Irreconcilable (adjective)

1. Incapable of being reconciled or resolved.
2. Incompatible, incongruous, or fundamentally different.

Origin:

Derived from the latin word 'irreconciliabilis' meaning impossible to reconcile.

Examples:

1. The differences between the two sides seemed irreconcilable, and a resolution seemed impossible.
2. The irreconcilable differences between the couple led to their divorce.
3. The irreconcilable conflict between the two nations has been ongoing for decades.
4. The irreconcilable contradictions in the defendant's testimony raised suspicions about their guilt.
5. The irreconcilable ideologies of the two political parties make cooperation difficult.