Rigid Borrowing and Spectral Reconstruction A: A Program for the Classification of Real Anosov Diffeomorphisms from the Perspective of the Rigid-Borrowing–Cayley–Dickson Ladder at the T0 Layer

The algebraic classification of real Anosov diffeomorphisms is open in the general case, while the complex version has been proved in codimension one and hascompleted machine formal verification (preprint). This gap can be explained at thelevel of algebraic structure: the real field lies at the bottom of the Cayley–Dicksonladder, its automorphism group is trivial, and it lacks algebraic closedness; the complex field and the quaternion division algebra provide stronger skeletal resources.This paper formalizes “rigid borrowing” as a layered program: partial lifting functors on the subcategory of almost-complex-compatible objects serve as channels,topological compatibility, analytic integrability, and algebraic spectral conditionsserve as three levels of obstruction, and sign recovery from periodic spectral sequences serves as the data bridge. For hyperbolic torus automorphisms, the signedspectral sequence satisfies a linear recurrence whose order is determined by the signof the determinant; the dynamically observable fixed-point count gives only absolute values: unsigned data can still fit the recurrence exactly and uniquely recoverthe absolute value of the trace, but the sign of the trace cannot be determined fromthe data. Sign recovery is therefore a sign-determination step, not a prerequisitefor fitting. In the two-dimensional case, the branch structure of sign recovery canbe completely classified: when the determinant equals minus one, a matrix andits negative generate exactly the same unsigned periodic sequence, the candidateset is always the two branches {±|τ|}, and generative verification cannot disambiguate them; when the determinant equals plus one, the candidate set has at mosttwo elements, and growth-rate pruning together with generative verification candisambiguate. Accordingly, spectral data lock the conjugacy invariants up to atmost finitely many branches; the construction of the conjugacy and branch disambiguation belong to the suspension segment. The three levels of obstructions havedifferent dimensions, so cross-level summation is invalid; additivity can be statedonly within each level. The program is organized in three layers—proved, assembled, and conjectural—and finite decidability at the analytic layer and single-layerquantitative control are independently testable progress indicators.