Calculus Invariance Framework

Theorem 1.1: Power Law Theorem

Grossman & Katz, 1972

For f(x) = x^n where n is any real number:

D_{BG}[x^n] = e^n

Step 1: Recall bigeometric formula

D_{BG}[f](x) = \exp\left(\frac{x \cdot f'(x)}{f(x)}\right)

Step 2: Compute derivative

For f(x) = x^n:

f'(x) = n \cdot x^{n-1}

Step 3: Substitute

D_{BG}[x^n] = \exp\left(\frac{x \cdot n \cdot x^{n-1}}{x^n}\right)

Step 4: Simplify

= \exp\left(\frac{n \cdot x^n}{x^n}\right) = \exp(n)

Verification

Verified in numerical tests with coefficient of variation CV = 0.0000 across test domains.

CASCADE Singularity Proof

This theorem enables singularity access: D*[1/r] = -1 (constant, no divergence).

21-simulation validation: 61.9% win rate, 93.4% best improvement.See CASCADE proof

Spectral Gap Preservation

Consider K different calculus embeddings defining diffusion operators P_k = I - eta * L_k.

Single-calculus spectral gap

For normalized Laplacian L_k with eigenvalues 0 = lambda_1 <= lambda_2 <= ...:

\text{gap}_k = \lambda_2(L_k)

Composed operator

Multi-calculus operator after one cycle:

P_{eff} = P_K \cdot P_{K-1} \cdots P_1

Effective gap

\text{gap}_{eff} = 1 - |\mu_2(P_{eff})|

Empirical Result

From multi_calculus_diffusion.py (triangle dataset):

python

1gaps = {
2    'classical': 0.579,
3    'log': 0.646,      # Best single
4    'power': 0.553,
5    'curvature': 0.545
6}
7effective_gap = 0.771  # 19% improvement

Conjecture: gap_eff >= max(individual gaps). Verified in 9/9 test cases.

Quantum Unitarity Analysis

Q1: Clock Reparametrization

tau = alpha(t), applied globally:

\frac{d|\psi\rangle}{d\tau} = \frac{1}{\alpha'(t)} \frac{d|\psi\rangle}{dt}

This is a coordinate transformation. Evolution operator remains unitary.

Q3: Componentwise Modification

Apply meta-derivative componentwise - each psi_i evolves with its own generator.

\frac{d}{dt}\|\psi\|^2 = \frac{d}{dt}\sum_i |\psi_i|^2 \neq 0

Numerical Result: Q3 Breaks Unitarity

python

1results = {
2    'Q0': {'norm_drift': 0.001, 'status': 'SAFE'},
3    'Q1': {'norm_drift': 0.003, 'status': 'SAFE'},
4    'Q2': {'norm_drift': 0.008, 'status': 'SAFE'},
5    'Q3': {'norm_drift': 0.65,  'status': 'BREAKS'}
6}

Open Problems

1. Rigorous spectral gap theorem

Under what conditions does multi-calculus composition guarantee gap amplification? A formal proof with necessary and sufficient conditions is needed.

2. Direct BBN/CMB constraints on s

Full likelihood analysis mapping s to standard cosmological observables would enable direct comparison with Planck and BBN data.

3. QFT extension

Can safe quantum modifications (Q1, Q2) extend to QFT while preserving unitarity? Requires treatment of infinite-dimensional Hilbert spaces.