Grammatical structure of "cease to exist"
Merriam-Webster lists cease to exist as an example of a transitive use of the verb: https://www.merriam-webster.com/dictionary/cease
Dictionary.com doesn't include the category transitive/intransitive verb, but it lists cease to exist as a use of the verb without an object (thus making it intransitive?): http://www.dictionary.com/browse/cease
I'm pretty rusty on my verb-verb interactions. I have: cease (transitive verb) to exist (infinitive acting as direct object).
My answer will answer your main question, and then provide a hint of direction to your further research.
The phrase {cease to exist} can be structurally perceived either
Ceasing to-exist. Where the phrase {to-exist} is taken as a composite verb. In this context, {cease} is used transitively.
Ceasing in order to exist. e.g. He ceased, to exist. In this context, {cease} is used intransitively.
The phrase {to exist} is due to a class of verb conjugate (actually, absence of conjugation) known as infinitives.
Infinitives are found in every language that has temporal verb inflection (i.e. time-based inflection or tenses). Note that {temporal} means time-based, rather than being temporary.
Infinitives are a subclass of nonfinites.
Nonfinites have the fundamental property of not being inflected temporally. Various subclasses of nonfinites are also non-conjugated in number, gender or any of the various categories of grammatical elements.
In English, a phrase or word can either be
- inherently structured as nonfinite
- incidentally used as non-finite due to an adjunct or auxiliary finite-verb, where rules against redundancy in the pairing of the two verbs strip off the finite mode of the main verb.
- e.g., He will eat his hat.
The existence and usage of nonfinites is due to functional modularity (aka -laziness is the mother of inventive necessity). Functional modularity is excessively used in software programming, hardware, IC design, manufacturing processes, such that a particular concept is modularised into a function that would require/allow minimum or zero modification/inflection to be reused/deployed anytime and almost anywhere. Statelessness is a form of nonfiniteness.
Participles, gerunds and verbal-nouns are nonfinites (Difference between Brand (noun) and Branding (gerund?)).
Nonfinites facilitate virtuous laziness. For example, these nonfinite phrases do not need to be inflected temporally and/or numerically:
- He came here to die. They are coming here to die. She will come here to die. We might come here to die.
- The painting was yours. Now the painting is mine. Soon, the painting shall be hers.
A rather interesting class or plane of nonfinites are subjunctives. Imagine the mathematics of real and imaginary numbers, where in applied mathematics a number is usually a combination of real and imaginary. Subjunctives are the imaginary conjugate storylines to the real world - they operate in the imaginary realm. Like their mathematical counterpart, they are a continuum, with infinite possibilities of combination between real and imaginary worlds.
However, in those deprecating traditional schools of grammar, they had been struggling to deal with that continuum by classifying subjunctives into distinct buckets of quantization, buckets that they keep on increasing, like propositional, wishfulness, doubtfulness, contingency, impossibility, narcissistic, concession, exhortative, imperative, blah, blah, blah. Not aware that subjunctivity is actually a continuum.
The new mathematically precise perspective is not simply {transitiveness} of a verb, but its {valency} https://en.wikipedia.org/wiki/Valency_(linguistics).