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.