Answer
I say, and with God's success: The intention in ablution is a Sunnah and it is not a condition for the validity of the ablution; thus, the validity of the ablution does not depend on the intention. This is because the reward is contingent upon the intention by consensus, so the reward must be estimated, or something that includes the reward must be estimated, such as the ruling of actions by intentions. If the reward is estimated, it is evident; and if the ruling is estimated, it is of two types: worldly, like validity, and otherworldly, like the reward, and the otherworldly is intended by consensus.
This statement does not apply to pure acts of worship; because the purpose of them is the reward, if they lack the intended purpose, they cannot be valid; because they were not legislated except as acts of worship. Unlike ablution, which is not an intended act of worship, but was legislated as a condition for the permissibility of prayer. If it lacks the intended purpose — that is, the reward — it ceases to be an act of worship, but this does not necessitate the invalidity of its validity. Because if ablution lacks intention, its validity remains in the sense that it is the key to prayer, as in the saying of the Prophet (peace be upon him): "The key to prayer is purity, and its prohibition is the Takbir, and its permissibility is the Tasleem," as mentioned in Sunan al-Tirmidhi 1: 9, 2: 3, and in Al-Mustadrak 1: 223.
And because the nature of water is to remove and purify, using it necessitates the attainment of purity even if it lacks intention; because the nature of a thing does not depart from it: like fire, its nature is to burn, it burns if it finds a combustible place, and no one would say that his beard does not burn in fire if he does not intend it. Likewise, food and water, for their use necessitates quenching thirst and satiation without the addition of something else. And because God Almighty commanded us to perform ablution, which is washing and wiping, each of which is a specific term for a known meaning, which is flowing and touching, and there is nothing in them that indicates intention. Therefore, requiring intention is an addition to the text, and this is not permissible by analogy and the report of a single narrator; this is because ablution is a condition for prayer, and conditions are considered based on their existence regardless of the intention behind them. It has become like the act of rushing to Friday prayer, as each of them is a means, and then rushing by any way that is suitable for performing Friday prayer. Likewise, ablution for prayer, and likewise in other conditions: like purifying the garment, the place, and covering the private parts, for intention is not required in any of them, unlike Tayammum, as its nature is impure, not purifying, thus it requires intention. Refer to: Sharh al-Waqayah pp. 82-83, and Fath Bab al-‘Inayah 1: 55, and God knows best.