Your math is not quite right. The formula is that, if you are single, take half your social security plus all your other income; if that total is more than $25,000, then part of your social security is taxable. Your other income is always taxable, except that it may be covered by standard deduction, which was $12,950 for 2022 and will be $13,850 for 2023. In your case, the result of the formula is $6000, so your social security is not taxable, and even though the $1000 is considered taxable income, it is less than the standard deduction, so no tax will be owed (as long as you file single, head of household, or married filing jointly.)
However, if you happen to be married and file separately from your spouse ("married filing separately" status) then your social security income is automatically taxable and your standard deduction is zero. So you need to file and pay tax.
Lastly, if you got a 1099-G from the lottery office, you need to file a return and report the income even though no tax will be owed. The IRS will come looking for the money if you don't report it.
