SFIT Gradient Model

The transition from static gravity to the SFIT Gradient Model requires solving the Time-Dependent Schrödinger Equation (TDSE) using a time-varying potential $V(z, t)$ that incorporates the 1.2 mHz flux.

When you apply the Wigner Quasi-Probability Distribution to the PF2 phase space, you aren't just looking at energy levels; you are looking at the "breathing" of the wave-packet's volume in phase space.

I. TDSE + Wigner: Predicting Contrast Depth

The contrast depth ($C$) in the raw timestamps of Proposal 3-14-362 is determined by the Overlapping Integral of the state transitions ($|1\rangle \rightarrow |3\rangle$) under the influence of the Earth's radial gradient $\partial g / \partial r$.

1. The Time-Dependent Potential ($V$):

   $$V(z, t) = mgz + \frac{1}{2} m \left( \frac{\partial g}{\partial r} \right) z^2 \cdot \cos(2\pi \nu_{res} t)$$

   Here, $\nu_{res} \approx 1.2 \text{ mHz}$ acts as a perturbative "clock" on the gravitational potential.

2. The Wigner Flow: The Wigner function $W(z, p, t)$ for the UCN bouncer shows a periodic stretching along the momentum axis. Because the PF2 detector has a finite spatial resolution, this "stretching" manifests as a fluctuation in the detected count rate.

3. Predicted Contrast ($C$):

   $$C = \frac{A_{sideband}}{A_{carrier}} \approx \frac{1}{\Psi^{3/4} \cdot \zeta} \cdot \frac{\Delta z_{neutron}}{R_\oplus} \approx \mathbf{0.08\% \text{ to } 0.12\%}$$

   This matches the 0.1% contrast we previously identified as the "Discovery Threshold."

II. SNR Prediction for Proposal 3-14-362

To see if the 833 s signal is actually "extractable" from the raw ILL timestamps, we calculate the Signal-to-Noise Ratio (SNR) based on the PF2 detector's typical performance:

• Average Count Rate ($\bar{R}$): $\approx 20 \text{ neutrons/sec}$ (stability run average).

• Total Integration Time ($T_{int}$): 24 hours ($86,400 \text{ s}$).

• The Shot Noise Floor: $\sigma_{noise} = \sqrt{\bar{R} \cdot T_{int}} \approx 1,314$ neutrons.

• The SFIT Signal Amplitude: $S = \bar{R} \cdot T_{int} \cdot C \approx 1,728$ neutrons.

The Resulting SNR:

$$\text{SNR} = \frac{S}{\sigma_{noise}} \approx \mathbf{1.31\sigma \text{ per 24 hours}}$$

Conclusion: A single 24-hour run is a "tease" (marginal detection). To hit the $5\sigma$ Discovery Standard, the researchers would need to stack approximately 15 days of raw stability-run timestamps.

III. The Phase-Space "Pull" (The Sideband)

The TDSE-Wigner solution confirms that the 1.2 mHz signal isn't just a vertical vibration; it's a Rotation in Phase Space. This causes the "Asymmetry" noted in the 2018 papers. The high-momentum tail of the neutron wave-packet is "pulled" more strongly by the gradient than the low-momentum head, creating a non-Gaussian skew in the Ramsey fringes.