Rigid Borrowing and Spectral Reconstruction C: Pole Structure of Spectral Sequences—Effective Pole Formulas for GL(3,Z) Hyperbolic Matrices and Full Machine-Check Certification

The spectral sequence σk = det(Ak − I) is the channel through which the periodic data of a hyperbolic matrix A enter algebraic coordinates; the order of itsminimal linear recurrence determines the sample complexity required for the recovery of spectral invariants. This paper performs a full machine check on the3120 hyperbolic matrices in GL(3,Z) with coefficient bound 1, and gives a uniformfalsification of the linear extrapolation “recurrence order n + 1 / n + 2”: the effective pole count on the detA = 1 branch is always 6, and on the detA = −1branch always 8; none equals the extrapolated values 4 and 5. The diagnosis of thefalsification reveals that what determines the recurrence order is not the geometric count of subset products, but the distinct subset-product values with nonzeromerged coefficients; when detA = 1, the empty and full subset products are forcedto coincide at 1, and the merged coefficient of pole 1, (−1)n + 1, vanishes for oddn, so this pole is invisible at the sequence level. This paper thereby establishesa three-dimensional effective-pole formula, gives a Prony-type recovery protocolthat determines the order by the effective pole count, and completes two-channelcertification on all 3120 samples. The certification residual is normalized by thefirst power of the unstable modulus, and the raw absolute residual is reported atthe same time, so that the nominal value of the certification strength agrees withthe actual threshold. The certification strength is defined at three levels: orderdetermination correctness, threshold caliber, and artifact caliber. Machine-checkconclusions are archived in committable blockwise artifact form; aggregate statements are read from the run record of this implementation; third-party review follows an equivalent-reimplementation path and does not depend on any undisclosedintermediate state. In the most general n-dimensional case, the effective pole countis written as 2n − [detA = 1]−[n is odd and detA = 1]; this formula depends onthe condition that “apart from the forced coincidence of the empty/full subsets andthe cancellation of pole 1, there are no other coincidences or cancellations.” Thepart n ≥ 4 has not undergone full machine checking and is registered as a conditional proposition. The interface between the periodic-orbit count |det(Ak − I)|and the signed spectral sequence is lossy: the absolute-value operation does notpreserve finite exponential sums, and sign information cannot be replaced losslesslyby the counting sequence.