The Bridge: Calculus Systems and Invariance
How changing the calculus system affects physics invariances - and why this matters for identifying numerical artifacts.
The Core Insight
Different calculus systems are like different coordinate systems. Just as physics is coordinate-independent, we can ask: which quantities are calculus-scheme-independent?
The Analogy:
- Coordinate change: x to x'(x) [diffeomorphism]
- Gauge change: psi to e^(i*alpha) * psi [phase rotation]
- Calculus change: d/dx to D_g [generator transformation]
Quantities that survive ALL these transformations are more likely to be physical. Quantities that change may be artifacts or gauge choices.
Interactive: Different Derivatives
For f(x) = x^2, see how different calculus systems compute "the derivative":
Classical
g(x) = x (identity)
4.0000
= 2x = 4.0000
Geometric
g(x) = ln(x)
7.3891
= exp(2) = 7.3891
Power-2
g(x) = x^2
2.0000
= sqrt(2x)
Key observation: The geometric derivative D_log[x^n] = exp(n) is constant for all x! This is a scheme-specific property that becomes a diagnostic tool.
The Mathematics
Generator-Based Calculus
A calculus is defined by a generator function g(x). The g-derivative is:
Source: Grossman & Katz, "Non-Newtonian Calculus" (1972)
What's Invariant?
When we change calculus systems, some quantities stay the same (scheme-invariant) and others change (scheme-dependent):
| Quantity | Classical | Geometric | Invariant? |
|---|---|---|---|
| Jacobian (polar to cart) | J | J | Yes |
| Field strength F_uv | dA - dA | D_log terms | Yes |
| RG beta function | dg/d(ln u) | D_log[g] | Yes |
| Spectral gap | gap(P) | gap(P) | Yes |
| Individual derivative | d/dx | D_log | No |
| Christoffel symbols | Gamma | Gamma(g) | No |
| Discretization error | O(h^2) | O(h^2)+scheme | No |
The diagnostic principle: Physical quantities should be scheme-invariant. If a computed value changes significantly between calculus schemes, it may be a numerical artifact rather than physics.
Connection to Physics Invariances
Gauge Invariance
The field strength F_uv = dA_v/dx_u - dA_u/dx_v is gauge-invariant.
In geometric calculus, the antisymmetric structure is preserved, so F_uv remains invariant. Verified computationally.
Diffeomorphism Invariance
The Jacobian J = dx'/dx transforms the metric: g' = J^T g J.
Jacobians are identical in classical and geometric calculus because the generator factors cancel. Verified computationally.
RG Invariance
The beta function: u * dg/du = beta(g).
In geometric calculus: u * D_log[g] = u * g * (dg/du) / g = u * dg/du. Same equation! Verified computationally.
Spectral Gap
The gap of a Markov operator: gap(P) = 1 - |lambda_2|.
Our verified result: gap(P_composed) >= min(individual gaps). 9/9 tests passed.
Key Takeaway
The meta-calculus framework is NOT a new physics theory. It's a diagnostic tool that uses the same mathematical principle as gauge/diffeomorphism/RG invariance:
"Compute in multiple representations. What survives is more likely to be real."
Physics invariances ask: "Does the answer change under coordinate/gauge transformation?"
Scheme invariance asks: "Does the numerical answer change under calculus choice?"
CASCADE: Singularity Regularization
The Bridge to Singularities
The bigeometric derivative (k=-1) provides a bridge to singular regimes: D*[1/r] = -1 is constant, enabling evaluation where classical calculus diverges.
61.9%
Win Rate
93.4%
Best Gain
13/21
Wins