ON POLYNOMIAL STRUCTURES OVER SPHERES

The paper proves global injectivity of the spectrum map on low-rank bundles for closed negatively curved manifolds with simple length spectrum: Theorem 1.1 asserts that for ranks r ≤ q_F(n) (an explicit rank threshold defined in (3.6)), the spectrum map S is injective on AF_{≤q_F(n)}; see the statement and discussion around Theorem 1.1 and its corollary from Theorem 4.5, which establishes injectivity of the geodesic Wilson loop and hence of S via the wave trace formula . The proof uses: (i) the Duistermaat–Guillemin trace formula to extract Tr(Hol(c)) from the spectrum when the length spectrum is simple (equation (4.1)) ; (ii) a representation-theoretic step showing equality of Wilson loops implies isomorphism of the associated Parry-monoid representations, followed by a non-abelian Livšic theorem to produce a smooth flow-invariant transfer map p on SM between the two connections ; and (iii) the finiteness-of-Fourier-degree result on negatively curved manifolds, plus the non-existence of low-rank polynomial maps Sn→SO(r),U(r), to force p to be fiber-constant and descend to M, giving gauge equivalence (and equality of bundles) . The candidate model claimed the result was likely open and required a new vanishing theorem for twisted conformal Killing tensors; however, the paper circumvents that route entirely by a different mechanism (polynomial structures over spheres), so the model’s “likely open” assessment is incorrect.