The point is that they've established a relationship between o16 levels and temperature, so if you've got twice the o16 then say it was 25% warmer (made up ratio, I haven't read the study).

This doesn't tell us what the air temperature was, but it does tell us what it wasn't (IE upper and lower bounds).

When you have several of these proxies it helps narrow down the temperature range (think how your god works better when you have more satellites).

Now if you know that the last seven days are the hottest on record and you know from your proxies that you are outside of temperatures of the past 100k years then it's a pretty safe bet to state that we're at the hottest time in the past 100k years.

There is no melodrama or lying in this fact, unfortunately.

It seems the temperature has been slightly hotter about 6500 years ago for a period of around 2 centuries with temperature estimated between +0.8 and +1.8 °C compared to 19th century, but this is subject to debate, (see for example https://www.nature.com/articles/s41597-020-0530-7).

Before that, we have to go back to a period where most Homo Sapiens were living in Africa about 125,000 years ago, where warming was likely +0.5 to +1.5°C compared to the same 19th century baseline.

Regardless if there was periods much hotter in the long past, the big difference with today's situation is the rate at which this warming is taking place. For example, for the "6500 years ago" period, it took about 3000 years of warming to go from +0 to it's maximum (which is between +0.8 and +1.8 °C). Today we are at about +1.1°C and it took us only 100 years, through fossil fuels burning and farming to reach that and most of which happened in the past 50 years.

Sources:

• IPCC WG1, figure2_1
• https://www.nature.com/articles/s41597-020-0530-7

Also, about oxygen 16 and oxygen 18:

The water remaining in the ocean develops increasingly higher concentration of heavy oxygen compared to the universal standard, and the ice develops a higher concentration of light oxygen. Thus, high concentrations of heavy oxygen in the ocean tell scientists that light oxygen was trapped in the ice sheets. The exact oxygen ratios can show how much ice covered the Earth. Sources:

• https://earthobservatory.nasa.gov/features/Paleoclimatology_OxygenBalance which is based on https://hal.science/hal-03334828/file/jgrD1994Jouzel25791.pdf
• You may also find this wikipedia article useful https://en.wikipedia.org/wiki/%CE%9418O#Extrapolation_of_temperature