Prefix Sum Range Query in PHP

<?php
declare(strict_types=1);
// Prefix Sum Range Query in PHP
//
// PURPOSE
// Prefix sums turn a range sum into a subtraction. The preprocessed prefix at index i stores
// the sum of all elements before i.
//
// METHOD
// Build prefix[i+1] = prefix[i] + a[i]. For inclusive bounds left and right, subtract
// prefix[left] from prefix[right+1]. Build once and reuse the prefix for many queries in a
// real application.
//
// COMPLEXITY
// O(n) preprocessing and O(n) space; each query after preprocessing is O(1). This self-
// contained function includes preprocessing on every call.
//
// CONTRACT
// 0 <= left <= right < n. Returns the inclusive range sum. All prefix totals must fit.
// Public APIs infer array lengths except in C, which accepts pointer/length pairs. Vertex
// counts and matrix dimensions remain explicit. Arrays must contain signed 32-bit integers.
// Sorted order, correct tree topology, and caller-owned data-structure invariants are
// preconditions; checking them on every query would change query complexity.
// Scalar and shape checks are included. Value-range checks are included where needed to
// prevent invalid indexing or nontermination. Allocation sizes are capped at 4,000,000
// workspace entries. Arithmetic outside Python raises an error on 32-bit overflow; Python
// uses arbitrary precision. Native allocation failures propagate. In-place outputs may be
// partly updated after arithmetic failure. These are single-threaded, in-memory
// implementations.
//
// LANGUAGE DETAILS
// PHP 8.1+ on a 64-bit runtime: packed arrays, strict scalar types, read-only inputs by
// value using copy-on-write, references only for outputs, and checked 32-bit arithmetic.
// Invalid input and overflow throw exceptions. PHP arrays have higher overhead than native
// packed buffers in C/Java.
//
// EXAMPLE
// Expected return: 6.
// Run: php example.php
// Tests for this exact source artifact run before publication. The accompanying source
// archive includes unit cases, runtime results, and source hashes.

function alg_require(bool $valid): void { if (!$valid) throw new InvalidArgumentException("invalid algorithm input"); }
function alg_checked(int|float $v): int { if ($v < -2147483648 || $v > 2147483647 || floor($v) != $v) throw new OverflowException("32-bit result overflow"); return (int)$v; }
function alg_add(int $a,int $b): int { return alg_checked($a+$b); }
function alg_sub(int $a,int $b): int { return alg_checked($a-$b); }
function alg_mul(int $a,int $b): int { return alg_checked($a*$b); }
function alg_size(int $n): int { alg_require($n>=0 && $n<=4000000); return $n; }

function range_sum(array $a, int $left, int $right): int {
    alg_require(array_is_list($a));
    $n = count($a);
    alg_require((($n >= 0) && ($n <= 1000000)));
    alg_require((count($a) == $n));
    alg_require((($left >= 0) && ($left <= $right) && ($right < $n)));
    $prefix = array_fill(0, alg_size(alg_add($n, 1)), 0);
    $i = 0;
    while (($i < $n)) {
        $prefix[alg_add($i, 1)] = alg_add($prefix[$i], $a[$i]);
        $i = alg_add($i, 1);
    }
    return alg_sub($prefix[alg_add($right, 1)], $prefix[$left]);
}

$a = [2, 4, -1, 3];
$left = 1;
$right = 3;
$result = range_sum($a, $left, $right);
echo $result, PHP_EOL;