Oh boy, this'll be fun! Living Time is basically the idea that time itself is... well, [i]living[/i]. Because of Precursor technology being based on neural physics, the Forerunners believed that the universe itself was a living being of sorts and that it was enriched by life's "interaction with the cosmos", as is stated in Cryptum. It's largely what forms the Forerunner principle of the Mantle. Death, war, competition and the struggles of life were regarded as being as much a natural part of the universe as peace and harmony. What the Mantle called for action against was needless conflict and excessive actions of cruelty which the Forerunners believed destabilised the flow of Living Time. In Primordium, we learn that Living Time is something that the Forerunners have some capacity to 'see' in a way that's kind of like precognition - the Librarian was most prominent for this, as the Lifeshaper. Living Time is imagined by the Forerunners as being like a river with many twisting streams that they call 'World Lines'. When two or more of these streams join together, it's called a Synchron - forces that bind others together. In Cryptum, Bornstellar feels a Synchron with Chakas and Riser - that their fates have been purposefully bound together for the adventure they're on as they set out to awaken the Didact. The name 'Forerunner' itself pays homage to Living Time. As we know, the word 'forerunner' means 'one who precedes another', who we know to be humanity in the modern Halo universe because we are meant to inherit their Mantle of Responsibility. As such, the Forerunners saw themselves as being only a stage of Living Time, that they would pass their burden of guardianship onto others someday, and the cycle would repeat across eternity. That is the basic concept of Living Time in a nutshell.