0. Why are there laws in the universe?
Because there are things like laws in the universe, we can understand it.
And we are satisfied. We want to understand the universe.
Will that ever succeed?
In the first place, the universe might not be something we can comprehend.
Even so, we believe that the universe is understandable.
It is like a baseless prayer.
This is my humble wish.
If so, the fundamental question would be as follows:
If the universe is an understandable entity, what conditions must it satisfy?
1. The Requirements of Existence
First, states or phenomena in the universe must correspond to some kind of numbers.
Although we do not know what kind of numbers they are, in order to apply any sort of law, states must correspond to something like numbers.
This can be rephrased as the requirement of describability.
Second, the states of the universe possess some algebraic structure.
A law means predictability under fixed conditions.
Conditions correspond to certain numbers, and to those numbers, certain operations—that is, algebra—must correspond.
This can be rephrased as the requirement of predictability.
Third, that algebra must possess divisibility.
Our universe exists with remarkable stability within the observable range.
If the algebra inherent in the universe were one that outputted zero mid-calculation, the universe would not be nearly as stable.
In algebra, the property where operation results never become zero is called divisibility.
This can be said to be the requirement of the permanence of the universe.
Adolf Hurwitz proved that there are only four algebras satisfying these three requirements: real numbers, complex numbers, quaternions, and octonions.
Alas, the universe has only four choices.
And as an observational fact, strange particles called elementary particles exist in our universe.
Let us make this the fourth requirement.
That is, fourth, the algebra of the universe must be able to describe the symmetry of elementary particles.
The only algebra with the potential to describe these strange particles is the octonions.
To guarantee complete divisibility, the algebra permitted in the universe must consist solely of multiplication.
The entity holding such conditions is closed octonions—purely imaginary $S^7$.
$S^7$ contains no zero to begin with.
2. The Geometry of $S^7$ and the Birth of Physics
$S^7$ contains three associative imaginary axes and four non-associative imaginary axes.
The three associative imaginary axes possess properties akin to space.
The three-dimensional space we know well must relate to these three associative imaginary axes.
Considering that $S^7$ is closed, it corresponds to spacetime metrics.
The three associative imaginary axes of $S^7$ determine the directions of three-dimensional space.
$S^7$ has a fatal weakness: it is non-commutative and non-associative, imposing severe constraints on operations.
Physics cannot be constructed with an algebra where one cannot move.
Now, what is the true nature of what we call physics? Phenomena we consider as physics boil down to the fact that interactions between two states serve as the minimal unit. In the world of $S^7$, does there not exist an operation that guarantees at least associativity?
Emil Artin stated: "Even if the entire algebra is non-associative, the subalgebra generated by any 'two elements ($a$ and $b$)' satisfying specific conditions is associative."
In $S^7$, that condition is sharing a single imaginary axis.
Sharing a single imaginary axis in $S^7$ corresponds to connecting two states with a null condition.
In special relativity, a null condition means connecting two points in spacetime with a single ray of light.
It is the condition for "observation," and it is also the condition for "interaction."
Of course you knew that, because observation is carried out through interaction.
And! From the algebra between two points connected by Artin's theorem, non-associativity is removed, creating room for physics to be established.
But! It is still non-commutative.
In a non-commutative algebra, operations occur through coproducts.
That is! Between two points connected by the simultaneity of special relativity, a coproduct algebra is established!
Coproduct algebra is otherwise known as quantum algebra.
Oh! This is truly a miracle!
Special relativity restores physics to $S^7$ and leaves behind quantum mechanics.
3. Relativity, Entanglement, and Light
Quantum mechanics carved out from $S^7$ provides the story that "two states are connected and interact via the concept of simultaneity in special relativity."
In other words, two states classified into cause and effect are simultaneous from the perspective of interaction, explaining that the temporal order impressed upon us by the distinction between cause and effect does not actually exist.
Because two states in four-dimensional spacetime have different proper times, their intersection angles with the interaction segment differ.
This difference in angles gives rise to the distinction between cause and effect.
Since special relativity denied the existence of a common clock between two locations, discussing the temporal precedence or succession between distant points was fundamentally impossible to begin with.
When viewing another location from one point, we were merely labeling it as "past" or "present" depending on the angle relative to our own proper time.
If we redefine the relationship between cause and effect in this way, quantum entanglement ceases to be a mysterious phenomenon at all.
When the angle of the interaction segment connecting two points is controlled by a special operation, we call it entanglement.
At the end of this scenario awaits a shocking conclusion.
Namely, light does not travel through space at the speed of light; it is not even a particle to begin with.
There is no actual entity moving.
Light is the bearer of interaction, and because we recognize that an interaction occurred, we perceive that light traveled between them.
Therefore, we have never once actually seen light.
What we recognize is solely the fact that an interaction took place.
Particles that transmit force might inherently be things like this.
4. Mass, Geometry, and History
Three axes of $S^7$ are associative, and the remaining four are non-associative.
The three associative ones are relatively free, but severe conditions are imposed on the four non-associative ones.
In ordinary interactions, the non-associative part behaves like a constant.
Since the norm of $S^7$ is fixed, the norm of the four non-associative axes acts as a "weight" relative to the norm of the three associative axes.
In other words, the norm of the four non-associative axes will appear as mass.
And the paired three associative axes correspond to momentum.
Spacetime may be born as the Fourier dual of $S^7$.
If so, the four non-associative axes will produce a slight distortion in spacetime—a distortion roughly proportional to their norm.
Could inertial mass and gravitational mass ever possibly differ?
Ruth Moufang pointed out that the subalgebra carved out by Artin's theorem possesses a discrete, cyclic associative law called a Moufang loop.
The Moufang loop has characteristics resembling rotation.
However, because that rotation exists outside the concept of time, it would appear as rotational motion exceeding the speed of light.
This would correspond to how the properties of elementary particles, such as electron spin, differ completely from ordinary physical laws.
What we can observe must correspond to the invariants of those rotation operations.
In $S^7$, Artin's cuts produce a special geometric arrangement.
Known as the 16-cell, it forms—through the Moufang loop—a miraculous figure called the 24-cell. $S^7$ contains $SU(3)$ symmetry, but when carved out by Artin's theorem, the symmetry becomes incomplete.
But! The 24-cell reconstructed by the Moufang loop contains $SU(3)$ in a discrete form.
$SU(3)$! Discrete! This is precisely the hallmark of elementary particles!
Although the interactions carved out by Artin's theorem are associative, $S^7$ itself remains non-associative.
But do not despair.
The non-associativity of $S^7$ explains a very important fact: namely, that the universe has history.
Non-associativity means things like $(AB)C \neq A(BC)$.
A state where a glass filled with milk is dropped on the floor is by no means the same as a state where the milk is spilled first and then the glass is dropped.
The universe clearly possesses such a property, and that property can be said to be none other than non-associativity.
5. This is just our wish.
Our discussion has expanded greatly, but please remember.
Our starting point was simply the conditions for the universe to be an understandable entity. That was my humble wish.
That was all.
To understand the universe is nothing less than continuing to hold onto that single, solitary wish.
