"Well, I don't remember."
Excerpt from 'Japanese Words Their Uses II: The Concise Guide to Japanese Vocabulary Grammar: Learn the Japanese Language Quickly and Effectively: 2':
To express the idea of 'retaining something that has been committed to memory,' one has to say oboete-iru rather than oboeru. American students of Japanese often make the error of identifying oboeru with 'remember' and make sentences such as (3), but (3) is a misrepresentation of (4) and cannot mean 'I don't remember his name.' / (3) Ano hito no namae wa oboemasen./ / (4) Ano hito no namae wa oboete-imasen./ I don't remember his name. Sentence (3) can be correct only in the sense of 'I won't (i.e., I refuse to) commit his name to memory.'
Please remember, to express current states in Japanese, we must use the progressive form ～ている. Progressive form represents both the continuous action as well as states (where in English normally we use present or perfect tense to represent).
(私は)覚えます I am going to remember. / (私は)覚えています I remember.
(私は)覚えません I am not going to remember. / (私は)覚えていません I do not remember.
映画が始まります The film is going to start. / 映画が始まっています The film has started.
映画は始まりません The film is not going to start. / 映画は始まっていません The film has not started.
車が止まります The car is going to stop. / 車が止まっています The car has stopped.
車は止まりません The car is not going to stop. / 車は止まっていません The car has not stopped.
If you really want to say "The car is stopping," 車が止まりかけています or 車は止まる途中です.
おぼえません is not accepted. 覚えていません is kind of progressive form (hopefully I am right - please correct me otherwise). Why is the progressive form needed here? Why is the "normal" negative form not accepted? Should I report it as another possible solution?
覚えません would not be correct. 覚えていません is correct. 覚えていません means that one 'is not remembering', while 覚えません means 'to not remember'. ～ていません is the present progressive, while ～ません is more the simple present(?) / negative infinitive(?). Sorry, grammar is not my strong suit.
Actions which are current and ongoing are expressed using the ～ていません form. ならっていません (I'm not learning), べんきょうしていません (I'm not studying). Basically, it translates to 'I'm not remembering', but this is how 'I don't remember' is expressed in Japanese. If you were to say おぼえません it would mean that you don't remember things in general.