Does MN9 imply a cyclic causation? How does it work?

It's not a cyclic causation, rather a linear regression leading to a much thicker ignorance. If you act following the taints, for example, sensual desire, a much thicker ignorance will arise. So, it's When A arise, B arise and when B arises then C arise. C is a much heavier ignorance that A.

As the Sutta you quoted explains the Noble Eightfold Path is the cessation of both ignorance and the taints.

The way leading to the cessation of the taints is just this Noble Eightfold Path; that is, right view, right intention, right speech, right action, right livelihood, right effort, right mindfulness and right concentration.

The way leading to the cessation of ignorance is just this Noble Eightfold Path; that is, right view, right intention, right speech, right action, right livelihood, right effort, right mindfulness and right concentration.

MN 9 includes both ignorance (avicca) & out-flows (asava) within the 1st condition of Dependent Origination however this does not infer any type of "cycle of causation". Instead, MN 9 is a common genre of Sariputta Abhidhamma found in the suttas which provides additional explanation to that provided by the Buddha.

The Buddha taught ignorance (non-knowing) is the 1st condition of Dependent Origination which conditions the 2nd condition of Dependent Origination, namely, sankhara. Sankhara is three-fold, namely, breathing in & out (kaya sankara), initial & continued thought (vaci sankhara) and perception & feeling (citta sankhara).

When these sankhara arise or are concocted (stirred-up) by ignorance, it is not merely ignorance preceding these sankhara. If it was only ignorance preceding these sankhara then the only type of distracting & discursive thoughts & memories (sankhara) would be thoughts of confusion.

But the Venerable Sariputta additionally points out that before sankhara arise, there are underlying tendencies (anusaya) that flow-out (asava) of the ignorant mind; and these out-flows (asava) also include defilements or drives of sensual desire and (past) becoming.

For example, if in meditation, an egotistical thought in relation to the past spontaneously arises, this is an out-flow of past becoming. Or if sexual urges spontaneously arise, this an an out-flow of sensual desire.

A child is born with in-born drives of sensual desire. The child has the natural urge to eat, be comforted, be touched, etc. These drives arise before the sankhara arise.

Thus, Sariputta is merely pointing out that the 1st link of ignorance is not merely not-knowing or blindness. The 1st link also includes the drives of sensual desire and (past) becoming.

Therefore, there is no cyclical causation involved at the 1st link. The fact that ignorance is also an asava shows there is no cyclical causation.

Including asava within the 1st condition adds a dynamic dimension to ignorance and describes how ignorance is not merely some static condition but describes now ignorance, sensual desire & past becomings erupt (asava) out of the mind & flood the mind.

When the mind of a child is screaming & crying due to its want for sensual pleasure to appease its sensual desire; naturally its mind is blinded by this sensual desire asava & is ignorant to the Four Noble Truths.

But this is not anything cyclical because when the child's sensual desire is appeased via getting fed or getting comforted, the ignorance remains in the mind.

Just because the asava of sensual desire may (temporarily) cease or the asava of past becoming cease, does not mean the asava of ignorance ceases. The asava of sensual desire & becoming may be stilled with samadhi but only vipassana can end the asava of ignorance.

The fact that the appeas*m*nt of sensual desire asava and past becoming asava may not end ignorance shows there is no cyclical causation between ignorance & the asava. Instead, ignorance & the asava are merely the same type of thing.

In AN 10.61, the Buddha makes clear that the five hindrances (which are asava) are the "food" ("ahara") of ignorance. By "food" or "nutriment", what is meant is as long as the five hindrances remain, ignorance will remain because the mind cannot see clearly when the five hindrances arise.

However, AN 10.61 makes it clear the five hindrances are not the preceding "cause" ("hetu") of ignorance. AN 10.61 merely states the five hindrances are a "condition" ("paccaya") that keeps ignorance alive:

Bhikkhus, this is said: ‘A first point of ignorance, bhikkhus, is not seen such that before this there was no ignorance and afterward it came into being.’ Still, ignorance is seen to have a specific condition.

Thus, the ending of the five hindrances in the 1st jhana does not automatically result in the ending of ignorance. The mind of a meditator can get stuck in jhana for a long time.

This shows there is no cyclical causation in the 1st condition of Dependent Origination as taught by Sariputta in MN 9 because, ultimately, the cessation of the only asava that can end ignorance is the cessation of the ignorance asava. Since the ending of ignorance (vipassana) is required to end ignorance, it sounds ridiculous to infer there is a circular causation of ignorance causing ignorance.

The idea of "circular causation" is "papanca"; of imputing upon the Dhamma something that does not exist or what ever taught.

When the Buddha spoke of "samsara" ("cycling"; "circling"), the Buddha spoke of ignorance (1st condition) and craving (8th condition) as comprising of samsara.

The 1st condition of Dependent Origination alone cannot be "samsara" or "circular causation".

Piya Tan writes that Bikkhu Ṇāṇananda wrote that it does involve a kind of cyclic:

⁷⁹ Ṇāṇananda thinks that these two lines on ignorance and the influxes contradict the two in §66. In the former, he influxes” precedes “ignorance,” while in the latter, it is the reverse; but in the dependent arising, we see ignorance always appearing first (Ñāṇananda 2003: 104 f). We see here a reciprocal conditioning of the influxes (āsava) and of ignorance involves a kind of cycle, so that the influxes of ignorance (avijjâsava) is the condition for the arising of ignorance, and ignorance is the condition (avijjā) for the arising of influxes of ignorance. “The implication of this tendency could be to highlight the tendency of ignorance to perpetuate itself, the influx of ignorance [avijjâsava] representing the worldling’s habitual tendency to ignore the true nature of reality, a habit kept alive by its own effects, ignorant thought and action” (Analayo 2005 at M 1:54). Ñāṇavīra notes: “avijjā…can have no anterior term that does not itself involve avijjā.” (1987:36)

Reading the original (i.e. page 104 onward of Nibbāna—the Mind Stilled), Bikkhu Ṇāṇananda says (my paraphrase) that the whole things is like watching a movie -- that ignorance is the dark theatre in/on which the movie is played, and the sankharas are the movie projected in the theatre; and with cessation of ignorance (light in the theatre) the sankharas no longer hold the spectator's attention.

Where the influxes come into that analogy seems to be that, in the original, it wasn't a movie but instead some kind of performance, a play or puppet show. So there are pauses in the production, while actors put on makeup, while the background is changed between scenes, and so on. It's due to the spectators influxes that the spectator enjoys watching the performance (perhaps, anticipating enjoyment) even during these pauses.

But if we are to understand the deeper implications of what the Buddha declared, with reference to the picture caraṇa, a film show or drama is produced, in the last analysis, by the spectator himself. When he goes to the cinema and the theatre, he takes with him the spices needed to concoct a film or a drama, and that is: the influxes, or āsavas. Whatever technical defects and shortcomings there are in them, he makes good with his influxes.

As we know, in a drama there is a certain interval between two scenes. But the average audience is able to appreciate even such a drama, because they are influenced by the influxes of sense desire, existence, and ignorance.

And

Now, for instance, let us think of an occasion when a film show is going on within the framework of darkness. In the case of a matinee show, doors and windows will have to be closed. Supposing the doors are suddenly flung open, while a vivid technicolour scene is flashing on the screen, what happens then? The spectators will find themselves suddenly thrown out of the cinema world they had created for themselves. Why? Because the scene in technicolour has now lost its colour. It has faded away. The result is dejection, disenchantment. The film show loses its significance.

That film show owed its existence to the dark framework of ignorance and the force of preparations. But now that the framework has broken down, such a vast change has come over, resulting in a disenchantment. Now the word rāga has a nuance suggestive of colour, so virāga, dispassion, can also literally mean a fading away or a decolouration. Here we have a possible instance of nibbidā virāga, disenchantment, dispassion, at least in a limited sense.