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

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Contributed by Zeng Chunping ↗

Version 1 / Oct 29, 2026 / CC BY 4.0

Abstract

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 j2j^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 {λ⋅O}\{\lambda \cdot \mathcal{O}\}: each direct summand of IQ≅⨁αFαI^{\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 ρ\rho. The main mathematical result of this paper is the exact closed form of this index, ρ=∣dK∣(n−1)dn−1/2\rho = |\mathfrak{d}_K|^{(n-1)d^{n-1}/2}, d=[K:Qp]d = [K:\mathbb{Q}_p], proved as an identity of positive integers (not merely a pp-adic valuation), where dK\mathfrak{d}_K is the discriminant of K/QpK/\mathbb{Q}_p. Its pp-adic corollary is vp(ρ)=(n−1)dn−12 vp(dK)v_p(\rho) = \frac{(n-1)d^{n-1}}{2}\, v_p(\mathfrak{d}_K): the loss equals n−12\frac{n-1}{2} times (number of components) times (the pp-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)(d-1) by the actual vp(dK)v_p(\mathfrak{d}_K). Substituting IUT's own parameters (n=j+1n = j+1, ℓ∗=(ℓ−1)/2≍h1/2\ell^* = (\ell-1)/2 \asymp h^{1/2}, v∈Vbadv \in V^{\mathrm{bad}}), the inflation is doubly exponential: log⁡log⁡ρ=Θ(h)\log\log\rho = \Theta(\sqrt{h}), i.e. ρ=exp⁡(exp⁡(Θ(h)))\rho = \exp(\exp(\Theta(\sqrt{h}))). Measured by the crossover scale h∗(κ)h^*(\kappa), the conclusion is robust: even if the target bound were as generous as h1000h^{1000}, the d=2d = 2 inflation overtakes it at h≈4×109h \approx 4\times 10^{9} and the d=4d = 4 inflation at h≈9×108h \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.

Tools used

Magma
Magma

References

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Version history

  1. v1Submitted by Zeng ChunpingInitial depositCurrentOct 29, 2026