BLACK AND GOLD III Spectral Fingerprint, Forced Grading, and the Exceptional-Point Dichotomy under the Common Integrity Gate

BLACK AND GOLD IIISpectral Fingerprint, Forced Grading, and the Exceptional-PointDichotomy under the Common Integrity Gate C. J. TullyIndependent Researcher, Engineer — Tamworth, NSW, AustraliaORCID: 0009-0007-5661-7332 October 2026 ABSTRACT This paper is the third of the black-and-gold series. It places the spectral fingerprint of the Ly-Device under thecommon Quasi-Equivalence Index gate established in Paper I. Conditional on the spectral operator having the exactgolden multiset {3, φ, φ, −1/φ, −1/φ}, the characteristic polynomial lies in &mathbb;Z[λ], power traces obey theLucas law, and the grading order is forced by the golden discriminant 5. The golden eigenvalues are the Gaussianperiods of &mathbb;Q}(ζ5) / &mathbb;Q}(√5). The remaining spectral invariant is the Jordan structure of thegolden subspace: either a diagonalizable doubled fusion module or a second-order exceptional point. Eitherresolution is consequential; the exceptional-point case supplies a candidate mechanism for uncloneability throughsquare-root amplification of disorder. Four decisive tests convert the conditional theorems into unconditional ones.The fingerprint is admitted into the series only when the resulting state continues to satisfy the hard gate. No claimis made that the spectrum has been shown to be exact, nor that uncloneability has been demonstrated.