The 'Managing a Stage Boundary' process in PRINCE2 is designed to confirm that the project remains viable and continues to align with organizational strategic objectives. This process involves reviewing the project's performance, updating the project plan, and reconfirming the business case to ensure that the project continues to deliver value as expected.
= PRINCE2 7 Foundation Training Manual and official PRINCE2 website documentation

In PRINCE2, the information about the quality requirements for a product is detailed in the product description. This document specifies the quality criteria, quality methods, and the responsibilities for achieving the required quality. References: PRINCE2 Foundation Handbook, Product Description section.

Maintaining the business case during the project lifecycle involves regularly updating it to reflect changes and actual experiences such as incurred costs. This ensures that the project remains viable and aligned with business objectives, as part of the continual justification process. The business case is dynamically managed to reflect any changes in the project's environment and is a fundamental part of the 'business case theme' in PRINCE2.
PRINCE2 7 Foundation source documents on business case management.

The Project Board acts as the primary communication channel between a project and corporate or programme management. This role involves ensuring that the project aligns with corporate objectives, communicating the needs and constraints of the project to higher management, and securing necessary resources and support.
References: PRINCE2 Foundation manual, PRINCE2 official resources

The purpose of the quality theme in PRINCE2 is to ensure that the project outputs meet the requirements that have been agreed upon. This theme guides the processes and activities required to verify that deliverables are fit for purpose and meet the standards expected by stakeholders.
= PRINCE2 7 Foundation handbook and online resources