Mathematical proof how inflation makes the rich richer and the poor poorer, and how deflationary Bitcoin fix it.

I'm not an expert in mathematics or economics, and I came here seeking your insightful thoughts on my ideas. Many of us have engaged in debates with people claiming that inflation is beneficial for society. I have always wanted to have mathematical refutation on this matter.

I came across this video, https://youtu.be/\_FuuYSM7yOo, and this article, https://www.jasoncollins.blog/posts/ergodicity-economics-a-primer, about the Kelly criterion. I do not have enough knowledge to understand this completely, but the visual presentation of the material tells me that this can be used to mathematically prove the thesis about the negarive impact of inflation on the welfare of society.

Allow me to briefly describe what I believe I understood from these materials. If a group of people plays an economic game, for example, "Invest in something or your money will lose value" under seemingly equal conditions, given certain parameters of the game, the average value of the final balances may exceed the initial average. However, the median value of the final balances will be lower than the initial median. In other words, a small portion of individuals will greatly enrich themselves while the majority will become poorer. There are parameters that could enrich everyone, but they must be well-defined and enforced universally, which is not feasible in real life.

Then, I asked GPT-chat about the target value of the Kelly criterion at which the median capital of the player group becomes lower than the initial average. The response was as follows:

"If the median f* (Kelly criterion for an individual player) for the player group is lower than the average value f*, it implies that more than half of the players invest a smaller share of capital compared to their share in the average capital. This results in the median capital of the player group being lower than the average value."

However, it is important to note that the poor do not invest money in this game to begin with. They simply lack the means and education to do so. They only bear losses in this game. Consequently, they will continue to grow poorer, even if the overall community becomes wealthier.

I would appreciate the insights of knowledgeable persons on my thoughts. Perhaps someone would be able to math describe the situation in a deflationary "Bitcoin game" based on the information provided.