As I understand it, certe can mean ‘at least’, as you write, but both certo and certe means ‘certainly’, albeit in slightly different ways.
Certe scio : Certainly I know it. (I affirm I surely know about it, in case someone thought that I was ignorant.)
Certo scio : I know it for certain. (I affirm I have in-depth knowledge of the matter.)
But I wouldn’t be surprised if you find examples where this distinction doesn’t seem to apply. Also, compare what Lewis & Short has to say about the matter. (I got more confused than I was before after reading it.)
Indeed, the ablative of “certum” and the adverb “certè” have come to be synonyms, but you recognise the same subtle difference in English (“with certainty” and “certainly”). Note: “I certainly know that” (the verb carries the weight and you can’t tell if “certainty” has special emphasis) and “I know that with certainty” (unambiguous special emphasis on the degree of knowing or “certainty”). L&S illustrate that as follows (but in Latin): “Certainly I know by Pollux, if I should believe or know for certain anything else whatsoever.” (You wouldn’t substitute “certainly” for “for certain” there in English and you wouldn’t in Latin either, apparently.)
Rectè dicitis “certo” (“certum” verbum casu ablativo) et “certè” (adverbium) synonyma facta esse, at subtilem differentiam notate, quae Anglicè quòque stat: “with certainty” et “certainly”. Locutio “with certainty” emphasin propriam habet quam sic verbis Plauti Lewis atque Short ostendunt: “certe edepol scio, si aliud quicquam est quod credam aut certo sciam, etc.”, Plaut. Am. 1, 1
“Send the judge money and you will surely loose your case”
Da iudici pecuniam et certe causam perdes = Certo, da iudici pecuniam et causam perdes > Da iudici pecuniam et certo causam perdes – unless you want to emphasize elsewhere that it is a relief to know with certainty that your case will be lost, in my opinion!!!
It is certainly possible that it is much a question of frequency, and that whatever difference has been deduced from that is more or less artificial. My main source for my statement was Antibarbarus der lateinischen Sprache by Krebs & Schmalz (Basel 1905):
certe > und > certo> . Jenes wird bei allen Verben angewandt, > certo > aber beschränkt sich in der bessern Prosa nur auf > scire; > der Unterschied zwischen > certe scio > u. > certo scio > wird dahin bestimmt, dass ich sage > certe scio, > wenn ich von > mir > versichern will, dass ich etwas weiss — > ja gewiss, in der Tat, wahrhaftig ich weiss es; > aber > certo scio, > wenn ich angebe, > wie > ich weiss, > mitt Gewissheit, > als etwas gewisses und von meiner Seite unbezweifeltes. Cicero bevorzugt > certo scio, > allein > certe scio > kann ihm nicht ganz abgesprochen werden … Bemerkenswert ist, dass Caesar > certo > ganz zurückgewiesen u. > certe > auch im Sinne von > certo > gebraucht hat; ihm folgen Verg. und Horaz.
which I would tentatively translate as (caveat lector!)
certe > and > certo> . Either is applied to all verbs, but in better prose > certo > is restricted to only > scire; > the difference between > certe scio > and > certo scio > is defined by that I say > certe scio > when I want to affirm for > my > part that I know something — > yes, certainly, in fact, truthfully do I know it; > but > certo scio, > when I state, > how > [in what manner?] I know, > with certainty, > as something certain and undoubtable as far as I am concerned. Cicero prefers > certo scio, > although > certe scio > can not totally “be denied him” … It is worth remarking, that Caesar rejects > certo > and uses > certe > also for the meaning[?] of > certo> ; Vergil and Horace follow him.
So, if we believe Krebs, we might as well forget about certo except possibly in the phrase certo scio, and hence write “…certe causam perdes”.
Adrianus, with “>”, do you mean “is better than, to be prefered before”?
I do mean “better than”, Alatius, but it’s just a personal feeling. By the way, my offering just repeats what you said above about the subtle difference, and tried to clarify what L&S say. Maybe I didn’t clarify anything.
Certè dicere “melius quam” volo, --dumtaxat mens mea est. Obiter, subtilia quod dixisti modò repetivi et verba L&S clara facere quaesivi, forsan sine eventu.
Assuming there is in fact a difference, perhaps “certo” could work as an ablative of characteristic (modifying whatever is known, etc.), while “certe” is simply an adverb?
I think that, most of the time there is NO difference in practice (as Lucus says), but some authors do draw a subtle distinction and it is a distinction similar to the one you say, Thesaurus. You get three different opinions on “certo” and “certe” when you look up Allen & Greenough, Lewis & Short and OLD (and Krebs & Schmalz as Alatius says --4 different opinions).
NULLAM differentiam plurimùm esse puto (ut ait Lucus), nisi emunctè apud aliquos scriptores qui sicut dicis, Thesaure, sensus distinguunt. De hac re, Allen & Greenough, Lewis & Short et OLD (et Krebs & Schmalz secundum Alatium) ipsi sibi singuli discrepant.