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L. F. gave a good review, but there is more:

1. Exceptions are not for programmer errors. That's what asserts are for.

2. Modularise your code:

   Extract down_heap() and up_heap() as free functions, making the algorithm available for anyone wanting to manipulate a heap.
   They should accept an iterator-range and a comparator, with the default being std::less<>().

3. Mark things noexcept if you can. Consider conditional noexcept too.

4. Also, make them constexpr if applicable.