A Parity Law for Polynomial Floor Sums

We prove a parity classification for a polynomial floor sum, where the exponent is an even integer greater than or equal to 4. The main result states that the sum is odd exactly when n is congruent to 3, 6, or 7 modulo 8. The proof is elementary and uses a pairing argument, the binomial theorem, and reduction modulo 2. We also provide computational verification for even exponents from 4 through 40 and values of n from 1 through 500. A separate literature and novelty check is included. To our knowledge, this parity classification has not previously been stated explicitly in the literature.