Is One Mind/One Truth the same or different meaning between “One Mind with Three Observations” (Yixin Sanguan 一心三觀) in Tiantai Buddhism (天台) and “One Mind or One Truth/Reality with Dharmadhātu” (一心法界 Yinxin Fajie or 一真法界 Yizhen Fajie) in Huayan Buddhism (華嚴 Avataṃsaka)?
Is One Mind/One Truth in Huayan Buddhism the same or different meaning between “One Mind with Dharmadhātu” (一心法界 Yinxin Fajie) and “One Truth/Reality with Dharmadhātu” (一真法界 Yizhen Fajie)?
一心三觀 means performing three contemplations, or seeing something from three viewpoints, in a single thought. The three viewpoints are emptiness (or supreme truth), conventionality, and the middle way. Chih-i’s philosophy made extensive use of this concept of analyzing things from three different philosophical points of view and then transcending these one-sided ways of seeing something by combining them all at once to arrive at actual reality. It’s bit like the way photons are defined in quantum physics: Photons can act like waves, or they can act like particles, but in reality photons are both and neither at the same time. This is similar to what Chih-i was trying to express: A reality that’s inherently difficult to define completely as it actually is.
Usually, what he was doing was trying to resolve the tension between the two truths of conventional and ultimate reality that became popular in later Indian Buddhist philosophy. He would add to these two truths a third truth of the middle way, and then assert that reality includes all three but is identical to none of them.
This is one of the ways Chinese Buddhists struggled with the discursive nature of Indian thought. Indians liked to “slice and dice” everything into discrete bits, but Chinese philosophy was syncretic in nature - they yearned to find the unity of things, not the differences between them. So, in digesting Indian Buddhist ideas like the two truths, they needed to add a level of unification for it to make sense to them. Chih-i was a meditator, so his thinking revolved around contemplation of truths, and he was very much trying to marry up his experiences as a meditator with Nagarjuna’s philosophy. He was tangentially influenced by Tathagatagarbha thought through the Nirvana Sutra. In the main, though, Nagarjuna and Kumarajiva were his primary Indian influencers. He was not a Yogacara Buddhist.
BTW, I strongly recommend Paul Swanson’s magnum opus, Clear Serenity, Quiet Insight, which is a complete and copiously annotated translation of Chih-i’s Mohazhiguan (lit. “Great Calming and Contemplation”). It’s a great way to explore Chih-i’s teachings.
Huayan writers were doing a similar thing - arriving at a syncretic, holistic vision of reality by both transcending and including different categories used to describe it. They spilled a fair amount of ink describing the “dharmadhatu” or “realm of things” as being both a whole unity and/or a collection of particular things. They were famous for saying that the universe is contained in a grain of sand and vice versa because the universe is simultaneously one thing and an infinite number of things which exist relative to each other. This is classic Chinese thinking. The logic is similar to what Chih-i was doing with Nagarjuna’s two truths. Different points of view that are real but different must be incomplete, and the actual reality something that contains all of them.
So, the one mind, or a single thought, is likely linked to the notion of Tathagatagarbha “the womb of the thus-come-one” (Buddha-Matrix, Buddha-Embryo). This is about Buddha-nature within all sentient beings.
Yes, Mohe Zhiguan (摩訶止観 Mahā-śamatha-vipaśyanā), the most influential meditation text of the Tiantai school.
In Theravada there is no “single thought,” conventional and ultimate are accepted as separate, coexisting concurrently and dealt with appropriately in full knowledge of their difference:
"An arahant monk,
one who is done,
effluent-free, bearing his last body:
Would he say, ‘I speak’?
Would he say, ‘They speak to me’?”
“An arahant monk,
one who is done,
effluent-free, bearing his last body:
He would say, ‘I speak’;
would say, ‘They speak to me.’
knowing harmonious gnosis
with regard to the world,
he uses expressions
just as expressions.”
—Samyutta Nikaya 1.25
The arahant continues to engage in appropriate attention:
“An arahant should attend in an appropriate way to these five clinging-aggregates as inconstant, stressful, a disease, a cancer, an arrow, painful, an affliction, alien, a dissolution, an emptiness, not-self. Although, for an arahant, there is nothing further to do, and nothing to add to what has been done, still these things — when developed & pursued — lead both to a pleasant abiding in the here-&-now and to mindfulness & alertness.”
—Samyutta Nikaya 22.122
Therefore practitioners should strive to separate conventional from ultimate reality that is the issue, and the main thing preventing it is awe of the “All” (conventional reality).
Maybe it means something like this: "As the bringer of light who has pierced the truth, you’ve seen what lies beyond all realms. When you saw and realized this for yourself, you taught it first to the group of five. (SN8.8)
No, it just means seeing a point of view that includes but isn’t the same as the two truths individually. It has nothing to do with Buddha nature, per se. It’s more like an attempt to describe the moment of awakening, if anything.