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:

f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}

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:

D_{BG}[x^n] = e^n \quad \text{(constant on } x > 0\text{)}

This means: the framework detects power laws and can switch to log-space arithmetic. Near a power-law singular point the log-slope is constant; the singularity itself is still there, but some stored ODE runs need far fewer function evaluations in log variables.

Stored result (results/tractability_benchmarks.json): Lane-Emden n=3-5 needs 7.7-8.5x fewer right-hand-side function evaluations in the stored log-space configuration; Bessel J0 needs 2.9x fewer but is about 10x less accurate. These ratios compare the two stored solver setups, not the coordinate alone: The classical x-space runs are capped at max_step = 0.01 by the script (about 2,000 RK45 steps over x up to 20, roughly 12,000 evaluations); the log-space runs are not capped. Much of the reduction comes from that cap, not from the 2/x term alone. The classical xi_1 is the first nonpositive output sample on a fixed output grid; the log-space xi_1 is a linearly interpolated zero crossing. Part of the error difference comes from these two estimators, not from the coordinate. Wall-clock speedups across the 8 benchmarks range 2.69-23.7x (mean 13.2x, machine-dependent).

What This Does Near Singularities

L_{BG}[1/r] = -1,\quad D_{BG}[1/r] = e^{-1}\quad (r>0)

This is a bounded log-slope diagnostic on r > 0. It does not remove the singularity: 1/r still diverges as r goes to 0.

CASCADE synthetic toy benchmark (not physics results) →

Converting Between Representations

Classical, geometric and bigeometric derivatives are different representations of the same local information, related by explicit formulas on their domains. For f > 0, D_G[f] = exp(f'/f), so f' = f ln(D_G[f]). For x > 0 and f > 0, D_BG[f] = exp(x f'/f), so f' = (f/x) ln(D_BG[f]). No extra weights are needed to pass between them.

Michael Grossman and Robert Katz introduced the non-Newtonian calculi (1972). Jane Grossman's published meta-calculus (1981) uses weight functions to define a different derivative; it is not required for the conversions above.

The weighted meta-derivatives and k-parameter families used on this site are project extensions for numerical experiments, not a mathematical necessity.

Three Calculus Frameworks

Classical (Newton-Leibniz)

f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}

Linear functions have constant derivative. Standard framework for most physics.

Geometric

D_G[f](x) = \lim_{h \to 0} \left[\frac{f(x+h)}{f(x)}\right]^{1/h}

Exponential functions have constant derivative. Useful for multiplicative processes.

D_G[e^x] = e (constant)

Bigeometric

D_{BG}[f](x) = \lim_{h \to 0} \left[\frac{f(x \cdot e^h)}{f(x)}\right]^{1/h}

Power functions have constant derivative. The Power Law Theorem (Grossman & Katz, 1972).

D_BG[x^n] = e^n (constant on x > 0)

Measured Local Results

Conditional

Spectral-Gap Boundary

3 distinct baseline compositions satisfy the proposed bound (the stored file lists 5 checks; the others repeat a case); a broader guarded audit found 9 counterexamples among 88 valid cases.

Verified

Constancy Properties

D_BG[x^n] = exp(n) holds exactly on x > 0 for every real n (analytic identity; see the proofs page).

Analogy

Connection to Symanzik Improvement

An analogy, not a demonstrated equivalence: the framework parallels the Symanzik improvement program in lattice QCD (mixing discretization schemes to reduce artifacts).

Limitations

Exploratory

Multi-Scale Hypothesis

Archived k(L) fit used 8 synthetic scale points. A fresh common-bounds control lowers R^2 to 0.4370, so it is not a physical lookup rule.

Archived

Cosmological Tension Claims

RETRACTED. The archived k = -0.7 fit conflicted with BBN constraints (no stored artifact). Post-hoc curve fitting with no predictive power.

Not Validated

Physical = Scheme-Invariant

Interesting diagnostic idea but not proven. Needs rigorous connection to established physics principles.