The climax of the Core probably is when they are in the core and decide that due to some flaws, they have to change the plan. They discover DESTINI, a weapon that could propagate earthquakes through the Earth's core, but it has unintentionally stopped its rotation instead. Zimsky reveals the government will use it again to attempt a restart of the core. Keyes is convinced it will have disastrous results, so he has Finch hack into DESTINI's systems- to try to buy time. While working to set up their detonations. Keyes and Zimsky realize they have miscalculated the necessary yield, but Zimsky gets trapped in a detaching compartment. He reveals that they need to use the ship's nuclear fuel Pod to achieve the sufficient yield, which will save the planet but leave Keyes and Childs without power.
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The climax is where Natasha divulges in her "dream" that she witnessed a murder.