How to state that you “rigorified” an existing paper without sounding arrogant or condescending? I am writing a paper aimed at a physics journal which extends an old paper (not by me), which introduced a mathematical formalism for a physical problem in a rather handwaving way and without addressing most of its mathematical background.
In addition to exentding this formalism, my paper elaborates on this mathematical background and relatedly introduces the formalism in a ”more rigorous” way.
I consider the latter one of my paper’s key features for the following reasons:


One might consider the formalism and the methods derived from it not to be properly substantiated in the old paper.
Though the mathematical background has no other application so far, it is interesting for its own sake and perhaps from a philosophical point of view.
My approach to the formalism might be more accessible to some people and make them give a better understanding of what they are doing when applying the formalism and related methods.


As these points are rather opinion-based, I do not intend to elaborate on them in the paper – at most I might shortly mention the didactical aspects in the conclusion.
(I mainly mentioned them here to give you some idea what I am talking about.)
However, I do want to briefly mention in the abstract or the introduction that I “rigorified” the formalism, where briefly can be anything between one word and two sentences.
For example, my abstract could look like:


  [Old paper] introduced [old concept], which is useful for [application].
  We extend this concept to [new concept] and also rigorify its mathematical background.


My problem is that rigorify its mathematical background is far from what I actually want to say.
I am therefore looking for a way to say this without seeming arrogant or condescending on the old paper and in particular without implying that the old paper “did not do its mathematics properly”.

I intentionally do not give my best solutions for now, as I do not want to induce any bias on them, in case I may underestimate them.

I am also torn between Academia and English Language &amp; Usage for posting this question.
It sounds to me like you are viewing the older paper through the lens of the type of paper you would have written in the first place and are finding it wanting.  It also sounds like you are viewing your own work as "fixing the flaws" of the older paper, almost as if you want to replace the older paper in your mind with your paper and pretend it had been written that way all along.  More quantitatively you have your eye on the mark that the older paper should have hit, you are filling in the gap between the actual old paper and your eyeballed mark, and since you are measuring from the eyeballed mark you are giving the difference a minus sign.  This framing seems to be behind most of your problems.  Try recasting the entire thing more positively in your own mind.

1) Someone else published an inspirational paper way back when.  This paper introduced some formalism and concentrated on its application to a physical problem.

2) Your paper gives much deeper attention to the mathematical aspects of the formalism, while also extending the formalism.  Since 1) showed that the formalism is interesting and useful, your work has evident value.  

There are no "minus signs" in the above description: 1) + 2) = your eyeball mark.  

Unless there are actual mathematical errors in the older paper, you don't need to say that you are fixing or "rigorifying" (not really a word, by the way) the older paper.  By the way, I don't know whether you've heard my rant about this use of the word "rigor" in mathematics and its applications.  As an adult mathematician I have become increasingly skeptical of mathematical "rigor": the other paper either made mistakes (which you will need to correct), made claims which were unjustified or insufficiently justified (in which case you make clear that what you are contributing is the justification and not the claims themselves) or they weren't doing mathematics at all (which is fine: that's what you're adding).  Adding rigor must mean one of the things above, right?  It is not some generic ingredient that you can sprinkle more or less liberally over a piece of "unrigorous mathematics" and make it rigorous.  

In summary: unless there are some clear mistakes, you don't need to bill your work as fixing their mathematics.  You can bill it either as adding the math or adding more math.  Both of these are good things added to other good things: no problem.