Headlamps lumens questions

Hi there,
Just stumble upon the latest version of the Petzl Actik core that claims 600 lumens...
I own one of the first version that I think is 250 lumens, then I bought the next one that is supposedly 450 lumens and now they have this new one...
My main use is alpine start and route finding.
My question is, is there any "official/organized" way of mesuring brightness or the manufacturers can claim whatever they want?
It may be interesting to have more light in the same form factor but maybe at some point (where?) it doesn't make sense to go higher....I mean, it seems to me that I m doing fine with 450 (claimed) lumens but maybe I m missing out...

nb: for the scope of this discussion I m putting aside caving where I my sure brighter is better

Lumens are tested using an integrating sphere.

Some guy on Candle Power Forums from way back in 2010: https://www.candlepowerforums.com/threads/does-doubling-the-lumens-double-the-visible-light.261857/

the perceived light intensity is proportional to the square root of lumen output. So if you double the lumens, the light is only intensified by a factor of square root 2 or 1.41 times as bright. It should also throw 1.41 times further also, as long as you compare the same light with different outputs as the beam angle is very important here. A smaller beam angle will throw farther than a wide beam angle.

If you quadruple the lumens, the light is intensified by a factor of root 4 or twice as much.

It is also notable that lumens isn't the only important thing. Reflector, lens, orange peal treatment of reflector, frosting of lens, etc... will all make a difference in how the beam spreads out. Some applications you may want an even wide beam others you may want spot and spill, like 80/20 flood/spot.

Also, color temperature. Bluer will have greater lumens per watt, but neutral or warmer, like specifically labelled high CRI will make it much easier to differentiate colors, even if the stated lumen output between two such LEDs is lower on the high CRI one.

LEDs are more efficient at low currents. High current / high output generates heat and lowers efficiency.

You could even get way into the weeds. Deep yellow cuts through dust and snow better than other colors, etc etc etc...

Lumens don’t matter, battery size does— the ANSI standard measures runtime until brightness drops to 10% of the peak/stated value, and many headlamps don’t even meet that. With the exception of some high end lights from zebralight, fenix, et al which have constant regulated output, you only get the advertised output on a full battery.

There isn’t tremendous variation in LED efficiency, any light with the same battery capacity will have roughly equivalent lumen-hours of output, just with different maximum output and set levels.