Are the Pali words "cause" ("hetu'), "condition" ("paccaya"), "nutriment" ("ahara") and "root" ("mula") synonyms? Are the Pali words "cause" ("hetu'), "condition" ("paccaya"), "nutriment" ("ahara") and "root" ("mula") always synonyms?
The Pali words "cause" ("hetu') is NOT a synonym of "condition" ("paccaya"). To explain this, let me take the example in paticca samuppada (PS). The PS cycle that describes the “cause and effect” in Buddha Dhamma starts as: “avijja paccaya sankhara, sankhara paccaya vinnana, vinnana paccaya nama rupa, nama rupa paccaya salayatana,….”.

One may tend to think that “avijja paccaya sankhara” means “avijja causes sankhara” or “ignorance causes one to acts that generate bad kamma”, but it is not so. All of us ordinay people have avijja, but that does not mean that we always act accordingly. Most of the time we act appropriately or morally. However, if avijja is there, it is likely that at times one will act with avijja and do inappropriate or immoral things.

Thus PS does not refer to a “direct link”. Any effect must have a cause. But there can be possible causes without leading to any effects. Otherwise, Nibbana would not be possible. For causes to bring about corresponding effects, suitable conditions must be present. That is what ‘paccaya’ means.

Similarly the Pali words "nutriment" ("ahara") is NOT a synonym of "root" ("mula"). In Pali language, “mula” is a root, so that “mūlika héthu” means “root causes”. To explain this, let me take the example of a tree. The roots (mūlika hetu) are critical for the tree’s survival, but the “patta” also plays an important role for the tree’s growth. The “patta” are essential for the tree’s survival, just as the roots of the tree are. One could kill a tree simply by a process called “girdling” where those phloem cells are removed. Girdling, also called ring-barking is the complete removal of a strip of bark. from around the entire circumference of either a branch or trunk of a woody plant. Girdling results in the death of the area above the girdle over time. A branch completely girdled will fail and when the main trunk of a tree is girdled, the entire tree will die.

So, the tree’s survival depends on not only its roots, but also its “patta” containing those critical phloem cells. Thus, CONDITIONS are as important as ROOT CAUSES, and both these taken together is "nutriment" ("ahara") for the tree.