It was 800 metres finals day, but to my surprise, there were three races scheduled, not two.

On inquiring, I learnt there had been a shake-up: the "men's" and "women's" classifications were no more; instead we had a low-testosterone final, a medium-testosterone final and an open final.

This, I was told, was how the International Association of Athletics Federations (IAAF) eventually responded to the successful challenge in 2019 by the South African athlete Caster Semenya of its new eligibility regulations.

My visit coincided with a historic evening: both the low- and medium-testosterone designations, being quite new, were won in world record times.

It will never happen, of course.

We have grown accustomed to men and women competing separately in foot-races, as in other forms of elite sports competition requiring the expenditure of great physical effort and/or strength.

Any other template for individual, as opposed to team or mixed relay, contests would be just too unsettling.

Yet, classification by testosterone-level seems in some ways a preferable, even a more logical, reaction to the awkward untidiness of real life than obliging some women competitors artificially to reduce and then maintain their natural testosterone levels below a given threshold, or to compete against men.

I was struck by a comment made by Seb Coe, the former 800m champion and current IAAF President, in an interview with my former insidethegames colleague Nick Butler, who is now in Germany with the excellent ARD.

According to Lord Coe: "The dominant factor in the outcome of any competition is testosterone."

He went on: "It is the dominant feature and if you have women, legal women, with testes that are producing levels of testosterone that are in the male ranges, then you clearly have a massive advantage and a huge disadvantage to biological females that are competing at those distances."

I am slightly sceptical that the prime determinant of competitiveness can be boiled down to a single hormone.

But Coe is evidently far better qualified to pronounce on this than me, and if testosterone is, as he says, the dominant factor, then using anything else to determine who can compete fairly against whom seems likely to leave you in exactly the sort of quandary that the IAAF has found itself in over recent years.

For reasons that have to do understandably with deep-seated sports traditions and social mores, gender seems to be acting as a sort of classification proxy for testosterone levels.

The difficulty with this is that some, to borrow Coe's phrase, "legal women" have natural testosterone levels, as he says, "in the male ranges".

So, if you stick with gender as your classification criterion, and if Coe is right about the hormone's dominant role in determining competitiveness, you can only establish a level playing-field by excluding some "legal women" from your women’s discipline - that or obliging them artificially to bring down their testosterone levels in order to render themselves eligible for the category labelled with their gender.

And the difficulty with this is that it looks uncomfortably close to a mere sports governing body, albeit a prominent one, attempting to define what a woman is. 

These are the sort of deep waters that even the most eminent and respected Supreme Court imaginable would have to handle with the most exquisite kid gloves.

Make testosterone not gender your classification criterion, and you jettison tradition, history, romance, but you avoid all that.

It would be an extraordinary step to take, jumping out of the frying-pan into the fire as some might argue, and I just cannot foresee circumstances under which sports leaders would take it - even if Semenya wins her landmark case.

But if they really deem it necessary to level the playing-field in this way, then replacing gender classifications might be judged a more direct and ultimately a more respectful way of going about it.