The origin of the "auxilliaire de négation pas" is not clear. I still found a page in Google Cache for Finnish student learning french. It also seems to be kind of confirmed by the Wiktionary. To be noted, the real negation is the "adverbe ne" and not the auxiliary.

[FR] Le mot pas est au départ un simple nom qui servait à renforcer la négation. Il subsiste d’autres noms de ce type, mais nettement moins fréquents : ne … point (= ne … pas), utilisé dans langue soutenue ou dans un usage régional ; ne … mot, ne … goutte sont littéraires et se sont conservés dans quelques expressions plus ou moins figées :

[EN] The word pas was originally a simple noun which was used to reinforce the negation. It persists other noun of this type, but less frequent : ne … point (= ne … pas), used in formal language or regional usage; ne … mot, ne … goutte are literary and have been conserved in fixed expressions.

Il n’y a point d’autre solution.

Il n’a dit mot.

On n’y voit goutte.

Je n’y comprends goutte.

Grevisse:

Primitivement, la négation portant sur un verbe est exprimée par l’adverbe ne, et celui peut encore suffire dans certaines circonstances. Le plus souvent cependant, on recourt à la fois à ne et à un auxiliaire, adverbe, déterminant ou pronom. Ces auxiliaires, qui n’avaient pas de sens négatif à l’origine ont fini par prendre eux-même ce sens négatif et par pouvoir s’employer négativement sans ne.

[…]

La forme atone ne [de non qui provient du latin non], négation ordinaire dans la langue du moyen âge, a été de très bonne heure renforcée par des noms désignant une petite quantité, une petite étendue, une chose de valeur insignifiante : pas, point, mie (= miette), goutte, mot, noix, pois, espi, dé, bouton, denier, pomme, […] Les plus fréquents de ces compléments ont perdu leur valeur propre pour devenir de simples auxiliaires de la négation. Le choix ne s’est pas opéré de la même façon partout : il y a des dialectes où l’auxiliaire normal est point ; d’autres où c’est mie ; etc.

Il y a 16 pages qui suivent pour décrire les manières de nier un verbe dans mon édition du bon usage plus 6 qui traitent du ne explétif. J’ai eu l’idée un temps de les résumer, j’abandonne.

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English summary: in old French, ne was used alone, and it still can be sometimes. Soon additional words were used for emphasis. Nowadays those additional words often must be used. While they didn’t have a negative meaning at the start, they acquired one and may now be used in some contexts without ne in their negative meaning.

There are 16 pages describing the way to negate a verb in my edition of le bon usage, plus 6 describing the ne explétif where ne isn’t used to negate. I played with the idea of summarizing them, but dropped it.

(Copying parts of another answer of mine.)

Originally, ne is what makes the negation. The etymology of “pas” and “jamais” are exactly the opposite of their current meaning because of this. The only remainder (and cause of ambiguity is “plus”). There is a direct parallel with not and ever/more in English.

For example, “Je (ne) mange jamais de bonbons” (“ne” is rarely used in speech) would mean “I (n)ever eat sweets.”. No one today would understand jamais in this context to mean ever or always: it’s always understood as never. However, there are cases where the “ever” meaning is still used. For example: “Il court plus vite que jamais.” (More than ever).

I’m not aware of any similar example with “pas” (although my understanding is that it comes from similar Latin roots and constructs).

There is still an ambiguity when you say “J’ai plus de bonbons.“

• If you pronounce “plusse”, it’s meant as it’s written (“I have more sweets.”)
• If you pronounce “plu”, ne is implied and should be written “Je n’ai plus de bonbons.” (i.e. no more).

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In addition, linking ne to the verb allows you to negate verbs in a way that can be quite tricky in English. For example:

• “Je peux ne pas faire quelque chose.” -> “(I may) (not do something)”, which is completely the opposite of “I may not do something” (i.e. I’m not allowed to).
• “Ca peut ne pas marcher pour telle ou telle raison.” -> “(It can) (not work)”, whereas “can not” (or cannot) would have a completely different meaning.
• In English, you also get split infinitives (at least in common speech) quite regularly because of this. “Je fais ceci pour ne pas faire ca.” would often be said (incorrectly) as “I do this to not do that.”

In the oldest time, you could have the “strong” form non, hence nonobstant, nonchalant, non-recevoir… But the most common form of the negation was in fact ne alone. Eventually emphatic elements were added, which varied depending on context: Je ne mange mie (“I don’t eat a crumb”), je ne vois goutte (“I can’t see a drop [of water]”), je n’avance pas (“I don’t move a step”), je ne dis mot (“I don’t speak a word”)…. most of those are still considered valid literary usage.

Eventually the “auxiliary” (as Grevisse calls them) to the negation became obligatory in many contexts. Grevisse (Le Bon usage., 14e ed., §§1012-1019) does not explain exactly why pas evolved from being used with verbs of movement to become the most common of these auxiliaries, then the default one. The TLFi (Trésor de la Langue Française informatisé), however, notes that the usage dates right back from the 12th century, so although it was not the standard of negation at the time, it is a feature of French almost from its birth.

Some dialects have gone in different directions and have other adverbs as the default auxiliary. If you want to get into the real nitty-gritty, you’ll probably need to look at more technical literature like the bibliography in the TLFi’s entry

From a linguistic viewpoint, the negation isn’t split, merely transformed immediately before final utterance. Ne is phonologically a single word, and historically a single syntactic structure, as is pas. Today ne . . . pas, while two words, is a single syntactic structure. The redundancy improves intelligibility, and the separation serves to delimit the verb or verbal. My French is passable, but I am not a native speaker, and will have to depend on parallels in English.

Two analogous structures exist in English: the progressive and the perfect. Applying each in turn to they go yields they are going and they have gone. In both cases the verb is encapsulated within a single auxiliary structure, as French verbs are encapsulated in ne . . . pas. Linguists write this as a transformation of something called a K-terminal string:

they + pres. + (be + ing) + go –> they + (pres. + be) + (ing + go) –> they are going

they + pres. + (have + part) + go –> they + (pres. + have) + (part + go) –> they have gone

You’ve got to have tense, and be takes tense, and the subject is they, so it’s are. The verb is go, and all verbs take ing, so it’s going. The structure (have + participle) works the same way.

The foregoing may look a little brain-damaged, but it’s remarkably efficient. The whole rule for auxiliary in English (neglecting passive and subjunctive, which are sentence transformations) is:

Aux => (tense) + {modal} + {have + part} + {be + ing}

where the parentheses indicate that tense is obligatory and the curly braces indicate that everything else is optional.

So with two tenses, past and present, and five modals–will, shall, can, may, must–you can generate a pageful of ways to link a verb with its auxiliaries, including the more recondite forms such as would have been giving. Note that this rule doesn’t link the whole verbal to the subject–that’s a different animal called the s-form rule, which is applied over the auxiliary.

To negate an affirmative sentence in modern English, the transformation is quite simple: apply not after the first word in the auxiliary. It goes there because the auxiliary delimits the truth of the whole sentence, both in time and condition. In the cases of simple past and present, when you haven’t got an auxiliary form for not to follow, you can put it after a be, but not a verb. You have to supply a do for the not to follow. If you simply apply the Do transformation to an affirmative sentence, that will emphasize its truth value, but a do followed by a not serves merely as a sort of pre-affirmation.

Pas du tout and similar forms like pas de problem don’t negate sentences, so the structure ne . . . pas is not required. The postfixed emphatic pas is analogous to the prefixed English emphatic do, and the archaic use of ne without pas is analogous to archaic English forms like “Say not the struggle naught availeth.”

I lighted upon this question at r/asklinguistics:

TomSFox.
25 points 2 years ago

There is a phenomenon called the Jespersen’s Cycle, whereby the force of the negative adverb weakens over time and has to be reinforced by an additional expression, for example by saying something like “not one bit” (as in, “I did not like it one bit.”) or, as was the case in Latin, “not one step” (“nec … passum”), which over time evolved into “ne … pas” in French.

McWhorter, J. PhD Linguistics (Stanford) expounds on pp. 26-27 of The Power of Babel (2003). I post a screenshot; I don’t know how to replicate its formatting with SE’s tools.

It’s a manifestation of Jespersen’s cycle.

Old English and Old French negated words with a simple particle, inherited directly from Proto-Indo-European:

• (Old French) io ne marche
• (Old French) io ne meniu
• (Old English) ic ne cnāwe

But the speakers of Middle French and Middle English came to think of the negation particle as being too difficult to hear, and so augmented the negation by adding an additional word:

• (Middle French) je ne marche pas
• (Middle French) je ne menje mie
• (Middle English) i ne knowe naht

Originally, pas was used with movement words, and other words were used in other contexts, like mie ‘crumb’ and goutte ‘drop’. But eventually, pas was generalized as the negative word, and became mandatory, while English dropped the original negation entirely.

• (Modern French) Je ne marche pas.
• (Modern French) Je ne mange pas.
• (Early Modern English) I know not.

The standardized varieties of French preserve this situation, while the colloquial varieties follow English down the cycle by dropping ne entirely. (Although English took a detour by requiring not to be attached to do.)

• (Colloquial French) Je marche pas.
• (Colloquial French) Je mange pas.
• (Modern English) I do not know. -> I don’t know.

The same thing happened in all other Germanic languages, ich weiß es nicht, ik weet het niet, jeg vet ikke, and it’s easy to imagine a future form of French losing ne entirely, colloquial or not. But who knows.