To rearrange a positive integer array such that the sum of the product of adjacent elements is maximized, you can follow these steps:

Sort the array in non-ascending order. This will place the largest elements at the beginning of the array.

Pair adjacent elements starting from the first element (index 0) and multiply them.

Let's write a JavaScript function to achieve this:

`javascript````
function rearrangeForMaxProductSum(arr) {
// Step 1: Sort the array in non-ascending order
arr.sort((a, b) => b - a);
// Step 2: Pair adjacent elements and calculate the sum of their products
let maxProductSum = 0;
for (let i = 0; i < arr.length - 1; i++) {
maxProductSum += arr[i] * arr[i + 1];
}
return { rearrangedArray: arr, maxProductSum };
}
// Example usage:
const inputArray = [4, 2, 3, 5, 1];
const result = rearrangeForMaxProductSum(inputArray);
console.log('Original Array:', inputArray);
console.log('Rearranged Array:', result.rearrangedArray);
console.log('Max Product Sum:', result.maxProductSum);
```

The function `rearrangeForMaxProductSum`

will sort the input array in non-ascending order and then calculate the sum of the product of adjacent elements. The rearranged array will have the largest elements first, which will maximize the sum of the product of adjacent elements.

Note that this approach doesn't guarantee a mathematically proven maximum sum, but it usually provides a good result for many cases.