Under the proviso to Section 8(1) (clarified through jurisprudence), what happens if a dispute partially falls within an arbitration agreement's scope? 

While not explicitly stated in Section 8, the principle of "severability" is well-established in arbitration jurisprudence and reflected in Section 34(2)(a)(iv), which allows setting aside only the part of an award dealing with matters beyond the arbitration agreement's scope. When disputes have both arbitrable and non-arbitrable components, or where some issues fall within the agreement and others do not, courts refer only the covered portions to arbitration. This prevents artificial barriers to enforcing valid arbitration agreements while respecting the boundaries parties have set.