Proofs and Derivations

Step-by-step mathematical proofs with full working.

Theorem 1.1: Power Law Theorem

Grossman & Katz, 1972

For f(x) = x^n on x > 0, where n is any real number:

DBG[xn]=enD_{BG}[x^n] = e^n

Step 1: Recall bigeometric formula

DBG[f](x)=exp⁡(x⋅f′(x)f(x))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⋅xn−1f'(x) = n \cdot x^{n-1}

Step 3: Substitute

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

Step 4: Simplify

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

Verification

The steps above are an exact symbolic identity on x > 0 for every real n; no numerical test is needed or claimed.

Scope

This is a derivative identity on x > 0: L_BG[1/r] = -1 and D_BG[1/r] = exp(-1) for r > 0. It is a bounded log-slope diagnostic; 1/r itself still diverges as r goes to 0.

Spectral-Gap Boundary

Consider K finite-difference discretizations L_k of the 1-D Laplacian on the same grid (n = 50, h = 1/51; simulations/spectral_gap_verification.py).

Single-calculus spectral gap

Sort the absolute eigenvalues |lambda_1| <= |lambda_2| <= ... of the (unnormalized) matrix L_k:

gapk=∣λ2(Lk)∣−∣λ1(Lk)∣\text{gap}_k = |\lambda_2(L_k)| - |\lambda_1(L_k)|

Composed operator

The script composes schemes as a weighted sum (uniform weights by default), not a product:

Leff=∑kwkLk,∑kwk=1L_{eff} = \sum_k w_k L_k, \quad \sum_k w_k = 1

The composed gap is the same eigenvalue spacing of L_eff. These gaps are not 1 - |mu_2| of a Markov operator; they scale with 1/h^2.

Empirical Result

From public/spectral_gap_verification.json (laplacian section):

python
1component_gaps = [29.562, 29.788]  # 3-point, 5-point
2composed_gap = 29.678               # "3pt + 5pt", uniform weights
3# Between the components; no amplification

The baseline cases satisfy the minimum-gap bound, but the guarded grid audit (results/spectral_gap_boundary_moo.json: 88 valid of 1000 cases) found 9 counterexamples, so the bound is refuted as a general theorem.

Quantum Unitarity Analysis

Q1: Clock Reparametrization

tau = alpha(t), applied globally:

d∣ψ⟩dτ=1α′(t)d∣ψ⟩dt\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.

ddt∥ψ∥2=ddt∑i∣ψi∣2≠0\frac{d}{dt}\|\psi\|^2 = \frac{d}{dt}\sum_i |\psi_i|^2 \neq 0

Historical Note: Q3 Breaks Unitarity

Historical values from an earlier experiment; no current artifact reproduces this table. The current committed scan (public/quantum_phase_corrected.json) finds 0 of 1000 configurations unitary (minimum norm drift 2.56e-4).

python
1# historical, not reproduced by a current artifact
2results = {
3    'Q0': {'norm_drift': 0.001, 'status': 'SAFE'},
4    'Q1': {'norm_drift': 0.003, 'status': 'SAFE'},
5    'Q2': {'norm_drift': 0.008, 'status': 'SAFE'},
6    'Q3': {'norm_drift': 0.65,  'status': 'BREAKS'}
7}

Open Problems

1. A scoped spectral-gap theorem

What structural assumptions, if any, make the minimum-gap bound provable? 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.