Discretization of SU(2) and the Orthogonal Group Using Icosahedral\n Symmetries and the Golden Numbers
Bibliographic record
Abstract
The vertices of the four dimensional $120$-cell form a non-crystallographic\nroot system whose corresponding symmetry group is the Coxeter group $H_{4}$.\nThere are two special coordinate representations of this root system in which\nthey and their corresponding Coxeter groups involve only rational numbers and\nthe golden ratio $\\tau$. The two are related by the conjugation $\\tau\n\\mapsto\\tau' = -1/\\tau$. This paper investigates what happens when the two root\nsystems are combined and the group generated by both versions of $H_{4}$ is\nallowed to operate on them. The result is a new, but infinite, `root system'\n$\\Sigma$ which itself turns out to have a natural structure of the unitary\ngroup $SU(2,\\mathcal R)$ over the ring $\\mathcal R = \\mathbb\nZ[\\frac{1}{2},\\tau]$ (called here golden numbers). Acting upon it is the\nnaturally associated infinite reflection group $H^{\\infty}$, which we prove is\nof index $2$ in the orthogonal group $O(4,\\mathcal R)$. The paper makes\nextensive use of the quaternions over $\\mathcal R$ and leads to highly\nstructured discretized filtration of $SU(2)$. We use this to offer a simple and\neffective way to approximate any element of $SU(2)$ to any degree of accuracy\nrequired using the repeated actions of just five fixed reflections, a process\nthat may find application in computational methods in quantum mechanics.\n
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How this classification was reachedexpand
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 | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.001 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, 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".