Rigid Borrowing and Spectral Reconstruction B: Spectral Reconstruction and Conditional Rigidity of Real Anosov Systems—A Reconstruction Pipeline from Periodic Data to Algebraic Coordinates

The algebraic classification of real Anosov diffeomorphisms remains open inthe general case. The constructions of Farrell–Gogolev and Farrell–Jones on exotic and fake tori show that the correct target statement of this problem mustbe classification-type, rather than a Boolean statement of “whether the system issmoothly conjugate to an algebraic model.” Within the class of systems homotopic to hyperbolic torus automorphisms, this paper establishes a reconstructionpipeline from the periodic-point cardinality Fix(fk) to the signed spectral sequenceσk =det(Ak−I), and registers the premise assumptions, data channels, and authorized outputs of each step in the pipeline according to condition-depth stratification.The core logical distinction is as follows: under H0 (f is homotopic to a hyperbolicA∈GL(n,Z), and the Franks–Manning topological conjugacy h◦f = A◦h holds),the topological counting bridge directly gives Fix(fk) = |σk|, and this step does notrequire the additional regularity of “sufficient smoothness to make periodic-orbitcounts conserved”; signed spectral recovery is an independent data hypothesis H1,σ,which requires signs, fixed-point indices, branches, or a sign oracle given by additional geometric or regular data. Without H1,σ, the absolute-value folding channelcan output only candidate pole subsets and finite ambiguity; it cannot claim torecover tr A, detA, or χA, and it cannot claim that the absolute-value zeta and thespectral recurrence polynomial P(z) interlock. The synthetic layer gives verifiableuniqueness conditions: given a branch and the effective pole set determined by themerged coefficients, if the exact rational Hankel matrix Hr is nonsingular, then therecurrence coefficients a = −H−1 r br are unique; for n = 2, the two branches recoverthe characteristic polynomial under window lengths 6 and 8, respectively; for n = 3,the pole count is registered under the independent-report convention as r = 6 whendetA = +1 and r = 8 when detA = −1, while blockwise reruns and checkpointmerge verification in this draft must be registered separately; for n ≥ 4, only conditional propositions are given. H2 (the effective Newlander–Nirenberg threshold)is used only at the CM candidate-coordinate output layer; when H2 breaks, thecomplexified arithmetic statement is retained. The paper does not construct a conjugacy, does not write the recovery of χA as a unique matrix A, and does not writevalidity within an enumeration as a general theorem.