What it is supposed to be forPhysics

Everything up to here is settled mathematics. This page is not. It is a research programme with real results and real problems, and the problems are on the page.

Where the line is. The division algebras, triality, the magic square and E8's root system are nineteenth- and twentieth-century mathematics, proved and not in dispute. Whether nature uses any of it is an open question, and the models below are proposals under construction — Lisi's own summary of them, in his §10, is a list of what each one still cannot do.

The shape of the idea

The Standard Model of particle physics is a list: twelve force carriers, a Higgs, and three copies of a set of twelve matter particles. The copies are identical except for mass. Nobody knows why there are three.

The grand-unified idea is that the forces are one force seen through a broken symmetry, and the standard big symmetry for that is so(10) — 45 dimensions, with one generation of matter and its antimatter fitting exactly into a 16-component spinor. That part is respectable and decades old.

The exceptional idea goes one further: put the forces and the matter in the same Lie algebra, as different parts of one connection. Bosons in the even half, fermions in the odd half. For that you need an algebra with the right two-way split, and the magic square hands you three candidates Lisi 2026 §10.

algebra	dim	fixed part	dim	each moving part	dim
`su(3)`	8	u(1) + u(1)	2	1 + 1 + 1	3
`sp(3)`	21	so(4) + u(1)	7	3 + 4	7
`f4`	52	so(7) + u(1)	22	7 + 8	15
`e6`	78	so(8) + u(1) + u(1)	30	8v + 8s+ + 8s-	24
`e7`	133	so(10) + su(2) + u(1)	49	10 + 2x16	42
`e8`	248	so(14) + u(1)	92	14 + 64	78

The three-way split of each triality algebra, from Lisi eq. (6.1). Fixed part plus twice a moving part equals the whole, in every row — checked at build time.

Six stacked bars, each split into a fixed part and two equal moving parts. — Fixed part in amber, the two moving parts in teal and rose. The proposal is that generations live in the parts that move.

Three candidates, three catches

e6 — 78 dimensions

The smallest one that works. so(10) and its 45 gauge directions fit, one generation of left-handed matter fits in a 16, one generation of right-handed in the other 16, and there are six directions of charge left over. The catch: one generation. Triality is barely used. It is a tidy repackaging of a grand unified theory, not an explanation of three T. Dray 2010.

e7 — 133 dimensions

This is where the ℂ ⊗ ℍ ⊗ 𝕆 programme lands — Dixon's algebra, and Furey and Hughes' work on generations G. M. Dixon 1994 N. Furey 2022 N. Furey 2025. Three generations do fit, related by triality. The catch: you cannot have all three generations and both handednesses and the antiparticles at once. You must pick. Taking three generations forces a complex form of e7, which then misbehaves for the gauge fields, and the gravitational frame is missing altogether Lisi 2026 §10.

e8 — 248 dimensions

The whole thing: gravity's spin connection, the frame, the Higgs, the SO(10) gauge fields, and three generations of fermions related by triality. This is the 2007 paper A. G. Lisi 2007 brought up to date. The 2026 version puts the three generations in the moving parts of the three-way split, so all three carry identical charges — which is what the Standard Model actually shows Lisi 2026 §10.

The catches, plural. The compact version is Euclidean: no time direction, so it has to be part of something bigger to describe the world we are in. The natural two-way split of e8 does not line up with that particular assignment of fermions. And the three generations are not independent of each other, so the model predicts generational mixing it then has to account for.

The standing objection. Distler and Garibaldi proved in 2010 that you cannot embed three generations of Standard Model fermions in E8 in the way the 2007 paper needed — there is always an unwanted mirror generation J. Distler 2010. Lisi's answer in §10 is not a denial: it accepts the mirror fermions and reinterprets them as the second and third generations, reached by triality rotation. That is a real answer and it has a real cost, which is the generational mixing above. This remains contested. If you read one criticism, read that one.

What would settle it

Nothing on this page is a prediction yet in the sense that would settle anything. The state of play, as the paper itself puts it in its discussion, is that the mathematical scaffolding is now written down in enough detail that model-building can proceed — the point of the 2026 paper is the scaffolding, not a finished theory Lisi 2026 §12.

What makes it worth watching is the coincidence count. Eight is the last dimension with a division algebra. Eight is the only dimension with triality. Triality has order three; matter comes in three generations; the charges of the three generations are identical, which is what a symmetry of order three would produce. None of that is proof. It is a set of numbers that keep lining up.