What is the ruling on one who experiences the emission of gas without sound or smell during ablution and prayer, while being certain of its emission? Is he required to perform ablution for every prayer, or should he act upon the Hadith: "He should not leave (his prayer) unless he hears a sound or perceives a smell"? And why has his supplication for the removal of this affliction not been answered?

As long as you do not suffer from continuous flatulence, and what you mentioned only occurs during ablution or prayer, then it is a whisper from Satan. So pray normally and do not pay attention to it until it gradually disappears. Continue to supplicate, for Allah answers prayers as He promised: "And your Lord says, 'Call upon Me; I will respond to you'" and "And when My servants ask you concerning Me, then [answer them], I am indeed near. I respond to the invocation of the supplicant when he calls upon Me. So let them respond to Me [by obedience] and believe in Me that they may be rightly guided." The answer may be by hastening what is sought, or reserving it for the Hereafter, or averting evil. Allah knows best what is in your interest, and you must have a good opinion of Him, and know that there is good in what He chooses for you, for "And perhaps you dislike a thing and it is good for you; and perhaps you love a thing and it is bad for you. And Allah knows, while you know not." And the hadith "He should not leave [his prayer] until he hears a sound or finds a smell" means certain knowledge of the emission of wind, not necessarily hearing or smelling it.