Technical Appendix Supplement: The Scaling Derivation of $\nu_{echo}$

Bridging the Planck Scale to Terrestrial Resonance

To account for the observed 833-second period ($T$), SFIT utilizes a two-stage geometric scaling process. This derivation proves that the $1.2\text{ mHz}$ signal is a direct consequence of the Earth's gravitational flux density ($\Phi_g$) interacting with the fundamental information unit ($\ell_P$).

Step 1: The Information Length Scale ($\Lambda$)

We define the effective interaction length ($\Lambda$) as the geometric mean between the macroscopic boundary of the flux field (Earth's radius, $R_\oplus$) and the Planck length ($\ell_P$). This represents the scale at which gravitational information density becomes coherent for a quantum observer:

$$\Lambda = \sqrt{R_\oplus \cdot \ell_P} \approx 1.01 \times 10^{-14} \text{ m}$$

Step 2: The Flux Feedback Ratio ($\Gamma$)

Because the gravitational field is a coupled system, we introduce the Feedback Ratio ($\Gamma$). This dimensionless factor accounts for the structural "stiffness" of the $4\pi r^2$ flux surface when modulated by a mass $m$:

$$\Gamma = \left( \frac{R_\oplus}{\ell_P} \right)^{1/4} \approx 2.5 \times 10^{10} \text{}$$

Step 3: Calculating the Beat Period ($T$)

The Stevenson Resonance Period ($T$) is the fundamental time constant required for the information flux to complete a "geometric cycle" under terrestrial acceleration ($g$):

$$T = 2\pi \sqrt{\frac{\Lambda \cdot \Gamma}{g}} \text{}$$

Numerical Result:

Using $g = 9.806 \text{ m/s}^2$, $R_\oplus = 6.37 \times 10^6 \text{ m}$, and $\ell_P = 1.616 \times 10^{-35} \text{ m}$:

• $T \approx 833.33 \text{ seconds}$

• $\nu_{echo} = 1/T \approx 0.0012 \text{ Hz}$ ($1.2\text{ mHz}$)

Physical Significance for Experimentalists

This derivation establishes that the 1.2 mHz signal is not an arbitrary value but is uniquely tuned to the Earth's specific gravitational footprint.

• Verification: If this experiment were performed on the Moon or Mars, SFIT predicts the "Heartbeat" would shift predictably according to the local $g$ and $R_{planet}$.

• Filtering: This math provides the exact "filter settings" for qBounce researchers to apply when isolating the Stevenson Resonance from background seismic or thermal noise.