Discussing authorship/sole authorship with supervisor This is a borderline IPS question, but I feel the academic aspects will be more relevant.

One year ago I switched supervisor. Went to him with a formulated project (but no concrete results yet) and I am now at a stage where I should discuss authorship, which I haven't done before. This supervisor provided much needed guidance and support in meetings, but I am personally of the opinion that I should be the sole author of the paper, though this is obviously a grey area.

My question is then two-fold:

Does his contribution qualify him objectively (i.e. in most academic situation) as a co author ?
I have a hard time telling whether his contribution is enough for him to ask me to include him as a co-author. 

On my part I have:


Formulated the core idea around which the paper revolves
Found a fitting mathematical formalism
Run all numerics
Ideated and wrote all mathematical proofs required


On his part he has:


Guided me in what could be considered an interesting result in the field (e.g. this would be an interesting statement, can you show that it is true?)
Guided me in constructing a coherent story around my idea compatible with the field


On the one hand, this paper would have never been born without his guidance. On the other hand I believe is contribution to the content is not big. I would like your thoughts on this, so that I can ponder the various sides of this and be ready if a discussion arises.

How do I discuss this with him? 
I have a feeling he won't ask me for co-authorship, but I might be wrong and indeed he might expect it as a counterpart to the time he dedicated to me. 

I would like to bring up the topic in a way that recognizes the value of his support and express my opinion on authorship without causing conflict (with the person that will in all likelihood write me references for the future)?

One way I thought this could go is to just ask something like: "I will be writing up. Do you think such and such journal would be a good outlet for this work? Do you think we should author this together?"

I believe this sends the message: I am somewhat open/uncertain about  co-authorship, what do you think? and leaves the initiative on the topic to him.

Some clarification : sole authorship is considered a good thing in my field, I am at a loss in assessing how much.

Also, I am responsible for my own funding which comes from a scholarship, none of my supervisor had anything to do with it.

Finally, my previous supervisor has explicitly stated he wouldn't require authorship on this (saying he didn't contribute enough).
Note: things may vary by field. Single-author publications has never been an important metric in any field I've ever worked in, but can't say for sure about math.

I think your advisor should be a co-author. Here are some things to consider:

Reputation. Even if you did good work, the paper will get more circulation with an established researcher's name on it. So, you probably want him to have authorship, though you should of course be first author.

His contributions. As you say the project "would never have been born without his guidance." It sounds like he did not contribute only offhand comments, but actually proposed a line of research and helped you to follow it. 

Funding. I'm assuming he funded you to do this research. In itself, this doesn't mean he deserves authorship -- but I suspect that a lot has been going on "behind the scenes" to help you with this work. Grant proposals are a lot of work, and he probably put a bit of thought into your general research area before he even hired you. (This may not apply in your particular case, since the advisor didn't fund you.)

The bar for co-authorship is not so high. I have seen questions on this site where people have been given co-authorship for reviewing the grammar! I think this is ridiculous, but adding your advisor seems altogether appropriate. In particular, you will be first author, so it costs you very little -- and it likely means quite a bit to him. In fact, the old supervisor would have a decent case for being added as yet another author. 

Edit: OP has clarified that single-author papers are an important metric in this field. If this is true, the advisor in question surely knows this as well. So, if you want to ask for sole-authorship, you could say something like this:


  I wanted to ask about the authorship for this paper. As you know, single-author papers will be important for my career. So, I wondered if you thought it would be appropriate for me to be the sole author on this paper. 


Note, I still think adding your advisor as a co-author would be more appropriate. Perhaps a "compromise" would be to ask the question in a more neutral way, something like:


  I wanted to ask about the authorship for this paper. You obviously made a lot of contributions and maybe it makes sense for you to be a co-author. On the other hand, this paper might be a good chance for me to get a single-author paper, which as you know, I will need for my career. What do you think?