13.7.  The internal algebra $A_{F}$ derived from P1–P3

This section closes Conjecture 1, stated formally as §19.5: P1–P3, with the maximal-division-algebra (Dixon) hyperspinor envelope (a consequence of P1, Computation 168) of §1.5, imply that the internal algebra of PST’s foundational object is $A_{F}=\mathbb{C}\oplus\mathbb{H}\oplus M_{3}(\mathbb{C})$. The closure is derived from P1–P3 with no further appeal to the Chamseddine–Connes–Marcolli classification. The argument has five pieces. The rigorous decomposition is delivered in Computations 16 and 19 below.

13.7.1.  Piece 1: the chirality grading

Computation 3 showed that on $D$ distinction bits with $D$ odd, the substrate’s complementation-and-parity real structure gives $\gamma_{K}=Z^{\otimes D}$ with $\gamma_{K}^{2}=+I$, sign triple $(\varepsilon,\varepsilon^{\prime},\varepsilon^{\prime\prime})=(+,+,-)$ matching Connes’ Lorentzian real K-theory (KO) dimension six. Computation 19, section 6 of the script closes the identification of $\gamma_{K}$ with the Cl(0,6) chirality grading. In the Jordan–Wigner Majorana representation [177] of $D$ bits, the six fermion Majoranas $\chi_{a},\chi^{\prime}_{a}$ ($a=1,2,3$) generate $\mathrm{Cl}(0,6)$ on the $2^{D}=8$-dimensional Hilbert space, and the volume element

$$\omega_{\mathrm{JW}}\;=\;\chi_{1}\chi^{\prime}_{1}\chi_{2}\chi^{\prime}_{2}\chi_{3}\chi^{\prime}_{3}$$ (170)

satisfies $\omega_{\mathrm{JW}}^{2}=-I$. Computation 19, section 6 of the script verifies

$$\gamma_{K} \;=\;Z\otimes Z\otimes Z\;=\;i\,\omega_{\mathrm{JW}}$$ (171)

$$\quad(\bigl{\lVert}\gamma_{K}-i\omega_{\mathrm{JW}}\bigr{\rVert}=0.0\;\text{to machine precision}).$$

The substrate’s parity-selected real structure is the $\mathrm{Cl}(0,6)$ chirality grading at the operator level, with no convention-dependent step intervening. Hence the substrate naturally provides the chirality projector

$$f\;=\;\tfrac{1}{2}(I+\gamma_{K})\;=\;\tfrac{1}{2}(I+i\omega),\qquad\bar{f}\;=\;\tfrac{1}{2}(I-i\omega),$$ (172)

a rank-4 projector onto the chiral half $H_{+}$ of the modal Hilbert space $H_{F}=\mathbb{C}^{8}=H_{+}\oplus H_{-}$.

13.7.2.  Piece 2: $f$ as the canonical idempotent

$\mathrm{Spin}(6)$ acts on $\mathrm{Cl}(0,6)\otimes\mathbb{C}\cong M_{8}(\mathbb{C})$ via the spinor representation, and on $\mathrm{Cl}(0,6)$ by conjugation through its embedding in $\mathrm{Cl}(0,6)^{\mathrm{even}}$ as the bivector Lie algebra. Computation 18 verifies that under this action $\omega$ is the unique (up to scalar) Spin(6)-invariant element of positive grade. Concretely, in the $\mathrm{Spin}(6)$ decomposition of $\mathrm{Cl}(0,6)$ by grade

$$\mathrm{Cl}(0,6)\;\cong\;\mathbf{1}\oplus\mathbf{6}\oplus\mathbf{15}\oplus\mathbf{20}\oplus\mathbf{15}\oplus\mathbf{6}\oplus\mathbf{1},$$ (173)

the singlet count is exactly $(1,0,0,0,0,0,1)$ (Computation 18 §6): only the identity (grade 0) and the volume $\omega$ (grade 6) are Spin(6)-invariant. Computation 19, section 4 of the script then shows that on $H_{F}=\mathbb{C}^{8}=H_{+}\oplus H_{-}$ the Hermitian Spin(6)-invariant endomorphisms form a 2-real-dimensional space spanned by $\{I,\,\gamma=i\omega\}$. This is Schur’s lemma applied to the two irreducible Spin(6) factors $H_{+}\cong\mathbf{4}$ and $H_{-}\cong\overline{\mathbf{4}}$ of $\mathrm{SU}(4)\cong\mathrm{Spin}(6)$. Hence $f=(I+i\omega)/2$ and $\bar{f}$ are the unique non-trivial Spin(6)-invariant idempotents in $\mathrm{Cl}(0,6)\otimes\mathbb{C}$: the two chirality projectors, whose canonical rank-1 refinements are Furey’s primitive idempotents.

13.7.3.  Piece 3: $\mathrm{SU}(3)_{c}\times\mathrm{U}(1)$ from the chain algebra

Given the chirality projector $f$, Furey 2014 [121] works in the chain algebra $\mathbb{C}\otimes\overleftarrow{\mathbb{O}}\cong\mathrm{Cl}(0,6)\cong M_{8}(\mathbb{C})$ and identifies:

• generators of $\mathrm{SU}(3)$ within the chain algebra acting on $\mathbb{C}\otimes\mathbb{O}$, under which the eight-dim spinor space carries one generation’s worth of fermion content under $\mathrm{SU}(3)_{c}\times\mathrm{U}(1)$ (one $\mathrm{SU}(3)$ triplet, one anti-triplet, and two singlets, per equation (176); the six-triplet, six-singlet count is the decomposition of the full chain-algebra module $\mathcal{S}_{\mathrm{full}}$, three minimal left ideals carrying the three generations; the decomposition of the chain algebra itself is equation (167));

• a $\mathrm{U}(1)$ hypercharge generator $\mu$ giving the hypercharges $-1/2$, $+1/6$, $-1/6$, $+1/2$ on the four chiral subspaces $S_{\pm\pm}$.

What Furey 2014 does not derive in her paper is the $\mathrm{SU}(2)_{L}$ factor of $A_{F}$; her “Open Questions” section (p. 4) flags this explicitly and points to the Dixon algebra $\mathbb{C}\otimes\mathbb{H}\otimes\mathbb{O}$ as the appropriate setting for the full electroweak structure.

13.7.4.  Piece 4: $\mathrm{SU}(2)$ from the Lorentzian Dirac sector

The substrate’s Mosco limit (Computation 4) delivers a 4-d Lorentzian spacetime $M=\mathbb{R}\times S^{3}$ with signature $(+,-,-,-)$ (one timelike + three spacelike directions; Computation 2), whose local Lorentz Clifford algebra is, by Cartan classification, $\mathrm{Cl}(1,3)\cong M_{2}(\mathbb{H})$. Dirac spinors are accordingly $\mathbb{H}^{2}$-valued, and right-multiplication by $\mathbb{H}$ on the spinor module is an internal symmetry (it commutes with the external Cl(1,3) Lorentz action); this is the classical structural observation of Dixon 2010 [146] (“isospin SU(2) arises from the subset of unit-quaternion elements of the right-$\mathbb{H}$ action”). Computation 66 verifies the four structural facts explicitly: (a) the Cl(1,3) anticommutators hold on $\mathbb{H}^{2}\cong\mathbb{R}^{8}$; (b) right-$\mathbb{H}$ satisfies the quaternion algebra; (c) right-$\mathbb{H}$ commutes with Cl(1,3); (d) right-$\mathbb{H}$ is chirality-preserving (commutes with the Lorentz pseudoscalar). The exponentiated unit-quaternion group $\mathrm{Sp}(1)\cong\mathrm{SU}(2)$ is the internal symmetry of the emergent Lorentzian Dirac sector.

Vectorial vs chiral. The internal $\mathrm{SU}(2)$ delivered by Piece 4 is vectorial: by construction it commutes with the Lorentz pseudoscalar $\gamma^{5}$ and therefore acts identically on left- and right-handed chiral subspaces. The Standard-Model $\mathrm{SU}(2)_{L}$ acts chirally (on left-handed fermions only); identifying Piece 4’s vectorial $\mathrm{SU}(2)$ with $\mathrm{SU}(2)_{L}$ requires a parity-breaking selection that singles out one chirality. This selection is supplied by PST’s directed modal threshold (§14.4): the directed crossing $\varepsilon=|T(C)|-\tau>0$ orients the $S^{3}$ volume form and breaks $\mathrm{SO}(4)\cong\mathrm{SU}(2)_{L}\times\mathrm{SU}(2)_{R}$ down to the left factor, making the vectorial Piece 4 $\mathrm{SU}(2)$ chiral on the threshold-selected sector. The chiral identification therefore rests on Piece 4 (the algebraic existence of the internal SU(2)) plus the directed-threshold parity argument (the chirality selection); both are structural to PST.

13.7.5.  Piece 5: Dixon hyperspinor synthesis

Dixon 2010 [146] (page 9) shows that the full Dixon algebra $T=\mathbb{C}\otimes\mathbb{H}\otimes\mathbb{O}$ is the spinor space of $\mathrm{Cl}(0,9)\cong M_{16}(\mathbb{C})$ and synthesises the $\mathrm{Cl}(0,6)$ chain-algebra structure of Piece 3 with the right-$\mathbb{H}$ Dirac-spinor structure of Piece 4 into a single algebra whose internal $A_{F}$ is $\mathbb{C}\oplus\mathbb{H}\oplus M_{3}(\mathbb{C})$ (with the standard $\mathrm{SU}(3)_{c}\times\mathrm{SU}(2)_{L}\times\mathrm{U}(1)_{Y}$ identification on one generation, replicated three times via Furey’s six-triplet decomposition). The Dixon algebra is the tensor of the four real normed division algebras $(\mathbb{R},\mathbb{C},\mathbb{H},\mathbb{O})$ classified by Hurwitz. Using the full tensor $\mathbb{C}\otimes\mathbb{H}\otimes\mathbb{O}$ is a maximal-structure selection, the same instruction as the $V_{7}=\mathrm{Im}(\mathbb{O})$ choice flagged in §1.5 rather than a separate one: it is a consequence of P1 rather than a selection principle beside it, a proper sub-tensor requiring a datum P1 does not supply (Computation 168) (other division-algebra Standard-Model programmes use proper subalgebras, e.g. $\mathbb{C}\otimes\mathbb{O}$ à la Furey, or $\mathbb{H}\otimes\mathbb{O}$). Under this selection the synthesis lands on $A_{F}=\mathbb{C}\oplus\mathbb{H}\oplus M_{3}(\mathbb{C})$.

13.7.6.  Conjecture 1, closed (within the Dixon synthesis)

Combining Pieces 1–5:

$$\mbox{\hyperref@@ii[P:1]{P1}--\hyperref@@ii[P:3]{P3}}+\mbox{Mosco convergence}+\mbox{Dixon synthesis}$$ (174)

$$\qquad\Longrightarrow\;A_{F}\;=\;\mathbb{C}\oplus\mathbb{H}\oplus M_{3}(\mathbb{C})$$

The chain of derivations is:

1. P1–P3 specify the Boolean substrate, the Bernoulli measure, the asymmetric tension, and the LG modal potential.

2. Computation 3’s parity-selected real structure on the odd-$D$ sub-limit delivers $\gamma_{K}^{2}=+I$, sign $(+,+,-)$, KO-6.

3. The canonical anticommutation relations (CAR) algebra of the substrate at $D$ bits is $\mathrm{Cl}(0,2D)$; for $D=3$ this is the eight-dimensional $\mathrm{Spin}(6)$ spinor representation, on which $M_{8}(\mathbb{C})$ acts by left multiplication (Computation 16).

4. In the Jordan–Wigner representation, $\gamma_{K}=i\omega_{\mathrm{JW}}$ exactly (Computation 19, section 6 of the script, equation (171)). Hence $f=(I+\gamma_{K})/2$ is the substrate’s chirality projector (of rank 4 on $H_{F}=\mathbb{C}^{8}$); Furey’s primitive idempotent is its canonical rank-1 refinement, the vacuum idempotent inside the chiral half.

5. Computation 18 and Computation 19, section 4 of the script certify that $f$ is the canonical Spin(6)-invariant choice (uniqueness up to chiral swap).

6. Furey 2014 [121] delivers $\mathrm{SU}(3)_{c}\times\mathrm{U}(1)$ on one generation of fermion content via the $\mathrm{Cl}(0,6)$ chain-algebra structure (not the full $A_{F}$).

7. The emergent Lorentzian Dirac sector $\mathrm{Cl}(1,3)\cong M_{2}(\mathbb{H})$ delivers the internal $\mathrm{SU}(2)$ via right-$\mathbb{H}$ on the spinor module (Computation 66).

8. Dixon 2010 [146]’s hyperspinor framework synthesises the chain-algebra structure of (6) with the Dirac-spinor structure of (7) and identifies $A_{F}=\mathbb{C}\oplus\mathbb{H}\oplus M_{3}(\mathbb{C})$ as the internal algebra acting on the combined hyperspinor.

Step 4 (equation (171)) is the bridge from the substrate’s parity grading to Furey’s idempotent. Step 7 (Computation 66) is the bridge from emergent Lorentzian spacetime to the internal $\mathrm{SU}(2)$. The derivation invokes Dixon’s hyperspinor framework rather than the Chamseddine–Connes–Marcolli axiomatic classification of finite spectral triples; the two routes arrive at the same algebra by different structural paths.

The closure rests on three pieces of standard mathematical content, all well established:

• The CAR algebra of $D$ fermion bits is isomorphic to $\mathrm{Cl}(0,2D)$ via the Jordan–Wigner Majorana representation.

• Furey 2014 [121]’s $\mathrm{SU}(3)\times\mathrm{U}(1)$ action on the $\mathrm{Cl}(0,6)$ chain algebra $\mathbb{C}\otimes\mathbb{O}$, partitioning the chain algebra into six $\mathrm{SU}(3)$ triplets and six singlets identifiable as three generations of fermion content under $\mathrm{SU}(3)_{c}\times\mathrm{U}(1)$.

• Dixon 2010 [146]’s hyperspinor framework $T=\mathbb{C}\otimes\mathbb{H}\otimes\mathbb{O}$, in which the Cl(0,6) chain algebra and the emergent Lorentzian Dirac sector’s right-$\mathbb{H}$ Sp(1) (Piece 4) combine to give the full internal algebra $A_{F}=\mathbb{C}\oplus\mathbb{H}\oplus M_{3}(\mathbb{C})$.

None is a PST-specific assumption. With PST read as a derivation of physics from P1–P3 plus the Connes-spectral-triple framework, including the emergent Lorentzian Dirac sector that supplies the $\mathbb{H}$ factor, the algebra identity is unconditional given the Dixon / maximal-tensor selection of §1.5 (the selection principle flagged above). The two structural items that accompany the closed Conjecture 1 have different status. The Dirac-convergence step is closed at the relaxed-criterion level (§7.1, Chapter 19). The three-generation count $N_{\mathrm{gen}}=3$ is read structurally from Furey 2014’s six-$\mathrm{SU}(3)$-triplet decomposition of $\mathrm{Cl}(0,6)$, with the colour-versus-generation distinguishability caveat inherited rather than resolved: inside PST it is settled in the negative, $\hat{\tau}^{\perp}$ being the irreducible $\mathbf{3}$ of $\mathrm{SU}(3)_{c}$ (§13.5, §14.2, Computation 175).

The remainder of this subsection lays the closure out explicitly: the minimal left ideal of $\mathrm{Cl}(0,6)$, the commutant calculation, the emergent $\mathrm{Cl}(1,3)$ contribution, the Dixon synthesis, and the three-generation reading, step by step. The mathematical content is well-established in [121, 146, 49]; the substrate-side identifications are PST-specific.

13.7.7.  $\mathrm{Cl}(0,6)$ representation theory

The real Clifford algebra $\mathrm{Cl}(0,6)$ has dimension $2^{6}=64$ over $\mathbb{R}$. Complexified, $\mathrm{Cl}(0,6)\otimes_{\mathbb{R}}\mathbb{C}\cong M_{8}(\mathbb{C})$, the full $8\times 8$ matrix algebra over $\mathbb{C}$. In the standard Cayley basis $\{e_{1},\ldots,e_{6}\}$ of $\mathrm{Im}(\mathbb{O})$, the Clifford generators $\gamma_{i}=L_{e_{i}}$ (left octonionic multiplication by $e_{i}$) satisfy $\{\gamma_{i},\gamma_{j}\}=-2\delta_{ij}$ on the spinor module $\mathbb{C}\otimes_{\mathbb{R}}\mathbb{O}\cong\mathbb{C}^{8}$ ([121], Computations 19, 16). The chirality grading

$$\omega\;=\;\gamma_{1}\gamma_{2}\gamma_{3}\gamma_{4}\gamma_{5}\gamma_{6}$$ (175)

satisfies $\omega^{2}=-1$, so $i\omega$ has eigenvalues $\pm 1$ and defines the chirality projector $f=\tfrac{1}{2}(I+i\omega)$, of rank 4 on the spinor module $\mathbb{C}^{8}=H_{+}\oplus H_{-}$. The substrate’s parity-selected real structure on an odd number of distinction bits delivers the KO-$6$ Connes sign triple $(+,+,-)$ exactly (Computation 3), and in the Jordan–Wigner representation $\gamma_{K}=i\omega$ to machine precision (Computation 19): $f$ is the substrate’s own parity-grading projector seen through $\mathrm{Cl}(0,6)\otimes\mathbb{C}$.

13.7.8.  The minimal left ideal

Within the chiral half selected by $f$, the canonical primitive idempotent $p$ (Furey’s vacuum idempotent, the Jordan–Wigner ladder product; rank 1, with $p=fp=pf$, unique up to the inner action certified in Piece 2) generates a minimal left ideal of $\mathrm{Cl}(0,6)\otimes\mathbb{C}\cong M_{8}(\mathbb{C})$: $\mathcal{S}=\bigl{(}\mathrm{Cl}(0,6)\otimes\mathbb{C}\bigr{)}p$. As a $\mathbb{C}$-vector space, $\mathcal{S}\cong\mathbb{C}^{8}$; viewed inside $M_{8}(\mathbb{C})$ via the rank-$1$ projector $p$, $\mathcal{S}$ is a single column of $M_{8}(\mathbb{C})$, its generator lying in the chiral half selected by $f$.

Decomposing $\mathcal{S}$ under $\mathrm{Stab}_{G_{2}}(e_{7})=\mathrm{SU}(3)\subset G_{2}=\mathrm{Aut}(\mathbb{O})$ gives the matter-content split

$$\mathcal{S}\;=\;\mathbb{C}\;\oplus\;\mathbb{C}^{3}\;\oplus\;\overline{\mathbb{C}^{3}}\;\oplus\;\mathbb{C}.$$ (176)

The two singlets are the neutrino and electron; the triplet and anti-triplet are the up- and down-quark colour triplets of one generation, with electric-charge content matching the Standard Model hypercharges (Furey 2014 §2; Computation 16).

13.7.9.  Furey’s chain-algebra action and the commutant

Furey’s construction acts the chain algebra $\mathcal{C}$, the algebra generated by left multiplication on $\mathbb{C}\otimes\mathbb{O}$, which is $\mathrm{Cl}(0,6)\otimes\mathbb{C}$, on the minimal left ideal $\mathcal{S}$ via the chain product: $x\in\mathcal{C}$ acts as $\psi\mapsto x\cdot\psi$ in $M_{8}(\mathbb{C})$. The chain product is associative on $\mathcal{C}$ (it is just matrix multiplication on $M_{8}(\mathbb{C})$), even though octonionic multiplication on $\mathbb{O}$ alone is not.

The commutant of this action is small, and it is worth being exact about it. $\mathcal{S}$ is a minimal left ideal of $\mathcal{C}\cong M_{8}(\mathbb{C})$, hence its defining irreducible representation, so by Schur

$$\mathrm{Comm}(\mathcal{C},\mathcal{S})\;=\;\mathrm{End}_{\mathcal{C}}(\mathcal{C}p)\;\cong\;p\,\mathcal{C}\,p\;=\;\mathbb{C},$$ (177)

one complex dimension, acting as the overall $\mathrm{U}(1)$ phase on $\mathcal{S}$. Nothing non-trivial commutes with the whole of $M_{8}(\mathbb{C})$. An earlier version of this passage gave the right-hand side as $\mathbb{C}\otimes M_{3}(\mathbb{C})$, nine dimensions; that is withdrawn.

$M_{3}(\mathbb{C})$ arises instead as an enveloping algebra rather than a commutant, and the distinction costs nothing because the object is the same one the chapter already uses. With $\omega=L_{e_{1}}\cdots L_{e_{6}}=L_{e_{7}}$ and the chirality $\gamma=i\omega$ of (175), the octonionic $\mathrm{SU}(3)=\mathrm{Stab}_{G_{2}}(e_{7})$, the centraliser of $J=L_{e_{7}}$ within $G_{2}$, acts on the chiral half $\mathcal{S}_{+}\cong\mathbb{C}^{4}$ of (176) as $\mathbf{1}\oplus\mathbf{3}$. The unital algebra it generates there is

$$\langle\,\mathbb{1},\ \Lambda_{a}\,\rangle\;\cong\;\mathbb{C}\oplus M_{3}(\mathbb{C}),$$ (178)

ten complex dimensions, with two-dimensional commutant as Schur requires (Computations 16, 17). That is the $\mathrm{U}(1)\oplus M_{3}(\mathbb{C})$ part of $A_{F}$, with $\mathbb{C}$ the overall phase and $M_{3}(\mathbb{C})$ acting on the colour triplets $\mathbb{C}^{3}\oplus\overline{\mathbb{C}^{3}}$. Restricting to unimodular unitaries gives $\mathrm{SU}(3)_{c}\times\mathrm{U}(1)$ on one generation, with hypercharge assignments forced by the decomposition of (176) (Furey 2014 §3–4). This is the substrate-side delivery of $\mathrm{SU}(3)_{c}\times\mathrm{U}(1)$; $\mathrm{SU}(2)_{L}$ is not yet present and requires the emergent-spacetime contribution below.

13.7.10.  $\mathrm{Cl}(1,3)\cong M_{2}(\mathbb{H})$ and the internal $\mathrm{SU}(2)$

The Mosco limit of the substrate’s Dirichlet form yields emergent spacetime $M=\mathbb{R}\times S^{3}$ of KO-dimension four (Computation 4; C1 of §7.1, a theorem under A1–A7 rather than under P1–P3 alone). The local Clifford algebra of $M$ is

$$\mathrm{Cl}(1,3)\;\cong\;M_{2}(\mathbb{H}),$$ (179)

by the Cartan classification: signature $(1,3)$ gives $M_{2}(\mathbb{H})$ (not $M_{4}(\mathbb{R})$ as for signature $(3,1)$). The spinor module is $\mathbb{H}^{2}\cong\mathbb{R}^{8}$ on which $\mathrm{Cl}(1,3)$ acts by left $\mathbb{H}$-multiplication and matrix action.

Right multiplication by $\mathbb{H}$ commutes with $\mathrm{Cl}(1,3)$ acting from the left (associativity of $\mathbb{H}$ on itself, distinct from non-associativity of $\mathbb{O}$). The right-$\mathbb{H}$ action defines an internal symmetry of the spinor module not part of the Lorentz group:

$$R_{q}\,:\,\psi\mapsto\psi\cdot q,\qquad q\in\mathbb{H},\quad[R_{q},\mathrm{Cl}(1,3)]=0.$$ (180)

Restricting to unit quaternions $|q|=1$ gives $\mathrm{Sp}(1)\cong\mathrm{SU}(2)$. Computation 66 verifies the Clifford anticommutators, centraliser, $\mathfrak{su}(2)$ generators $\{R_{iH}/2,R_{jH}/2,R_{kH}/2\}$, and chirality preservation. This is the substrate-derived $\mathrm{SU}(2)$.

Identifying this spacetime-side internal $\mathrm{SU}(2)$ with the SM electroweak $\mathrm{SU}(2)_{L}$ factor of $A_{F}$ requires combining with the substrate-side $\mathrm{Cl}(0,6)$ chain algebra above through the Dixon hyperspinor synthesis. Chirality of the resulting $\mathrm{SU}(2)_{L}$ is sourced from the directedness of the modal threshold crossing, $\varepsilon=|T(C)|-\tau>0$ (§13.4), which selects the left chiral isometry factor of $S^{3}\cong\mathrm{SU}(2)$ over the right one.

13.7.11.  Dixon synthesis: $T=\mathbb{C}\otimes\mathbb{H}\otimes\mathbb{O}$

Dixon’s hyperspinor framework [146] constructs a single $64$-complex-dimensional module

$$T\;=\;\mathbb{C}\otimes\mathbb{H}\otimes\mathbb{O}$$ (181)

on which all three Hurwitz-classified normed division algebras act in their respective representations:

• $\mathbb{C}$ acts as scalar multiplication (overall phase, the $\mathrm{U}(1)$ of the Furey commutant).

• $\mathbb{H}$ acts by right-multiplication, generating the internal $\mathrm{Sp}(1)\cong\mathrm{SU}(2)$ from the emergent Dirac sector.

• $\mathbb{O}$ acts by left-multiplication through the Furey chain algebra, generating $\mathrm{SU}(3)_{c}\times\mathrm{U}(1)$.

The associativity needed for each action to define a representation follows from each algebra acting on its own tensor factor: the non-associativity of $\mathbb{O}$ is sequestered to the $\mathbb{O}$ factor, while the $\mathbb{H}$ and $\mathbb{C}$ actions commute with it. Adopting the full Dixon envelope $\mathbb{C}\otimes\mathbb{H}\otimes\mathbb{O}$ rather than $\mathbb{O}$ alone is the maximal-tensor selection principle of §1.5.

13.7.12.  The algebra identity $A_{F}=\mathbb{C}\oplus\mathbb{H}\oplus M_{3}(\mathbb{C})$

The full internal algebra of the Connes Standard-Model spectral triple is the unimodular-unitary group of the algebra $A_{F}$ acting on the Dixon hyperspinor $T$ via the actions above. Tracking each contribution:

• The $\mathbb{C}$-action contributes the $\mathbb{C}$ summand of $A_{F}$ (overall $\mathrm{U}(1)$).

• The right-$\mathbb{H}$ action contributes the $\mathbb{H}$ summand of $A_{F}$ (internal $\mathrm{SU}(2)$ on the Dirac sector).

• The left-$\mathbb{O}$ chain-algebra action contributes the $M_{3}(\mathbb{C})$ summand of $A_{F}$ (the enveloping algebra gives $M_{3}(\mathbb{C})$ by (178)).

Together,

$$A_{F}\;=\;\mathbb{C}\;\oplus\;\mathbb{H}\;\oplus\;M_{3}(\mathbb{C}).$$ (182)

The unimodular unitary group of $A_{F}$ is $\mathrm{U}(1)\times\mathrm{SU}(2)\times\mathrm{SU}(3)$, the Standard-Model gauge group. Up to the $\mathrm{SU}(2)$-doublet hypercharge assignment (forced by photon masslessness, §13.4) and the chirality of $\mathrm{SU}(2)_{L}$ (sourced by the directedness of the threshold crossing, $\varepsilon>0$, §13.4), this is the full SM gauge content on one generation.

13.7.13.  Three generations from six $\mathrm{SU}(3)$ triplets

Furey 2014 §5 observes that the full chain-algebra module decomposes under $\mathrm{SU}(3)$ into six colour triplets, one generation being one triplet and one anti-triplet, and reads the six as three generations. In PST the count is not read from the module but derived from the directional content of $V_{7}$: the three quaternionic sub-algebras of $\mathbb{O}$ containing $\hat{\tau}$ index three disjoint per-generation sectors, each carrying the sixteen chiral states of one Standard-Model generation (Computations 46–48), and colour and generation are one three rather than two, $\hat{\tau}^{\perp}$ being the irreducible $\mathbf{3}$ of $\mathrm{SU}(3)_{c}$ (Computation 175). The argument, its inputs and its scope (what is structural, what the Yukawa and Cabibbo–Kobayashi–Maskawa (CKM) structure leaves to $T(C)$, and why six of the eight gluons are generation-changing, Computation 106) are given once in §14.2 and §14.3.

13.7.14.  The gauge group as substrate automorphisms

Combining the colour factor with the $\mathrm{U}(1)_{Y}$ and $\mathrm{SU}(2)_{L}$ factors derived from the vacuum topology (§13.4), the unimodular unitary group of $A_{F}$ is the Standard Model gauge group:

$$G_{\mathrm{SM}}\;=\;\mathrm{SU}(3)_{c}\times\mathrm{SU}(2)_{L}\times\mathrm{U}(1)_{Y},$$ (183)

as established by Glashow, Weinberg, and Salam [79, 77, 78].

The four fundamental forces are all contained within the emergence chain:

• Gravity, the gradient term of $\mathcal{F}$ projected to the Einstein–Hilbert action; $G_{\mu\nu}$ uniquely fixed by diffeomorphism invariance and Lovelock’s theorem [22] (§8.6).

• Electromagnetism, the $\mathrm{U}(1)_{\mathrm{em}}$ gauge theory of the unbroken combination $Q=T_{3}+Y/2$ left by the electroweak breaking on $\mathcal{V}_{\mathrm{EW}}\cong S^{3}$ (§13.4); the photon is its connection on the $\mathrm{U}(1)$ bundle over $M$. Gauging the Goldstone phase of $\mathcal{V}\cong S^{1}$ by itself gives a massive vector, the $Z$ (§13.3).

• The weak force, the $\mathrm{SU}(2)_{L}$ gauge theory from the $S^{3}\cong\mathrm{SU}(2)$ vacuum manifold; left-chiral because the threshold crossing is directional; $W^{\pm}$ and $Z$ masses from the Higgs mechanism on $\mathcal{V}_{\mathrm{EW}}$.

• The strong force, the $\mathrm{SU}(3)_{c}$ gauge theory from the $M_{3}(\mathbb{C})$ colour factor of the internal algebra $A_{F}$; eight massless gluons; confinement from the non-Abelian running coupling.