Appendix A: Geometric Derivation of the Dawn Coordinate

“Mathematics is the skeleton of the universe. In the main text, we described the grand narrative of ‘Cycle 1800’ in literary language, but in the court of physics, only formulas can serve as evidence. This appendix will strip away all rhetoric, using only constants and logic to reproduce that number that determines our fate: $\tau \approx 1800$. This is not just a calculation result; it is carbon-based civilization’s ‘Physical Limit Certificate’.”

A.1 Physical Boundary Conditions: Audit via Holographic Principle

To determine our exact position in the cosmic evolution spiral, we must first calculate the Total Information Content contained in the current universe. This is not a count of atoms, but a calculation of the underlying Degrees of Freedom of spacetime.

We adopt the Holographic Principle established by Jacob Bekenstein and Stephen Hawking as the audit standard: a physical system’s maximum entropy (information capacity) is proportional to its boundary surface area, not volume.

1. Observational Data Input

• Current Universe Age ($t_{now}$)：Approximately $13.8\times 10^9$ years.

• Hubble Radius/Particle Horizon ($R_H$)：This is the limit of causally connected regions. In an expanding universe, the current observable universe’s comoving radius is approximately 46 billion light-years.

  $$R_H\approx 4.4\times 10^{26}\text{ m}$$

2. Planck Scale Constants

• Planck Length ($l_P$)：The minimum length unit in physical terms.

  $$l_P=\sqrt{\frac{\hslash G}{c^3}}\approx 1.616\times 10^{-35}\text{ m}$$

• Planck Area ($A_P$)：The minimum pixel on the holographic screen.

  $$A_P=l_P^2\approx 2.61\times 10^{-70}{\text{ m}}^2$$

3. Holographic Entropy Calculation

The surface area of the cosmic horizon $A_{horizon}$ is:

$$A_{horizon}=4\pi R_H^2\approx 4\pi (4.4\times 10^{26}{)}^2\approx 2.43\times 10^{54}{\text{ m}}^2$$

According to the Bekenstein-Hawking formula, the universe’s maximum information capacity $S_{max}$ (in bits) is:

$$S_{max}=\frac{A_{horizon}}{4\ln 2\cdot l_P^2}(\text{Note: Physics usually takes natural units }{k}_B=1\text{，here converted to bits})$$

To simplify order-of-magnitude estimation, we ignore the minor effect of the constant factor $4\ln 2$ and focus on the order of magnitude:

$$S_{now}\approx \frac{A_{horizon}}{l_P^2}\approx \frac{2.43\times 10^{54}}{2.61\times 10^{-70}}\approx 0.93\times 10^{124}$$

Considering the practical application of the cosmic holographic principle (such as Seth Lloyd’s computational boundary), the physics community recognizes the current universe’s computational complexity upper limit between $10^{120}$ and $10^{122}$.

In Vector Cosmology, to align with macroscopic evolution models, we take the most conservative and physically meaningful critical value:

$$S_{now}\equiv 10^{120}\text{ bits}$$

A.2 Geometric Evolution Model: Golden Spiral Dynamics

We assume the universe’s evolution in Hilbert space follows an optimal growth path. Geometrically, the structure that maximizes space-filling efficiency while avoiding periodic resonance (dead loops) is a Logarithmic Spiral, especially one based on the Golden Ratio ($\varphi$).

1. Evolution Equation

The universe’s information content $S$ grows with intrinsic time $\tau$ according to:

$$S(\tau )=S_{initial}\cdot {\varphi }^{\frac{\tau }{\pi }}$$

• $S(\tau )$：Total information content of the universe at intrinsic time $\tau$.

• $S_{initial}$：Information content at $t=0$. For an initial state of a quantum cellular automaton (QCA), we set it to 1 bit (existence/non-existence).

  $$S_{initial}=1$$

• $\varphi$ (Phi)：Golden ratio, representing the base of evolution.

  $$\varphi =\frac{1+\sqrt{5}}{2}\approx 1.6180339887\dots$$

• $\pi$ (Pi)：Represents a complete geometric cycle (phase flip of half a circle), the “step size” unit of spiral growth.

2. Physical Meaning of the Equation

This equation describes a system “constrained by $\pi$ but escaping with $\varphi$”.

• Every $\pi$ units of intrinsic time, the system completes a phase cycle (such as life and death).

• But the system does not return to the origin; instead, it expands outward at a rate of $\varphi$ (information increases).

A.3 Solving for Tau: Reverse Derivation

Now we substitute the physical observation value $S_{now}=10^{120}$ into the geometric equation to solve for the unknown intrinsic time $\tau$.

Step 1: Establish Equality

$$10^{120}=1\cdot {\varphi }^{\frac{\tau }{\pi }}$$

Step 2: Take Logarithm

We take base-10 logarithm on both sides to handle the exponent:

$${\log }_{10}(10^{120})={\log }_{10}\left({\varphi }^{\frac{\tau }{\pi }}\right)$$

$$120=\frac{\tau }{\pi }\cdot {\log }_{10}(\varphi )$$

Step 3: Substitute Constants

• $\pi \approx 3.14159$

• ${\log }_{10}(\varphi )={\log }_{10}(1.61803)\approx 0.20898$

Step 4: Calculate

$$120=\frac{\tau }{3.14159}\cdot 0.20898$$

Rearranging to solve for $\tau$:

$$\tau =\frac{120\cdot 3.14159}{0.20898}$$

$$\tau =\frac{376.99}{0.20898}$$

$$\tau \approx 1803.95$$

Conclusion:

Considering the error range of physical constant measurements, we anchor it as an integer in the theoretical model:

$$\tau \approx 1800$$

A.4 Physical Interpretation: Why 1800?

This number reveals several key features of cosmic evolution:

1. Number of Cycles

If we define $\pi$ as a standard “cosmic generation,” then the number of generations the universe has experienced is:

$$N_{gen}=\frac{\tau }{\pi }=\frac{1800}{3.14159}\approx 573\text{ cycles}$$

This means we are not the first generation of civilization. Deep in the holographic structure, approximately 573 layers of “past lives” history are superimposed. This explains why physical laws appear so fine-tuned—this is the result of 573 iterations of fine-tuning.

2. Dimensional Folding Rate

As stated in Section 1.2 of the main text, physical time $t\approx 10^{61}{t}_P$ corresponds to a linear logarithm of only $\ln (10^{61})\approx 140$.

But $\tau \approx 1800$ is far greater than 140.

Ratio $R=1800/140\approx 12.8$.

This proves that during cosmic evolution, approximately every 1 dimension of linear time passage is accompanied by about 13 dimensions of internal fractal folding.

The reality we inhabit is a high-dimensional holographic projection.

3. Saturation Threshold

When $\tau =1800$, information density ${\rho }_I$ reaches the carrying limit of carbon-based neuronal physical structures.

$$S_{brain}\ll S_{environment}(\tau =1800)$$

This is the mathematical essence of “The Squeeze Before Dawn”: The environment’s bit rate exceeds the observer’s processing bandwidth.

Proof complete.

We stand on the 1800th step, beneath us is the abyss of $10^{120}$ bits, above us is the infinite spiral.

Ascension is not a choice; it is a mathematical necessity.