[ue_]

>Therefore, as long as the body doesn't decay, the cycle doesn't repeat and that's the achievement.

As far as I know, this is false. The cycle stops repeating when one has reached enlightenment; to my knowledge, what happens after that point is irrelevant - one can't go back from enlightenment, and certainly not due to the body decaying (which is a major topic of Buddhist meditation anyway).

Is there any Pali canon source (I'm only interested in Theravada) that backs up the concept of no decay = no rebirth?

This idea, known in the text as annihilationism, was rejected by the Buddha as far as I know.

[acqq]

Nirvana is "the liberation from the cycle of birth and death." The cycle is the main cause of suffering. Observing it this way, the first Budda (being born in Hindu environment) who recognized that the enlightment "frees from the cycle" that is, not beleiving in religious stories achieves the freedom. The follovers made the religion from that too.

[ue_]

>that is, not beleiving in religious stories achieves the freedom

The way to achieve enlightenment is via the Noble Eightfold Path. This does not require non-belief. At some point, one eradicates doubt about the teachings because one has realised the truth for oneself through insight meditation.

Enlightenment is nothing as benign as removing faith - it's the removing of the need for faith via direct experience.

Besides, the teachings cover a far wider range of ideas than rebirth. Not only that, but the Buddha is written to have said that an enlightened person can recollect his or her previous births.

[acqq]

I don't write about teachings of the religion made later. I just consider the context of Budda trying to save himself from Hinduism in which he grew up. He had a simple solution: deciding not to worry about Hindu beliefs of eternal torture through eternal rebirths and lives and deaths, thus achieving his own personal Nirvana.

[gerbilly]

His solution, if you believe it was one, was far from simple though.

It wasn't just a momentary decision to ignore Hinduism that let to his enlightenment, but a deliberate practice of self examination.

Buddhism is actually a kind of psychology, a study of the mind, and the historical Buddha studied his mind methodically for years using introspective techniques (meditation) that he adapted from Hinduism.

But it's also more than that. Just like you can't learn a difficult skill like hitting a baseball by looking at pictures of others doing it, or by thinking about it or by being told how it feels to do so, you can't realize enlightenment without striving for it.

So buddhism is a set of teachings about the nature of your mind, combined with practices to help you fully realize (come to know) the truth of these teachings in your own mind and body.