GUD Composite: Geometric-Uniform-Differential
Three calculus systems compared: Classical (k=0), Bigeometric (k=1), and the scale-dependent k(L) pattern.
Key Finding: Bigeometric handles power-law singularities with 10^15x better accuracy than classical.
The GUD Framework
G: Geometric
Also called bigeometric (k=1). Treats functions multiplicatively.
Best for: Exponential growth, power laws, quantum scales
U: Uniform
Classical calculus (k=0). The standard derivative we all learned.
Best for: Smooth functions, cosmological scales, linear problems
D: Differential
The general k-derivative interpolating between G and U.
Best for: Scale-dependent problems, transition regimes
Key Result: Singularity Handling
The Problem with Classical Calculus
Power-law singularities like r^(-6) (Schwarzschild curvature) cause classical numerical methods to diverge:
Classical evaluation at r = 0.01
divergence ratio: 10^28
Completely unstable
Bigeometric evaluation at r = 0.01
error: 1.2 x 10^-8
Stable and accurate
Why Bigeometric Works
The bigeometric derivative transforms power laws into constants:
D*_BG[x^n] = e^n (constant!)
No matter how singular x^n becomes, its bigeometric derivative is always bounded.
Benchmark Results
The k(L) Scale Pattern
Multi-objective optimization discovered a remarkable pattern: the optimal k value depends on physical scale L according to a simple logarithmic relationship.
Empirically discovered formula:
k_optimal(L) = -0.0137 * log10(L) + 0.1593
R-squared = 0.71, p-value = 0.008
Quantum Scales
L < 10^-6 m
k > 0.2
Bigeometric dominates
Transition
10^-6 m < L < 10^7 m
0.05 < k < 0.2
Mixed regime
Cosmological
L > 10^7 m
k -> 0
Classical emerges
Key insight: This pattern was NOT forced - the optimizer discovered it independently while minimizing error and maximizing stability. Classical calculus naturally emerges at large scales, while NNC with higher k is optimal at small scales.
Specific Benchmarks
Lane-Emden Stellar Structure (n=5)
8.8x speedupModels stellar polytropes with singular behavior at the core.
6,026
Classical steps
686
Bigeometric steps
836
k(L) Nuclear steps
Power-Law Integration (n=-6)
40,000x accuracyTests integration of Kretschmann-type r^(-6) singularities.
4910%
Classical error
0.12%
Bigeometric error
1730%
k(L) Planck error
Connection to CASCADE
The GUD framework provides the theoretical foundation for CASCADE's three-phase algorithm:
- Phase A (k-discovery): Finds optimal k using the GUD composite space
- Phase B (bound transform): Uses bigeometric transform to regularize singularities
- Phase C (interior search): Runs standard MOO in the transformed space
Related Pages
Symbolic Verification
Algebraic proof that NNC preserves physics invariants (10/10 tests)
Interactive Benchmarks
Run your own comparisons between Classical, Bigeometric, and k(L)
k(L) Formula
Detailed derivation of the scale-dependent k pattern
MOO Invariance
Why NNC transformations preserve Pareto optimality