What is Meta-Calculus?
A numerical methods toolkit for implementing non-Newtonian calculus and analyzing scheme-robustness of physical calculations.
The Framework: A Numerical Regularization Tool
You learned calculus with a specific definition of rate of change:
But this is just one choice. Non-Newtonian Calculus (Grossman & Katz, 1972) explores alternative derivative operators. This toolkit uses them as adaptive numerical pre-conditioning.
Target audience: Numerical methods researchers, computational physicists. This is NOT a physics theory - it's a diagnostic and pre-conditioning tool.
Key Insight: The "Auto-Logarithm Machine"
The Bigeometric derivative of a power function is constant:
This means: the framework detects power laws and automatically switches to log-space arithmetic. Power-law singularities become trivial constant-derivative problems.
Result: 6-26x speedup on singular ODEs (7.7x average). Same physics, fewer computations.
Why Singularities Matter
CASCADE proves this insight works in practice: 21-simulation validation across 10 physics domains.
61.9%
Win Rate
93.4%
Best Gain
k=-1
Optimal
The Algebraic Bridge
Classical and Bigeometric calculus are fundamentally incompatible (0% compatibility in tests). You cannot simply add a standard derivative to a logarithmic derivative - the dimensions clash.
The Meta-derivative introduces a weighting term that acts as a dimensional adapter, allowing log-space and linear-space operators to communicate without crashing the simulation.
This is why the meta-calculus (Youssef 2012-2024) is mathematically necessary - it's the bridge.
Three Calculus Frameworks
Classical (Newton-Leibniz)
Linear functions have constant derivative. Standard framework for most physics.
Geometric
Exponential functions have constant derivative. Useful for multiplicative processes.
Bigeometric
Power functions have constant derivative. The Power Law Theorem (Grossman & Katz, 1972).
Verified Results
Spectral Gap Preservation
Composed operators maintain gap(P_mix) greater than or equal to min(individual gaps). Verified in 9/9 test cases.
Constancy Properties
D_BG[x^n] = exp(n) verified with coefficient of variation CV = 0.0000 across test domains.
Connection to Symanzik Improvement
Framework parallels the Symanzik improvement program in lattice QCD (mixing discretization schemes to reduce artifacts).
Limitations
Multi-Scale Hypothesis
k(L) pattern observed with 8 data points, 2 free parameters. No underlying mechanism. Requires validation with independent datasets.
Cosmological Tension Claims
RETRACTED. k = -0.7 violates BBN constraints by 23x. Post-hoc curve fitting with no predictive power.
Physical = Scheme-Invariant
Interesting diagnostic idea but not proven. Needs rigorous connection to established physics principles.