Beal Conjecture Replication Results
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[PASS] MATH_entropy_bounds
  H=1.039721, p sum=1.0000000000

[PASS] MATH_symmetric_entropy
  H(1/3,1/3,1/3)=1.0986122887, log(3)=1.0986122887

[PASS] MATH_constrained_equilibrium
  H(1/3)=1.098612 >= H(other) for all test distributions: True

[PASS] MATH_hessian_negative
  d^2H/dp_i^2 = -1/p_i < 0 at all test points: True

[PASS] MATH_lagrangian
  p*=exp(lambda-1)=0.3333333333, 3p*=1.0000000000

[PASS] MATH_unconstrained_off_simplex
  3/e=1.103638, |3/e - 1|=0.1036 > 0.1

[PASS] NUM_known_solution
  2^3+2^3=2^4: eq=True, gcd=2: True, H=1.039721

[PASS] NUM_congruence
  S=16, C^z=16, S mod 2^4=0

[PASS] NUM_diophantine_deviation
  Solutions found: 6, all p3=0.5 (not 1/3): True

[PASS] NUM_no_counterexamples
  Triples scanned: 10, N=20, solutions: 7, counterexamples (gcd=1): 0

[PASS] NUM_congruence_all
  Congruence A^x+B^y = 0 (mod p^z) for all primes p|C: True

[PASS] NUM_reproducibility
  Two identical scans match: True


Overall: PASS
