Why is a sotāpanna defined as unable to take another teacher and unable to commit matricide?

What is "Entering the Stream"

Quoting from the Saṃyutta Nikāya

holding masses, ( matter, feelings, perceptions, intentions, and consciousness).

“Monks, when the noble disciple knows the arising, fading, satisfaction, danger, and the escape from these five holding masses as they really are,it is said he has entered the Stream of the Teaching, would not fall from it, would directly aim extinction.”

What are the "Rewards"

Quoting from the Anisansa Sutta: Rewards

One is certain of the true Dhamma.
One is not subject to falling back.
There is no suffering over what has had a limit placed on it.
One is endowed with uncommon knowledge.
One rightly sees cause, along with causally-originated phenomena.

What are the "Impossibles" that "Stream enterer" cannot do.

What are the signs (or proofs) that one has entered the stream?
Quoting from the Aṅguttara Nikāya

Impossible

The view about a self, (Identical View),
doubts,(About Teacher, Doctrine and Community),
grasping to the idea of breaking the rules and greed (Accept another teaching),
hate and delusion (Karma) which lead to hellish results.

– Third on impossibilities

It is impossible that one come to right view should. deprive the life of mother, father, of a noble one, with a defiled mind cause the blood of the Thus Gone One to spill, split the Community of bhikkhus, or appoint another, as teacher.

What is the reason?

The one come to right view, has thoroughly seen the reason that pleasantness and unpleasantness arises on account of some reason.

The reason why a stream-winner cannot commit matricide etc. is because there is no such stress with him that capable enough to matricide etc.

Most normal persons when get angry they behave without self control. A sothapanna has no wrong view such as "The view about a self, doubts, grasping to the idea of breaking the rules and greed, hate and delusion which lead to hellish results." So there is no reasons to get angry.

I have an armchair philosophy answer to this one, (i.e. I can't cite any one on it, but it makes sense).

One of the following is different than the other.

Murdering one's own mother. Murdering one's own father. Murdering an arahant. Maliciously injuring the Buddha to the point of drawing blood. Deliberately creating a schism in the monastic community.

Killing parents is incredibly rare-- a quick google is showing small 2 digit numbers for typical large countries. In the time when these rules were made up, the population was probably too small to find more than one or two a year or decade. Arahats and Buddhas were exceedingly rare-- even if you wanted to kill one, there weren't any available.

The bit about wounding was because at one point someone made up the "fact" that you couldn't kill a Buddha, so in that case, you could only wound them.

But schism? It was a something that the leaders of Sanghas worried about. Someone who was intent on committing a crime, could commit this one, the rest are just not something anyone does or can do. This list of rules essentially says, schism is as bad as the worst imaginable crime. This list is normally combined with the penalty of never being able to reach enlightenment. The particular enumerated crimes are rhetorical flourishes.

(Taking up a new teacher is just a variation on the rule against schism, I don't see it as often.)

In sum, the whole point of this list of rules is to say "Schism is the worst crime imaginable" It is a self serving rule that would have been made by a leader worried about challenges to his authority. Me not being a monk, when I first saw this rule (which gets told to lay followers as well), I thought, how quaint, a list of rules that are near impossible for me to personally violate.

A stream winner is not capable of creating schism and taking another teacher, because a stream winner has verified confidence (i.e. conviction resulting from experiential realization) in the Buddha, the Dhamma (his teachings) and the Sangha (the monastic community).

From the Vera Sutta:

"And which are the four factors of stream-entry with which he is endowed?

"There is the case where the disciple of the noble ones is endowed with verified confidence in the Awakened One: 'Indeed, the Blessed One is worthy & rightly self-awakened, consummate in knowledge & conduct, well-gone, an expert with regard to the world, unexcelled as a trainer for those people fit to be tamed, the Teacher of divine & human beings, awakened, blessed.'

"He is endowed with verified confidence in the Dhamma: 'The Dhamma is well-expounded by the Blessed One, to be seen here & now, timeless, inviting verification, pertinent, to be realized by the wise for themselves.'

"He is endowed with verified confidence in the Sangha: 'The Sangha of the Blessed One's disciples who have practiced well... who have practiced straight-forwardly... who have practiced methodically... who have practiced masterfully — in other words, the four pairs, the eight individuals [1] — they are the Sangha of the Blessed One's disciples: worthy of gifts, worthy of hospitality, worthy of offerings, worthy of respect, the incomparable field of merit for the world.'

The Sangha referred above is composed of eight types of individuals:

The four pairs are (1) the person on the path to stream-entry, the person experiencing the fruit of stream-entry; (2) the person on the path to once-returning, the person experiencing the fruit of once-returning; (3) the person on the path to non-returning, the person experiencing the fruit of non-returning; (4) the person on the path to arahantship, the person experiencing the fruit of arahantship. The eight individuals are the eight types forming these four pairs.

It doesn't say that those are offences. It says that a stream winner is not capable of these acts. Since a stream winner is fully convinced in the Buddha and his teachings, he is not capable of taking another teacher. A stream winner has also reached the level of spiritual maturity that makes him incapable of committing matricide, patricide and murder of an enlightened person.

Not just matricide, patricide and killing arahant, according to the Cakkhu Sutta, the stream winner "is incapable of doing any deed by which he might be reborn in hell, in the animal womb, or in the realm of hungry shades." Also in this verse, The Buddha said "the sotapannas do not kill, they do not wish others to get killed."

I have a theory or opinion as follows (not a canonical explanation). "Stream-winner" is one of the stages of enlightenment. Essential or defining characteristics of a stream-winner are:

• Understands the Buddha's teaching sufficiently well
• Will therefore become enlightened within a few lifetimes (e.g. within seven lifetimes) furthermore none of those lifetimes will be in hell.

The "impossibilities" which were listed follow from these characteristics:

• Because they understand Dhamma sufficiently well, they won't split the Sangha and won't seek another teacher. It may be a slightly circular definition, in that the ariya-sangha is defined as being composed of stream-winners etc. (i.e. those who attained those various stages enlightenment).
• Because they're enlightened they won't do things which will cause them to be reborn in hell; matricide is an example of an action which will cause a person to be reborn in hell; so the syllogism is that they won't commit something like matricide.

There may well be (and presumably is somewhere) a commentary, but I don't know it. Shrawaka's answer shows the list is taken from the Anguttara Nikaya ... that doesn't add much context except what we already know i.e. that it's a list of six things (because the Anguttara Nikaya consists of lists).