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

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 of regions scaled independently on each direct summand of I^Q. 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 = |d_K|^{(n-1)d^{n-1}/2}, d = [K:Q_p], proved as an identity of positive integers (not merely a p-adic valuation), where d_K is the discriminant of K/Q_p. Its p-adic corollary is v_p(rho) = ((n-1)d^{n-1}/2)·v_p(d_K): the loss equals (n-1)/2 x (number of components) x (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(d_K).

Substituting IUT's own parameters (n = j+1, l* = (l-1)/2 ~ h^{1/2}, v in V^{bad}), the inflation is doubly exponential: log log rho = 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 ~ 4 x 10^9 and the d = 4 inflation at h ~ 9 x 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.