Six Obstructions to Native Hecke Extraction on the SCT Critical Sector

Research Note 10 in the "Geometry of the Critical Line" programme This note proves six independent obstructions (Kills #60–64 plus a structural incompatibility theorem) showing that Hecke-type extraction operators cannot be faithfully represented on the SCT critical sector's archimedean carrier alone. The central result: the irrational rotation parameter α = 2ln2/π that makes the geometric carrier work is the same irrationality that kills every native Hecke construction. The carrier and the extractor cannot coexist on the same space — this split is identified as the archimedean/non-archimedean split in the adelic framework. The note also establishes a conditional KMS recursion: on any abstract algebra satisfying isometry and scaling relations, the KMS state at β = 1/2 gives φ(μ_p^{*r} μ_p^r) = p^{-r/2}, recovering the Dirichlet series amplitude from thermal equilibrium alone. Part of a 46-paper open-access programme on the geometry of the Riemann zeta function's critical line, anchored by the SCT 5-Manifold and the cover equation Φ + e^{iπ − 1/Φ} = 0. The programme does not claim to prove the Riemann Hypothesis.