Math for college dependent student costs - I understand it as support, but want to see if math works otherwise

I am reposting this as a standalone question, there are several resources that say the money I pay as a parent for my dependent college student is support and not a gift, but there is sometimes ambiguity in the response, so to be doubly sure I wanted to check the logic of the math below using what I understand is considered a legitimate "layered" approach.

 

Here is a hypothetical situation and I'd like someone to check the logic:

College costs for dependent student:

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Tuition: $10,000; Qualified Education costs (would be in Box 1 of 1098-T) 

Mandatory Fees: $5,000; Qualified Education costs (would be in Box 1 of 1098-T) 

Housing and Food: $15,000; Unqualified education costs (would not be in Box 1 of 1098-T)

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Total: $30,000

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Unrestricted Scholarship (can be used for any of above): $5,000

Accounting wise, the school lumps expenses and payments, including the scholarship in one general account, but will report the scholarship in Box 5 of the 1098-T

 

I have checked online, and it says as long as a person makes the payment directly to the school, the tuition is never considered a gift, it also says that a parent has leeway in terms of how the unrestricted scholarship is allocated when performing tax calculations regardless of how the school actually did it.

 

So, to avoid any need to fill out a 709 I assume that the scholarship is applied to Fees first, then Tuition, such that:

Tuition: $10,000 - not a gift, not reportable as gift

Fees: $5,000 - $5,000 scholarship = $0 - therefore no money to report 

Housing and Food: $15,000 - while this is considered as support when used on a dependent and not a gift, if it was considered a gift, it is below the $19,000 gift threshold so no 709 needs to be filed

 

Is this good math?  I did research and asked several AI's and I am told this is completely legitimate way to approach the problem.  I also had several AI's say that since the housing and food is considered support when the college student is my dependent, the IRS wouldn't expect it to be reported as a gift even if it was over $19,000