N Queens Backtracking in PHP

<?php
declare(strict_types=1);
// N Queens Backtracking in PHP
//
// PURPOSE
// N-queens places one queen in each row with no shared columns or diagonals.
//
// METHOD
// Try columns row by row. Keep occupied-column and diagonal marker arrays for constant-time
// safety checks. Backtrack when no column remains and count complete placements.
//
// COMPLEXITY
// Exponential search, bounded by O(n!) candidate placements with additional column-scan
// overhead; O(n) auxiliary space.
//
// CONTRACT
// 0<=n<=12. Returns solution count. The empty board has one placement. This implementation
// avoids recursion.
// 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: 1.
// 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 n_queens(int $n): int {
    alg_require((($n >= 0) && ($n <= 1000000)));
    alg_require((($n >= 0) && ($n <= 12)));
    $columns = array_fill(0, alg_size(alg_add($n, 1)), 0);
    $rising = array_fill(0, alg_size(alg_add(alg_mul(2, $n), 1)), 0);
    $falling = array_fill(0, alg_size(alg_add(alg_mul(2, $n), 1)), 0);
    $placement = array_fill(0, alg_size(alg_add($n, 1)), 0);
    $row = 0;
    $col = 0;
    $count = 0;
    if (($n == 0)) {
        return 1;
    }
    while (($row >= 0)) {
        $col = $placement[$row];
        if (($col > 0)) {
            $columns[alg_sub($col, 1)] = 0;
            $rising[alg_sub(alg_add($row, $col), 1)] = 0;
            $falling[alg_add(alg_sub($row, $col), $n)] = 0;
        }
        while ((($col < $n) && (($columns[$col] != 0) || ($rising[alg_add($row, $col)] != 0) || ($falling[alg_sub(alg_add(alg_sub($row, $col), $n), 1)] != 0)))) {
            $col = alg_add($col, 1);
        }
        if (($col == $n)) {
            $placement[$row] = 0;
            $row = alg_sub($row, 1);
        } else {
            $placement[$row] = alg_add($col, 1);
            $columns[$col] = 1;
            $rising[alg_add($row, $col)] = 1;
            $falling[alg_sub(alg_add(alg_sub($row, $col), $n), 1)] = 1;
            if (($row == alg_sub($n, 1))) {
                $count = alg_add($count, 1);
            } else {
                $row = alg_add($row, 1);
                $placement[$row] = 0;
            }
        }
    }
    return $count;
}

$n = 0;
$result = n_queens($n);
echo $result, PHP_EOL;