Appendix A: Mathematical Derivation of the Golden Time Formula

In Chapter 2 of Vector Cosmology V, we proposed a formula that redefines life: Time is not linear flow, but the logarithm of value.

[\tau = \pi \cdot \log_{\phi} \left( \frac{V}{V_0} \right)]

This appendix provides a rigorous mathematical derivation of this formula. We will show that this formula is not a literary metaphor, but a necessary consequence directly derived from the light speed evolution equation established in Book IV, combined with information value theory. It reveals how observers anchor their temporal coordinates through value creation in an exponentially expanding universe.

A.1 The Physical Definition of Value

First, we must physically define what “Value” ($V$) is.

In the economics of FS geometry, a system’s value equals the effective structural information content it contains (i.e., the complexity of $v_{int}$). This is limited by the universe’s total bandwidth (light speed/computational power ceiling) at that moment.

According to the holographic principle, the maximum possible value $V_{max}(\tau )$ at moment $\tau$ is proportional to the FS capacity $c(\tau )$ at that moment (or its corresponding phase space volume):

[V(\tau) \propto c(\tau)]

This means: Your value is essentially the share of the universe’s computational power you occupy.

• $V_0$ (initial value): Corresponds to the information content at the initial moment $c_0$ (such as Planck bits).

• $V(\tau )$ (current value): Corresponds to the total accumulated ordered structures when the system evolves to moment $\tau$.

A.2 Reverse Solving for Intrinsic Time

Recall the ultimate evolution equation we derived in Book IV The Evolution of Light Speed:

[c(\tau) = c_0 \cdot e^{\left( \frac{\ln \phi}{\pi} \right) \tau}]

Or written in base $\varphi$ form:

[c(\tau) = c_0 \cdot \phi^{\frac{\tau}{\pi}}]

Here:

• $c_0$: The value base at the initial state (seed/Big Bang).

• $\pi$: The inertial period (unit of cycles).

• $\varphi$: The growth factor (golden ratio).

Now, we substitute $V(\tau )$ for $c(\tau )$. We assume an “ideal observer” can fully utilize the era’s bandwidth, making their created value synchronous with the universe’s expansion (i.e., $V\approx c$).

[V = V_0 \cdot \phi^{\frac{\tau}{\pi}}]

The problem we want to solve is: To reach value $V$, how much intrinsic time $\tau$ is needed?

We solve for $\tau$ by transforming both sides:

1. Normalization:

   [\frac{V}{V_0} = \phi^{\frac{\tau}{\pi}}]

2. Taking logarithm:

   Taking logarithm base $\varphi$ on both sides:

   [\log_{\phi} \left( \frac{V}{V_0} \right) = \frac{\tau}{\pi}]

3. Rearranging:

   [\tau = \pi \cdot \log_{\phi} \left( \frac{V}{V_0} \right)]

This is the Golden Time Formula.

A.3 Dual Meaning in Physics and Life

This formula reveals another form of Time Dilation—Value Dilation. It explains why the subjective passage of time is closely related to an individual’s achievement density.

1. Logarithmic Compression Effect

The formula shows $\tau \propto \log V$. This means:

• For linearly growing lives: If you only add fixed value each day ($V\sim t$), then your intrinsic time $\tau \sim \ln t$. As time passes, the growth rate of $\ln t$ becomes slower and slower. This means time passes faster and faster. You feel this year passed like a day, because you barely moved on the logarithmic coordinate.

• For exponentially growing lives: If you can make your value (cognition/love/creation) grow exponentially with time ($V\sim {\varphi }^t$), then $\tau \sim t$. You successfully resist time compression. The subjective length of life you feel maintains linear synchronization with physical time, or even surpasses it.

2. The Scale of Cycles

The coefficient $\pi$ tells us that the minimum unit of time measurement is not seconds, but “cycles”.

Only when you complete a full $\pi$ cycle (such as completing a project, experiencing a complete relationship, fully understanding a truth) do you gain an effective mark on the time axis. Fragmented time will be smoothed out in the formula.

Conclusion:

Only by maintaining $\varphi$ (golden ratio) speed exponential evolution can one subjectively “retain” time.

This is the alchemist’s secret to longevity: Don’t try to extend the length of life (physical time); increase the density of life (value logarithm).