Universal subspaces for local unitary groups of fermionic systems
Bibliographic record
Abstract
Let $${\mathcal{V}=\wedge^{N} V}$$ be the N-fermion Hilbert space with M-dimensional single particle space V and 2N ≤ M. We refer to the unitary group G of V as the local unitary (LU) group. We fix an orthonormal (o.n.) basis |v 1⟩,...,|v M 〉 of V. Then the Slater determinants $${e_{i_1,\cdots,i_N}:= |{v_{i_1}\wedge v_{i_2} \wedge\cdots\wedge v_{i_N}}\rangle}$$ with i 1 < ... < i N form an o.n. basis of $${\mathcal{V}}$$ . Let $${\mathcal{S}\subseteq\mathcal{V}}$$ be the subspace spanned by all $${e_{i_1,\cdots,i_N}}$$ such that the set {i 1,...,i N } contains no pair {2k−1,2k}, k an integer. We say that the $${|{\psi}\rangle \in\mathcal{S}}$$ are single occupancy states (with respect to the basis |v 1⟩,...,|v M ⟩). We prove that for N = 3 the subspace $${\mathcal{S}}$$ is universal, i.e., each G-orbit in $${\mathcal{V}}$$ meets $${\mathcal{S}}$$ , and that this is false for N > 3. If M is even, the well known BCS states are not LU-equivalent to any single occupancy state. Our main result is that for N = 3 and M even there is a universal subspace $${\mathcal{W}\subseteq\mathcal{S}}$$ spanned by M(M−1)(M−5)/6 states $${e_{i_1,\ldots,i_N}}$$ . Moreover, the number M(M−1)(M−5)/6 is minimal.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.002 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.002 | 0.003 |
| Scholarly communication | 0.002 | 0.002 |
| Open science | 0.001 | 0.003 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.007 | 0.001 |
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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".