<p>It is difficult to read what your question is about. Your introduction is:</p>
<blockquote>
<p>I really like the truncated normal but I want to make it discrete. I need a discrete distribution and I hate approximations (the best fit for my data is truncated normal but my data are discrete [0,25]) I am fixated on the kernel and the nice relations and ratios of the different values.</p>
</blockquote>
<p>So you hate approximations... but what is the question?</p>
<blockquote>
<p>I already answered why. Most people when they have such data use Normal or Truncated Normal at best.</p>
</blockquote>
<p>This doesn't clarify at all why you want to discretize the normal distribution and what the problem is.</p>
<hr />
<p>The problem is very general/broad and cannot be easily answered without the question having more focus.</p>
<p>There might be a reason why you 'really like the truncated normal'. But if you leave it open for others to guess why this is the case, then this information is not very  useful.</p>
<ul>
<li>If you want some discretized truncated normal distribution, what are the properties of the truncated normal that should remain when the distribution is discretized?</li>
<li>How do you want the discretization to be done? The underlying mechanism in your problem, why it is that a normal distribution is a good model, will  influence  the way to do the discretization.</li>
</ul>
<p>If you explain the problem and the motivation then all such considerations do not need to be guessed.</p>
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<p>And to be honest. I did not read the second half of the question. It is a big block of text which might become more readable if you split it into multiple paragraphs.</p>
