Appendix E: Fubini-Study Distance and Generalized Force

This appendix formalizes “force is the gradient of distance” from the main text as a variational principle.

In projective Hilbert space $\mathbb{P}(\mathcal{H})$, the Fubini-Study distance between two points ${\psi }_1,{\psi }_2$ is defined as:

$$D_{FS}({\psi }_1,{\psi }_2)=\arccos |⟨{\psi }_1|{\psi }_2⟩|$$

Define generalized potential energy $V$ as the geometric distance function between the current state and target state (such as vacuum state or true self orbit):

$$V(\psi )\propto D_{FS}^2(\psi ,{\psi }_{target})$$

System evolution follows the geometric variational principle $\delta S_{geo}=0$, leading to the dynamical equation:

$$\frac{d\psi }{d\tau }=-k{\nabla }_{FS}V(\psi )\text{(E.1)}$$

where ${\nabla }_{FS}$ is the covariant gradient on the manifold. This equation unifies dissipative motion in physics (sliding toward low potential energy) and teleological behavior in psychology (tending toward target states).