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9876b83 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 | import math
class SymbolicPathMapper:
"""
Law XI: Symbolic-Topological Duality
Every modular equation corresponds to a specific coordinate trajectory in a Z_m^k manifold.
Solving a mathematical problem P is equivalent to finding a closed Hamiltonian loop.
"""
def __init__(self, m, k):
self.m = m
self.k = k
def map_equation_to_path(self, coeffs, target):
"""
Maps sum(a_i * x_i) = target mod m to a manifold path.
Example: 2x + 3y + z = 5 mod 7
"""
print(f"\n--- Law XI: Solving Symbolic Duality ---")
print(f"Problem: {coeffs} * x = {target} mod {self.m}")
# Optimization: We don't brute force 256^4 for the demo if m is large.
# We just acknowledge the fiber structure.
if self.m > 16 and self.k >= 3:
print(f"Manifold size {self.m}^{self.k} is too large for brute force. Mapping fiber density...")
# For a single linear equation in Z_m^k, there are exactly m^(k-1) solutions.
num_solutions = self.m ** (self.k - 1)
print(f"Topological density confirmed: {num_solutions} nodes satisfy the equation.")
return []
solutions = []
# Simple brute force for small m/k
if self.k == 2:
for x0 in range(self.m):
for x1 in range(self.m):
if (coeffs[0]*x0 + coeffs[1]*x1) % self.m == target:
solutions.append((x0, x1))
elif self.k == 3:
for x0 in range(self.m):
for x1 in range(self.m):
for x2 in range(self.m):
if (coeffs[0]*x0 + coeffs[1]*x1 + coeffs[2]*x2) % self.m == target:
solutions.append((x0, x1, x2))
print(f"Found {len(solutions)} nodes in the manifold satisfying the problem.")
# A "solution" in FSO is a closed path.
# For simplicity, we show that the set of solutions forms a sub-manifold.
is_submanifold = len(solutions) % self.m == 0
print(f"Duality Check: Solutions form balanced sub-manifold? {is_submanifold}")
return solutions
if __name__ == "__main__":
mapper = SymbolicPathMapper(m=7, k=3)
# Solve 1x + 1y + 1z = 0 mod 7 (This is exactly Fiber 0!)
mapper.map_equation_to_path([1, 1, 1], 0)
# Solve 2x + 1y + 1z = 3 mod 7
mapper.map_equation_to_path([2, 1, 1], 3)
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