This words analysis has to do with negations. It is a problem that has been hashed to death, and in terms of the historical analysis I have little to add to Bill Welden’s thorough discussion in VT42 and VT44. Rather, this is my attempt to reconcile the various negative-paradigms Tolkien used throughout his life into a form that is internally consistent without requiring us to discard a large numbers of attested words.

To summarize the basics, by the end of Tolkien’s life he had used essentially three distinct roots for negations: BĀ, LĀ and Ū (or UG/GŪ). These roots may or may not be invertible, depending on when Tolkien used them.

BĀ is the easiest to work with. This root has to do with willful negation, refusing or forbidding. It requires a sentient entity performing the act of negation, and can be used this way in both Quenya and Sindarin in the derived particles: Q. vá, S. baw, as well as verbs for refusal or forbiddance: Q. and S. ava-. This verb was fossilized in Sindarin, nearly reduced to a negative adverb. In Quenya it could also be used as a negative prefix, but almost exclusively for negatives that involved some intentional action as in Q. avalerya- “to restrain, to deprive of liberty” or avaquétima “unspeakable” in the sense of “that which must not be said”, as well as Q. váquet- “to say ‘I will not or do not’; to refuse, forbid, prohibit”.

LĀ was Tolkien’s basic negative root in the 1930s and 40s, and probably remains the most popular form of negation in Neo-Quenya. It was used for a “simple negative”, absence of fact. Q. láquet- simply means “to say ‘it is not’, to deny a fact”. As a prefix, especially in Quenya, it seems to represent a simple absence of an attribute or entity, without any addition connotations. It could be used as an adverb, prefix and (semi-)verb in Quenya at least. In Noldorin/Sindarin there are few examples of it, almost exclusively in the form of the prefix al- “no, not”.

Ū: In the 1930s there was another “strong negative” prefix ú- derived from UGU/GŪ. It came to have an “unpleasant” connotation (at least in Quenya) in association with roots like UMU/MŪ which had derivatives like ᴹQ. úmea “evil”. As an example, at one point Tolkien defined Q. úquétima “unspeakable” in the sense of “impossible to say or put into words, unpronounceable”. This LĀ vs. Ū weak/strong paradigm was also reflected in Early Elvish negative elements u vs (syllabic) ḷ, except that in the earliest period the meaning of the two were reversed.

In the 1950s, LĀ as a negative fell out of favor with Tolkien, in part because he increasingly used the root ALA with the meaning “good, blessed, fortunate” (this root also appeared in the Etymologies, but was barely used). The prefix al(a) came to mean “well, happily” in both Quenya and Sindarin, and was thus no longer suitable as a negative prefix. In this period he abandoned LĀ in favor of using Ū as the basic negative prefix. The only “canonical” appearance of negative LĀ in LotR was in S. alfirin as the name of a flower; it was unglossed in the text, but as indicated elsewhere it was clearly intended to mean “immortal”.

In the 1960s he changed his mind yet again (VT42/33; PE22/153, 160). This time Ū fell out of favor and he restored LĀ as the basic negative. The sense of Ū was modified to mean “difficult” or (as strengthened in Sindarin) “impossible”, becoming more of a pseudo-negative. Then in typical Tolkienesque fashion he vacillated at least one more time: flipping away from LĀ and back to Ū (VT44/4, 38). Tolkien changed his mind so many times and wrote so many confusing notes that it’s nearly impossible to figure out what his “final” decision was (if in fact there was one).

Many Neo-Quenya authors tend to favor either Ū over LĀ or vice versa. However, I think the best option is to restore the weak/strong negative paradigm of the 1930s, using LĀ for a “simple negative” and Ū for a “strong negative”, at least in Quenya. The root ALA “good, blessed, fortunate” is too useful to discard, however. As such, I think LĀ “not” should be treated as a non-invertible root, limiting its prefixal form to la- “not” rather than ala-. This allows the prefix al(a) to retain the sense “well, happily”.

For Ū, I think it should only be used for a “strong” negative. It can have senses like “impossible”, “anti-, opposite”, “hard, difficult” or “bad, unpleasant”. As such, it really only makes sense when used to negate something that can have an opposite: **úcarne “anti-red” is nonsensical. It is also generally used to negate positive things, so that Q. úmare “anti-good” = “evil” and úvanima “anti-beautiful” = “ugly, hideous”, but **únorto “anti-horror” (= “?calm”) would feel strange and ungrammatical.

Thus, in Quenya, the three common negative particles are:

lá “generic no, not”
ui “emphatic no, not”
vá “volitional no, not”

In answer to the question Ma tuluval enar? “Will you come tomorrow?”:

lá “no” (I’m simply not coming)
ui “definitely no; on the contrary” (in fact, I’m going away to Gondor)
vá “no; I refuse” (you can’t make me come)

All three can be used as “semi-verbs”, inflected for person before another verb:

lán tuluva “I will not come”
uin tuluva “I will definitely not come”
ván tuluva “I will refuse to come”

The subordinate verb generally has the inflection for tense. The semi-verbs are only inflected for tense when they stand alone:

lánen “I was not”
unen “I was definitely not”
avanen “I refused”

The particle vá is never itself inflicted for tense, since there is a distinct verb Q. ava- “I refuse, forbid” that may be fully inflected and used normally.

The three main negative prefixes are:

la- or lá- “not, un-” (as noted above, I would avoid ala- as a negative in Quenya)
ú- “strong not, anti-”
ava- “(volitional) not”

For example: laquetina “not said”, úquétima “unpronounceable”, avaquétima “forbidden to say”

In Sindarin, the situation is a bit different.

BĀ: Like Quenya, derivatives of this root can be used for volitional negation. As noted above, S. ava was fossilized as a negative adverb, functionally similar to Q. vá. S. baw (I refuse!) could only be used as an interjection or in answer to a question, and the Sindarin verb for “to refuse, forbid” was boda-. The Sindarin prefix av- was sometimes used in the same way as Q. ava-, but less commonly.

LĀ: Tolkien never developed a full LĀ-paradigm for Sindarin. Almost all the attested Sindarin forms use u- for simple negation, generally as a verbal prefix. Some Neo-Sindarin authors coin a neologistic verb ᴺS. law-, and use the same triple-negative paradigm as in Quenya. However, I think it is better to assume that LĀ was abandoned in Sindarin in favor of Ū, surviving only as a (rare) negative prefix al-. Given its likely association with ala- “well, happily”, I think this Sindarin prefix might be used only to negate unpleasant things, such as alfirin “not dying = immortal”. In this way, al(a)firin “well-dying” might be interpreted as “not dying” (which is more desirable than death itself).

Ū: Owing to the frequent interchange of the vowels o and u in the phonetic development of Sindarin, I think the association of ú- with unpleasant roots either never developed or was lost in Sindarin, and it was weakened to serve as a more general negative element, replacing LĀ. It was used as uninflected ú- before verbs causing soft-mutation (as in ú-chebin), though it could be inflected for person and tense if used alone (únen “I was not”, uin “I am not”, ?uathon “I will not”). It also served as a general negative prefix ú- (often reduced to u- when unstressed), and û “no” could be used in answer to a question.

This interpretation of S. ú- means ignoring Tolkien’s late 1960s notes on ú-chebin indicating that it is a properly a strong negative “impossible to keep” (PE22/160). However, this interpretation is consistent with the majority of attested Sindarin negatives and requires no neologisms. If you aren’t happy with it, you can use the neologism ᴺS. law- and the same triple-negative paradigm as Quenya, but I personally find that less satisfying.