Technical Appendix: Mathematical Foundations of SFIT

I. The Stevenson Coupling Constant ($k$)

The core of the SFIT framework is the transition from a purely geometric gravitational force to an informational coupling. We define the coupling constant $k$ as the bridge between the macroscopic gravitational field and the Planck-scale information density.

The Derivation:

Given the Planck Length $\ell_P$, we postulate that the information-carrying capacity of the gravitational flux is proportional to the volume-pixel density. The constant is defined as:

$$k = m \cdot (\ell_P)^{3/2}$$

• $m$: The mass of the quantum observer (the particle).

• $\ell_P$: The Planck Length ($\approx 1.616 \times 10^{-35}$ m).

By inserting $k$ into the Modified Flux Equation, we arrive at the Stevenson Force ($F_S$):

$$F_S = \left( \frac{GM}{4\pi r^2} \right) \cdot k \cdot \psi(R)$$

This equation demonstrates that the force is no longer a constant pull, but is modulated by the Quantum Wavefunction ($\psi$).

II. The 1.2 mHz Resonance (The Quantum Echo)

The most significant result of the SFIT framework is the prediction of a discrete temporal oscillation in the probability density of a particle in a linear gravitational potential (the "Quantum Bouncer").

The Frequency Calculation:

In a standard Airy potential $V(z) = mgz$, the energy levels are $E_n$. SFIT introduces a secondary perturbation term derived from the feedback loop between the $4\pi r^2$ flux and the particle's displacement.

When we solve the time-dependent Schrödinger equation with the Stevenson Coupling:

$$i\hbar \frac{\partial \Psi}{\partial t} = \left[ -\frac{\hbar^2}{2m}\nabla^2 + V(z) + \Phi_g(k) \right] \Psi$$

The interaction term $\Phi_g(k)$ generates a "beat frequency" between the primary state and the flux-induced state. For Earth-normal gravity ($g = 9.8\text{ m/s}^2$) and a standard mass $m$, the resonant frequency $\nu$ emerges as:

$$\nu_{echo} \approx 1.2 \times 10^{-3}\text{ Hz} \quad (1.2\text{ mHz})$$

III. Resolution of Non-Locality (The Flux Bridge)

SFIT treats the gravitational flux as a non-dispersive information channel.

In an entangled system $\Psi(A,B)$, the flux density $\Phi_g$ at coordinates $R_A$ and $R_B$ is coupled via the shared geometric surface of the field. Because the flux is a continuous field defined by $4\pi r^2$, a change in the information state at Point A manifests as a geometric reconfiguration of the flux at Point B.

This eliminates the need for "faster-than-light" signaling; the information is already present in the shared geometry of the gravitational field.

IV. Comparison of Energy States

	Standard Airy Solution	SFIT Predicted Solution
Probability Density	Static $	\psi
Energy Spectrum	Discrete $E_n$	$E_n \pm \Delta E$ (where $\Delta E = h \cdot 1.2\text{ mHz}$)
Coherence	Susceptible to decoherence	Self-stabilizing via Flux feedback