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math.NT — Number Theory

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

Contributed by Zeng Chunping

We develop a self-contained lattice-theoretic framework for the abc conjecture. For coprime positive integers $a+b=c$ with $k=\omega(ab)$ , the framework rests on a weighted arithmetic derivative $d^\psi$ whose additivity equation $d^\psi(a+b)=d^\psi(a)+d^\psi(b)$ is solvable modulo the modulus $g_c=\gcd_{p\parallel c}\bigl(\frac{c}{p}v_p(c)\bigr)$ , and on a Wronskian identity $W=a\,d^\psi(b)-b\,d^\psi(a)=M_{ab}\,H(u)$ , where $M_{ab}=ab/\operatorname{rad}(ab)$ and $H$ is an explicit linear form. A divisibility lemma $M_aM_bM_c\mid W$ and an elementary upper bound yield the master inequality $c\le\mu_{ab}\cdot\frac{\Omega(ab)}{2}\cdot\operatorname{rad}(abc)$ , in which the defect factor $\mu_{ab}$ is the shortest non-kernel weight of an explicit solution lattice $\Lambda$ of determinant $g_c/g$ . Writing $\eta=\log\mu_{ab}/\log c$ , 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 $\eta\le\frac1k+o(1)$ , whence $\limsup_{c\to\infty}q(a,b,c)\le\frac{k}{k-1}\le2$ , i.e.\ $c\lesssim\operatorname{rad}(abc)^{2+o(1)}$ . On the subclass $a+b=r^e$ the method closes unconditionally (Zsigmondy for $r\ge3$ ; cyclotomic/LTE control of valuations for $a=1$ ). The residual family is localized by a power--congruence bridge: $q^V\parallel b$ forces $e$ into a single residue class modulo $d\,q^{V-s}$ ( $d=\operatorname{ord}_q(r)$ , $s$ 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 $\eta$ is at most $\frac1k+o(1)$ and attains $\frac1k$ on the family $c=r^e$ , so the full abc conjecture ( $q\le1+\varepsilon$ ) is structurally unreachable inside this framework. A final section determines the reach and limits of the method elementarily: almost all $c$ have every coprime decomposition with $q\le1+o(1)$ (density theorem, by one-variable counting, exceptional set $\ll X^{1/2+o(1)}$ ), while any argument within the method class $\mathcal M$ cannot prove $\limsup q\le\alpha$ for any $\alpha<2$ (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.

A mix of human-written and AI-generated textHuman understanding: all partsPillai equationWronskianabc conjecturedensity theoremerratalattice methodmaster inequalityobstacle theoremweighted arithmetic derivative

Oct 29, 2026PDF ↗View work →

hexagon:2610.00646v1

math.NT — Number Theory

An Integral-Structure Inflation in the Hull Operation of IUT Corollary 3.12

Contributed by Zeng Chunping

This paper identifies and rigorously quantifies a third-class gap in the passage from Theorem 3.11 to Corollary 3.12 in S. Mochizuki's inter-universal Teichmuller theory (IUT). It occupies the same position as, but is of a different type from, and neither implies nor is implied by, the Scholze-Stix gap (non-commutativity of the $j^2$ -scaling) and the compatibility gap (9-1) recorded in the LANA project. The holomorphic hull of IUT III, Remark 3.9.5, is the minimum over the constrained family $\{\lambda \cdot \mathcal{O}\}$ : each direct summand of $I^{\mathbb{Q}} \cong \bigoplus_{\alpha} F_{\alpha}$ is scaled independently. This operation "boxes" the image of the tensor-product integral structure and introduces an index $\rho$ . The main mathematical result of this paper is the exact closed form of this index, $\rho = |\mathfrak{d}_K|^{(n-1)d^{n-1}/2}$ , $d = [K:\mathbb{Q}_p]$ , proved as an identity of positive integers (not merely a $p$ -adic valuation), where $\mathfrak{d}_K$ is the discriminant of $K/\mathbb{Q}_p$ . Its $p$ -adic corollary is $v_p(\rho) = \frac{(n-1)d^{n-1}}{2}\, v_p(\mathfrak{d}_K)$ : the loss equals $\frac{n-1}{2}$ times (number of components) times (the $p$ -adic valuation of the local discriminant). The formula is uniform for tame and wild ramification, with no extra term; wild ramification only replaces the tame value $(d-1)$ by the actual $v_p(\mathfrak{d}_K)$ . Substituting IUT's own parameters ( $n = j+1$ , $\ell^* = (\ell-1)/2 \asymp h^{1/2}$ , $v \in V^{\mathrm{bad}}$ ), the inflation is doubly exponential: $\log\log\rho = \Theta(\sqrt{h})$ , i.e. $\rho = \exp(\exp(\Theta(\sqrt{h})))$ . Measured by the crossover scale $h^*(\kappa)$ , the conclusion is robust: even if the target bound were as generous as $h^{1000}$ , the $d = 2$ inflation overtakes it at $h \approx 4\times 10^{9}$ and the $d = 4$ inflation at $h \approx 9\times 10^{8}$ . This paper is not a verdict on IUT. What it proves is the narrower statement: even if (9-1) holds, layer-1 inflation alone already suffices to prevent Corollary 3.12 from yielding abc.

Primarily human-written textHuman understanding: all partsinter-universal Teichmuller theory，abc conjecture，holomorphic hull

Oct 29, 2026PDF ↗View work →

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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

An Integral-Structure Inflation in the Hull Operation of IUT Corollary 3.12

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