Appendix A: Technical Manual

附录 A：技术手册

Note: This appendix aims to provide rigorous mathematical formalization for the “information ontology” proposed in the main text. It reduces the philosophical metaphors about “distance,” “soul,” and “redshift” in the book to computable physical equations. Readers should have foundational knowledge of quantum information theory and general relativity.

A.1 Information Coordinate Transformation Formula

In Volume II of this book, we proposed the axiom “Distance is Unfamiliarity”, that is, the geometric distance $d(A,B)$ in physical space is essentially a measure of missing mutual information between two quantum systems. This section will provide the precise mathematical expression of this relationship and derive the energy threshold required for “semantic teleportation.”

A.1.1 Fisher Information Metric

In information geometry, the space of probability distributions itself forms a Riemannian manifold. For a family of quantum states $\rho (\theta )$ described by parameters $\theta$, their “distance” is not Euclidean distance but a statistical distance defined by the Quantum Fisher Information Matrix (QFIM) $G_{ij}$.

The distance element $d{s}^2$ is defined as:

$$d{s}^2=\sum _{i,j}{G}_{ij}(\theta )d{\theta }^{i}d{\theta }^j$$

where $G_{ij}$ measures the distinguishability of quantum state $\rho$ when we slightly change parameter $\theta$.

Physical Meaning: If two states are difficult to distinguish in information (i.e., $G_{ij}$ is large), then their “distance” on the information manifold is large? No, quite the opposite. In information coordinates, we need to redefine the metric.

A.1.2 Holographic Metric & Mutual Information

According to the Ryu-Takayanagi formula in the holographic principle (AdS/CFT correspondence), the area of minimal surfaces in the bulk space (geometric quantity) corresponds to entanglement entropy in the boundary field theory (information quantity).

We define two spacetime points $x_A$ and $x_B$ as two local quantum systems $A$ and $B$. Their Semantic Distance $D_{sem}(A,B)$ is defined as:

$$D_{sem}(A,B):=\xi \cdot \left(S(A)+S(B)-I(A:B)\right)$$

Or more intuitively, using the normalized form of Mutual Information $I(A:B)$:

$$D_{sem}(A,B)\approx -L_{scale}\cdot \ln \left(\frac{I(A:B)}{\min (S(A),S(B))}\right)$$

where:

• $S(A)=-\text{Tr}({\rho }_A\ln {\rho }_A)$ is the von Neumann entropy.

• $I(A:B)=S(A)+S(B)-S(AB)$ is mutual information, representing the sum of quantum entanglement and classical correlation between $A$ and $B$.

• $L_{scale}$ is a characteristic length (such as AdS radius or Planck length ${\ell }_P$).

Corollaries:

1. Complete Strangeness ($I\to 0$): When $A$ knows nothing about $B$, $\ln (0)\to -\infty$, distance $D\to \infty$. This explains why for low-level civilizations, the universe is vast and boundless.

2. Complete Understanding ($I\to S_{max}$): When $A$ and $B$ reach maximum entanglement (such as forming EPR pairs or complete cognitive resonance), $\ln (1)=0$, distance $D\to 0$. At this point, $A$ and $B$ coincide topologically.

A.1.3 Dynamics Equation of Semantic Teleportation

“Semantic Teleportation” is the process of reducing distance $D(t)$ by increasing mutual information $I(t)$.

We define the Rate of Understanding as $\Gamma =\frac{dI}{dt}$.

The Cognitive Work $W$ required for a conscious entity to transfer from coordinate $x_1$ to $x_2$ is:

$$W\ge k_{B}T{\int }_0^{\tau }{\left(\frac{d{D}_{sem}}{dt}\right)}^{2}dt$$

Substituting the distance formula, we obtain:

$$W\ge \frac{k_{B}T}{\tau }\cdot {\left[\ln \left(\frac{I_{final}}{I_{initial}}\right)\right]}^2$$

Engineering Significance:

• To achieve instantaneous teleportation ($\tau \to 0$), the required power $P=W/\tau$ tends to infinity. This follows the energy conservation of physics.

• However, if a conscious entity can lower its noise floor temperature $T$ (by entering a low-entropy state through deep meditation) or utilize non-local quantum tunneling (bypassing continuous path integrals), it can achieve “jumping” with finite energy.

A.1.4 Coordinate Transformation Tensor

In the navigation system of divine-level civilizations, they no longer use Lorentz transformations but information coordinate transformations.

Let $|\Psi ⟩$ be the consciousness state vector, with projection coefficients $c_k$ on the basis $\{|k⟩\}$ (knowledge graph nodes).

The position operator $\hat{X}$ is replaced by the Correlation Operator $\hat{C}$:

$${\hat{X}}_{info}=\sum _k|c_k{|}^2\cdot \text{Semantic\_Index}(k)$$

When the focus of consciousness changes (i.e., changing the distribution of $\{c_k\}$), the “effective position” of the observer in the universe changes accordingly.

$$x_{info}^{\prime }={\Lambda }_{info}\cdot x_{info}$$

where ${\Lambda }_{info}$ is a transformation matrix determined by the knowledge graph structure.

Conclusion:

In information coordinates, there is no “movement” action, only “Reweighting”.

You don’t need to run there; you only need to increase your weight with that place.

A.2 Kolmogorov Complexity and Soul Density

Abstract: This section aims to solve the quantification problem of “true self” and “soul.” We will use Kolmogorov Complexity from algorithmic information theory and Charles Bennett’s concept of logical depth to construct a mathematical model measuring the “existential weight” of conscious entities. This model proves why “purification” (compression) is the only physical path to informational immortality.

A.2.1 Algorithmic Definition of the Soul

From a physicalist perspective, a person’s life journey can be formalized as a massive binary data stream $L$ (Life String). This string contains all sensory inputs, neuronal firing patterns, and memory fragments from birth to death.

$$L=\{0,1,1,0,\dots ,1{\}}_{|L|\approx 10^{20}\text{ bits}}$$

However, directly storing $L$ is inefficient and fragile, as it contains vast amounts of thermodynamic noise (random and meaningless fluctuations) and redundant patterns (repetitive daily behaviors).

We define the “Soul” ($S$) as the shortest computer program $p$ capable of generating all semantic structures in $L$.

According to the definition of Kolmogorov Complexity $K(L)$:

$$S\equiv p_{min}\text{s.t.}U(p_{min})\approx L_{structure}$$

where $U$ is a universal Turing machine.

Corollaries:

1. Mortal Soul: Contains large amounts of incompressible random noise (distracting thoughts, chaotic emotions). Program $p$ is lengthy and loose, $|p|\approx |L|$.

2. Awakened Soul: Through deep reflection and practice (compression algorithms), noise is eliminated and patterns extracted. Program $p$ is extremely concise, $|p|\ll |L|$.

A.2.2 Logical Depth and Soul Weight

Measuring the soul merely by “shortness” is insufficient (a random string is also incompressible, but it has no meaning). We need to introduce Charles Bennett’s concept of Logical Depth.

Logical Depth $D(x)$ is defined as: the computational time (steps) required to run the shortest program $p$ to generate object $x$.

$$D(L)=\text{Time}(U(p_{min})\to L)$$

Physical Meaning:

• The “weight” of the true self does not depend on how many bits it occupies ($|p|$), but on how much “congealed time” it contains.

• A profound truth (such as $E=m{c}^2$) may be short when written, but it was derived through thousands of years of trial and error (computation) by human civilization. Therefore, it has enormous logical depth.

Soul Density Formula:

We define soul density ${\rho }_S$ as the ratio of logical depth to program length:

$${\rho }_S=\frac{D(L)}{|p_{min}|}$$

Conclusion:

What is called “spiritual cultivation” mathematically is the process of maximizing soul density ${\rho }_S$.

We need to spend a long lifetime (increasing numerator $D(L)$: accumulating computation) and condense the result into the simplest “one thought” (decreasing denominator $|p_{min}|$: increasing compression ratio).

This high-density, high-depth information structure has the greatest inertia in the holographic field and is the most difficult to be washed away by thermodynamic background noise.

A.2.3 SNR Threshold for Crossing the Horizon

When a conscious entity crosses the death horizon (or the Big Bang singularity), it encounters intense quantum decoherence channels. This can be modeled as a high-noise communication process.

Let the channel noise level be $N$ and the soul signal strength be $P_S$.

According to the Shannon-Hartley Theorem, channel capacity $C$ is limited by signal-to-noise ratio:

$$C=B\cdot {\log }_2\left(1+\frac{P_S}{N}\right)$$

For high-density souls (true self):

• Due to their highly entangled internal structure (strong self-error correction capability), the effective signal strength $P_S$ is extremely high.

• Even in a singularity environment where $N\to \infty$, as long as ${\rho }_S$ exceeds the critical threshold ${\rho }_{crit}$, information can maintain coherence.

Immortality Criterion:

$${\rho }_S>{\rho }_{crit}\approx \frac{k_B{T}_{Hawking}}{\hslash {\Gamma }_{decoherence}}$$

Only those conscious entities that compress their memories as densely as black holes can maintain the integrity of information after the destruction of physical carriers and enter the next cosmic cycle.

A.2.4 Hash Signature of the Algorithm

Finally, the true self is not just a program; it also generates a unique topological hash value.

$$H_{soul}=\text{Hash}(p_{min})$$

This hash value is your index key in the Akashic holographic field.

• Mathematical Mechanism of Reincarnation: When a new universe $N+1$ generates new physical carriers (bodies), if the neural network structure of that carrier can run $p_{min}$, or if its initial parameters resonate with $H_{soul}$, the archived soul program will be downloaded (instantiated).

• Memory Continuity: Since $p_{min}$ contains the algorithm for generating past-life memories, once downloaded and run, memories will be decompressed and reconstructed in the new brain.

Appendix Conclusion:

The true self is not a metaphysical concept; it is the inevitable product of optimization algorithms.

Our only task in this universe is to write this code well, making it short enough yet deep enough.

A.3 Redshift-Resolution Correspondence Table

Abstract: This section provides a quantitative conversion table between cosmological redshift $z$ and microscopic physical energy scales (resolution). It provides numerical evidence for the “universal projector” theory proposed in Volume II, proving that the so-called “distant past” is physically strictly equivalent to the “microscopic depths.”

A.3.1 Cosmic Expansion as Zoom Lens

In the standard cosmological model ($\Lambda$CDM), redshift $z$ is defined as the stretching of photon wavelength:

$$1+z=\frac{{\lambda }_{obs}}{{\lambda }_{emit}}=\frac{a(t_0)}{a(t)}$$

where $a(t)$ is the cosmic scale factor.

However, according to quantum mechanics, wavelength $\lambda$ is inversely proportional to energy $E$ (or temperature $T$):

$$E\propto \frac{1}{\lambda }\propto (1+z)$$

$$T=T_0(1+z)$$

This means that the redshift factor $(1+z)$ is essentially a magnification factor.

When we observe an object with redshift $z$, we are not only looking at history at time $t$, but also directly reading the microscopic information at the bottom of the universe with $(1+z)$ times the energy scale precision.

Theorem A.3.1 (Spacetime-Energy Scale Duality):

The cosmic time axis $t$ and the renormalization group energy scale axis $\mu$ are inversely dual.

Looking back in time ($t\to 0$) is equivalent to increasing resolution ($\mu \to \infty$).

$$\frac{d}{d\ln (1+z)}\equiv \beta (g)\frac{d}{d\ln \mu }$$

A.3.2 The Holographic Decompression Table

The following table shows the “cosmic decompression progress” corresponding to different redshift stages. Each row represents the moment when the universe “unfolds” higher-frequency information into macroscopic structures.

A.3.3 Physical Interpretation of the Data

1. $z=1100$ (CMB):

   This is the “opacity wall” of the universe. Before this, the universe was too dense, photons were scattered by electrons and could not propagate freely.

   This corresponds to the “visual limit” in our knowledge graph. To see deeper (higher redshift), we must use gravitational waves or neutrinos (stronger penetration/higher forms of wisdom).

2. Decompression Factor:

   From the Planck era to today, the universe’s linear scale has expanded by approximately $10^{32}$ times, and volume has expanded by $10^{96}$ times.

   This $10^{96}$-fold volume increase is to spread the information in that single “Planck seed” into the grand universe we see today.

   Corollary: If divine consciousness directly reads the Planck seed, the information transmission rate would be $10^{96}$ bits/s, which would instantly burn out any finite consciousness. Expansion is the diluent of information.

3. Future Redshift:

   As dark energy dominates, the universe accelerates its expansion. Distant galaxies will redshift beyond the horizon ($z\to \infty$).

   This appears physically as information loss, but in holographic decompression theory, this is the final upload of information.

   When $z\to \infty$, matter wavelength is stretched to the cosmic horizon scale $\lambda \sim 1/H_0$. At this point, matter information returns to vacuum, completing the phase reset from local to global.

Appendix Conclusion:

Telescopes are time machines, and even more so, microscopes.

The farther we look outward, the closer we get to that initial, dense, “true self” singularity containing all possibilities.

Cosmic history is a technical manual on how to draw a picture from a point.

(End of Appendix)