K-Theory of Non-Commutative Spheres Arising from the Fourier Automorphism
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Bibliographic record
Abstract
Abstract For a dense G δ set of real parameters θ in [0, 1] (containing the rationals) it is shown that the group K 0 (A θ ⋊ σ ) is isomorphic to , where A θ is the rotation C * -algebra generated by unitaries U , V satisfying VU = e 2πiθ UV and σ is the Fourier automorphism of A θ defined by σ(U) = V, σ(V) = U −1 . More precisely, an explicit basis for K 0 consisting of nine canonical modules is given. (A slight generalization of this result is also obtained for certain separable continuous fields of unital C * -algebras over [0, 1].) The Connes Chern character ch: K 0 (A θ ⋊ σ ) → H ev (A θ ⋊ σ ) * is shown to be injective for a dense G δ set of parameters θ. The main computational tool in this paper is a group homomorphism T : K 0 (A θ ⋊ σ ) → obtained from the Connes Chern character by restricting the functionals in its codomain to a certain nine-dimensional subspace of H ev (A θ ⋊ σ ). The range of T is fully determined for each θ. (We conjecture that this subspace is all of H ev .)
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Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.003 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.001 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from it