Rigid Borrowing and Spectral Reconstruction F: Synthetic Decision of Integrability Obstacles—Existence of Complex Structures in Real Four-Dimensional Linear Systems, Mixed Lattice Phenomena, and Perturbation Boundaries

The decision problem for commuting complex structures is a central step in theintegrability analysis of real four-dimensional linear systems. At the synthetic layer,this paper establishes an existence criterion centered on the real Jordan pairing condition: for A ∈ GL(n,R), there exists J satisfying J2 = −I and AJ = JA if andonly if, at each real eigenvalue, real Jordan blocks of the same size occur in pairs;the non-real eigenvalue part is automatically satisfied by the conjugate structureand imposes no additional condition. The criterion requires neither semisimplicity nor hyperbolicity. The classification of four-dimensional hyperbolic semisimpleGL(4,Z) matrices is thereby unified as follows: if the characteristic polynomial hasno real roots, the system is integrable; if the characteristic polynomial is the squareof a monic integer quadratic factor with real roots and constant term ±1, the system is integrable; all other semisimple cases are non-integrable. Two verificationsamples respectively block two overly strong assertions, namely “no real root isnecessary” and “even algebraic multiplicity is sufficient”: a B ⊕B type matrix hasreal roots yet is integrable, while a companion q2 type matrix has even algebraicmultiplicities but is non-integrable because its real Jordan blocks are not paired.The synthetic layer also gives independent decision boundaries for roots of unity,hyperbolicity, and GL(4,Z) membership: a root of unity is not a certificate for thenonexistence of a commuting complex structure; hyperbolicity excludes eigenvaluesof modulus one but does not exclude real roots; samples whose constant term is not±1 do not belong to the torus automorphism ledger. The perturbation boundaryis stratified by three types of algebraic events: realification of the discriminant,coalescence of real roots, and Jordan degeneration; spectral silence defined only byconservation of the characteristic polynomial cannot cover the Jordan degenerationchannel. Full enumeration of GL(4,Z), classification for n ≥ 6, nonconstant J tensor fields, and the quantitative form of an effective Newlander–Nirenberg thresholdare outside the coverage of the synthetic layer.