How do the admission procedures for new States reflect the underlying values of the Constitution?
The Constitution of the United States only talks about the process for admitting new states in Article IV, Section 3. It really does not say much about it at all. The only thing that really reflects the underlying values of the Constitution to any great extent is the part about making states within existing states or making states by using parts of existing states. In such cases, the states in question must agree and so must Congress. This reflects the underlying idea of federalism in which both the state and the national governments have their own sets of powers.
We could also say that the process for allowing new states into the Union (which is not explicitly described in the Constitution) reflects the values of that Constitution. In general, states are allowed into the Union when they have enough people and when a vote by their population indicates a desire to enter. When that happens, the area is directed to write a constitution and is admitted if the constitution is acceptable to Congress. This reflects the basic underlying value of popular sovereignty. That is, it reflects the idea that our government is supposed to reflect the will of the people.