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verify_fso.py

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+import math
+
+def gcd(a, b):
+    while b:
+        a, b = b, a % b
+    return a
+
+def phi(n):
+    result = n
+    p = 2
+    temp_n = n
+    while p * p <= temp_n:
+        if temp_n % p == 0:
+            while temp_n % p == 0:
+                temp_n //= p
+            result -= result // p
+        p += 1
+    if temp_n > 1:
+        result -= result // temp_n
+    return result
+
+def test_theorem_2_1(m):
+    """Theorem 2.1: N_b(m) = m^{m-1} * phi(m)"""
+    nb_theoretical = (m**(m-1)) * phi(m)
+    # Brute force count for small m
+    count = 0
+    for i in range(m**m):
+        b = []
+        temp = i
+        for _ in range(m):
+            b.append(temp % m)
+            temp //= m
+        if gcd(sum(b), m) == 1:
+            count += 1
+    print(f"Theorem 2.1 (m={m}): Theoretical={nb_theoretical}, Empirical={count}")
+    return nb_theoretical == count
+
+def test_theorem_3_1(m, k):
+    """Theorem 3.1: |M_k(G_m)| = phi(m) * [N_b(m)]^{k-1}"""
+    nb = (m**(m-1)) * phi(m)
+    mk_theoretical = phi(m) * (nb**(k-1))
+    print(f"Theorem 3.1 (m={m}, k={k}): Theoretical={mk_theoretical}")
+    return mk_theoretical
+
+def test_theorem_4_1(m, k):
+    """Theorem 4.1: Obstruction when m is even and k is odd"""
+    if m % 2 == 0 and k % 2 != 0:
+        print(f"Theorem 4.1 (m={m}, k={k}): Obstruction Predicted.")
+        return True
+    return False
+
+def test_law_vi_2d(m):
+    """Law VI: 2D Torus is universally solvable for all m"""
+    # Sum of two r_c must be m.
+    # For m=4, r=(1, 3) works. (1+3=4, gcd(1,4)=1, gcd(3,4)=1)
+    # For m=3, r=(1, 2) works. (1+2=3, gcd(1,3)=1, gcd(2,3)=1)
+    # Coprimality is possible for both odd and even m when k=2.
+    if m % 2 == 0:
+        # Sum of two odd numbers is even. r1=1, r2=m-1 are both odd.
+        r1, r2 = 1, m-1
+        valid = (gcd(r1, m) == 1 and gcd(r2, m) == 1)
+    else:
+        # Sum of an odd and an even is odd.
+        # But we need r1, r2 to be coprime to m.
+        # For m=3, r1=1, r2=2 works.
+        r1, r2 = 1, m-1
+        valid = (gcd(r1, m) == 1 and gcd(r2, m) == 1)
+    print(f"Law VI (m={m}, k=2): r=({r1}, {r2}), coprimality={valid}")
+    return valid
+
+def test_spike_construction(m):
+    """Theorem 5.1 & 5.3: Spike Construction (1, m-2, 1)"""
+    if m % 2 == 0:
+        return False
+    r = (1, m-2, 1)
+    sum_b0 = 2 % m
+    sum_b1 = (m-1) % m
+    sum_b2 = (m-1) % m
+    valid0 = gcd(sum_b0, m) == 1
+    valid1 = gcd(sum_b1, m) == 1
+    valid2 = gcd(sum_b2, m) == 1
+    print(f"Spike (m={m}): r={r}, sums=({sum_b0}, {sum_b1}, {sum_b2}), valid=({valid0}, {valid1}, {valid2})")
+    return valid0 and valid1 and valid2
+
+if __name__ == "__main__":
+    print("--- FSO Mathematical Verification ---")
+    for m in [3, 4, 5]:
+        assert test_theorem_2_1(m)
+    assert test_theorem_3_1(3, 3) == 648
+    assert test_theorem_4_1(4, 3) == True
+    assert test_theorem_4_1(3, 3) == False
+    for m in [3, 4, 5, 6, 100, 101]:
+        assert test_law_vi_2d(m)
+    for m in [3, 5, 7, 9, 11, 13, 101]:
+        assert test_spike_construction(m)
+    print("--- All Tests Passed Successfully ---")