Write the Euler characteristic of the smooth hypersurface $X_d^n$ in its binomial form, $\chi_n(d) = \sum_{j=1}^{n+1} (-1)^{j+1}\binom{n+2}{j+1} d^j$ (the polynomial of $EulerCharacteristic$ divided by $d$). Its coefficients are not all even, so evenness of $\chi_n(d)$ is not a polynomial identity; it is a parity fact.

The first statement: for every odd dimension $n$ and every integer $d$, $\chi_n(d)$ is even. For threefolds this is what makes the mirror-symmetry formula $h^{2,1} = b_3/2 - h^{3,0}$ with $b_3 = 4 - \chi$ an integer for every degree; the second statement is that instance.

The remaining two statements concern surfaces in $\mathbb{P}^3$, where the literature gives $\chi = d^3 - 4d^2 + 6d$, $h^{2,0} = \binom{d-1}{3}$ and $h^{1,1} = \tfrac{2d^3}{3} - 2d^2 + \tfrac{7d}{3}$. The third statement is that $h^{1,1}$ is an integer for every $d$, i.e. $3 \mid 2d^3 - 6d^2 + 7d$; the fourth is that the three formulas satisfy the Betti relation of a simply connected surface, $\chi = 2 + 2h^{2,0} + h^{1,1}$, as a polynomial identity in $d$ (multiplied by $3$ and with $6\binom{d-1}{3} = (d-1)(d-2)(d-3)$).