Cayleyan Sn-encoded SU(2)×Sn↓G embeddings: Nuclear spin permutation symmetries via polyhedral lattice-point models, for modulo-i χ(Ci(Sn↓G)) combinatorial invariance sets
Bibliographic record
Abstract
The complete nuclear permutational (CNP) statistics of SU(2)×𝒮n spin ensembles of forms [A]n/[AX]n for cage molecules [i.e., exclusive (not mixed-isotope) isotopomers] are shown to yleld totally analytic invariance sets on the basis of vertex-point spins (or more generally Schur function labels) on regular polyhedral lattice-point models and their modulo-i [of Ci (𝒮n↓𝒢)] algebras in the specific cases discussed. This occurs only when (i.e., iff) they correspond to the criterion for Cayleyan group embedding or index n=/𝒢/. Such realizations correspond to 𝒮n symbolic (lattice-based) encodings, well known in cybernetics. Hence exclusively combinatorial invariance descriptions [within (Voronoi) vertex-point lattice geometric models], within j≡0 mod(i) (equivalence modulo-i) for each of i indices of the class operators (of the embedded group) Cis, else a null factor, arise for certain specific automorphic CNP/NMR spin symmetries; to date these are shown to be limited to some five main Cayleyan types of SU(2)×𝒮n↓𝒢 embedding. Both the question of the sufficiency of Cayley criterion for dual group embeddings beyond SU(m≥3)×𝒮n and that of the further role of Kostka coefficient hierarchy in the natural embedding process, are reviewed here and more extensively in related work [Eur. Phys. J., B 11, 177 (1999); Int. J. Quant. Chem., 78(1), 5–14 (2000)]. A brief comment on the value of Yamanouchi chain-based system invariants reaffirms the central role of the 𝒮n group and its encodings in subduction processes associated with spin algebras and in certain fundamental Liouvillian (bosonic) mapping processes [Physica, A198, 245 (1993)]. In highlighting these linkages between subtopics, we demonstrate the ultimate consequences of Balasubramanian's use of automorphisms, based on the {Jij} zeroth-order structure, in NMR [J. Chem. Phys., 78, 6258 (1983)], or equally his use of cycle-index methods in CNP statistics. © 2000 John Wiley & Sons, Inc. Int J Quant Chem 78: 71–82, 2000
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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 | 0.000 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.002 | 0.001 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.005 | 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".