13.5.  Colour $\mathrm{SU}(3)_{c}$: octonionic origin and Casimir decomposition

Colour $\mathrm{SU}(3)_{c}$ is the symmetry of the $M_{3}(\mathbb{C})$ summand of the internal algebra $A_{F}=\mathbb{C}\oplus\mathbb{H}\oplus M_{3}(\mathbb{C})$. The underlying Furey-2014 foundation, the action of the octonion chain algebra $\overleftarrow{\mathbb{O}}$ on one-generation matter content delivering the $\mathrm{SU}(3)$ structure, is verified in Computation 16. The Standard Model gauge group is the unimodular unitary group of $A_{F}$ (Chapter 19), with $\mathrm{SU}(3)_{c}$ the unimodular unitaries of the $M_{3}(\mathbb{C})$ factor, $\mathrm{SU}(2)_{L}$ the unit quaternions of $\mathbb{H}$, and $\mathrm{U}(1)_{Y}$ the surviving abelian factor. Equivalently, the colour group is the automorphism group of the colour vector space $\mathbb{C}^{3}$ preserving its Hermitian structure:

$$\mathrm{Aut}\!\bigl{(}\mathbb{C}^{3},\,\langle\cdot,\cdot\rangle\bigr{)}\;=\;\mathrm{U}(3)\;=\;\mathrm{U}(1)\times\mathrm{SU}(3)_{c},$$ (161)

with the overall $\mathrm{U}(1)$ removed by unimodularity. The eight gluons arise as the inner fluctuations of the Dirac operator on the $M_{3}(\mathbb{C})$ factor (Computation 6). This section presents the colour $\mathrm{SU}(3)_{c}$ argument at its three depths: the algebraic statement above, the $G_{2}$-stabiliser-of-$V_{7}$ source of the $M_{3}(\mathbb{C})$ factor, and the rigorous Casimir decomposition that verifies the irrep content exactly.

13.5.1.  Octonionic origin: the $G_{2}$ stabiliser of a $V_{7}$ direction

The colour factor descends from the octonions. The exceptional group $G_{2}=\mathrm{Aut}(\mathbb{O})$, the automorphism group of the octonions $\mathbb{O}$, acts irreducibly on the direction algebra $V_{7}\cong\mathbb{R}^{7}$, and the stabiliser of a fixed unit vector in $V_{7}$ is $\mathrm{SU}(3)$ (Freudenthal–Tits). Under $\mathrm{SU}(3)\subset G_{2}$ the seven directions of $V_{7}$ split as

$$\mathbf{7}\;\longrightarrow\;\mathbf{1}_{\mathbb{R}}\oplus\mathbf{3}_{\mathbb{C}}\;\cong\;\mathbb{R}\oplus\mathbb{C}^{3},$$ (162)

where $\mathbf{3}_{\mathbb{C}}$ is the colour triplet on which $\mathrm{SU}(3)_{c}$ acts. Equivalently, the complexified octonions $\mathbb{C}\otimes\mathbb{O}$ generate $\mathrm{Cl}(6)\cong M_{8}(\mathbb{C})$, whose minimal left ideal carries one generation’s colour and electric-charge content (§14.1), and whose three fermionic ladder operators $\alpha_{k}=\tfrac{1}{2}(L_{2k-1}+iL_{2k})$ ($k=1,2,3$) build one generation’s colour triplet (the Gunaydin–Gursey/Furey construction; Computation 16). The colour multiplicity is the complex dimension of this triplet,

$$N_{c}\;=\;\dim_{\mathbb{C}}(\mathbf{3}_{\mathbb{C}})\;=\;3,$$ (163)

the rank of the $M_{3}(\mathbb{C})$ colour factor.

Consequences of $N_{c}=3$. (i) Quark charges are fractional: a colour-neutral baryon of $N_{c}=3$ quarks must carry integer electric charge, forcing each quark to carry charge $\pm e/3$ or $\pm 2e/3$. (ii) Colour confinement at low energies follows from the non-Abelian structure of any $\mathrm{SU}(N_{c})$ gauge theory with $N_{c}\geq 2$, inherited by PST from the standard quantum chromodynamics (QCD) argument [81, 82, 80].

Colour 3 versus generation 3. The three complex directions of $\mathbf{3}_{\mathbb{C}}$ are the three colours of a single generation, not three generations: they are the three colour charges on which $\mathrm{SU}(3)_{c}$ acts. The threefold generation replication is a structurally distinct count: in PST it arises from the $V_{7}$-valued directional structure of P1–P3 and the $\hat{\tau}$-selection at modal sublimation (§14.2, Computations 46–48). The three quaternionic sub-algebras of $\mathbb{O}$ containing the complex sub-algebra $C_{\hat{\tau}}=\mathrm{span}\{1,\hat{\tau}\}$ are identified with three generations of matter, with $\hat{\tau}$ acting within each generation as the chiral preferred direction. PST therefore derives one three, not two: $\hat{\tau}^{\perp}$ is three-dimensional over $\mathbb{C}$, and that single fact is the source of both the colour rank $N_{c}=3$ and the generation count $N_{\mathrm{gen}}=3$. The assignment of the three octonionic sectors to generations versus colour states is the colour-versus-generation distinguishability caveat of the octonionic-Standard-Model programme (Furey 2014 [121], Open Questions section, p. 4): on $\mathbb{O}$ alone the three sectors mix under colour $\mathrm{SU}(3)$ because $\mathrm{Stab}_{G_{2}}(\hat{\tau})\cong\mathrm{SU}(3)$ acts on $\hat{\tau}^{\perp}\cong\mathbb{C}^{3}$ as the fundamental.

This book previously claimed to resolve that on the Dixon side, and the claim is withdrawn. The argument was that the ambient $\mathbb{C}\otimes\mathbb{H}\otimes\mathbb{O}$ makes the three generation sectors orthogonal (Computation 48). They are orthogonal, exactly; that is not the issue. Orthogonality of subspaces is not invariance under a group, and Computation 175 measures both facts side by side: the sectors are mutually orthogonal to machine zero and colour leaks out of them at $\mathrm{O}(1)$. Worse for the enlargement, colour acts on the ambient only through the $\mathbb{O}$ factor, as $I\otimes g$, so $\lVert(1-I\otimes P)(I\otimes g)(I\otimes P)\rVert=\lVert(1-P)\,g\,P\rVert$: the leakage on the $64$-real-dimensional ambient equals the leakage on $\mathbb{O}$ alone, to $0$. The ambient supplies room and no invariance.

And within $\hat{\tau}^{\perp}$ no such splitting can exist at all. The commutant of $\mathrm{Stab}_{G_{2}}(\hat{\tau})$ there is exactly $\mathbb{C}$, so by Schur the representation is the irreducible fundamental $\mathbf{3}$ and has no proper invariant subspace (Computation 175). The obstruction is therefore a theorem rather than the dimension count the register used to state it: a $\mathbf{3}_{\text{colour}}\otimes\mathbf{3}_{\text{gen}}$ would need nine complex dimensions against three, and irreducibility additionally forbids the direct-sum splitting the Fano lines actually realise. PST inherits this question rather than closing it; the full account is given in §14.2.

13.5.2.  Rigorous Casimir decomposition

The qualitative octonionic-origin story above admits a rigorous, fully checked algebraic version. The Lie group $G_{2}=\mathrm{Aut}(\mathbb{O})$ acts on the direction algebra $V_{7}\cong\mathbb{R}^{7}$; fixing a unit direction $e_{7}\in V_{7}$ singles out the stabiliser $\mathrm{Stab}_{G_{2}}(e_{7})=\mathrm{SU}(3)$, the octonionic colour group. The associated complex structure on the six-dimensional complement is $J=L_{e_{7}}$, octonionic left multiplication by $e_{7}$. In the Cayley convention of Computation 16 (Chapter 14 uses the Fano-plane convention of Computation 46), $J$ pairs the basis directions as $(e_{1},e_{3})$, $(e_{2},e_{6})$, $(e_{4},e_{5})$ (Computation 16, section 16 of the script), and the $J=+i$ eigenspace is the SU(3) triplet

$$T\;=\;\mathrm{span}_{\mathbb{C}}\{u_{1},u_{2},u_{3}\},$$ (164)

$$u_{1}=\tfrac{1}{\sqrt{2}}(e_{1}-ie_{3}),\;u_{2}=\tfrac{1}{\sqrt{2}}(e_{2}-ie_{6}),\;u_{3}=\tfrac{1}{\sqrt{2}}(e_{4}-ie_{5}),$$ (165)

with its $J=-i$ conjugate $\bar{T}=\mathrm{span}_{\mathbb{C}}\{\bar{u}_{1},\bar{u}_{2},\bar{u}_{3}\}$ giving the anti-triplet. The complex octonion space therefore decomposes under SU(3) as

$$\mathbb{C}^{8}\;\cong\;\langle e_{0}\rangle\,\oplus\,\langle e_{7}\rangle\,\oplus\,T\,\oplus\,\bar{T}\;=\;\mathbf{1}\,\oplus\,\mathbf{1}\,\oplus\,\mathbf{3}\,\oplus\,\overline{\mathbf{3}}.$$ (166)

The structural claim about §13.5’s qualitative identification is then a sharp Clebsch–Gordan computation [171], fully verified in Computation 17. Lifting each Gell-Mann [167] $\lambda_{a}$ to an $8\times 8$ matrix $\Lambda_{a}$ that acts as $\lambda_{a}$ on $T$, as $-\lambda_{a}^{T}$ on $\bar{T}$ (the conjugate representation), and as zero on the two singlets, the SU(3) commutation relations $[\Lambda_{a},\Lambda_{b}]=2if_{abc}\Lambda_{c}$ hold to machine precision. The adjoint action of $\mathrm{SU}(3)$ on $\overleftarrow{\mathbb{O}}\cong M_{8}(\mathbb{C})$ is then $\mathbf{8}\otimes\overline{\mathbf{8}}$, and the $(\mathbf{1}+\mathbf{1}+\mathbf{3}+\overline{\mathbf{3}})\otimes(\mathbf{1}+\mathbf{1}+\overline{\mathbf{3}}+\mathbf{3})$ Clebsch–Gordan expansion gives

$$M_{8}(\mathbb{C})\;=\;6\cdot\mathbf{1}\,\oplus\,5\cdot\mathbf{3}\,\oplus\,5\cdot\overline{\mathbf{3}}\,\oplus\,2\cdot\mathbf{8}\,\oplus\,\mathbf{6}\,\oplus\,\overline{\mathbf{6}}.$$ (167)

Dimension check: $6\cdot 1+5\cdot 3+5\cdot 3+2\cdot 8+6+6=64$.

To verify equation (167) numerically, Computation 17 constructs the SU(3) quadratic Casimir $C_{2}=\sum_{a}(\mathrm{ad}\,\Lambda_{a})^{2}/4$ as a $64\times 64$ Hermitian matrix and diagonalises it. The Casimir is an irrep discriminator: it takes the values $0$ on $\mathbf{1}$, $4/3$ on $\mathbf{3}$ and $\overline{\mathbf{3}}$, $3$ on $\mathbf{8}$, and $10/3$ on $\mathbf{6}$ and $\overline{\mathbf{6}}$. The numerical Casimir spectrum is

$C_{2}$ value	irrep dimension	expected mult.	found mult.
$0$	$1$	$6$	$6$
$4/3$	$3$	$30$	$30$
$3$	$8$	$16$	$16$
$10/3$	$6$	$12$	$12$

to machine precision, total $64=64$. The decomposition of equation (167) is therefore not a conjecture but an exact Clebsch–Gordan identity verified at the irrep-multiplicity level.

13.5.3.  What the decomposition does and does not settle

The verified decomposition of equation (167) settles the gauge-group structural derivation: the colour SU(3) is the unique octonionic Lie subgroup of $G_{2}$, it acts on PST’s substrate Clifford algebra $\mathbb{C}\otimes\mathrm{Cl}(0,6)\cong M_{8}(\mathbb{C})$ through the natural representation (166), and the irrep content of the substrate algebra matches the Clebsch–Gordan expectation exactly. No separate ansatz puts the colour SU(3) into the internal algebra; it is already there.

This is the structural closure of the gauge-group portion of the algebra-identity question. It does not by itself fix $N_{\mathrm{gen}}=3$ generations: equation (167) gives the colour decomposition of $M_{8}(\mathbb{C})$ but is silent on the generation count. The generation count is fixed separately, by the $V_{7}$-directional structure of P1–P3 and the $\hat{\tau}$-selection at modal sublimation. Three quaternionic sub-algebras of $\mathbb{O}$ contain $\hat{\tau}$, and they deliver three disjoint generation sectors of $\mathbb{C}\otimes\mathbb{H}\otimes\mathbb{O}$, each carrying one generation’s Standard-Model matter content (§14.2, Computations 46–48).

13.5.4.  The strong sector: what is derived and what is inherited

The preceding derivation is narrower than the phrase “colour from the substrate” suggests, and the boundary is worth drawing explicitly, because the strong sector is where a reader is most likely to expect a claim the theory does not make.

What is derived. The gauge group and its rank: $\mathrm{SU}(3)_{c}$ as the stabiliser $\mathrm{Stab}_{G_{2}}(\hat{\tau})$ of the threshold direction in $V_{7}$, acting on $\hat{\tau}^{\perp}\cong\mathbb{C}^{3}$ as the fundamental, whence $N_{c}=3$ by equation (163). That is a structural result and it is where the substrate does work.

What is inherited. Everything dynamical. Asymptotic freedom, confinement, the running of $\alpha_{s}$, the hadron spectrum and the whole apparatus of non-perturbative calculation are standard quantum chromodynamics, taken over unmodified. The derivation above says as much of confinement, which follows from the non-Abelian structure of any $\mathrm{SU}(N_{c})$ theory with $N_{c}\geq 2$ and not from P1–P3. PST supplies the group on which QCD is built; it does not re-derive QCD, and nothing in this work should be read as claiming a substrate account of confinement. Such an account would be a separate and considerably harder result.

What the comparison is, so the boundary is not read as a deficit. No framework derives the group. The Standard Model posits $\mathrm{SU}(3)_{c}$ and fixes $N_{c}=3$ by measurement, and grand unified theories posit a larger group that contains it. A structural derivation of the group and its rank from the threshold direction is therefore more than the incumbent offers, not less, and the dynamical half is inherited by every framework alike.

What follows for lattice results. Lattice QCD is the principal non-perturbative method in this sector, and it is worth stating plainly what its results do and do not bear on here. They are computations within the theory PST reproduces, not tests of the derivation above. A lattice determination that revised a hadronic quantity, or that resolved a discrepancy between lattice and data-driven methods, would revise QCD phenomenology and leave the group-theoretic result untouched: the stabiliser calculation does not depend on any hadronic number. The corollary is equally plain, and cuts the other way: agreement between lattice QCD and experiment is not evidence for PST either.

The one channel by which a strong-sector number reaches a PST prediction, and its size. The strong coupling enters this work at a single load-bearing place, the three-loop error budget for the Higgs self-coupling test (Computation 202), where the parametric uncertainty on $\lambda_{\mathrm{SM}}(M_{*})$ decomposes as $m_{t}\;{\pm}0.00061$ (its pole-mass ambiguity included), $m_{h}\;{\pm}0.00025$ and $\alpha_{s}\;{\pm}0.00018$. The strong coupling is the smallest of the three, by a factor of three against the top mass. A percent-level revision to a hadronic input therefore does not move the comparison that §19.7 reports. The residual there is dominated by the top and Higgs masses, and it is those, not $\alpha_{s}$, whose tightening will sharpen or break the test.

What would be a problem. A demonstration that the strong interaction is not described by an $\mathrm{SU}(3)$ gauge theory, or that the colour multiplicity is not three, would remove the structure this section derives and with it the octonionic reading of the internal algebra. Neither is an open question, which is why the derivation is a consistency result rather than a prediction: PST recovers a gauge group already established, and the weight it can carry is correspondingly limited.