When will a chi-squared statistic equal zero? You

Introduction:
The chi-squared test is a non-parametric hypothesis test that is useful in case of a frequency data.
There are three main types of chi-squared tests- chi-squared test of independence to test for independence/association between two categorical variables each with at least two categories, chi-squared test of homogeneity to test whether two populations are homogeneously distributed across various categories of a categorical variable of interest, and the chi-squared test of goodness of fit to test whether the distribution of a sample fits into a pre-defined population.
Explanation:
The data on which the chi-squared test is applied is usually represented in the form of a two-dimensional contingency table, where the rows correspond to the categories of one of the categorical variables, and the columns correspond to the categories of the other categorical variable.
The formula of the test statistic for any of the chi-squared tests discussed above is:
${\displaystyle {\chi }^2=\sum _i\sum _j\frac{{(O_{ij}-E_{ij})}^2}{E_{ij}}}$
wher
${\displaystyle O_{ij}}$ is the observed frequency in cell (i,j);
${\displaystyle E_{ij}}$ is the expected frequency in cell (i,j).
Note that the cell (i, j) corresponds to the ${\displaystyle i^{th}}$ category of the row categorical variable, and the ${\displaystyle j^{th}}$ category of the column categorical variable.
Now, the expected frequencies will always be non-negative integers, that is, at least 0, as it will be obtained by multiplying the total sample size by the hypothesized proportion for the particular cell, both of which are non-negative integers.
The observed frequencies will also always be non-negative integers.
Clearly, both ${\displaystyle O_{ij}}$ and ${\displaystyle O{E}_{ij}}$ are real quantities.
Being the sum of squared differences of real quantities, the numerator will always be non-negative.
Since both the numerator and the denominator of the test statistic are non-negative real quantities, the test statistic itself must be a non-negative real quantity.
In order for the test statistic to be zero, the numerator must be zero, that is, only when ${\displaystyle O_{ij}=E_{ij}}$, for each i, j.
Now, the observed and expected frequencies are equal for each cell if the two categorical variables are perfectly independent of each other (for independence/association test), or the two populations are perfectly homogeneous (for homogeneity test), or the data perfectly fit the hypothesized population (for goodness of fit test).