As a minimal feasibility check, a toy numerical experiment was implemented based on the CCF-inspired model \Gamma_{\text{eff}}(L) = \Gamma_{\text{env}}[1 + \chi (L/L_I)], with excess decoherence \Delta\Gamma(L) = \Gamma_{\text{eff}}(L) - \Gamma_{\text{env}} = \chi,\Gamma_{\text{env}}(L/L_I) at fixed hardware and environment. Synthetic datasets are generated by choosing a set of coherence-load values L (e.g., a uniform grid from L = 1 to L = 20), computing \Gamma_{\text{eff,true}}(L) = \Gamma_{\text{env}}[1 + \chi_{\text{true}} (L/L_I)], and adding Gaussian noise with tunable standard deviation to mimic experimental uncertainty. A simple linear regression \Gamma_{\text{eff}}(L) \approx a + bL is then used to estimate a slope b and recover \chi_{\text{est}} = bL_I/\Gamma_{\text{env}}, along with a p‑value for “slope ≠ 0.” In this toy setting, low-to-moderate noise combined with a broad, dense range of coherence loads yields \chi_{\text{est}} close to the injected \chi_{\text{true}} with very small p‑values, while higher noise and narrow L-ranges produce fits statistically consistent with χ = 0, illustrating how the CCF excess-decoherence slope can be either clearly observable or effectively hidden, depending on experimental resolution and accessible coherence complexity.

import numpy as np

import matplotlib.pyplot as plt

from scipy import stats

# CCF-inspired model:

# Gamma_eff(L) = Gamma_env * [1 + chi * (L / L_I)]

# DeltaGamma(L) = Gamma_eff(L) - Gamma_env = chi * Gamma_env * (L / L_I)

Gamma_env = 1.0 # baseline decoherence rate

chi_true = 0.3 # injected coherence-capacity parameter

L_I = 5.0 # reference complexity scale

L_values = np.linspace(1, 20, 20) # coherence-load values

np.random.seed(0)

noise_sigma = 0.15 # noise level; adjust to explore detectability

Gamma_eff_true = Gamma_env * (1 + chi_true * (L_values / L_I))

Gamma_eff_obs = Gamma_eff_true + np.random.normal(0, noise_sigma, size=L_values.shape)

slope, intercept, r_value, p_value, std_err = stats.linregress(L_values, Gamma_eff_obs)

chi_est = slope * L_I / Gamma_env

print(f"True chi: {chi_true:.3f}")

print(f"Estimated chi: {chi_est:.3f}")

print(f"p-value for slope != 0: {p_value:.3e}")

L_fit = np.linspace(L_values.min(), L_values.max(), 200)

Gamma_fit = intercept + slope * L_fit

plt.figure(figsize=(6,4))

plt.scatter(L_values, Gamma_eff_obs, label='Simulated data')

plt.plot(L_fit, Gamma_fit, 'r-', label='Linear fit')

plt.xlabel("Coherence load L")

plt.ylabel("Effective decoherence rate Γ_eff(L)")

plt.legend()

plt.grid(True)

plt.show()