Q:

Scope of negation with absolute constructions

In Latin and Greek, when a negator appears in an absolute construction (ablative absolute, genitive absolute), it is generally taken to negate the predicate within that construction:

hostibus impetum facientibus "the enemy attacking"
hostibus non impetum facientibus "the enemy not attacking" 

Do we ever get instances of a negator applying not to the predicate, but to the whole absolute construction? A made-up example of what I mean:

Caesar se recepit non hostibus impetum facientibus, sed fessis suis. "Caesar retreated not because the enemy were attacking, but because his men were tired."

Here non is semantically outside the absolute construction rather than applying to a constituent within it.
The same question could be asked of Greek genitive absolutes; in fact this question occurred to me reading an answer by Draconis to a Greek question, in which he translates οὐ παραμιγνυμένου αὐτῷ κασσιτέρου as "not because it's mixed with tin" (οὐ negating the whole construction) rather than "because it's not mixed with tin" (οὐ negating only the predicate).
My sense is that for both Latin and Greek, this kind of usage is either nonexistent or very rare, and that meanings like these would be expressed with some other construction (e.g. non quod, οὐχ ὅτι). Thoughts?
ETA: prodded by the examples in Mitomino's answer, I'm now thinking this is really a question about information structure rather than negation. Maybe what's odd about my made-up example above is not that it's negated, but that it's acting as the focus of the sentence. In that case the question can be recast as "Can an absolute construction be focused?" Answers to this question would be equally welcome.

A:

What follows is not an answer but just some initial thoughts related to your question. My first impression/intuition is like the one you express at the end of your post. I'd be surprised to find examples that follow the specific schema you suggest (i.e., "non AA sed AA") in a classical author like Cicero. However, I must also say that I would be less surprised to find this sort of data in Tacitus. In the latter (but, as far as I know, not in the former) one can find, for example, causal ablative absolutes (turbatis omnibus) coordinated with causal clauses introduced by the conjunction quod:

Nec dissolutio nauigii sequebatur, turbatis omnibus et quod plerique ignari etiam conscios impediebant (Tac. Ann. 14.5). 

Given the existence of this sort of coordination, one could perhaps also find a sequence like "non + Ablative Absolute sed quod ..." in Tacitus (but, as noted above, according to {my intuition/my readings}, not in Cicero). If so, notice that the scope of non would be on the full Ablative Absolute (and not only on the predicate included in it).  

I've just taken a look at Pinkster's (2015) Oxford Latin Syntax (OUP). When dealing with the scope of negation in ablative absolutes, he gives the following relevant information:
"Most often the negation concerns only one or more words in the clause to which it belongs. The constituent in question is either a contrastive topic or a focus (...). In example (k), the ablative absolute clause me ... auctore is the scope of non (note quidem) (...) As the examples show, the scope may be on a nuclear constituent (...) or on an adjunct (me auctore). The scope may concern one word or a subordinate clause. Ex. (k) illustrates an ablative absolute clause" (Pinkster 2015: 676-677; bold mine: Mitomino). 
(k) Non me quidem / faciet auctore hodie ut illum decipiat (Pl. St. 602-3) 
Unfortunately, as you have seen, the discussion above is exemplified with a sort of lexicalized ablative absolute: me auctore. More interestingly, on page 715 Pinkster gives an intriguing example, where the negation, due to its external position in the coordination, could be claimed to take scope on the full Ablative Absolute (and not necessarily only on the predicate included in it). I add Pinkster's translation below.

...cum C. Sulpicius et C. Licinius Calvus consules in Hernicos exercitum duxissent neque inventis in agro hostibus Ferentinum urbem eorum vi cepissent, revertentibus inde eis Tiburtes portas clausere (Liv. 7.9.1).

'(...) the consuls Gaius Sulpicius and Gaius Licinius Calvus led an army against the Hernici, and not finding the enemy abroad, captured their city of Ferentinum by assault. As they were returning thence, the men of Tibur closed their gates against them'.

