We would rewrite our laws of thermodynamics is guess?

Well, you can’t be in a verified calorie deficit and gain weight, outside of extreme water retention. Thats the definition of a calorie deficit.

But there are vanishingly small numbers of people who gain weight eating a very small amount of food who have hormonal imbalances that make that happen. Theres a much larger number of people who forget to count the handful of crisps or nuts or chocolates in their diet.

Same thing that would happen if someone proved in a lab they're filling a cup while removing more water than what they're putting in. I.e. it won't happen.

In reality the body is a lot more complex than a cup of water, and it's possible you gain weight on calories deficit by accumulating water or feces. But you WILL be losing fat and/or muscle, otherwise we will use you as an infinite source of energy.

Depends on the deficit amount, accuracy of the scales, and accuracy of the record keeping.

People tend to overestimate the Kj they burn while underestimating the amount they eat.

We would have to ask them where they hid the water bottle.

They might be increasing in weight due to absorbing more water into their body

Lead tire balancing weights have zero calories.

He would just have proved he was drinking water with something salty.

Your description of circumstances are lacking, but what you imply is impossible and ridiculous.

By definition, that wouldn't be a deficit. You could have a "predicted" calorie deficit that ends up being off by some percentage. The models for energy expenditure typically just use pretty simple demographic info like BMI, sex, age, and activity level. If someone burned less calories than predicted, that basically means that they are less fit than the average person of their demographic cohort.

You could use more advanced models with more information, but they would still be predictions. Drugs also come into play: uppers like caffeine, nicotine, amphetamines, etc, increase the amount of activity in your body so you are literally warmer from burning more calories, everything else equal.

Then the researchers would record their findings dutifully and continue the experiment as normal.

There are any number of reasons why a person might gain weight temporarily while on a calorie deficit.

Most obviously, some non-caloric material is accumulating in their body, like water. Or poop. Bodyweight can swing 5 kg or more daily depending on these sorts of factors.

It is also possible that the calorie "deficit" is not actually a deficit.

The charts and online calculators that you can find to figure out your daily calorie expenditure are extremely unreliable, as how many calories a person burns per day is highly individual. So if "daily calories burned" is based off a generic calorie calculator, then the most likely explaination is that this person just burns fewer calories than whatever the calculator says.

Even if the person's calorie expenditure was accurately measured before the experiment, the daily total calorie burn a person experiences is highly susceptible to change - especially in circumstances like intentional calorie restriction. When you restrict calories, you lose weight, which means your body has less tissue to maintain, which lowers BMR. It also means you weigh less, so weight-bearing exercise expends fewer calories. When you eat less food, it takes fewer calories to digest what you do eat. And also, most peoples bodies respond to calorie restricion by reducing non-exercise activity thermogenesis - the random, subconscious movements you do throughout the day that your body does in order to maintain a particular body composition in the presence of excess calories.

But if we suppose that we are conducting this experiment over a long time, and the participant has all their food measured out so we know exactly what they are eating, and their calorie expenditure is tracked continuously in a rigorous way, and they consistently gain weight, then presumably the scientists would request the person stay in the experiment longer, and see if they wanted to participate in further experiments, because they would be an extraordinarily interesting case study that could broaden our knowledge on where a human might gain weight from in the absence of excess calories. Because unless they are secretly inserting steel bars under their skin in the dead of night, gaining weight on a real calorie deficit does not happen, as it would violate the first law of thermodynamics.

NOT possible.

obviously,

That is achievable...

Work out while very fat.

Fat loss brings less weight loss than acquired muscles, that are much more dense.

And this can happen very well in a calorie deficit diet.

Its a specific situation, but happens all the time when you start exercising, and people get confused, why I am gaining weight?

Just converting fat to muscle (so to speak, ofc)

It's easy enough. Love, betablockers and the menopause.