Two mathematical defects in the IT³ "Perez Hourglass" architecture (Zenodo 21792530): a metric signature change at the poles, and failure of the stated O_h invariance

Abstract

Comment on V. Logvinovich, “The Macroscopic Vacuum Architecture: Perez Hourglass Topology, Counter-Rotating Bowls, and m=6⊕12 Fractal Superposition in the Solar System (IT3 Framework v6.0)”, Zenodo, 2026-08-04, doi:10.5281/zenodo.21792530. Two equations displayed in the target manuscript do not have the properties the manuscript asserts for them. (1) The metric is not globally Lorentzian. The line element of §2.1, explicitly declared to define a Lorentzian metric tensor, has determinant det g = −c²r⁴(sin²θ − ε²). With the value εtopo = 0.010586 adopted in the same section, the metric is degenerate on the cones |sin θ| = ε — where it has no inverse, hence no Levi-Civita connection and no field equations — and has signature (2,2) inside them: two open polar caps of angular radius arcsin(ε) = 0.6065°, within which there are two independent timelike directions. Because metric signature is a pointwise diffeomorphism invariant (Sylvester's law of inertia), this is not an artifact of spherical polar coordinates and is not removable by any change of chart. (2) The stated Oh symmetry does not hold. The potential Φtopo = Φ₀[1 + ε₆Y6,6 + ε₁₂Y12,12] of §2.3 is not invariant under the octahedral group Oh, contrary to the Theorem stated immediately beneath it. The four-fold rotation C₄z ∈ Oh acts as Y6,6 → −Y6,6, so the Reynolds projection of Y6,6 onto the trivial representation A1g vanishes identically: the m=6 term carries no Oh-invariant content at all, and no choice of ε₆ ≠ 0 repairs it. Independently, Oh contains no six-fold rotation axis (crystallographic restriction) whereas the displayed field has exact six-fold azimuthal periodicity; and the 12 extrema of Re Y6,6 are coplanar on the equator, whereas the 12 cuboctahedron vertices they are identified with lie at three distinct colatitudes (45°, 90°, 135°). For accuracy, the comment records that the m=12 term is not equally affected: since 12 ≡ 0 (mod 4), Y12,12 survives C₄z and has non-zero A1g projection, though it is still not invariant on its own. Available repairs for both defects are given. Scope. These results falsify two specific stated claims — that the §2.1 line element is Lorentzian, and the §2.3 Theorem on Oh symmetry and the cuboctahedral node count. They do not address the empirical portion of the target manuscript (the MPCORB analysis, the √3 scaling ladder, the reported clustering statistics, the K-PHAM operator), which does not pass through either defective equation and is neither endorsed nor contested here. The identical Φtopo equation and cuboctahedron claim appear verbatim in doi:10.5281/zenodo.20931136; the analysis of §3 applies unchanged there. Reproduction. Three short Python scripts (sympy/numpy/scipy, seconds to run) are attached and reproduce every numerical value quoted.