Nuclear

EstablishedLight 3+anti-3 frozen [E]; e_i to six quarks CLOSED F; FL-HEAVY parked. · structure under packages; see /status residuals.

Octonion Flavour Assignments

Light G2 structure freezes 3+anti-3 as lobe/channel geometry readable as (u,d,s). Quarks are not free e_i. Heavy multiplet membership is parked.

Last updated: 9 August 2026

Reading this entry

The octonion page freezes the light flavour slots as the 3⊕3̄ of SU(3) inside G₂. Up, down, and strange are 50/50 mixtures of physical and extra imaginary units, with Cartan charges Q = I₃ + Y/2, and the heavy flavours sit on the S⁷ latitude ladder.

What this is used for

The mass-splitting asymmetry reads G₂ = Aut(𝕆) from this stabiliser. Quark-model and mass-hierarchy pages take the light Cartan table, with I₃, Y, and Q, as the flavour-slot dictionary for u, d, and s.

Walk through

The observable chart is S³, and the division-algebra arena is S⁷ inside Im(𝕆) × time, with imaginary units (e₁, …, e₇). The automorphism group is the 14-dimensional group G₂, and fixing e₄ gives SU(3) = Stab(e₄). The remaining span is P = span(e₁, e₂, e₃, e₅, e₆, e₇), which decomposes as 3 ⊕ 3̄ of SU(3). Isospin I₃ = L_z rotates inside (e₁, e₂, e₃), and hypercharge Y rotates that triple into (e₅, e₆, e₇). The light slots are then u with I₃ = +1/2, Y = +1/3, Q = +2/3; d with I₃ = −1/2, Y = +1/3, Q = −1/3; and s with I₃ = 0, Y = −2/3, Q = −1/3, each a 50/50 mixture, under the charge rule Q = I₃ + Y/2.

Structure package: Claims here are geometric/packaging results under named premises. They do not by themselves close absolute α, continuum hold, or the full SM mass spectrum. See /status.

1. The 8D Octonionic Arena

While describes our observable 3D spatial chart, the full division-algebra spatial universe is embedded within an 8-dimensional space . The octonions are the largest normed division algebra, with 7 imaginary basis units .

The automorphism group of the octonions is the 14-dimensional exceptional Lie group . Fixing the Cayley-Dickson doubling unit defines the stabilizer subgroup , providing the exact geometric origin for quantum chromodynamics and quark flavour.

2. The Light $3 \oplus \bar3$ Representation

Under , the remaining 6 imaginary basis units span , which decomposes cleanly into the ** representation** (Günaydin & Gürsey 1973):

Decomposition of Octonion Basis under $Stab(e_4)$

3. Light Flavour Cartan Quantum Numbers

Flavour	Isospin $I_3$	Hypercharge $Y$	Charge $Q = I_3 + Y/2$	Physical / Extra Weight
u (up)	+1/2	+1/3	+2/3	50% (e₁,e₂) / 50% (e₆)
d (down)	-1/2	+1/3	-1/3	50% (e₁,e₂) / 50% (e₆)
s (strange)	0	-2/3	-1/3	50% (e₃) / 50% (e₇)

Note: Heavy quark generations do not reside inside this light representation (dimension 3 is filled by ). They join via the latitude generation ladder and multiplet packaging.

STATUS: [M] MODELLED PACKAGE

The light quark representation is an established geometric result [E] derived from the subgroup of . Naïve 7-to-6 basis vector bijections are mathematically closed [CLOSED F].

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