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Modelled[E] four-square theorem and the Hurwitz order. Hadron seats use α and m_e as inputs.Quaternion Seats on S³
Every natural number is a sum of four squares. The Hurwitz order has integer norms. Three typed quaternions evaluate to masses near π⁰, K⁰, and η(1405) once an input unit is supplied.
Last updated: October 2026
Two squares on the Gaussian lattice
A Gaussian integer is a point of . Its norm is a sum of two integer squares. Fermat's theorem states that a natural number is a sum of two squares if and only if every prime congruent to 3 mod 4 appears with even exponent.
Of the integers from 1 to 100, 57 fail that test. The examples begin 3, 6, 7, 11, and 12. The same test rejects 15, 51, and 408:
- , and 3 appears once.
- , and both 3 and 17 appear once.
- , and 3 and 17 each appear once.
Four squares
Lagrange's four-square theorem states that every natural number is a sum of four integer squares. The integer obstruction counted above is empty in . Three representations, written in nonnegative nondecreasing order, are , , and . The theorem builds no icosian. An icosian coordinate lies in , with .
The Lipschitz order is inside the quaternions. The Hurwitz order is the set of quaternions whose four coordinates all lie in , or all lie in . A Hurwitz norm is an integer. The 24 Hurwitz units are the binary tetrahedral group inside the 120 vertices of the 600-cell. That vertex set is recorded on the 600-cell sections page.
The input unit
The mass unit used for the evaluations below is
and are inputs. The formula does not derive either of them. The products are at the typed squared length .
Three typed evaluations
The coordinate has squared length . Its coordinates mix and , so the point lies in and outside the Hurwitz order. Because a Hurwitz norm is an integer, is not a Hurwitz norm. The neighbouring integer norms are 3 and 4:
- , which is 14.131 MeV below the PDG mass 134.977 MeV.
- , which is 4.564 MeV above that mass.
The typed half-integer point itself evaluates to
Residual against 134.977 MeV: +0.0988%. One integer representation of the numerator 15 is , and is a different mass.
The Lipschitz quaternion has norm 51. Up to signs and order there are three nonnegative nondecreasing representations: , , and .
Residual against the PDG mass 497.611 MeV: +0.1309%. The target ratio is 50.867, and 51 is a neighbour of that ratio.
The Lipschitz quaternion has norm 408. Up to signs and order there are seven nonnegative nondecreasing representations. One of them is . The other six are , , , , , and .
Residual against 1409 MeV: +0.0212%. The neighbouring norm 409 gives . The PDG interval quoted for is MeV, and 1411.025 lies inside that interval.
The 240 roots of E₈
The root system has 240 roots. In the normalisation where each root has squared length 2, the inner product of two distinct roots lies in . The binary icosahedral group has 120 elements. Those are two integers. Writing them as a split into materialised hadrons and a sterile complement is a label, not a constructed bijection.
Conway and Sloane realise the icosian ring as up to scale by reading eight rational coordinates out of the quaternion norm . This page does not build that map. Concatenating each of the 120 unit icosians with its image under the Galois automorphism of produces 120 vectors in . Each has squared length 2, and the set of inner products includes . That set is a different configuration from the roots.
The same automorphism keeps 24 of the units inside the 600-cell and sends 96 outside it. The intersection of the vertex set with its image has 24 quaternions. The union has 216. The public statement of the inventory is unchanged: the particle halo is retired as inventory, on the dark-matter page. These roots are not a replacement inventory.