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Variants on the $abc$-Conjecture using Alternative Quality Metrics

T0 review · 2 major / 2 minor · reviewed 2026-06-27 · grok-4.3

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Pith's one-line read Quality metrics based on the doubly geometric mean of prime factors in abc-triples yield families and asymptotic bounds analogous to the abc-conjecture.

desk verdict New quality metrics based on doubly geometric means are defined, but the phase-transition claims rest on an unverified transfer of Szpiro heuristics without any reduction shown. read the letter →

arxiv 2606.08416 v1 pith:VXZAM555 submitted 2026-06-07 math.GM

Akilan Sankaran This is my paper · ORCID
classification math.GM
keywords abc-conjecturequalitymetricsdoublygeometricmeanabc-triplesFreycurvesSzpiroratiophasetransitionsasymptoticbounds
topics The abc Conjecture
open problems The abc Conjecture
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper defines new classes of quality metrics for abc-triples, each built from the doubly geometric mean of the prime factors. It locates families of triples that score high under these metrics and derives several asymptotic results that mirror the form of the abc-conjecture. Sharp phase transitions in metric behavior are identified when prime smoothness is parametrized, using heuristics drawn from the Szpiro ratio on associated Frey curves. Efficient algorithms with sub-linear runtime are given for locating high-quality triples.

What carries the argument

Quality metrics defined from the doubly geometric mean of the prime factors of abc-triples, which identify high-quality families and support asymptotic bounds.

What would settle it

A computation or proof showing that no families of abc-triples under these metrics follow the claimed asymptotic quality bounds, or that the predicted phase transitions do not occur at the smoothness parameters given by the Szpiro heuristics.

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Extended reading notes

Core claim

By measuring abc-triples with quality metrics based on the doubly geometric mean of their prime factors, families of high-quality triples are identified that satisfy asymptotic bounds analogous to the abc-conjecture. Sharp phase transitions are established for families of such metrics within specified parametrizations for smoothness of primes, using heuristics from the Szpiro ratio for associated Frey curves. Algorithms are implemented to determine triples with high qualities in sub-linear runtime.

Load-bearing premise

That the doubly geometric mean defines a quality metric whose high-quality families obey asymptotic bounds analogous to the abc-conjecture, and that Szpiro-ratio heuristics for Frey curves transfer to these new metrics.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

2 major / 2 minor

Summary. The paper introduces new classes of quality metrics for abc-triples based on the doubly geometric mean of their prime factors. It claims to identify families of high-quality triples under these metrics that yield asymptotic results analogous to the abc-conjecture, to derive sharp phase transitions for families of such metrics (parametrized by prime smoothness) via Szpiro-ratio heuristics on associated Frey curves, and to give sub-linear runtime algorithms for locating high-quality triples.

Significance. If the asymptotic analogies and phase-transition claims hold with the stated rigor, the work would supply concrete variants of the abc-conjecture that are analytically independent and potentially falsifiable. The algorithmic contribution would also be of separate interest. However, the manuscript supplies no explicit reduction showing that the Szpiro heuristic remains valid under the new quality functions, which undercuts the claimed sharpness of the phase transitions.

major comments (2)
  1. [phase transitions (abstract and corresponding section)] The section developing phase transitions states that these are obtained “using heuristics from the Szpiro ratio for associated Frey curves,” yet provides no derivation or invariance argument relating the doubly geometric mean quality function to the conductor or minimal discriminant in a manner that preserves the Szpiro constant. Without such a map, the transfer of the heuristic is an unverified analogy rather than a justified correspondence, and this directly supports the central claim of sharp phase transitions.
  2. [asymptotic results] The asymptotic results analogous to abc are asserted to follow from families of high-quality triples under the new metrics, but the manuscript does not exhibit the explicit height or radical bounds that would make these statements load-bearing analogues rather than reparametrizations of known abc data.
minor comments (2)
  1. [definitions] Notation for the doubly geometric mean is introduced without a displayed equation number or comparison table against the classical radical and quality functions.
  2. [algorithms] The abstract claims “sub-linear runtime” but the complexity analysis section does not state the precise dependence on the smoothness parameter or on the size of the triples.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments. We address each major comment below and indicate planned revisions.

read point-by-point responses

  1. Referee: [phase transitions (abstract and corresponding section)] The section developing phase transitions states that these are obtained “using heuristics from the Szpiro ratio for associated Frey curves,” yet provides no derivation or invariance argument relating the doubly geometric mean quality function to the conductor or minimal discriminant in a manner that preserves the Szpiro constant. Without such a map, the transfer of the heuristic is an unverified analogy rather than a justified correspondence, and this directly supports the central claim of sharp phase transitions.

    Authors: We agree that the manuscript applies the Szpiro heuristic by direct analogy to the new quality function without supplying an explicit invariance argument or reduction relating the doubly geometric mean to the conductor and minimal discriminant. The phase transitions are obtained under this heuristic assumption for the parametrized families. In revision we will add a clarifying subsection that explains the rationale for the analogy (the doubly geometric mean remains a multiplicative function of the prime factors and correlates monotonically with the standard radical for the smoothness classes considered) while explicitly noting the heuristic character and the absence of a full map. This will qualify the sharpness claim accordingly. revision: partial

  2. Referee: [asymptotic results] The asymptotic results analogous to abc are asserted to follow from families of high-quality triples under the new metrics, but the manuscript does not exhibit the explicit height or radical bounds that would make these statements load-bearing analogues rather than reparametrizations of known abc data.

    Authors: The asymptotic statements are derived from concrete parametric families of triples in which the new quality exceeds fixed thresholds, yielding infinitely many such triples. To make the load-bearing character explicit we will revise the relevant sections to display the specific families, the associated height and radical controls, and the precise growth statements that follow for the new metric. These are not mere reparametrizations; the change of quality function produces distinct distribution statements that we will highlight with the added bounds. revision: yes

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity detected

full rationale

The abstract defines new quality metrics via the doubly geometric mean, then identifies families yielding high quality under those metrics and states asymptotic results analogous to abc. The phase-transition claims invoke Szpiro-ratio heuristics on Frey curves as an external input rather than deriving them from the new metric by construction. No quoted equations reduce a claimed prediction to a fitted parameter, no self-citation chain is load-bearing, and no ansatz is smuggled. The derivation therefore remains self-contained against the supplied text; the skeptic concern is one of unverified transfer, not circular reduction.

Assumptions & free parameters 0 free parameters · 2 assumptions · 1 invented entities

The central claims rest on the definition of new quality metrics and the transfer of Szpiro-ratio heuristics; no explicit free parameters are named, but the metrics themselves are invented entities whose validity is assumed.

assumptions (2)
  • standard math Standard properties of prime factorizations and geometric means hold for abc-triples
    Invoked to define the doubly geometric mean quality metrics.
  • domain assumption Heuristics from the Szpiro ratio for Frey curves apply to the new quality metrics
    Used to develop sharp phase transitions for families of metrics.
invented entities (1)
  • New classes of quality metrics based on doubly geometric mean of prime factors
    purpose: To create variants on the abc-conjecture with analogous asymptotic behavior
    Introduced as the core new construction in the abstract.

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Cite this review

Pith. "Pith review of Variants on the $abc$-Conjecture using Alternative Quality Metrics." pith.science (2026). https://pith.science/paper/VXZAM555

@misc{pith2026260608416,
  author       = {Pith},
  title        = {Pith review of: Variants on the $abc$-Conjecture using Alternative Quality Metrics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VXZAM555}},
  note         = {Machine review of arXiv:2606.08416}
}
abstract

The $abc$-conjecture (Masser and Oesterle) has remained open for decades. By measuring $abc$-triples using a particular quality metric, the conjecture may be framed as seeking the asymptotic distribution of triples of sufficient quality. We create new classes of quality metrics to develop variants on the $abc$-conjecture, with each metric based upon the doubly geometric mean of the prime factors of triples. We investigate the behavior of the resulting class of quality metrics; by determining families of triples that yield high quality, we establish several asymptotic results that are analogous to the $abc$-conjecture for our metrics. We also develop sharp phase transitions for the behavior of families of such quality metrics within specified parametrizations for smoothness of primes in $abc$-triples, using heuristics from the Szpiro ratio for associated Frey curves. Finally, we implement algorithms to determine triples with high qualities with sub-linear runtime, an asymptotic speedup over na\"ive approaches. Our analysis offers robust variations of, and connections to, the $abc$-conjecture that offer independent questions of analytical interest.

Figures

Figures reproduced from arXiv: 2606.08416 by the authors.

Figure 1
Figure 1. qDGM evaluated on all abc-triples (a, b, c) with c ≤ 100. Red “x”s are DGM-hits, and black “x”s give the highest-quality triples for a valid abc-triples upon a given value of c. Lemma 3.3. For any fixed integer s ∈ Z + with distinct prime divisors of the form p1, . . . , pωs , we must have that Pωs i=1 ln(pi) ≤ ln(s), with equality if, and only if, s is a squarefree integer. Proof. Write the prime factorization of s… view at source ↗
Figure 2
Figure 2. Comparative attainment of high-quality triples through distinct methods, enu￾merated in Section 3.2. The Mersenne method yields significantly higher quality triples due to known enumerated Mersenne primes. Noting that ln(q) + ln(r) = ln(qr) = ln(c), AM-GM applied to ln(q), ln(r) gives p ln(q) ln(r) ≤ ln(c)/2. Hence, together with p < c, we have ln(2) · ln(p) · ln(q) · ln(r) ≤ ln(2) 4 (ln(c))3 . Therefore, taking fou… view at source ↗
Figure 3
Figure 3. Illustration of qC(a, b, c; α, 1) for α = 0, 0.5, 1, with highest-qualities for a given c interpolated. b c qC (·; 0, 1) qC (·; 0.5, 1) qC (·; 1, 1) 8 9 2.518 1.792 1.259 80 81 4.106 2.678 1.369 242 243 4.494 3.017 1.498 512 513 4.768 3.239 1.589 4374 4375 6.746 4.253 1.687 8191 8192 — — 1.803 62207 62208 — — 1.810 65536 65537 — — 2.000 131071 131072 — — 2.062 [PITH_FULL_IMAGE:figures/full_fig_p019_3.png] view at source ↗

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