We would like to argue that the above induction can therefore be performed as-is, meaning that the system has stabilized by the time each node has generated its first pulse.
However, there is a subtlety: it could happen that a spurious message that is still in transit at time $0$ overwrites the state of node $(1,0)$ after it received the first message from $(0,0)$.
Node $(1,0)$ then behaves as if the first message of $(0,0)$ arrived later, at the exact same time as the spurious message.
Because also such a spurious message is delivered within at most $d$ time, we can re-interpret this as a longer delay of still at most $d$ of the first message sent by node $(0,0)$.
Note that this modification reduces the difference between the reception times of the first and second pulse from node $(0,0)$ at node $(1,0)$ by up to $u$, but the separation remains at least $\Lambda-u\ge \Lambda-d$, i.e., the second message is not received before $(1,0)$ generates its first pulse.
We can apply the same scheme to nodes $2,\ldots,D$, resulting in the desired bound on the stabilization time.

$\localskew_0\le \kappa/2$ with probability $1-o(1)$.
It is self-stabilizing with stabilization time $\Lambda D$.

We remark that for a general base graph $G$, ensuring a small local skew is non-trivial.
However, so long as $|V|$ is small enough such that faults on layer $0$ occur with probability $o(1)$, one is free to fall back on a non-fault-tolerant GCS algorithm.
This achieves $\localskew_0\in O(\kappa \log D)$, which does not increase the asymptotic local skew bound of the pulse forwarding scheme.

§ FULL PULSE FORWARDING ALGORITHM

Discrete GCS at node $(v,\ell)$, $\ell>0$. The parameters $\Lambda$, and $\kappa$ will be determined later, based on the analysis.

$H_{\min},H_{\own},H_{\max} := \infty$
$\{v,w\}\in E$
received pulse from $v_{\ell-1}$ and $H_{\own}=\infty$
$H_{\own} := H_{v,\ell}(t)$
for some $\{v,w\}\in E$ received pulse from $(w,\ell-1)$ and $r_w=0$
$r_{w'}=0$ for all $\{v,w'\}\in E$
$r_{w'}=1$ for all $\{v,w'\}\in E$
$H_{\min}<\infty$ and $H_{v,\ell}(t)\ge \min\{H_{\max}+\kappa/2+\vartheta\kappa,2H_{\own}-H_{\min}+2\kappa)\}$
$H_{v,\ell}(t)= H_{\max}+\kappa/2+\vartheta \kappa$
wait until $H_{v,\ell}(t)=H_{\max}+3\kappa/2+\Lambda-d$
$\Cor_{v,\ell}:=\min_{s\in \N}\{\max\{H_{\own}-H_{\max}+4s\kappa,H_{\own}-H_{\min}-4 s\kappa \}\}-\kappa/2$
$\Cor_{v,\ell} < 0$
$\Cor_{v,\ell} := \min\set{H_{\own}-H_{\min} + 3\kappa/2, 0}$
$\Cor_{v,\ell} > \vartheta\kappa$
$\Cor_{v,\ell} := \max\set{H_{\own}-H_{\max} - 3\kappa/2,\vartheta\kappa}$
wait until $H_{v,\ell}(t) = H_{\own} + \Lambda - d - \Cor_{v,\ell}$
broadcast pulse

A basic requirement for the algorithm to work correctly is that $(v,\ell)$ receives the $k$-th pulses of all correct predecessors within its $k$-th iteration of the main loop of <Ref>.

For all $k\in \N$ and $(v,\ell)\in V_{\ell}$, $\ell>0$, node $(v,\ell)$ receives the $k$-th pulses of all correct predecessors within its $k$-th iteration of the main loop of <Ref>.

We show by induction on $\ell\in \N_{>0}$ and $k\in \N_{>0}$ that $(v,\ell)$ broadcasts the $k^{th}$ pulse after receiving the $k$-th pulse from all correct $(w,\ell-1)$ satisfying that $((w,\ell-1),(v,\ell))\in E$, but before receiving the $(k+1)$-th pulse from such a node.
Moreover, for all $k\ge 2$, $t_{v,\ell}^k-t_{v,\ell}^{k-1}=\Lambda$.

For the induction on $\ell$, we use $\ell=0$ as base case, requiring only that nodes generate pulses at frequency $1/\Lambda$.
As delays and clock speeds do not change, this holds true.
For the step from $\ell-1\in \N$ to $\ell$, we perform the induction on $k$.
Suppose that the claim holds for all $k'<k\in \N_{>0}$ and consider the $k$-th iteration of the outer loop at $(v,\ell)$.

* The inner loop terminated because $H_{v,\ell}(t)=H_{\max} +\kappa/2+\vartheta \kappa$. Then a message from each node $(w,\ell-1)$, $\{v,w\}\in E$, has been received in the current loop iteration. By the induction hypotheses for layer $\ell-1$ and pulse $k-1$, respectively, for correct such nodes this is the $k$-th pulse message.

We need to show that the $k$-th message from $(v,\ell-1)$ is received in time; the induction hypothesis guarantees that it is not received too early. As the minimum degree of $G$ is $2$, at least one node $(w,\ell-1)$, $\{v,w\}\in E$, is correct. If $(v,\ell-1)$ is correct, too, it sent its pulse message at the latest at time $t_{w,\ell-1}+\localskew_{\ell-1}$. By the bounds on message delay and clock speed, this message is received at a local time

\begin{equation*}
H\le H_{\max}+\vartheta(\localskew_{\ell-1}+u)\le H_{\max}+\Lambda-d<H_{v,\ell}(t_{v,\ell}^k).
\end{equation*}

* The inner loop terminated because $H_{v,\ell}(t)=2H_{\own}-H_{\min}+2\kappa$. As $H_{\min}<\infty$, also $H_{\own}< \infty$. Using that $H_{\min}\le H_{\max}$, we get that

\begin{align*}
\Delta&:=\min_{s\in \N}\{\max\{H_{\own}-H_{\max}+4s\kappa,H_{\own}-H_{\min}-4 s\kappa \}\}- \frac{\kappa}{2}\\
&\le \max\{H_{\own}-H_{\max},H_{\own}-H_{\min}\}- \frac{\kappa}{2}\\
&\le H_{\own}-H_{\min}-\frac{\kappa}{2}
\end{align*}

and hence $\Cor_{v,\ell}\le H_{\own}-H_{\min}+3\kappa/2\le 3\kappa/2$. It follows that

\begin{equation*}
H_{v,\ell}(t_{v,\ell}^k)\ge \max\{H_{\min},H_{\own}\}+\Lambda-d-\frac{3\kappa}{2}.
\end{equation*}

We distinguish two subcases.

* $(v,\ell-1)$ is correct. Then by the bounds on message delay and clock speed, for each correct $(w,\ell-1)$, $\{v,w\}\in E$, its $k$-th pulse message is received at a local time

\begin{equation*}
H\le H_{\own}+\vartheta(\localskew_{\ell-1}+u)\le H_{\own}+\Lambda-d-\frac{3\kappa}{2}<H_{v,\ell}(t_{v,\ell}^k),
\end{equation*}

where the last step uses <Ref>.

* $(v,\ell-1)$ is faulty, implying that all $(w,\ell-1)$, $\{v,w\}\in E$, are correct. Then by the bounds on message delay and clock speed, for each correct $(w,\ell-1)$, $\{v,w\}\in E$, its $k$-th pulse message is received at a local time

\begin{equation*}
H\le H_{\min}+\Lambda-d-\frac{3\kappa}{2}<H_{v,\ell}(t_{v,\ell}^k),
\end{equation*}

where we use that in order to guarantee that $\Lambda-d \ge \vartheta(2\localskew_{\ell-1}+u)$ (i.e., <Ref>), this must also hold in an execution that differs by $(v,\ell-1)$ being correct; in such an execution, we have that

\begin{equation*}
\max_{\{v,w\}\in E}\{t_{w,\ell-1}\}-\min_{\{v,w\}\in E}\{t_{w,\ell-1}\}\le \max_{\{v,w\}\in E}\{t_{w,\ell-1}\}-t_{v,\ell-1}+t_{v,\ell-1}-\min_{\{v,w\}\in E}\{t_{w,\ell-1}\}\le 2\localskew_{\ell-1}.
\end{equation*}

It remains to show that $(v,\ell)$ generates its pulse before receiving a $(k+1)$-th pulse message from a correct predecessor.
We distinguish two cases.

* $(v,\ell-1)$ is not faulty. Then the earliest local time $H$ at which $(v,\ell)$ has received a $k$-th pulse from a correct predecessor is bounded from below by

\begin{equation*}
H\ge H_{\own}-\vartheta(\localskew_{\ell-1}+u).
\end{equation*}

As delays and clock speeds do not change, the earliest message reception time for a $(k+1)$-th pulse from a correct predecessor is $\Lambda$ time later. Hence, it is sufficient to show that $H_{v,\ell}(t_{v,\ell}^k)\le H+\Lambda$. We distinguish three subcases.

* The inner loop terminated because $H_{v,\ell}(t)=H_{\max} +\kappa/2+\vartheta \kappa$ and at local time $H_{\min}$ a message from a correct predecessor $(w,\ell-1)$, $\{v,w\}\in E$, was received by $(v,\ell)$.

\begin{equation*}
H_{\own}+\vartheta(\localskew_{\ell-1}+u)+2\kappa \ge 2H_{\own}-H_{\min}+2\kappa\ge H_{\max}+\frac{\kappa}{2}+\vartheta\kappa.
\end{equation*}

and, by <Ref>,

\begin{align*}
H_{v,\ell}(t_{v,\ell}^k) &=H_{\max}+\frac{3\kappa}{2}+\Lambda-d\\
&\le H_{\own}+\vartheta(\localskew_{\ell-1}+u)+2\kappa+\Lambda-d\\
&\le H_{\own}-\vartheta(\localskew_{\ell-1}+u)\\
&\le H+\Lambda.
\end{align*}

* The inner loop terminated because $H_{v,\ell}(t)=H_{\max} +\kappa/2+\vartheta \kappa$ and at local time $H_{\max}$ a message from a correct predecessor $(w,\ell-1)$, $\{v,w\}\in E$, was received by $(v,\ell)$. Therefore,

\begin{equation*}
H_{\own}+\vartheta(\localskew_{\ell-1}+u) \ge H_{\max}
\end{equation*}

and, by <Ref>,

\begin{align*}
H_{v,\ell}(t_{v,\ell}^k) &=H_{\max}+\frac{3\kappa}{2}+\Lambda-d\\
&\le H_{\own}+\vartheta(\localskew_{\ell-1}+u)+\frac{3\kappa}{2}+\Lambda-d\\
&\le H_{\own}-\vartheta(\localskew_{\ell-1}+u)\\
&\le H+\Lambda.
\end{align*}

* The inner loop terminated because $H_{v,\ell}(t)=2H_{\own}-H_{\min}+2\kappa$ and $\Cor_{v,\ell}\ge 0$. By <Ref>, then

\begin{equation*}
H_{v,\ell}(t_{v,\ell}^k)=H_{\own}+\Lambda-d-\Cor_{v,\ell}\le H_{\own}+\Lambda-d\le H+\Lambda.
\end{equation*}

* The inner loop terminated because $H_{v,\ell}(t)=2H_{\own}-H_{\min}+2\kappa$ and $\Cor_{v,\ell}< 0$. Then

\begin{equation*}
\Cor_{v,\ell}= H_{\own}-H_{\min}+\frac{3\kappa}{2}
\end{equation*}

\begin{equation*}
\end{equation*}

Since $H_{\min}$ is bounded from above by the earliest local reception time of a message from a correct node $(w,\ell-1)$, $\{v,w\}\in E$, we have that

\begin{equation*}
H_{\min}\le H_{\own}+\vartheta(\localskew_{\ell-1}+u).
\end{equation*}

By <Ref>, we conclude that

\begin{equation*}
H_{v,\ell}(t_{v,\ell}^k)\le H_{\own}+\vartheta(\localskew_{\ell-1}+u)-\frac{3\kappa}{2}+\Lambda-d<H+\Lambda.
\end{equation*}

* $(v,\ell-1)$ is faulty. Then $H=H_{\min}$. Checking all cases in a similar fashion, we see that

\begin{equation*}
H_{v,\ell}(t_{v,\ell}^k)\le H_{\max}+\frac{3\kappa}{2}+\Lambda-d.
\end{equation*}

Using that <Ref> must also apply in an execution where $(v,\ell-1)$ is not faulty and hence $\max_{\{v,w\}\in E}\{t_{w,\ell-1}\}-\min_{\{v,w\}\in E}\{t_{w,\ell-1}\}\le 2\localskew_{\ell-1}$, it follows that

\begin{align*}
H_{v,\ell}(t_{v,\ell}^k)&\le H_{\max}+\frac{3\kappa}{2}+\Lambda-d\\
&\le H_{\min}+2\vartheta(\localskew_{\ell-1}+u)+\frac{3\kappa}{2}+\Lambda-d\\
&\le H_{\min}+\Lambda\\
&\le H+\Lambda.\qedhere
\end{align*}

We are now ready to show that <Ref> is equivalent to <Ref> in the absence of faults.

Suppose that for $(v,\ell)\in V_{\ell}$, $\ell>0$, and the predecessors of $(v,\ell)$ are correct.
Then running <Ref> instead of <Ref> results in the same pulse times of node $(v,\ell)$.

Assume towards a contradiction that the claim is false.
Denote by $t_{v,\ell}^k$ and $(t_{v,\ell}^k)'$ the pulse times of <Ref> and <Ref> in executions with identical delays, clock speeds, and behavior of faulty nodes.
W.l.o.g., let $t_{v,\ell}^k$ be minimal with the property that $t_{v,\ell}^k\neq (t_{v,\ell}^k)'$.

Consider the $k$-th loop iteration of <Ref> at node $(v,\ell)$.
We distinguish cases according to why the inner loop terminated.

* The inner loop terminated because $H_{v,\ell}(t)=H_{\max} +\kappa/2+\vartheta \kappa$. Then in <Ref>, we have that

\begin{equation*}
H_{\own}\ge H_{\max}+\frac{\kappa}{2}+\vartheta \kappa,
\end{equation*}

implying that

\begin{align*}
\Delta&:=\min_{s\in \N}\{\max\{H_{\own}-H_{\max}+4s\kappa,H_{\own}-H_{\min}-4 s\kappa \}\}- \frac{\kappa}{2}\\
&\ge \min_{s\in \N}\{H_{\own}-H_{\max}+4s\kappa\}- \frac{\kappa}{2}\\
&\ge H_{\own}-H_{\min}-\frac{\kappa}{2}\\
&\ge \vartheta \kappa.
\end{align*}

Hence, <Ref> computes

\begin{equation*}
\Cor_{v,\ell}=H_{\own}-H_{\max}-\frac{3\kappa}{2}
\end{equation*}

and generates its $k$-th pulse at local time

\begin{equation*}
\end{equation*}

a contradiction.

* The inner loop terminated because $H_{v,\ell}(t)=2H_{\own}-H_{\min}+2\kappa$. As $H_{\min}<\infty$, also $H_{\own}< \infty$ for <Ref>. We distinguish two subcases.

* In <Ref>, we have

\begin{equation*}
\Delta:=\min_{s\in \N}\{\max\{H_{\own}-H_{\max}+4s\kappa,H_{\own}-H_{\min}-4 s\kappa \}\}- \frac{\kappa}{2}<0.
\end{equation*}

Then the same holds in <Ref>, as there $H_{\max}$ is either identical to that of <Ref> of $\infty$. Hence, both algorithms compute $C_{v,\ell}=\min\{H_{\own}-H_{\min}+3\kappa/2,0\}$ and subsequently $H_{v,\ell}(t_{v,\ell}^k)=H_{\own}+\Lambda-d-\Cor_{v,\ell}=H_{v,\ell}((t_{v,\ell}^k)')$, a contradiction.

* In <Ref>, we have

\begin{equation*}
\Delta:=\min_{s\in \N}\{\max\{H_{\own}-H_{\max}+4s\kappa,H_{\own}-H_{\min}-4 s\kappa \}\}- \frac{\kappa}{2}\ge 0
\end{equation*}

Let $s_{\min}\in \N$ be such that

\begin{equation*}
\Delta:=\max\{H_{\own}-H_{\max}+4s_{\min}\kappa,H_{\own}-H_{\min}-4 s_{\min}\kappa \}-\frac{\kappa}{2}.
\end{equation*}

If $\Delta=H_{\own}-H_{\min}-4 s_{\min}\kappa-\kappa/2$, the fact that $H_{\own}$ and $H_{\min}$ are identical in both algorithms, while $H_{\max}$ is either also identical or $-\infty$ in <Ref>, again leads to the contradiction $H_{v,\ell}(t_{v,\ell}^k)=H_{v,\ell}((t_{v,\ell}^k)')$. Hence, suppose that $\Delta=H_{\own}-H_{\max}+4s_{\min}\kappa-\kappa/2$ in <Ref>. Therefore,

\begin{align*}
0&\le \Delta \\
&\le \max\{H_{\own}-H_{\max}+4(s_{\min}-1)\kappa,H_{\own}-H_{\min}-4(s_{\min}-1)\kappa\}-\frac{\kappa}{2}\\
\end{align*}

\begin{equation*}
2H_{\own}-H_{\min}+2\kappa\ge H_{\own}+4s_{\min}\kappa-\frac{3\kappa}{2}\ge H_{\max}-\kappa<H_{\max}-\frac{\kappa}{2}-\vartheta\kappa.
\end{equation*}

This is a contradiction, as then the inner loop in <Ref> would have terminated at an earlier time.

§.§ Self-Stabilization

Making <Ref> self-stabilizing follows standard techniques.
Accordingly, we confine ourselves to a brief high-level discussion of how this is achieved.

The key observation is that self-stabilization can proceed layer by layer, where <Ref> shows that layer $0$ stabilizes fast enough.
Thus, we can assume that the correct nodes of the previous layer generate pulses at a stable frequency of $\Lambda$ satisfying the skew bounds obtained in the analysis.

This allows us to make sure that the timing of its listening loop aligns with the pulse signals from the previous layer: From all but one predecessor, the pulse signals must be received while the inner loop is running.
Moreover, the inner loop will terminate within $\Lambda$ time.
Instead of restarting the inner loop dependent on the own generated pulse, we can instead start a loop iteration when receiving the first pulse after a quiet period of, say, $\Lambda/10$ (where too frequent pulses of a faulty predecessor are filtered out).
As such a quiet period must occur by <Ref>, this will align the loop correctly with the $k$-th pulses of correct predecessors for some $k\in \N$.

Once the inner loop terminates, we look for the next quiet period, and start a new instance of the inner loop on reception of the next pulse from a predecessor, and so on.
Whenever the inner loop terminates correctly, i.e., not due to a timeout, we also compute the time to generate the next pulse as in <Ref>.
However, we do not wait until the pulse is generated before willing to start a new instance of the inner loop.
This way, we ensure that we do not miss the first pulse message of a correct predecessor for pulse $k+1$ in case the inner loop for pulse $k$ was misaligned.

§ OBTAINING THE FINAL SKEW BOUNDS

Recall that our model assumes that message delays and clock speeds do not vary.
If the behavior of faulty nodes is static, i.e., the timing of their output pulse messages is identical in each pulse as well, a stable input frequency of $1/\Lambda$ results in repeating the exact same message pattern with the same timing every $1/\Lambda$ time.
We can exploit this to bound $\localskew_{\ell,\ell+1}$ in terms of $\localskew_{\ell}$.

By <Ref>, for correct $(v,\ell+1)\in V_{\ell+1}$, $\ell\in \N$,

\begin{equation*}
\min_{\substack{((w,\ell),(v,\ell+1))\in E_{\ell}\\ (w,\ell)\notin F}}\{t_{w,\ell}^k\}+\Lambda-2\kappa \le t_{v,\ell+1}^k \le \max_{\substack{((w,\ell),(v,\ell))\in E_{\ell}\\ (w,\ell)\notin F}}\{t_{w,\ell}^k\}+\Lambda+2\kappa.
\end{equation*}

Because the behavior of fault nodes does not change between pulses, a simple induction shows that $t_{x,\ell'}^{k+1}=t_{x,\ell'}^k+\Lambda$ for all correct nodes $(x,\ell')\in V_{\ell'}$, $\ell'\in \N$.
In particular,

\begin{equation*}
\min_{\substack{((w,\ell),(v,\ell+1))\in E_{\ell}\\ (w,\ell-1)\notin F}}\{t_{w,\ell}^{k+1}\}-2\kappa \le t_{v,\ell+1}^k \le \max_{\substack{((w,\ell),(v,\ell+1))\in E_{\ell}\\ (w,\ell)\notin F}}\{t_{w,\ell}^{k+1}\}+2\kappa.
\end{equation*}

By <Ref>, $\localskew_{\ell}\in O(\kappa \log D)$.
Note that this bound applies uniformly over all executions.
Thus, even if $(v,\ell)$ is faulty, using that its neighbors are within distance $2$ of each other, it holds that

\begin{equation*}
\min_{\substack{((w,\ell),(v,\ell+1))\in E_{\ell}\\ (w,\ell-1)\notin F}}\{t_{w,\ell}^{k+1}\}-\max_{\substack{((w,\ell),(v,\ell+1))\in E_{\ell}\\ (w,\ell)\notin F}}\{t_{w,\ell}^{k+1}\}\in O(\kappa \log D),
\end{equation*}

by virtue of comparing to an execution in which $(v,\ell)$ is correct.
As $(v,\ell+1)$ was an arbitrary correct node, the claim of the theorem follows.

It remains to argue that some variation can be sustained.

The maximum length of a directed path in $H$ is bounded by $2\sqrt{n}$: at most $D\le \sqrt{n}$ hops in layer $0$, followed by at most $\sqrt{n}$ links from layer to layer.
Thus, accumulating all changes in timing due to link delay and clock speed variation along a path results in a deviation of $O((u+(\vartheta-1)(\Lambda-d))\log D=O(\kappa \log D)$.
This is trivial for layer $0$ and applies to pulse propagation through the layers as well, because our respective analysis relies on <Ref> and <Ref>.
In order to take into account a constant number of faulty nodes with arbitrary behavior, we reason analogously to the proof of <Ref>, i.e., rely on <Ref> as well.


# An Algebraic Roadmap of Particle Theories
Part II: Theoretical checkpoints

N. Furey
Iris Adlershof, Humboldt-Universität zu Berlin,
Zum Grossen Windkanal 2, Berlin, 12489
<EMAIL_ADDRESS>
HU-EP-23/65

###### Abstract

An optimal algebraic model of particle physics has a number of checkpoints to
pass. As a minimum, models should $\langle 1\rangle$ conform to the Coleman-
Mandula theorem (or establish a loophole), $\langle 2\rangle$ evade familiar
fermion doubling problems, $\langle 3\rangle$ naturally explain the Standard
Model’s chirality, $\langle 4\rangle$ exclude B-L gauge symmetry at low
energy, and $\langle 5\rangle$ explain the existence of three generations. We
demonstrate how the model introduced in [1] passes checkpoints $\langle
1\rangle,\langle 2\rangle,\langle 3\rangle,\langle 4\rangle$, and has yet to
cross $\langle 5\rangle$. We close by elucidating an unexpected appearance of
spacetime symmetries.

###### pacs:

112.10.Dm, 2.60.Rc, 12.38.-t, 02.10.Hh, 12.90.+b

This article relies on the results and conventions of [1].

##  I Five checkpoints

The last decades have seen a significant number researchers steadily unearth
more and more algebraic structure underlying the Standard Model of particle
physics, [1]-[43]. With recent meetings, [44], [45], [46], this community
continues to grow.

Needless to say, however, a full solution is still at large. Certain
checkpoints must necessarily be crossed before one can finally claim a
compelling algebraic Standard Model description. We find that five challenges
in particular recur in the literature again and again (including in our own
models). We search for a concise algebraic explanation for the Standard
Model’s structure that overcomes the following hurdles:

$\langle 1\rangle$ Heed or evade the Coleman-Mandula theorem. Internal and
spacetime symmetries should either commute on particle representations, or a
loophole should be established.

$\langle 2\rangle$ Solve the fermion doubling problem. The Hilbert space
corresponding to one generation of standard model fermions is locally
32$\hskip 1.42262pt\mathbb{C}$ dimensional. Likewise, under
$\mathfrak{spin}(10)\oplus\mathfrak{spin}(1,3),$ unconstrained
$(\mathbf{16},\mathbf{2})$ Weyl representations corresponding to one
generation of fermions is also locally 32$\hskip 1.42262pt\mathbb{C}$
dimensional.

$\langle 3\rangle$ Explain the Standard Model’s chirality. Here we mean
without implementing ad hoc projection operators, without fixing arbitrarily
chosen mathematical objects, without introducing other ad hoc constraints.

$\langle 4\rangle$ Exclude the extraneous $\mathfrak{u}(1)_{B-L}$ symmetry at
low energy. This additional gauge symmetry often appears in algebraic models.
Again, it should be eliminated on logical grounds, not explained away with an
arbitrarily chosen condition.

$\langle 5\rangle$ Explain the existence of three generations. One would
anticipate that these three generations should be linearly independent from
one another, or that an explanation should be provided as to why their linear
dependence is desired.

This list extends the one originally presented in [2]. In our recent work,
[2], [3], we demonstrated solutions for $\langle 1\rangle$ and $\langle
2\rangle$, but not $\langle 3\rangle$, $\langle 4\rangle$, $\langle 5\rangle$.
In earlier work, [41], it could be argued that we achieved solutions for
$\langle 1\rangle$ and $\langle 3\rangle$, but not $\langle 2\rangle$,
$\langle 4\rangle$, $\langle 5\rangle$.

Figure 1:  A particle roadmap described in [1] interconnecting six well-known
models: Spin(10), Pati-Salam (PS), Left-Right Symmetric (LR), Georgi-Glashow
(SU(5)), Standard Model pre-Higgs-mechanism (SM), Standard Model post-Higgs-
mechanism (UB).

The purpose of this current article is to work through the implications of our
general construction in [1]. We begin by reminding the reader of the detailed
particle roadmap, first constructed in [1], depicted here in Figure (1). The
majority of this article is then devoted to demonstrating that our new model
passes checkpoints $\langle 1\rangle$, $\langle 2\rangle$, $\langle 3\rangle$,
$\langle 4\rangle$, and has yet to overcome $\langle 5\rangle$. In this
context, we finally show how SL(2,$\mathbb{C}$) spacetime symmetries arise
readily from the maximal Clifford subspace commuting with Spin(10).

##  II An early checkpoint

In our previous paper, [1], we showed that it is possible to use the
multivector condition, $\Psi_{V}$, in order to arrive at the Lie algebra for
the Standard Model, $\ell_{\textup{SM}}$,

$\begin{array}[]{rccll}\ell_{\textup{SM}}\Psi_{\textup{L}}=\big{(}r_{j}i\Lambda_{j}&+&r^{\prime}_{m}L_{\epsilon_{m}}s&+&irY\big{)}\Psi_{\textup{L}}.\vspace{2mm}\\\
\mathfrak{su}(3)&\oplus&\mathfrak{su}(2)&\oplus&\mathfrak{u}(1)\end{array}$
(1)

Unlike in our earlier work, [3], this result now excludes an unwanted B-L
gauge symmetry at low energy. In [1], we defined the B-L operator as

$i(B-L)\Psi_{\textup{L}}:=\frac{1}{3}(L_{e_{1}}L_{e_{3}}+L_{e_{2}}L_{e_{6}}+L_{e_{4}}L_{e_{5}})\Psi_{\textup{L}}.$
(2)

Given that the multivector condition may be thought of as a consistency
condition when spinors are constructed from multivectors, this qualifies as a
natural explanation for the omission of B-L. Hence, we see that this model
passes checkpoint $\langle 4\rangle$.

##  III Fermion identification

###  III.1 The even semi-spinor

We may now apply the Standard Model symmetries above to $\Psi_{\textup{L}}$ so
as to label its states, and confirm that the electroweak sector is maximally
chiral (checkpoint $\langle 3\rangle$). We label these below, according to
their behaviour under $\ell_{\textup{SM}}$ of equation (1). The B-L operator
is also useful in this task, as it differentiates between sterile neutrinos
and sterile antineutrinos, where the Standard Model symmetries do not.

$\begin{array}[]{l}\Psi_{\textup{L}}\hskip 2.84526pt=\hskip
2.84526pt\frac{1}{2}\left(1+i\Gamma_{11}\right)\mathbb{C}l(10)\hskip
1.42262ptv\vspace{3mm}\\\ =\hskip 1.42262pt{\mathcal{V}}_{L}\hskip
1.42262pta_{4}a_{3}a_{2}a_{1}\hskip 1.42262ptv+\mathcal{E}^{-}_{L}\hskip
1.42262pta_{5}a_{3}a_{2}a_{1}\hskip
1.42262ptv+\overline{\mathcal{D}^{k}_{R}}\hskip
1.42262pt\epsilon_{ijk}a_{5}a_{4}a_{j}a_{i}\hskip 1.42262ptv\vspace{3mm}\\\
+\hskip 2.84526pt\overline{\mathcal{U}_{R}^{k}}\hskip
1.42262pt\epsilon_{ijk}a_{j}a_{i}\hskip 1.42262ptv+{\mathcal{U}}_{L}^{i}\hskip
1.42262pta_{4}a_{i}\hskip 1.42262ptv+{\mathcal{D}}_{L}^{i}\hskip
1.42262pta_{5}a_{i}\hskip 1.42262ptv+\overline{{\mathcal{E}}^{-}_{R}}\hskip
2.84526pta_{5}a_{4}\hskip 1.42262ptv\vspace{3mm}\\\ +\hskip
2.84526pt\overline{\mathcal{V}_{R}}\hskip 1.42262pt\hskip
1.42262ptv,\end{array}$ (3)

where $i,j,k\in\\{1,2,3\\}$. Here, the coefficients $\mathcal{V}_{L},$
$\mathcal{E}^{-}_{L}$, $\dots$ are suggestively named complex coefficients.
For example, $\mathcal{V}_{L}$ refers to the left-handed neutrino,
$\mathcal{E}^{-}_{L}$ refers to the left-handed electron,
$\overline{\mathcal{D}^{k}_{R}}$ refers to the anti-particle of the right-
handed down quark (which is itself left-handed), etc. This convention is
chosen so as to coincide with [2], [3], but differs from [47], [38].

Substituting in the division algebraic realizations of these states via
equations (25) and (28) of our previous paper, [1], gives $\Psi_{\textup{L}}$
as

$\begin{array}[]{lll}\Psi_{\textup{L}}&=&\hskip 1.42262pt\Large(\hskip
1.42262pt{\mathcal{V}}_{L}\hskip
1.42262pti\alpha_{1}\alpha_{2}\alpha_{3}L_{\epsilon_{\uparrow\downarrow}}R_{\epsilon_{\downarrow\downarrow}}\vspace{3mm}\\\
&+&\mathcal{E}^{-}_{L}\hskip
1.42262pti\alpha_{3}\alpha_{2}\alpha_{1}L_{\epsilon_{\downarrow\downarrow}}R_{\epsilon_{\downarrow\downarrow}}+\overline{\mathcal{D}^{k}_{R}}\hskip
1.42262pt\epsilon_{ijk}\alpha_{j}\alpha_{i}L_{\epsilon_{\uparrow\downarrow}}R_{\epsilon_{\downarrow\downarrow}}\vspace{3mm}\\\
&+&\hskip 2.84526pt\overline{\mathcal{U}_{R}^{k}}\hskip
1.42262pt\epsilon_{ijk}\alpha_{j}\alpha_{i}L_{\epsilon_{\downarrow\downarrow}}R_{\epsilon_{\downarrow\downarrow}}-{\mathcal{U}}_{L}^{i}\hskip
1.42262pti\alpha_{i}L_{\epsilon_{\uparrow\downarrow}}R_{\epsilon_{\downarrow\downarrow}}\vspace{3mm}\\\
&+&{\mathcal{D}}_{L}^{i}\hskip
1.42262pti\alpha_{i}L_{\epsilon_{\downarrow\downarrow}}R_{\epsilon_{\downarrow\downarrow}}+\overline{{\mathcal{E}}^{-}_{R}}\hskip
2.84526ptL_{\epsilon_{\uparrow\downarrow}}R_{\epsilon_{\downarrow\downarrow}}\vspace{3mm}\\\
&+&\hskip 2.84526pt\overline{\mathcal{V}_{R}}\hskip 1.42262pt\hskip
1.42262ptL_{\epsilon_{\downarrow\downarrow}}R_{\epsilon_{\downarrow\downarrow}}\Large)\hskip
2.84526pts^{*}S^{*},\end{array}$ (4)

where $i,j,k\in\\{1,2,3\\},$

$\begin{array}[]{ll}s:=\frac{1}{2}\left(1+iL_{e_{7}}\right),&S:=\frac{1}{2}\left(1+iR_{e_{7}}\right),\vspace{2mm}\\\
s^{*}:=\frac{1}{2}\left(1-iL_{e_{7}}\right),&S^{*}:=\frac{1}{2}\left(1-iR_{e_{7}}\right),\vspace{2mm}\\\
\epsilon_{\uparrow\uparrow}:=\frac{1}{2}\left(1+i\epsilon_{3}\right),&\epsilon_{\downarrow\uparrow}:=\frac{1}{2}\left(\epsilon_{2}+i\epsilon_{1}\right),\vspace{2mm}\\\
\epsilon_{\uparrow\downarrow}:=\frac{1}{2}\left(-\epsilon_{2}+i\epsilon_{1}\right),&\epsilon_{\downarrow\downarrow}:=\frac{1}{2}\left(1-i\epsilon_{3}\right),\end{array}$
(5)

and the $\alpha_{i}$ are defined so as to match previous articles, [2], [3],
[38], [39], [40]. Namely,

$\begin{array}[]{l}\alpha_{1}:=\frac{1}{2}\left(-L_{e_{5}}+iL_{e_{4}}\right),\vspace{2mm}\\\
\alpha_{2}:=\frac{1}{2}\left(-L_{e_{3}}+iL_{e_{1}}\right),\vspace{2mm}\\\
\alpha_{3}:=\frac{1}{2}\left(-L_{e_{6}}+iL_{e_{2}}\right).\end{array}$ (6)

###  III.2 The odd semi-spinor

It is clear from the $\mathbb{Z}_{2}$-grading inherent in the minimal left
ideal, $\Psi,$ that these left-handed particles, $\Psi_{\textup{L}},$ comprise
the even-graded subspace. How might we then interpret $\Psi$’s odd-graded
subspace?

It will be of little surprise to the reader that when we choose
$\mathfrak{spin}(10)$ acting on $\Psi$ to be represented by linear
combinations of Clifford algebraic bivectors, $\Gamma_{i}\Gamma_{j},$ then
this 32$\hskip 1.42262pt\mathbb{C}$ dimensional $\Psi$ decomposes as the
$\mathbf{16}\oplus\mathbf{16^{*}}$ of $\mathfrak{spin}(10).$ In terms of
division algebras, these $\mathfrak{spin}(10)$ Lie algebra elements may be
encoded as

$\begin{array}[]{ll}\ell_{10}^{D}:=&r_{ij}L_{e_{i}}L_{e_{j}}+r_{m}L_{\epsilon_{m}}+r^{\prime}_{m}L_{\epsilon_{m}}L_{e_{7}}R_{\epsilon_{3}}\vspace{2mm}\\\
&+\hskip
2.84526ptr_{mj}^{\prime}iL_{\epsilon_{m}}L_{e_{j}}+r_{j}^{\prime\prime}iL_{e_{j}}L_{e_{7}}R_{\epsilon_{3}},\end{array}$
(7)

for $i,j\in\\{1,\dots 6\\}$ with $i\neq j,$ $m\in\\{1,2,3\\},$ and
$r_{ij},r_{m},r^{\prime}_{m},r_{mj}^{\prime},r_{j}^{\prime\prime}\in\mathbb{R}$.
In this case, the odd-graded space is interpreted as a right-handed semi-
spinor:

$\begin{array}[]{ll}\Psi_{\textup{R}}&=\hskip
2.84526pt\frac{1}{2}\left(1-i\Gamma_{11}\right)\mathbb{C}l(10)\hskip
1.42262ptv\vspace{3mm}\\\ &=\mathcal{V}_{R}\hskip
1.42262pta_{5}a_{4}a_{3}a_{2}a_{1}\hskip 1.42262ptv\vspace{3mm}\\\ &+\hskip
1.42262pt\mathcal{E}^{-}_{R}\hskip 1.42262pta_{3}a_{2}a_{1}\hskip
1.42262ptv+\overline{\mathcal{D}_{L}^{k}}\hskip
2.84526pt\epsilon_{ijk}a_{4}a_{j}a_{i}\hskip 1.42262ptv\vspace{3mm}\\\
&+\hskip 1.42262pt\overline{\mathcal{U}_{L}^{k}}\hskip
2.84526pt\epsilon_{ijk}a_{5}a_{j}a_{i}\hskip
1.42262ptv+{\mathcal{U}_{R}^{i}}\hskip 1.42262pta_{5}a_{4}a_{i}\hskip
1.42262ptv\vspace{3mm}\\\ &+\hskip 1.42262pt\mathcal{D}_{R}^{i}\hskip
1.42262pta_{i}^{\dagger}\hskip 1.42262ptv+\hskip
1.42262pt\overline{\mathcal{E}^{-}_{L}}\hskip 1.42262pta_{4}\hskip
1.42262ptv+\overline{\mathcal{V}_{L}}\hskip 1.42262pta_{5}\hskip
1.42262ptv.\end{array}$ (8)

Again, we have made use of the Standard Model + B-L Lie subalgebra in order to
identify states. In terms of division algebras, $\Psi_{\textup{R}}$ becomes

$\begin{array}[]{lll}\Psi_{\textup{R}}&=&\hskip 1.42262pt\Large(\hskip
1.42262pt{\mathcal{V}}_{R}\hskip
1.42262pti\alpha_{3}\alpha_{2}\alpha_{1}L_{\epsilon_{\uparrow\downarrow}}R_{\epsilon_{\downarrow\uparrow}}\vspace{3mm}\\\
&+&\mathcal{E}^{-}_{R}\hskip
1.42262pti\alpha_{3}\alpha_{2}\alpha_{1}L_{\epsilon_{\downarrow\downarrow}}R_{\epsilon_{\downarrow\uparrow}}+\overline{\mathcal{D}^{k}_{L}}\hskip
1.42262pt\epsilon_{ijk}\alpha_{j}\alpha_{i}L_{\epsilon_{\uparrow\downarrow}}R_{\epsilon_{\downarrow\uparrow}}\vspace{3mm}\\\
&-&\overline{\mathcal{U}_{L}^{k}}\hskip
1.42262pt\epsilon_{ijk}\alpha_{j}\alpha_{i}L_{\epsilon_{\downarrow\downarrow}}R_{\epsilon_{\downarrow\uparrow}}+{\mathcal{U}}_{R}^{i}\hskip
1.42262pti\alpha_{i}L_{\epsilon_{\uparrow\downarrow}}R_{\epsilon_{\downarrow\uparrow}}\vspace{3mm}\\\
&+&{\mathcal{D}}_{R}^{i}\hskip
1.42262pti\alpha_{i}L_{\epsilon_{\downarrow\downarrow}}R_{\epsilon_{\downarrow\uparrow}}+\overline{{\mathcal{E}}^{-}_{L}}\hskip
2.84526ptL_{\epsilon_{\uparrow\downarrow}}R_{\epsilon_{\downarrow\uparrow}}\vspace{3mm}\\\
&-&\overline{\mathcal{V}_{L}}\hskip 1.42262pt\hskip
1.42262ptL_{\epsilon_{\downarrow\downarrow}}R_{\epsilon_{\downarrow\uparrow}}\Large)\hskip
2.84526pts^{*}S^{*}.\end{array}$ (9)

With this identification of $\Psi_{\textup{R}}$, the full minimal left ideal,
$\Psi=\Psi_{\textup{L}}\oplus\Psi_{\textup{R}},$ transforms reducibly as
$\mathbf{32}=\mathbf{16}\oplus\mathbf{16^{*}}$ under $\mathfrak{spin}(10)$. It
is therefore appropriate for use as a 32$\hskip 1.42262pt\mathbb{C}$
dimensional Hilbert space for the Standard Model at high energies. This
matches the representation space described in [47] and [38].

Readers may confirm through application of equation (1) and its right-handed
counterpart, that this new model is indeed chiral, and hence passes checkpoint
$\langle 3\rangle$.

Finally, we note for future reference that the maximal Lie subalgebra in
$\mathbb{C}l(10)$ commuting with elements $\ell_{10}^{D}$ is generated by two
objects over $\mathbb{C}$, namely $\\{1,\Gamma_{11}\\}$.

$[\hskip 1.42262pt\ell_{max},\hskip 2.84526pt\ell_{10}^{D}\hskip
1.42262pt]=0\hskip 11.38109pt\Rightarrow\hskip
11.38109pt\ell_{max}=c_{0}+c_{1}\Gamma_{11}$ (10)

for $c_{0},c_{1}\in\mathbb{C}$.

##  IV Spacetime symmetries as the complement of internal symmetries

Readers should note the unusual way in which local spacetime symmetries arise
in this model.

###  IV.1 The last division algebraic reflection

It can be seen that the Dirac spinor
$\mathbf{32}=\mathbf{16}\oplus\mathbf{16^{*}}$ is well-suited so as to
describe a fermionic Hilbert space, [47], [38]. However, these representations
do not match those of the unconstrained fields one would typically use to
build a Lagrangian. Under local symmetries $\left(\mathfrak{spin}(10),\hskip
2.84526pt\mathfrak{sl}(2,\mathbb{C})\right)$ the representation
$\mathbf{32}=\left(\mathbf{16},\mathbf{2}\right)$ would be more appropriate,
where $\mathfrak{sl}(2,\mathbb{C})$ enacts local spacetime symmetries, and our
fermions are represented as $2\hskip 1.42262pt\mathbb{C}$ dimensional Weyl
representations.

How does one bring about this $\left(\mathbf{16},\mathbf{2}\right)$
representation? It is important to note that the Coleman-Mandula theorem and
equation (10) imply that the $\ell_{10}^{D}$ of equation (7) do not provide
the $\mathfrak{spin}(10)$ action that we need. Instead, we will need a
$\mathfrak{spin}(10)$ action that fashions the 32 $\mathbb{C}$ dimensional
minimal left ideal into a $\mathbf{16}\oplus\mathbf{16}.$ In [2] such an
action was found, however, at the time we did not explain how the solution was
originally obtained.

In [1], we made use of four division algebraic reflections:
${*_{\overrightarrow{\mathbb{O}}}}$, $*_{\overleftarrow{\mathbb{O}}}$,
$*_{\overrightarrow{\mathbb{H}}}$, $*_{\overrightarrow{\mathbb{C}}}$ in
relation to the internal symmetries of our model. Readers may notice that
these four types of reflection correspond to the four types of multiplication
algebra that are expressible in terms of left multiplication. (Please refer to
Sections IV D and IV F of [1]). Beyond these four types of reflection, only
one remained, $*_{\overleftarrow{\mathbb{H}}}$, and its corresponding
multiplication algebra is not expressible in terms of left multiplication.

For concreteness, let us take $*_{\overleftarrow{\mathbb{H}}}\vspace{.5mm}$,
to define the map $R_{\epsilon_{1}}\mapsto R_{\epsilon_{1}},$
$R_{\epsilon_{2}}\mapsto-R_{\epsilon_{2}},$ $R_{\epsilon_{3}}\mapsto-
R_{\epsilon_{3}}.$ With this final involution at our disposal, we may
generalize the Majorana action defined in Section 3.5 of [38]. That is,
instead of relying on $\ell_{10}^{D}$ of equation (7), we will consider
elements of $\mathfrak{spin}(10)$ represented as

$\begin{array}[]{ll}&\ell_{10}^{D}\hskip 2.84526ptP\hskip 2.84526pt+\hskip
2.84526pt\left({\ell_{10}^{D}}\hskip
2.84526ptP\right)^{*_{\overleftarrow{\mathbb{H}}}}\vspace{2mm}\\\
=&\ell_{10}^{D}\hskip 2.84526ptP\hskip 2.84526pt+\hskip
2.84526pt{\ell_{10}^{D}}^{*_{\overleftarrow{\mathbb{H}}}}\hskip
2.84526ptP^{*}\vspace{2mm}\\\
=&r_{ab}L_{e_{a}}L_{e_{b}}+r^{\prime}_{mn}L_{\epsilon_{m}}L_{\epsilon_{n}}+r_{ma}^{\prime\prime}iL_{\epsilon_{m}}L_{e_{a}}\vspace{2mm}\\\
=&\ell_{10},\end{array}$ (11)

familiar from equation (34) of reference [1]. Here, the idempotent $P$ is
defined as $P:=\frac{1}{2}\left(1+i\Gamma_{11}\right).$

When we apply this $*_{\overleftarrow{\mathbb{H}}}\vspace{.5mm}$-invariant
action to the entire minimal left ideal, $\Psi=\mathbb{C}l(10)\hskip
1.42262ptv$, then this 32 $\mathbb{C}$ dimensional space transfoms as a
$\mathbf{16}\oplus\mathbf{16}$, as desired.

###  IV.2 An unexpected appearance of spacetime symmetries

With this new $*_{\overleftarrow{\mathbb{H}}}\vspace{.5mm}$-invariant
$\mathfrak{spin}(10)$ action, we may now ask again: What is the maximal Lie
subalgebra in $\mathbb{C}l(10)$ commuting with it? In contrast to equation
(10), we now find that

$[\hskip 1.42262pt\ell_{max},\hskip 2.84526pt\ell_{10}\hskip
1.42262pt]=0\hskip 11.38109pt\Rightarrow\hskip
11.38109pt\ell_{max}=c_{\mu}R_{\epsilon_{\mu}}$ (12)

for $\mu\in\\{0,1,2,3\\},$ and $c_{\mu}\in\mathbb{C}$.

In other words, the maximal space commuting with this version of
$\mathfrak{spin}(10)$ is given by the trivial centre of $\mathbb{C}l(10)$, and
$\mathfrak{sl}(2,\mathbb{C})$. Furthermore, this $\mathfrak{sl}(2,\mathbb{C})$
action behaves precisely as do local spacetime symmetries on Weyl
representations. It is perhaps intriguing that the maximal subspace commuting
with $\mathfrak{spin}(10)$ should almost exclusively act as a source for
spacetime symmetries.

Schematically, $*_{\overleftarrow{\mathbb{H}}}$ has allowed us to redefine

$\begin{array}[]{lll}\delta\Psi=\ell_{10}^{D}\hskip 1.42262pt\Psi&\hskip
7.11317pt\mapsto&\delta\Psi=\ell_{10}^{D}\hskip 1.42262ptP\hskip
1.42262pt\Psi+(\ell_{10}^{D}\hskip
1.42262ptP)^{*_{\overleftarrow{\mathbb{H}}}}\hskip
1.42262pt\Psi=\ell_{10}\hskip 1.42262pt\Psi\vspace{2mm}\\\
\Psi\sim\mathbf{16}\oplus\mathbf{16^{*}}&\hskip
7.11317pt\mapsto&\Psi\sim\mathbf{16}\oplus\mathbf{16}\vspace{2mm}\\\
\ell_{\textup{max}}\in\mathbb{C}\oplus\mathbb{C}&\hskip
7.11317pt\mapsto&\ell_{\textup{max}}\in\mathbb{C}\oplus\mathfrak{sl}(2,\mathbb{C}).\end{array}$
(13)

The new symmetry actions permit us to compile and re-identify our fermionic
states as

$\begin{array}[]{l}\Psi_{\textup{L}}^{\uparrow}+\Psi_{\textup{L}}^{\downarrow}\hskip
2.84526pt=\hskip 2.84526pt\mathbb{C}l(10)\hskip 1.42262ptv\vspace{3mm}\\\
=\mathcal{V}_{L}^{\uparrow}\hskip 1.42262pta_{5}a_{4}a_{3}a_{2}a_{1}\hskip
1.42262ptv+{\mathcal{V}}_{L}^{\downarrow}\hskip
1.42262pta_{4}a_{3}a_{2}a_{1}\hskip 1.42262ptv\vspace{3mm}\\\ +\hskip
1.42262pt\mathcal{E}^{-\downarrow}_{L}\hskip
1.42262pta_{5}a_{3}a_{2}a_{1}\hskip
1.42262ptv+\overline{\mathcal{D}^{k\uparrow}_{R}}\hskip
1.42262pt\epsilon_{ijk}a_{5}a_{4}a_{j}a_{i}\hskip 1.42262ptv\vspace{3mm}\\\
+\hskip 1.42262pt\mathcal{E}^{-\uparrow}_{L}\hskip
1.42262pta_{3}a_{2}a_{1}\hskip
1.42262ptv+\overline{\mathcal{D}_{R}^{k\downarrow}}\hskip
2.84526pt\epsilon_{ijk}a_{4}a_{j}a_{i}\hskip 1.42262ptv\vspace{3mm}\\\ +\hskip
1.42262pt\overline{\mathcal{U}_{R}^{k\downarrow}}\hskip
2.84526pt\epsilon_{ijk}a_{5}a_{j}a_{i}\hskip
1.42262ptv+{\mathcal{U}_{L}^{i\uparrow}}\hskip 1.42262pta_{5}a_{4}a_{i}\hskip
1.42262ptv\vspace{3mm}\\\ +\hskip
1.42262pt\overline{\mathcal{U}_{R}^{k\uparrow}}\hskip
1.42262pt\epsilon_{ijk}a_{j}a_{i}\hskip
1.42262ptv+{\mathcal{U}}_{L}^{i\downarrow}\hskip 1.42262pta_{4}a_{i}\hskip
1.42262ptv\vspace{3mm}\\\ +\hskip
1.42262pt{\mathcal{D}}_{L}^{i\downarrow}\hskip 1.42262pta_{5}a_{i}\hskip
1.42262ptv+\overline{{\mathcal{E}}^{-\uparrow}_{R}}\hskip
2.84526pta_{5}a_{4}\hskip 1.42262ptv\vspace{3mm}\\\ +\hskip
1.42262pt\mathcal{D}_{L}^{i\uparrow}\hskip 1.42262pta_{i}\hskip
1.42262ptv+\hskip 1.42262pt\overline{\mathcal{E}^{-\downarrow}_{R}}\hskip
1.42262pta_{4}\hskip 1.42262ptv\vspace{3mm}\\\ +\hskip
1.42262pt\overline{\mathcal{V}^{\downarrow}_{R}}\hskip 1.42262pta_{5}\hskip
1.42262ptv+\overline{\mathcal{V}^{\uparrow}_{R}}\hskip 1.42262pt\hskip
1.42262ptv,\end{array}$ (14)

where we use the convention that the bar over a particle flips the chirality
and $\sigma_{z}$ spin value. For example, the anti-particle of the $1/2\hskip
1.42262pt\sigma_{z}=+1/2$ sterile right-handed neutrino is denoted
$\overline{\mathcal{V}^{\uparrow}_{R}}$. It is itself left-handed, and has
eigenvalue -1/2 under the spin operator $R_{\sigma_{z}}:=R_{i\epsilon_{3}}$.

Rewriting in terms of division algebras, and regrouping, gives

$\begin{array}[]{lll}\Psi_{\textup{L}}^{\downarrow}+\Psi_{\textup{L}}^{\uparrow}&=&\hskip
1.42262pt\Large(\hskip 1.42262pt{\mathcal{V}}_{L}^{\downarrow}\hskip
1.42262pti\alpha_{1}\alpha_{2}\alpha_{3}L_{\epsilon_{\uparrow\downarrow}}R_{\epsilon_{\downarrow\downarrow}}\vspace{3mm}\\\
&+&\mathcal{E}^{-\downarrow}_{L}\hskip
1.42262pti\alpha_{3}\alpha_{2}\alpha_{1}L_{\epsilon_{\downarrow\downarrow}}R_{\epsilon_{\downarrow\downarrow}}+\overline{\mathcal{D}^{k\uparrow}_{R}}\hskip
1.42262pt\epsilon_{ijk}\alpha_{j}\alpha_{i}L_{\epsilon_{\uparrow\downarrow}}R_{\epsilon_{\downarrow\downarrow}}\vspace{3mm}\\\
&+&\hskip 2.84526pt\overline{\mathcal{U}_{R}^{k\uparrow}}\hskip
1.42262pt\epsilon_{ijk}\alpha_{j}\alpha_{i}L_{\epsilon_{\downarrow\downarrow}}R_{\epsilon_{\downarrow\downarrow}}-{\mathcal{U}}_{L}^{i\downarrow}\hskip
1.42262pti\alpha_{i}L_{\epsilon_{\uparrow\downarrow}}R_{\epsilon_{\downarrow\downarrow}}\vspace{3mm}\\\
&+&{\mathcal{D}}_{L}^{i\downarrow}\hskip
1.42262pti\alpha_{i}L_{\epsilon_{\downarrow\downarrow}}R_{\epsilon_{\downarrow\downarrow}}+\overline{{\mathcal{E}}^{-\uparrow}_{R}}\hskip
2.84526ptL_{\epsilon_{\uparrow\downarrow}}R_{\epsilon_{\downarrow\downarrow}}\vspace{3mm}\\\
&+&\hskip 2.84526pt\overline{\mathcal{V}_{R}^{\uparrow}}\hskip 1.42262pt\hskip
1.42262ptL_{\epsilon_{\downarrow\downarrow}}R_{\epsilon_{\downarrow\downarrow}},\vspace{3mm}\\\
&+&\hskip 2.84526pt{\mathcal{V}}_{L}^{\uparrow}\hskip
1.42262pti\alpha_{3}\alpha_{2}\alpha_{1}L_{\epsilon_{\uparrow\downarrow}}R_{\epsilon_{\downarrow\uparrow}}\vspace{3mm}\\\
&+&\mathcal{E}^{-\uparrow}_{L}\hskip
1.42262pti\alpha_{3}\alpha_{2}\alpha_{1}L_{\epsilon_{\downarrow\downarrow}}R_{\epsilon_{\downarrow\uparrow}}+\overline{\mathcal{D}^{k\downarrow}_{R}}\hskip
1.42262pt\epsilon_{ijk}\alpha_{j}\alpha_{i}L_{\epsilon_{\uparrow\downarrow}}R_{\epsilon_{\downarrow\uparrow}}\vspace{3mm}\\\
&-&\hskip 2.84526pt\overline{\mathcal{U}_{R}^{k\downarrow}}\hskip
1.42262pt\epsilon_{ijk}\alpha_{j}\alpha_{i}L_{\epsilon_{\downarrow\downarrow}}R_{\epsilon_{\downarrow\uparrow}}+{\mathcal{U}}_{L}^{i\uparrow}\hskip
1.42262pti\alpha_{i}L_{\epsilon_{\uparrow\downarrow}}R_{\epsilon_{\downarrow\uparrow}}\vspace{3mm}\\\
&+&{\mathcal{D}}_{L}^{i\uparrow}\hskip
1.42262pti\alpha_{i}L_{\epsilon_{\downarrow\downarrow}}R_{\epsilon_{\downarrow\uparrow}}+\overline{{\mathcal{E}}^{-\downarrow}_{R}}\hskip
2.84526ptL_{\epsilon_{\uparrow\downarrow}}R_{\epsilon_{\downarrow\uparrow}}\vspace{3mm}\\\
&-&\hskip 2.84526pt\overline{\mathcal{V}_{R}^{\downarrow}}\hskip
1.42262pt\hskip
1.42262ptL_{\epsilon_{\downarrow\downarrow}}R_{\epsilon_{\downarrow\uparrow}}\Large)\hskip
2.84526pts^{*}S^{*}.\end{array}$ (15)

At this point, we see that rewriting our Dirac action $\ell^{D}_{10}$ as the
Weyl action $\ell_{10}$ allows us to evade the fermion doubling problem. That
is, the model successfully incorporates $\mathfrak{sl}(2,\mathbb{C})$
spacetime symmetries, without doubling the spinor space. Hence, it passes
checkpoint $\langle 2\rangle$.

Finally, readers may observe above that internal symmetries are described
entirely using the left-multiplication algebra, while the local spacetime
symmetries are are described using right-multiplication. This model cleanly
separates the two. Again, this was an unexpected finding. It also demonstrates
that this model conforms to the Coleman-Mandula theorem, and hence passes
checkpoint $\langle 1\rangle$.

##  V Reintroduction of $\mathfrak{u}(n)$

It may at some point be useful to note that a $\mathfrak{u}(n)$ Lie subalgebra
is easily obtainable by a single constraint on $\mathfrak{so}(2n)$, similar to
the multivector condition that gave us $\mathfrak{su}(n)$. While the
multivector condition gave us $\mathfrak{su}(n)$, the alternative constraint

$\begin{array}[]{c}[r_{k_{1}k_{2}}\Gamma_{k_{1}}\Gamma_{k_{2}},\hskip
1.42262pt\Psi_{\textup{L}}]\hskip
2.84526pt=r_{k_{1}k_{2}}^{\prime}\Gamma_{k_{1}}\Gamma_{k_{2}}\Psi_{\textup{L}}\end{array}$
(16)

provides us with $\mathfrak{u}(n)$ for some
$r_{k_{1}k_{2}}^{\prime}\in\mathbb{R},$ not necessarily equal to
$r_{k_{1}k_{2}}.$

##  VI Conclusion

The purpose of this second article was to work through the implications of the
new model constructed in [1]. We listed a set of five checkpoints that serve
as minimum conditions that an ideal algebraic description of particle physics
should pass. Namely, such an ideal model should

$\langle 1\rangle$ Heed or evade the Coleman-Mandula theorem,

$\langle 2\rangle$ Solve the fermion doubling problem,

$\langle 3\rangle$ Explain the Standard Model’s chirality,

$\langle 4\rangle$ Exclude B-L gauge symmetry at low energies,

$\langle 5\rangle$ Explain the existence of three generations.

Then, we explained how [1] excluded B-L gauge symmetry and demonstrated the
correct chiral fermion representations, thanks to the multivector condition
(checkpoints $\langle 4\rangle$ and $\langle 3\rangle$).

Of significance was the serendipitous appearance of spacetime symmetries.
Namely, we constructed a Weyl $\mathfrak{spin}(10)$ representation from the
$*_{\overleftarrow{\mathbb{H}}}$-invariant part of the known Dirac
$\mathfrak{spin}(10)$ representation. It so happened that the maximal
$\mathbb{C}l(10)$ subspace commuting with this Weyl $\mathfrak{spin}(10)$
representation produced the correct $\mathfrak{sl}(2,\mathbb{C})$ spacetime
symmetries. This allowed us to cross checkpoint $\langle 2\rangle$.

Also fortuitous was the finding that internal symmetries can find their
definition using the pure left-multiplication algebra of
$\mathbb{R}\otimes\mathbb{C}\otimes\mathbb{H}\otimes\mathbb{O},$ while
spacetime symmetries find their definition using the pure right-multiplication
algebra. That is, internal and spacetime symmetries exhibit a clean
separation, [48], (checkpoint $\langle 1\rangle$).

At this point, we see that [1] has passed checkpoints $\langle 1\rangle$,
$\langle 2\rangle$, $\langle 3\rangle$, $\langle 4\rangle$. We do not attempt
here to solve the three-generation problem (checkpoint $\langle 5\rangle$),
but such steps are being made elsewhere. Interested readers are directed to a
new line of research, known as Bott Periodic Particle Physics, first
introduced in [42] and [43]. Alternatively, please see [49] and [50].

###### Acknowledgements.

These manuscripts have benefitted from numerous discussions with Beth Romano.
The author is furthermore grateful for feedback and encouragement from John
Baez, Sukṛti Bansal, John Barrett, Latham Boyle, Hilary Carteret, Mia Hughes,
Kaushlendra Kumar, Agostino Patella, Shadi Tahvildar-Zadeh, Carlos Tamarit,
Andreas Trautner, and Jorge Zanelli. This work was graciously supported by the
VW Stiftung Freigeist Fellowship, and Humboldt-Universität zu Berlin.

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