Measured Local ResultsLocal benchmark, single machine

Tractability Benchmarks

Testing whether a log-space coordinate transform makes near-singular problems cheaper to solve. 7/8 benchmarks meet the script's own criterion; Bessel J0 needs 2.9x fewer evaluations but is about 9.6x less accurate.

Log-space transform: 7.7x fewer evaluations on Lane-Emden n=3

The Tractability Conjecture

"Problems exist that are numerically intractable classically but tractable in NNC/GUD composite calculus."

This is a methodology claim, not a physics claim. We are not saying NNC reveals new physics. These runs test whether a change of coordinates gives computational savings for specific problem classes. None of the problems below is intractable classically; the classical runs all finish.

The Log-Space Transform

Approach

The bigeometric derivative measures multiplicative change, which suggests working in log coordinates. The benchmarks do not modify the equations: they transform coordinates, solve the SAME equation in u = ln x, and transform back.

Classical (x-space)

Lane-Emden has a 2/x coefficient (a regular singular point) at the origin:

\frac{d^2\theta}{dx^2} + \frac{2}{x}\frac{d\theta}{dx} + \theta^n = 0

The 2/x term forces small step sizes near x=0

Log-Space (u = ln x)

In log coordinates the 2/x coefficient is gone:

\frac{d^2\phi}{du^2} + \frac{d\phi}{du} + e^{2u}\phi^n = 0

SAME equation in new coordinates; both runs start at x = 1e-6

Why This Can Help

For power laws on x > 0, $D_{BG}[x^n] = e^n$ is constant because it measures multiplicative rates of change: power laws are linear in log-space. The transform u = ln(x) maps x=0 to u = -infinity, so uniform steps in u are geometrically graded steps in x. It is a coordinate choice; it does not remove a physical singularity, and it does not help every problem (Bessel J0 below).

Benchmark Results

Measured Results

Benchmark	Classical	Log-space	Reduction	Error (classical / log-space)	Accuracy ratio	Script criterion
Lane-Emden n=3	12,080 evaluations	1,562 evaluations	7.7x	9.6e-4 / 3.2e-5	30x better	met
Lane-Emden n=4	12,086 evaluations	1,496 evaluations	8.1x	4.0e-4 / 8.3e-5	4.8x better	met
Lane-Emden n=5	12,092 evaluations	1,430 evaluations	8.5x	1.0e-1 / 1.0e-1	not measured	met
Power-law n=-2	10,000 points	100 points	100x	2.5e+0 / 1.1e-3	2218x better	met
Power-law n=-3	10,000 points	100 points	100x	5.7e+0 / 4.5e-3	1263x better	met
Power-law n=-6	10,000 points	100 points	100x	1.6e+1 / 2.8e-2	559x better	not demonstrated
Kolmogorov Cascade	10,000 points	100 points	100x	8.2e+0 / 0	not measured	met
Bessel J0	12,182 evaluations	4,262 evaluations	2.9x	4.9e-6 / 4.7e-5	9.6x worse	met

Lane-Emden errors are relative errors in the first zero xi_1 against tabulated values; Lane-Emden n=5: n=5 has no finite first zero; the script records a placeholder error of 0.1 for both stable runs. Power-law errors are relative errors of a uniform-grid Simpson rule (classical) vs a log-uniform grid (transformed); the 10,000 vs 100 point counts are fixed by the script, not adapted. Kolmogorov: evaluates the identity D_BG[k^(-5/3)] = e^(-5/3); no numerical accuracy comparison.

ODE counts are solve_ivp right-hand-side evaluations. 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.

2.9x-8.5x

ODE evaluation reduction

100x

Point reduction (fixed by design)

Less accurate after transform

7/8

Meet script criterion

Benchmark Descriptions

1. Lane-Emden Equation (Stellar Structure)

Both solvers use the same adaptive integrator and tolerances; the log-space run solves $\phi'' + \phi' + e^{2u}\phi^n = 0$ and maps back.

Lane-Emden n=3

12,080 vs 1,562 evaluations

7.7x fewer

xi_1 error 9.6e-4 vs 3.2e-5

Lane-Emden n=4

12,086 vs 1,496 evaluations

8.1x fewer

xi_1 error 4.0e-4 vs 8.3e-5

Lane-Emden n=5

12,092 vs 1,430 evaluations

8.5x fewer

no finite xi_1 (stability check only)

2. Power-Law Integration

Integrating $f(x) = x^{n}$ from x = 1e-4 to 10 on a uniform grid vs a log-uniform grid:

\int x^n dx = \int e^{(n+1)u} du \text{ where } u = \ln x

Power-law n=-2

error 2.5e+0 vs 1.1e-3

meets criterion

Power-law n=-3

error 5.7e+0 vs 4.5e-3

meets criterion

Power-law n=-6

error 1.6e+1 vs 2.8e-2

not demonstrated (log-space error above 1%)

3. Kolmogorov Spectrum (identity check)

For $E(k) \sim k^{-5/3}$ on k > 0 the bigeometric derivative is a constant:

D_{BG}[k^{-5/3}] = e^{-5/3} \approx 0.189

This row evaluates an identity; it is not a turbulence simulation and its "error" fields are not an accuracy comparison. The script flags it as tractable unconditionally.

4. Bessel J0 (the counter-case)

The log-space run takes 12,182 vs 4,262 evaluations (2.9x fewer) but its maximum error is 4.7e-5 vs 4.9e-6 classically: about 9.6x less accurate. Fewer evaluations is not the same as a better answer.

What Would Falsify This

For the tractability claim to be meaningful, there must be ways it could fail:

1.No work reduction: If the transform required the same or more evaluations, tractability would be falsified.Result: 2.9x-8.5x fewer ODE evaluations (see the max_step note above); the 100x point reduction is set by the script.
2.Accuracy loss: If less work came with worse error, the trade-off would not be favorable.Result: observed for Bessel J0; not measurable for Lane-Emden n=5 and Kolmogorov Cascade.
3.Instability: If the transformed runs were less stable, they would not be practically useful.Result: 8/8 stable (vs 8/8 classical).
4.Problem-specific: If it only worked for cherry-picked problems, it would not be a general methodology.Result: 7/8 meet the script's criterion; not demonstrated: Power-law n=-6. All eight are power-law or origin-singular problems chosen to suit log coordinates.

Reproduce These Results

# Clone the repository
git clone https://github.com/DNYoussef/meta-calculus-portfolio
cd meta-calculus-portfolio

# Install dependencies
pip install numpy scipy

# Run benchmarks (writes results/tractability_benchmarks.json)
python simulations/tractability_benchmarks.py

# Expected evaluation/point counts (wall-clock times vary by machine):
# Lane-Emden n=3: classical 12080 evaluations -> log-space 1562 evaluations (7.7x)
# Lane-Emden n=4: classical 12086 evaluations -> log-space 1496 evaluations (8.1x)
# Lane-Emden n=5: classical 12092 evaluations -> log-space 1430 evaluations (8.5x)
# Power-law n=-2: classical 10000 points -> log-space 100 points (100x)
# Power-law n=-3: classical 10000 points -> log-space 100 points (100x)
# Power-law n=-6: classical 10000 points -> log-space 100 points (100x)
# Kolmogorov Cascade: classical 10000 points -> log-space 100 points (100x)
# Bessel J0: classical 12182 evaluations -> log-space 4262 evaluations (2.9x)

Mathematical Precedents

QFT Regularization

Dimensional regularization changes the framework to make divergent integrals finite. An analogy only: framework choice affects computability.

Distribution Theory

Dirac delta is undefined as a function but well-defined as a distribution. The right framework makes "impossible" objects tractable.

Tropical Geometry

Replacing (+, x) with (min, +) makes certain algebraic problems become linear. Framework choice enables new solution methods.