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A lattice-theoretic approach to the abc conjecture: Weighted arithmetic derivatives, a master inequality, and the structure of the residual family

A mix of human-written and AI-generated textHuman understanding: all partsmath.NT — Number Theory

Contributed by Zeng Chunping ↗

Version 1 / Oct 29, 2026 / CC BY 4.0

Abstract

We develop a self-contained lattice-theoretic framework for the abc conjecture. For coprime positive integers a+b=ca+b=c with k=ω(ab)k=\omega(ab), the framework rests on a weighted arithmetic derivative dψd^\psi whose additivity equation dψ(a+b)=dψ(a)+dψ(b)d^\psi(a+b)=d^\psi(a)+d^\psi(b) is solvable modulo the modulus gc=gcd⁡p∥c(cpvp(c))g_c=\gcd_{p\parallel c}\bigl(\frac{c}{p}v_p(c)\bigr), and on a Wronskian identity W=a dψ(b)−b dψ(a)=Mab H(u)W=a\,d^\psi(b)-b\,d^\psi(a)=M_{ab}\,H(u), where Mab=ab/rad⁡(ab)M_{ab}=ab/\operatorname{rad}(ab) and HH is an explicit linear form. A divisibility lemma MaMbMc∣WM_aM_bM_c\mid W and an elementary upper bound yield the master inequality c≤μab⋅Ω(ab)2⋅rad⁡(abc)c\le\mu_{ab}\cdot\frac{\Omega(ab)}{2}\cdot\operatorname{rad}(abc), in which the defect factor μab\mu_{ab} is the shortest non-kernel weight of an explicit solution lattice Λ\Lambda of determinant gc/gg_c/g. Writing η=log⁡μab/log⁡c\eta=\log\mu_{ab}/\log c, geometry-of-numbers input (Hermite's bound and an orthogonal covering-radius sandwich; an earlier ``min--max barrier'' is falsified in Theorem 4.5) yields, conditionally on an explicit gap hypothesis (⋆′)(\star') isolated in Section 7, the bound η≤1k+o(1)\eta\le\frac1k+o(1), whence lim sup⁡c→∞q(a,b,c)≤kk−1≤2\limsup_{c\to\infty}q(a,b,c)\le\frac{k}{k-1}\le2, i.e.\ c≲rad⁡(abc)2+o(1)c\lesssim\operatorname{rad}(abc)^{2+o(1)}. On the subclass a+b=rea+b=r^e the method closes unconditionally (Zsigmondy for r≥3r\ge3; cyclotomic/LTE control of valuations for a=1a=1). The residual family is localized by a power--congruence bridge: qV∥bq^V\parallel b forces ee into a single residue class modulo d qV−sd\,q^{V-s} (d=ord⁡q(r)d=\operatorname{ord}_q(r), ss the Wieferich depth), of convergent total density; the unique unclosed layer reduces to growth-difference Pillai-type equations. We also prove a structural ceiling: the main term of η\eta is at most 1k+o(1)\frac1k+o(1) and attains 1k\frac1k on the family c=rec=r^e, so the full abc conjecture (q≤1+εq\le1+\varepsilon) is structurally unreachable inside this framework. A final section determines the reach and limits of the method elementarily: almost all cc have every coprime decomposition with q≤1+o(1)q\le1+o(1) (density theorem, by one-variable counting, exceptional set ≪X1/2+o(1)\ll X^{1/2+o(1)}), while any argument within the method class M\mathcal M cannot prove lim sup⁡q≤α\limsup q\le\alpha for any α<2\alpha<2 (obstacle theorem), and the only remaining channel is equivalent to a sparse growth-difference Pillai core. All statements are proved in full or are explicitly labelled as working hypotheses; numerical cross-checks from two independent implementations are included, and no claim of a proof of the abc conjecture is made.

Provenance statement

The mathematical content originates from the author's working manuscripts (Volumes I--X and their revisions). An AI assistant (large language model) was used for: reconstructing the LaTeX source after the loss of the original files (rebuilt from the author's compiled PDF and verified against it); independent numerical verification in separate sandbox implementations (certification of the defect factor mu by exhaustive search, exact enumeration of exceptional sets up to 10^7); consolidation of the revision history (Appendix A, entries E1--E30); and implementing a certified external revision list of 13 items (corrected statements and proofs at Theorems 4.3, 7.1, 9.1, 9.8, 10.1, Lemma 9.5, and completed Table 4), every item verified by the author before inclusion. No theorem in the paper is due to the AI.

Tools used

SageMath
SageMath

References

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Version history

  1. v1Submitted by Zeng ChunpingInitial depositCurrentOct 28, 2026