The Explicit Inverse Period Map for U ⊕ E8(−1)-Polarized K3 Surfaces: Type II Monodromy, Collision Exponents, Exact Reconstruction, and Family-Deformation Structure

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Let (S=U⊕ E_8(-1)). We prove that Sakai’s eleven orthogonal modular forms are the coordinates of the inverse period map for (S)-polarized K3 surfaces. This gives a second proof of Sakai’s conjecture, first proved by Clingher, Malmendier and Williams, by a route independent of the Vinberg family. The geometric input is a general theorem: for a one-parameter Weierstrass family of elliptic K3 surfaces acquiring a single non-minimal ((4,6)) point with smooth residual fibre, the collision exponent read from the Newton polygon equals the content of the logarithm of monodromy. Applied at the one-dimensional cusp, it forces the Type II Fourier–Jacobi orders of the true coordinates, while the vanishing tori identify the double-curve modulus with (τ). Exact rational linear algebra, reproduced by an independent reconstruction with explicit integrality control, shows that the normalized 74-parameter boundary system has a unique solution, coefficientwise equal to Sakai’s. We also prove a reduced boundary criterion, classify two finite threshold certificates (222 and 252 profiles, with an exact half-space law), and determine the whole-family deformation spaces.