Rigid Borrowing and Spectral Reconstruction H: Degradation and Vacuumization of the Suspension Segment—Complete Classification of Jordan Multiplicities for Invariant Almost Complex Structures, Total Vacuum of the Two-Dimensional Channel, and a Full-Distribution Theorem

This paper studies the problem of invariant almost complex structures for torusautomorphisms and Anosov diffeomorphisms. Within the framework of this seriesof work, it performs degradation and vacuumization treatments for two suspensionhypotheses, respectively, and delivers a previously missing hard asset: a completelinear-algebraic classification of the existence of invariant almost complex structuresand a full-distribution theorem for such structures in the universe of block-diagonalinteger matrices. Theorem A gives the complete classification: a real linear operatorL admits a commuting complex structure J if and only if, for every real eigenvalue,its Jordan blocks, counted with multiplicity by eigenvalue and order, occur an evennumber of times; nonreal eigenvalues do not participate in this condition. Necessityis obtained via a complexification–realification argument on real generalized eigensubspaces; sufficiency is given by blockwise realification of nonreal blocks togetherwith a pairing construction for real blocks. Theorem B proves that the Nijenhuistensor of a constant almost complex structure on a torus vanishes identically: thepartial-derivative terms disappear because of constancy, and the algebraic termsare satisfied identically by J2 = −I; the integrability obstruction collapses entirelywithin the class of constant tensors, and the effective Newlander–Nirenberg threshold degenerates for this class. Theorem C establishes the two-dimensional totalvacuum: the spectrum of a hyperbolic two-dimensional torus automorphism consists of a pair of distinct real eigenvalues, and no invariant almost complex structureexists; for a two-dimensional Anosov system homotopic to the hyperbolic model,by genus-one uniformization and the affineness of holomorphic automorphisms ofcomplex tori, no smooth invariant almost complex structure exists either. TheoremD treats periodic-data rigidity in an honest-criterion form: the refutation side isfinitely checkable from the differential trace and determinant at a single periodicpoint; the affirmation side relies on citing the periodic-data rigidity theorem; theuniversal-verification side requires infinitely many data, and no claim is made aboutthe sufficiency of finite truncation. The distribution theorem gives exact counts inthe full universe of 53,824 four-dimensional block-diagonal integer matrices formedfrom all 232 blocks in GL(2,Z) with coefficient bound 3: 14,400 hyperbolic elements,1,376 admitting an invariant almost complex structure, and 13,024 not admittingone; all 200 hyperbolic matrices in the two-dimensional cross-check fail to admitone. All counts are produced by integer criteria, and per-sample certificates can beindependently rechecked without running code.