Sieve of Eratosthenes in C#

using System;
// Sieve of Eratosthenes in C#
//
// PURPOSE
// The sieve identifies all primes up to a limit by marking multiples of smaller primes. This
// example returns their count.
//
// METHOD
// Keep a composite marker array. For each unmarked p up to the square root, mark multiples
// starting at p squared; smaller multiples were already handled. Count the remaining
// unmarked values from two onward.
//
// COMPLEXITY
// O(n log log n) time and O(n) auxiliary space.
//
// CONTRACT
// n >= 0. Returns the number of primes <= n. The marker array requires memory proportional
// to n.
// 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
// C# / .NET 10: ReadOnlySpan<int> inputs, Span<int> outputs, primitive workspace arrays,
// Array.BinarySearch for LIS, and checked integer arithmetic. Invalid arguments and overflow
// throw exceptions.
//
// EXAMPLE
// Expected return: 4.
// Run: Use Program.cs in a net10.0 console project; dotnet run.
// Tests for this exact source artifact run before publication. The accompanying source
// archive includes unit cases, runtime results, and source hashes.

class Program {
    static void alg_require(bool valid) { if (!valid) throw new ArgumentException("invalid algorithm input"); }
    static int alg_size(int n) { alg_require(n>=0 && n<=4000000); return n; }

    static int sieve_count(int n) {
        checked {
            alg_require(((n >= 0) && (n <= 1000000)));
            int[] composite = new int[alg_size((n + 1))];
            int p = 2;
            int multiple = 0;
            int count = 0;
            while ((p <= (n / p))) {
                if ((composite[p] == 0)) {
                    multiple = (p * p);
                    while ((multiple <= n)) {
                        composite[multiple] = 1;
                        multiple = (multiple + p);
                    }
                }
                p = (p + 1);
            }
            p = 2;
            while ((p <= n)) {
                if ((composite[p] == 0)) {
                    count = (count + 1);
                }
                p = (p + 1);
            }
            return count;
        }
    }

    static void Main() {
        int n = 10;
        int result = sieve_count(n);
        Console.WriteLine(result);
    }
}