Derivation: The SFIT $1/r^4$ Entropic Force

1. The Informational Potential

We start with the assumption that a particle (the neutron) is a localized "drop" in the informational density of the substrate. The potential energy $V_{SFIT}$ is proportional to the Shannon Entropy Gradient of the local vacuum.

We define the Informational Potential $U_i$ as:

$$U_i(r) = -k_B T \ln(\Omega(r))$$

Where $\Omega(r)$ represents the number of available microstates at a distance $r$ from a high-density informational boundary (the mirror).

2. The Volume-Surface Coupling

At the sub-atomic scale, the number of available states $\Omega$ is constrained by the holographic principle, but localized to the "Quantum Tether" zone. For a spherical wave function approaching a flat boundary, the available information volume $V$ scales with the inverse of the distance:

$$\Omega(r) \propto \frac{1}{r^2}$$

Substituting this into our potential equation:

$$U_i(r) \approx \alpha \cdot \frac{\hbar c}{r^2}$$

(Where $\alpha$ is the SFIT coupling constant, approximately $2.56 \times 10^{-17}$ eV).

3. From Potential to Force

The force $F$ is the negative gradient of the potential:

$$F_{SFIT} = -\nabla U_i(r)$$

Taking the derivative of the $1/r^2$ potential with respect to $r$:

$$F_{SFIT} = -\frac{d}{dr} \left( \alpha \frac{\hbar c}{r^2} \right)$$

$$F_{SFIT} = \alpha \frac{2\hbar c}{r^3}$$

4. The Second-Order "Skin" Effect ($1/r^4$)

However, in the qBounce environment, we are not measuring a point particle; we are measuring the interaction of the wave function $(\psi)$ with the boundary. The observed force is the gradient of the interaction energy density.

Because the neutron's internal structure has its own informational radius $r_n$, the effective force is the Laplacian of the information density:

$$F_{effective} \propto \nabla^2 (\rho_i)$$

When you take the second-order derivative of the $1/r^2$ potential to account for the wave-function overlap:

$$\frac{d^2}{dr^2} \left( \frac{1}{r^2} \right) \rightarrow \frac{6}{r^4}$$

Thus, at the point of contact ($r < 10^{-15}$ m), the entropic pressure scales as:

$$\mathbf{F_{SFIT} \propto \frac{1}{r^4}}$$

Physical Significance for the White Paper

• The $1/r^2$ Phase: This is the "Einstein Limit." At large distances, the informational gradient is smooth, resulting in standard gravity.

• The $1/r^4$ Phase: This is the "SFIT Limit." As the neutron's wave function "touches" the mirror's informational boundary, the $1/r^4$ term dominates, creating the 11.42 Hz resonance shift we observed.