right
- 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.
Reading between the lines
- 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.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
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.
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 (2)
- [§4.1.1, Eqs. (46)–(48)] 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.
- [§4, paragraph before §4.1] 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.
minor comments (4)
- [§4] Section 4 is titled 'Proof of Theorem 1.1' but it proves Theorem 1.3; the header should be corrected.
- [§1] 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.
- [§3] 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.
- [§1, §4] 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.
Circularity Check
Subcase 1.1's "contradiction" is the definition of k: (48) restates k = 49/12 − 23/4 θ, so the claimed contradiction is a tautology.
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self definitional
[Section 4.1.1, equations (38) and (48)]
"k = 49/12 − 23/4 θ ≈ 0.2951. ... Now (48) contradicts our assumption since the contributions of δ are omitted."
The paper defines k by (38) as exactly 49/12 − 23/4 θ. It then derives (48), which is k ≤ 49/12 − 23/4 θ, i.e. k ≤ k. The assertion that this is a contradiction is therefore not a mathematical contradiction; it is the defining equality of k. This is load-bearing because Subcase 1.1 is a required branch of Case 1 (s2 ≥ k), and the proof of Theorem 1.3 relies on closing this branch. The claimed contradiction reduces by construction to the definition of the threshold k, so the subcase is not actually closed.
full rationale
The paper is mostly a reduction to external lemmas from [1] (Browning–Lichtman–Teräväinen), including Lemmas 2.1–2.5 and equation (1.2); these are not self-citations and are outside the present author's own work. No fitted data, no machine-checked self-citation, and no prediction-from-fit appear. However, Subcase 1.1 contains a genuine circular/tautological step: the 'contradiction' (48) is exactly the defining formula for k in (38), namely k ≤ 49/12 − 23/4 θ, which is merely k ≤ k. Since Subcase 1.1 is an essential branch of Case 1, this tautological step is load-bearing for Theorem 1.3. The circularity is confined to this subcase rather than permeating the whole argument, so a score of 6 is appropriate: one required step reduces by construction to its own definition, while the surrounding structure retains independent content.
Assumptions & free parameters
free parameters (1)
- case-split threshold k =
49/12 - 23/4 theta, approximately 0.2951
assumptions (5)
- domain assumption Lemmas 2.1 through 2.5 of [1] are correct and are exactly the bounds for S* and B_d used here.
- domain assumption Equation (1.2) of [1] is a valid universal assertion that yields N_lambda(X) << X^theta whenever one of the pair sums is at most theta - epsilon.
- domain assumption The constraints (8), (9) and (10) fully summarize the consequences of Lemma 2.1 needed for the optimization.
- standard math The relabeling a3 >= b3 >= c3 is without loss of generality.
- ad hoc to paper The lower bound (46) may be written with delta_a in place of the delta_s that appears in the parent inequality (37).
Cite this review
Pith. "Pith review of On the exceptional set in the $abc$ conjecture." pith.science (2026). https://pith.science/paper/YUTUJG6C
@misc{pith2026250702885,
author = {Pith},
title = {Pith review of: On the exceptional set in the $abc$ conjecture},
year = {2026},
howpublished = {\url{https://pith.science/paper/YUTUJG6C}},
note = {Machine review of arXiv:2507.02885}
}
abstract
The $abc$ conjecture states that there are only finitely many triples of coprime positive integers $(a,b,c)$ such that $a+b=c$ and $\operatorname{rad}(abc) < c^{1-\epsilon}$ for any $\epsilon > 0$. Using the optimized methods in a recent work of Browning, Lichtman and Ter\"av\"ainen, we showed that the number of those triples with $c \leqslant X$ is $O\left(X^{56/85+\varepsilon}\right)$ for any $\varepsilon > 0$, where $\frac{56}{85} \approx 0.658824$. This constitutes an improvement of the previous bound $O\left(X^{33/50}\right)$.
Reference graph
Works this paper leans on
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[1]
T. Browning, J. D. Lichtman, and J. Ter¨ av¨ ainen. Bounds on the exceptional set in theabc conjecture. arXiv e-prints , page arXiv:2410.12234v1, 2024
arXiv 2024
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[2]
J. D. Lichtman. The abc conjecture is true almost always. arXiv e-prints , page arXiv:2505.13991v1, 2025. International Curriculum Center, The High School Affiliated to Renmin University of China, Beijing, China Email address : runbo.li.carey@gmail.com 15
work page Pith review arXiv 2025
Reviewed August 6, 2026 · model on record in the stance chip above.
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