Octonion Jordan-Shadow decomposition applied to ML embeddings: a category-discrimination benchmark

We demonstrate that applying Jordan-Shadow octonion decomposition (aoi_collapse) to standard machine learning embeddings produces structurally discriminative features that outperform raw embedding cosine similarity by 8.83x for category classification. Method: 10 IMO-level mathematics problems from 4 categories (geometry, number theory, combinatorics, algebra) were embedded via mathstral:7b (4096D), folded to 24D, and decomposed through 24-dimensional octonion algebra into Jordan (symmetric), commutator (anti-symmetric), and associator (non-linear chaos) components. Results: Average same-category vs different-category cosine similarity ratios: - Raw 4096D embedding: 1.06x (near random) - 24D projection: 1.11x - Associator (personality): 8.83x - Commutator (control): 3.49x The associator — the non-associative interaction term (Jordan x Commutator) — captures structural relationships between problems that linear similarity on raw embeddings cannot detect. Two geometry problems with no shared vocabulary still cluster together after decomposition because the octonion non-associativity amplifies structural signal while attenuating surface-level noise. Builds on prior work: Zenodo DOI chain 10.5281/zenodo.18690444, 18722487, 18809406, 18809716. Correction (2026-07-29): The 8.83x figure in this record's title and abstract is an artifact of the scoring statistic and is withdrawn. The statistic is the ratio of mean same-category cosine similarity to mean different-category cosine similarity - a quotient of two means of a signed quantity, with a ceiling at 1/mean_diff and no invariance to an additive shift: sweeping a constant offset across every cosine moves the same measurement from 4.8 to 6580.6 to -23.0 while a kNN classifier on the identical features stays at 0.6780 throughout. Re-benchmarked at scale on 20 Newsgroups (6,000 documents, 20 categories, n = 2,000 per run, 5 seeds, 4 embedding families, 20,000 permutation nulls and 5 negative controls), the associator features score 63.32 +/- 60.67 on the original's own metric - larger than the claimed 8.83x - while reaching kNN accuracy of only 0.071 +/- 0.013 against a 0.049 chance rate for 20 balanced classes, against 0.664 +/- 0.010 for the raw embedding and 0.184 +/- 0.016 for a random 8-dimensional Gaussian projection of the same embedding. The raw embedding with its global mean removed is the best representation tested (kNN 0.673 +/- 0.011) and scores -17.8 on the original metric, and the permutation null for the ratio on 8-dimensional pure noise routinely exceeds 8.83 by chance. The mechanism is compression, not discrimination: sentence embeddings are anisotropic, so between-category cosines average +0.0348, while the collapse output is near-isotropic at +0.0010; the numerator falls 8.2x, the denominator falls 34.8x, and the shift-invariant separation (the same two means subtracted rather than divided) falls 6.9x, as expected from a lossy 384-to-24-to-8 compression. Under a scale-invariant metric a small real residual remains - kNN accuracy 0.083 against a 0.049 chance baseline - which is not zero and is not an order of magnitude. The original evidence base of 10 problems in 4 categories cannot support an order-of-magnitude claim in either direction.