If a firm agrees to settle a retiring or deceased partner's account by paying them a lump sum amount, what happens to the excess amount paid to them beyond what is due based on their capital account balance after necessary adjustments?

It is treated as their share of goodwill (known as hidden goodwill).

When a firm settles a retiring or deceased partner's account by paying them a lump sum amount, the payment is based on various factors such as the partner's capital account balance after making necessary adjustments for accumulated profits and losses, revaluation of assets and liabilities, etc. If the amount paid to the retiring or deceased partner exceeds what is due to them based on these adjustments, the excess amount is considered as their share of goodwill. In other words, the excess payment reflects the value attributed to the partner's reputation, customer relationships, or any other intangible factors that contributed to the firm's success. This excess amount is referred to as hidden goodwill because it is not explicitly recorded in the books of the firm but represents the value associated with the retiring or deceased partner's contribution. By treating the excess amount as the retiring or deceased partner's share of goodwill, it ensures a fair distribution of the firm's assets and recognizes the intangible value brought by the partner to the firm.