Correct syntax to render $block['content'] as theme of type table

UPDATE: I think I found the answer, see in the Comments section.

Summary: when rendering a table into a block, nothing is displayed. What am I doing wrong?

I've been researching the answer to my question for the past 48 hours, and have held off writing a comment or new forum question so as to not duplicate a question, or ask about something that is clearly documented elsewhere.

I am writing a module to display the results of a very simple query:

'SELECT fname, superlname, title, organization, contactid FROM {TABLE1} WHERE contactid > 0'

(I know the query works, as I am dpr()'ing this information and getting values back.)

I then put values into a header array, and foreach() the query results into another array. I then attempt to save this into $block['content'] and return this, but I get nothing on the screen.

I have dpr() the $block['content'] variable and I am seeing just :
<table></table>

So, clearly, I am sending in the header and rows information incorrectly, or it would be rendering properly in my block.
Also, if I change the query to:
'SELECT fname, superlname, title, organization, contactid FROM {TABLE1} WHERE contactid = 0'
(equal zero), then I fork over to the other part of the if statement, and the block properly displays "no contacts available."

function show_import_table_block_view($delta = '') {
  switch ($delta) {
    case 'show_import_table':
      $block['subject'] = t('All finalimport table rows.');
      if (user_access('access content')) {
        // Use our custom function to retrieve data.
        $result = show_import_table_contents();
        // Array to contain items for the block to render.
        // Iterate over the resultset and format as links.
       
       $header = array('fname', 'superlname', 'title', 'organization');
       $datarow = array();
       $allrows = array();
        foreach ($result as $row) {
          $title1 = substr($row->title, 0, 10);
          $organization1 = substr($row->organization, 0, 10);
          $datarow = array($row->fname, $row->superlname, 'title', 'organization');
          array_push($allrows, $datarow);
        }

    // No content in the last week.
        if (empty($allrows)) {
          $block['content'] = t('No contacts available.');  
        } 
        else {
          // Pass data through theme function.
          $superrows = array(
        array('row1col1', 'row1col2'),
        array('row2col1', 'row2col2'));
          $block['content'] = theme('table', $header, $superrows);
          dpr($header, $return = FALSE, $name = NULL);
          dpr($allrows, $return = FALSE, $name = NULL);
        }
      }
    return $block;
    $apple = 'Steve Jobs';
    dpr($apple, $return = FALSE, $name = NULL);
  }
  
}

I have been using the following two sources as syntax guidelines:
https://api.drupal.org/api/drupal/modules%21system%21theme.api.php/group...
https://api.drupal.org/api/drupal/includes%21theme.inc/function/theme_ta...