arXiv:2610.26490v1

Pasten's arithmetic derivatives and the decomposition into gains

math.NT11D7511J9711H0611C08

Paper metadata

Original submission: October 19, 2026
Claimed MSC: 11D75, 11J97, 11H06, 11C08
arXiv categories: math.NT
Paper license: arXiv non-exclusive distribution license ↗

Abstract

Pasten attached to each coprime triple $a+b=c$ a lattice of arithmetic derivations that obey the Leibniz rule and are additive on this one equation. An arithmetic Wronskian then gives an $abc$ -type inequality governed by the norm $\|ψ\|$ of the derivation, and Pasten showed that sufficiently small derivations exist under the $abc$ conjecture and give an $abc$ inequality with a fixed exponent. We determine exactly which values this Wronskian takes, and we show that its estimate with exact prime weights has the best possible coefficient. The quality $X=\log c/\log\operatorname{rad}(abc)$ factors into an approximation gain and a power gain for each chosen presentation of the triple, and the resulting inequality bounds $X$ and both gains. For coprime polynomials the same argument gives $G_a < 1$ and $G_p < 3$ . We also show that the largest absolute value of the derivation on $a$ , $b$ and $c$ exceeds $c/\operatorname{rad}(abc)$ whenever the derivation is additive on the triple with non-zero Wronskian, so that it gives no bound on the quality by itself, and we determine its minimum when $a=1$ . For Fermat triples the presentation by bases has approximation gain greater than $n/3$ , which recovers Pasten's conditional asymptotic Fermat theorem. Finally, for five triples we certify the minimal norms and find the inequality within $0.08$ of the quality, with equality for the triples $1+80=81$ and $1+2400=2401$ .

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Audited against arXiv v1

Not a correctness certificate. A green “no issues found” result may include yellow suspected typos or minor formal issues that do not affect substantive soundness. It means this audit found no unresolved substantive error under the stated criteria; it does not replace expert scrutiny or formal verification.

Current report

Detailed mathematical audit

Generated October 23, 2026

01Statements4 reported findingsNo statement issues found

The exact Wronskian image, optimal prime-weight coefficient, derivation-size bounds, and numerical minima are supported. The conditional consequences retain the small-derivative hypothesis and the finite exceptional sets; they do not assert an unconditional abc bound.

Lemma 4.1; Corollary 4.3No issue found

The divisibility factor is the exact positive generator of the Wronskian image

PDF pp. 3–5 · equations (4.3)–(4.8) · arXiv:2610.26490v1

Disjoint prime supports let the three derivative values range independently over $g_a\mathbb Z$ , $g_b\mathbb Z$ , and $g_c\mathbb Z$ . A unimodular column transformation sends the additivity row to $(\delta,0,0)$ ; invariance of the ideal of two-by-two minors then gives exactly $\Delta\mathbb Z$ as the image on its integer kernel. The three minors are $cg_ag_b$ , $bg_ag_c$ , and $ag_bg_c$ up to sign. The valuation argument proves $abc/\operatorname{rad}(abc)\mid\Delta$ , including $a=1$ , and the three Wronskian forms give the stated minimum defining $\sigma$ .

Proposition 4.4No issue found

The prime-weight norm coefficient and the two-factor defect identity are exact

PDF p. 5 · equations (4.9)–(4.10) · arXiv:2610.26490v1

The real optimization is over three independent intervals with $u+v=w$ . The three cases for their radii attain one of the three universal upper bounds for $|av-bu|$ , hence attain their minimum $ab\sigma$ . Each displayed extremizer is rational and can be realized by constant rational values on each prime block. Clearing denominators produces an integer derivation without changing the ratio. Finally $R\|\psi\|\sigma/(\nu c)=m_\psi/\theta_\psi$ , so taking logarithms gives exactly (4.10).

Corollary 3.4; Propositions 4.6, 5.1–5.2, 6.1No issue found

The polynomial, size, and conditional exponent consequences have the claimed scope

PDF pp. 2–3 and 6–8 · arXiv:2610.26490v1

The polynomial divisibility and degree inequalities give $N\leq D-1$ , hence $G_a<1$ and $G_p<3$ . For arithmetic triples, $|W^\psi|\leq cD_\psi$ gives the lower size bound, and the separate $a=1$ argument gives the strict inequality $D_\psi>c/R$ . With $a=1$ , the image formula reduces to $\operatorname{lcm}(g_b,g_c)$ and yields the prime-power norm bound. Absorbing the logarithm into $c^\delta$ under Conjecture 4.5 gives every exponent above $1/(1-\eta)$ outside a finite set. The bases presentation of a Fermat triple has $G_a>n/3$ , giving exactly the stated conditional exponent restriction.

Section 7; Table 1No issue found

All five displayed derivations and their optimality certificates check

PDF pp. 8–10 · equations (7.1)–(7.2) · arXiv:2610.26490v1

Direct evaluation verifies additivity, nonzero Wronskians of magnitudes $432$ , $1600$ , $27440$ , $91125$ , and $5508110403$ , the reported radicals and prime weights, and all rounded quality bounds. The first and third norm minima follow from Proposition 5.2; the second follows from $448x_2=x_3+75x_5$ ; and the fourth follows by rounding $243/11$ upward. For the last, norm at most $600$ forces the Wronskian multiple in (7.1) to be $\pm19683$ , and the residue condition then forces $|x_{23}|>600$ . The displayed norm- $601$ vector attains the bound.

02Proofs2 reported findingsNo proof issues found

The integer-kernel calculation, interval optimization, valuation bounds, and finite arithmetic certificates close the principal arguments. The proofs distinguish nonzero derivations from derivations with nonzero Wronskian and preserve this condition wherever it is required.

Sections 3–4No issue found in the reviewed proof

The polynomial and arithmetic Wronskian arguments use valid divisibility and rank calculations

PDF pp. 2–5 · arXiv:2610.26490v1

Coprimality makes the prime blocks independent and makes the root-multiplicity divisibilities multiply in the polynomial proof. The arithmetic image calculation correctly divides the minors' gcd by the first row's gcd, rather than merely computing a rational kernel. The possible zero block at $a=1$ causes no division by zero because the other two blocks are nonzero. The interval extremizers and rational rescaling prove attainment, rather than just an upper estimate.

Sections 5–7No issue found in the reviewed proof

The strict inequalities and the finite-exception conclusions propagate correctly

PDF pp. 6–9 · arXiv:2610.26490v1

For $a\geq2$ , $ab>c$ makes the size lower bound strict; for $a=1$ , additivity instead identifies $D_\psi=|W^\psi|$ and supplies strictness. Integer rounding of the Fermat exponent is performed only after choosing an interval containing no further integer. Each fixed additive triple has only finitely many possible exponents, so the finite exceptional set yields an asymptotic bound. The minimum-norm certificates require only the displayed exact equations and divisibility; they do not rely on an unverifiable exhaustive-search assertion.

03Novelty0 reported findingsNo novelty concerns found

No non-novelty findings.

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Paper: arXiv:2610.26490v1
Authors listed: R. Laniewski, K. Müller
Audit date: October 23, 2026

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Pasten's arithmetic derivatives and the decomposition into gains

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

The divisibility factor is the exact positive generator of the Wronskian image

The prime-weight norm coefficient and the two-factor defect identity are exact

The polynomial, size, and conditional exponent consequences have the claimed scope

All five displayed derivations and their optimality certificates check

The polynomial and arithmetic Wronskian arguments use valid divisibility and rank calculations

The strict inequalities and the finite-exception conclusions propagate correctly

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