with a linked account, why compute SP recommendation is empty but EC2 instance SP recommendation has a large number of recommendation items

The difference you're observing between Compute Savings Plan (SP) and EC2 Instance Savings Plan recommendations for a linked account can be attributed to how these recommendations are calculated and the nature of the two types of Savings Plans.

EC2 Instance Savings Plans are more specific and tied to particular instance families in a region. They offer a lower discount but provide more flexibility within the chosen instance family. This specificity allows for more granular recommendations, even at the linked account level, which is why you're seeing a large number of recommendation items.

On the other hand, Compute Savings Plans are more flexible and can apply to a wider range of compute services (including EC2, Fargate, and Lambda) across any instance family, size, or region. Due to this flexibility, Compute SP recommendations are often calculated at the management account level to maximize savings across all linked accounts in the organization.

When you're looking at recommendations for a single linked account, the Compute SP recommendation might be empty because:

1. The system is optimizing for overall organizational savings rather than individual account savings for Compute SPs.
2. The usage pattern in that specific linked account might not be sufficient or consistent enough to generate a meaningful Compute SP recommendation on its own.
3. The existing Compute SPs at the management account level might already be covering the eligible usage in that linked account.

It's important to note that Compute and EC2 Instance Savings Plans recommendations are created using the same set of usage data, but they are not meant to be taken together simultaneously. The empty Compute SP recommendation for the linked account doesn't necessarily mean there's no potential for savings; it might just be that the savings are better optimized at the management account level for Compute SPs.

For the most comprehensive view of potential savings, it's generally recommended to review Savings Plans recommendations at the management account level, as this takes into account usage across all linked accounts and can provide a more holistic approach to cost optimization.
Sources
Understanding your recommendation calculations - Savings Plans
Compute Savings Plan for Payer has higher estimated savings than Linked account option, why aren't they the same? | AWS re:Post
COST07-BP05 Perform pricing model analysis at the management account level - AWS Well-Architected Framework