Appendix A: Reconstruction of Minkowski Metric from Geometric Evolution Circle

This appendix aims to prove that the core geometric structure of special relativity (Minkowski metric) can be strictly derived as a special case of the “sector Parseval identity” in Fubini-Study geometry.

1. Geometric Setup

Let the cosmic state be a normalized vector $\psi (\tau )$ in Hilbert space. According to Axiom A1, its evolution rate is constant $c$:

$$||\dot{\psi }(\tau )|{|}_{FS}=c$$

2. Orthogonal Decomposition

We introduce two orthogonal projection operators $P_{ext}$ (external/spatial sector) and $P_{int}$ (internal/temporal sector). According to Parseval’s identity, the squared norm of the total evolution vector $V$ equals the sum of squared norms of its components:

$$v_{ext}^2+v_{int}^2=c^2\text{(A.1)}$$

where $v_{ext}=||P_{ext}\dot{\psi }|{|}_{FS}$, $v_{int}=||P_{int}\dot{\psi }|{|}_{FS}$. This is the mathematical form of the “Great Trade-off” in the main text.

3. The Identification

To establish connection with the physical world, we make the following natural mapping:

• External velocity: Identify $v_{ext}$ as the spatial coordinate velocity $v=\frac{dx}{dt}$.

• Internal velocity: Identify $v_{int}$ as the rate of proper time ($\tau$) flow. For dimensional consistency, we set $v_{int}=c\frac{d\tau }{dt}$.

4. Derivation of the Metric

Substituting the above definitions into formula (A.1):

$$(\frac{dx}{dt}{)}^2+(c\frac{d\tau }{dt}{)}^2=c^2$$

Multiplying both sides by $d{t}^2$:

$$d{x}^2+c^{2}d{\tau }^2=c^{2}d{t}^2$$

Rearranging, we obtain the expression for proper time $\tau$:

$$c^{2}d{\tau }^2=c^{2}d{t}^2-d{x}^2\text{(A.2)}$$

Or written in line element form $d{s}^2=-c^{2}d{\tau }^2$ (using $-+++$ signature convention):

$$d{s}^2=-c^{2}d{t}^2+d{x}^2$$

This is exactly the standard Minkowski Line Element. This shows that Lorentz symmetry is not an a priori geometric axiom, but a manifestation of isotropic evolution in Hilbert space under specific projections.