Pith Number

pith:KB66V7Z7

pith:2026:KB66V7Z7WZLAWKLAIZFQLZSWYT

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Non-crystallographic systems of integers over composition algebras

Daniele Corradetti

A weak golden octonion order arises from Cayley-Dickson doubling of the icosian ring, carrying a 240-element H4⊕H4 shell and remaining self-dual under the polar norm.

arxiv:2605.15075 v1 · 2026-05-14 · math.RA

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Claims

C1strongest claim

We construct a weak golden octonion order by Cayley--Dickson doubling of the icosian ring; the resulting free rank-8 Zφ-order has a 240-element finite shell of type H4⊕H4 and its multiplication is genuinely octonionic. We prove (i) that this weak double is self-dual with respect to the polar norm pairing, hence has no strict norm-integral overorder, and (ii) that the first trace-integral discriminant tower over it contains no octonion-stable nonzero isotropic gluing.

C2weakest assumption

The assumption that the Cayley-Dickson doubling applied to the icosian ring preserves a genuinely octonionic multiplication while producing exactly the claimed 240-element H4⊕H4 shell and satisfying the self-duality and discriminant-tower properties without additional hidden constraints on the order.

C3one line summary

A new free rank-8 Z[φ]-order is built by doubling the icosian ring, equipped with a 240-element H4⊕H4 finite root shell, shown to be self-dual under the polar norm and free of octonion-stable nonzero isotropic gluings in its first trace-integral discriminant tower.

References

24 extracted · 24 resolved · 0 Pith anchors

[1] Baake, A guide to mathematical quasicrystals, in Quasicrystals, eds 1998

[2] M. Baake and U. Grimm, Aperiodic Order, Cambridge University Press, 2013 2013

[3] L. Chen, R. V. Moody and J. Patera, Non-crystallographic root systems, Fields Institute Monographs, Volume 10, 1998, 135--178 1998

[4] J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, Springer, 1999 1999

[5] J. H. Conway and D. A. Smith, On Quaternions and Octonions, A K Peters/CRC Press, 2003 2003

Formal links

3 machine-checked theorem links

Cited by

1 paper in Pith

Integral Planes and Unit-Norm Polytopes

Receipt and verification

First computed	2026-05-17T21:40:26.009084Z
Last reissued	2026-05-17T21:57:19.316544Z
Builder	pith-number-builder-2026-05-17-v1
Signature	unsigned_v0
Schema	pith-number/v1.0

Canonical hash

507deaff3fb6560b2960464b05e656c4f5f37c32ee94ede8f19422ae42c5a848

Aliases

arxiv: 2605.15075 · arxiv_version: 2605.15075v1 · pith_short_12: KB66V7Z7WZLA · pith_short_16: KB66V7Z7WZLAWKLA · pith_short_8: KB66V7Z7

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curl -sH 'Accept: application/ld+json' https://pith.science/pith/KB66V7Z7WZLAWKLAIZFQLZSWYT \
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  | python3 -c "import sys,json,hashlib; b=json.dumps(json.loads(sys.stdin.read()), sort_keys=True, separators=(',',':'), ensure_ascii=False).encode(); print(hashlib.sha256(b).hexdigest())"
# expect: 507deaff3fb6560b2960464b05e656c4f5f37c32ee94ede8f19422ae42c5a848

Canonical record JSON

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    "abstract_canon_sha256": "c0d0024855128de337833a4f31224d4ed476c2675ef10dcf9c573f15122bc578",
    "cross_cats_sorted": [],
    "license": "http://creativecommons.org/licenses/by/4.0/",
    "primary_cat": "math.RA",
    "submitted_at": "2026-05-14T17:00:48Z",
    "title_canon_sha256": "c181ddac5251e00de9b9f6f4ff67f12ecb46788cb9f8d6c23b70e33c916cf9d2"
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