Because the company cash to buy the stocks should decrease the amount of capital cash on hand by the same amount resulting in no change in the capitalized value

Because the company cash to buy the stocks should decrease the amount of capital cash on hand by the same amount resulting in no change in the capitalized value of the stock. The stock doesn't just vanish. It is held and owned by the company. So no, each share doesn't represent a larger percentage of the company. Unless the company retires the shares.