HEXAGONbeta

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

Primarily human-written textHuman understanding: all partsmath.NT — Number Theorymath.AG — Algebraic Geometry

Contributed by Zeng Chunping

Version 2 / 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

  1. S. Mochizuki, Inter-universal Teichmüller Theory I: Construction of Hodge Theaters , Publ.\ Res.\ Inst.\ Math.\ Sci.\ 57 (2021), 3--207.
  2. S. Mochizuki, Inter-universal Teichmüller Theory II: Hodge--Arakelov-Theoretic Evaluation , Publ.\ Res.\ Inst.\ Math.\ Sci.\ 57 (2021), 209--401.
  3. S. Mochizuki, Inter-universal Teichmüller Theory III: Canonical Splittings of the Log-Theta-Lattice , Publ.\ Res.\ Inst.\ Math.\ Sci.\ 57 (2021), 403--626.
  4. S. Mochizuki, Inter-universal Teichmüller Theory IV: Log-Volume Computations and Set-Theoretic Foundations , Publ.\ Res.\ Inst.\ Math.\ Sci.\ 57 (2021), 627--723.
  5. P. Scholze, J. Stix, Why abc is still a conjecture , https://www.math.uni-bonn.de/people/scholze/WhyAbcIsStillAConjecture.pdf.link
  6. K. Kato, J. Commelin, Y. Hoshi, K. Kedlaya, A. Topaz, LANA Project Midterm Report (2026-07), https://github.com/katobungen/LANA_report_202607.link
  7. A. Grothendieck et al., Revêtements étales et groupe fondamental (SGA 1) ; cf.\ also J.-P. Serre, Local Fields , Grad.\ Texts in Math.\ 67, Springer.
  8. J.-P. Serre, Local Fields , Graduate Texts in Mathematics 67 , Springer, 1979 (Hilbert's ramification formula, Ch.\ IV).
  9. S. Mochizuki, I. Fesenko, Y. Hoshi, A. Minamide, W. Porowski, Explicit Estimates in Inter-universal Teichmüller Theory , Kodai Math.\ J.\ 45 (2022), 175--236 (RIMS Preprint 1933).
  10. Z. Zhou, Explicit abc-conjecture bounds , arXiv:2503.14510.arXiv
Show all 18 referencesShow fewer references
  1. E. M. Matveev, An explicit lower bound for a homogeneous rational linear form in logarithms of algebraic numbers II , Izv.\ Math.\ 64 (2000), 1217--1269.
  2. C. L. Stewart, K. Yu, On the abc conjecture, II , Duke Math.\ J.\ 108 (2001), 169--181.
  3. H. Pasten, Shimura curves and the abc conjecture , J.\ Number Theory 254 (2024); and Invent.\ Math. (2024).
  4. S. D. Sanyal, Irrational behavior of algebraic discrete valuations , arXiv:1311.6047 (Hilbert--Samuel functions in dimension 22 over a regular local ring can be non-polynomial).arXiv
  5. J.-H. Evertse, K. Győry, Unit Equations in Diophantine Number Theory , Cambridge Univ.\ Press, 2015.
  6. J. H. Silverman, Advanced Topics in the Arithmetic of Elliptic Curves , Grad.\ Texts in Math.\ 151, Springer (Tate curves and local heights).
  7. K. Joshi, Construction of Arithmetic Teichmüller Spaces III , arXiv:2401.13508.arXiv
  8. Y. Hoshi, Comments on the LANA Project Report (2026).

Version history

  1. v1 · 24 pagesSubmitted by Zeng ChunpingInitial depositSupersededOct 29, 2026
  2. v2 · 42 pagesSubmitted by Zeng ChunpingSubmitted as a new versionCurrentNov 06, 2026