Hello, fellow explorer! Today, we will unravel the mystery of "Recursion" — a concept as enthralling as the patterns formed by two mirrors facing each other. We aim to decipher recursion, understand its inner workings, and master its application in programming.

Consider a stack of books. Want the bottom one? You'll need to remove each book above it, one by one. It's a recurring action — an example of recursion. In programming, recursion involves a function calling itself repeatedly until a specific condition is met, similar to descending stairs one step at a time until you reach the ground.

Here's a simple PHP function illustrating recursion:

php
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1<?php
2function recursiveFunction($x) {
3  if($x <= 0){ // Termination condition --> base case
4    echo "Base case reached\n";
5  } else {
6    echo $x . PHP_EOL;
7    recursiveFunction($x - 1); // Recursive function call --> recursive case
8  }
9}
10
11recursiveFunction(5);
12
13/* Output:
145
154
163
172
181
19Base case reached
20*/
21?>

This function keeps calling itself with $x decreasing by one until $x <= 0, which is our base case. At this point, it stops the recursion.

The base case acts like a friendly signpost, telling the recursion when to stop. In our book stack example, reaching a point where no more books are left to remove serves as the signal. Similarly, $x <= 0 is our base case in our function. The base case is crucial as it prevents infinite recursion and related errors.

The recursive case is an essential aspect of recursion — the rule responsible for creating smaller versions of the original problem. Each call brings us a step closer to the base case. Let's use the process of calculating a factorial as an illustrative example.

To find a factorial, we multiply a number by the factorial of the number minus one and repeat this process until we get to one (our base case):

php
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1<?php
2function factorial($n) {
3  if($n == 1){ // base case
4    return 1; 
5  } else {
6    return $n * factorial($n - 1); // recursive case
7  }
8}
9
10echo factorial(3); // we expect 6 (3 * 2 * 1)
11?>

In this case, when we call factorial(3), it returns 3 * factorial(2), where factorial(2) returns 2 * factorial(1). As factorial(1) is a base case, it returns 1. As a result, the whole recursion chain returns 3 * 2 * 1.

To think recursively, visualize the problem like an onion. Peeling each layer brings you closer to the center. The center of the onion represents the base case, and the peeling process denotes the recursive case.

Remember that a complex problem often contains smaller, simpler sub-problems. You can trust these sub-problems to be solved correctly, culminating in an elegant solution.

Let's develop a function that calculates the sum of an integer's digits. Normally, it would involve using a while loop, but with recursion, it is done much more easily:

php
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1<?php
2function sumOfDigits($num) {
3  // Base case: if num is less than 10, return num itself
4  if($num < 10) {
5    return $num; 
6  } else {
7    return $num % 10 + sumOfDigits(intval($num / 10)); // Recursive case
8  }
9}
10
11echo sumOfDigits(12345); // Will print out 15 (1+2+3+4+5)
12?>

In this example, we use the same principle as with factorial calculation, but we pass intval($num / 10) to the next recursion call, chopping off the last digit every time.

Let's pause here. We've unveiled the concept of recursion, determined the roles of base and recursive cases, and written a simple recursive function in PHP. It's time to unlock its full potential through continuous practice, building a solid foundation for the upcoming lessons on sorting and searching algorithms. Remember — practice illuminates knowledge. Happy experimenting!