Number Scalpel
Direct Inspection
Khinchin-Compliant Giants (Transcendental / Random)
Periodic Quadratic Irrationals (Mean ≠ K₀)
Special Exceptions (Divergent / Terminating)
Geometric Mean (n=100)
2.68
Δ: -0.005 from K₀
Khinchin Limit K₀
2.685452
Theoretical Constant
Gauss-Kuzmin Fit
0.94
Distribution Match
Compliance Status
Compliant
Transcendental
[3; 7, 15, 1, 292, 1, 1, 1, 2, 1, 3, 1, 14, 2, 1, 1, 2, 2, 2, 2, ...]
Geometric Mean Convergence: π (Pi)
Step-by-step (∏ aᵢ)^(1/n)
Running Geometric Mean $\left(\prod a_i\right)^{1/n}$
Khinchin Constant $K_0 = 2.685452...$
Running Arithmetic Mean $\frac{1}{n}\sum a_i$
Universal Attractor: For almost all real numbers (except a set of Lebesgue measure zero), the geometric mean of continued fraction coefficients converges to 2.685452001... regardless of starting value!
Gauss-Kuzmin Probability Distribution
Empirical vs Theoretical
The probability that coefficient $a_k = k$ follows $P(a_k = k) = -\log_2\left(1 - \frac{1}{(k+1)^2}\right)$.
| Coeff (k) | Gauss-Kuzmin Law | Observed Freq in Current # |
|---|
Continued Fraction Recursive Tree
Algebraic Anatomy
Any real number $x$ decomposes into integer parts and reciprocal remainders:
x = a₀ + 1 / (a₁ + 1 / (a₂ + 1 / (a₃ + ...)))
π ≈ 3 + 1 / (7 + 1 / (15 + 1 / (1 + 1 / (292 + ...))))
Why Exceptions Fail:
- Quadratic Irrationals (e.g. $\phi = [1; 1, 1...], \sqrt{2} = [1; 2, 2...]$): Periodic coefficients cause geometric means to lock onto algebraic values like 1.0 or 2.0.
- Euler's $e = [2; 1, 2, 1, 1, 4, 1, 1, 6, ...]$: Arithmetic progression of coefficients causes geometric mean to grow like $(n!)^{1/n} \to \infty$.
- Rationals ($355/113$): Terminate with finite terms; geometric mean stops updating.
Monte Carlo Measure-Theoretic Demonstration (100 Simultaneous Real Numbers)
Lebesgue Measure = 1
Pick 100 random real numbers uniformly at random from $[0,1]$. While individual trajectories fluctuate wildly due to large coefficients (like 292 in $\pi$), their aggregate geometric means inevitably condense onto the singular line 2.685452...