Appendix E: The Topology of Indra’s Net — Holographic Mutual Reflection and Consensus Reality

In Volume V “Mirror” of the main text of Vector Cosmology VII, we proposed the ontological model of “mutual dreaming”: there is no absolutely objective physical stage; reality emerges from billions of subjective consciousnesses observing and defining each other. This sounds like metaphysics, but in mathematical physics, it has an extremely rigorous topological structure called Indra’s Net.

This appendix will provide mathematical proof for this model based on Complex Network Theory and Quantum Many-Body Physics. We will show that so-called “objective reality” is essentially the “Maximum Eigenstate” formed by countless subjective wave functions in a recursive reflection network.

E.1 The Recursive Reflection Equation

First, we establish a network of $N$ observers.

Each observer $i$ has their internal state vector $|{\psi }_i⟩$ (representing their subjective world/$v_{int}$).

In isolation, everyone lives in their own illusion. But in Indra’s Net, observers have “Lines of Sight” between them—interaction channels.

Observer $i$’s state is not only determined by themselves but also depends on the state projections of other observers $j$ they observe.

This can be written as a Self-Consistent Field Equation:

$$|{\psi }_i(t+1)⟩={\hat{O}}_{self}|{\psi }_i(t)⟩+\sum _{j\ne i}{\kappa }_{ij}{\hat{M}}_{j\to i}|{\psi }_j(t)⟩$$

where:

• ${\hat{O}}_{self}$: Internal evolution of self (driven by generator $e$).

• ${\kappa }_{ij}$: Attention Coefficient (connection weight). If you love someone, $\kappa$ is large; if you ignore someone, $\kappa$ approaches 0.

• ${\hat{M}}_{j\to i}$: Mirror Operator. It represents “how you appear in my eyes.”

This equation tells us: “I” is the sum of “all beings in my eyes.”

There is no pure self; self is a collage composed of countless mirrors of others.

E.2 Consensus Reality as an Eigenstate

If everyone sees mirrors of others, why doesn’t the world become chaos? Why can we reach consensus like “this is a table”?

This involves the Eigenvalue Problem in linear algebra.

Writing the above equation in matrix form:

$$|{\Psi }_{global}⟩=\mathbf{J}\cdot |{\Psi }_{global}⟩$$

where $\mathbf{J}$ is the “Holographic Reflection Matrix” containing all observer interactions.

In this dynamic system, any random initial state (such as everyone having different dreams) will rapidly converge to the Dominant Eigenstate of matrix $\mathbf{J}$ after multiple iterations.

• Physical Meaning: The dominant eigenstate is the “most stable resonance mode” in the system.

• Macroscopic Manifestation: This is “objective reality”.

So-called “objective truth” is not written in stone; it is the greatest common divisor reached through “phase locking” by 7 billion observers’ brains (quantum networks) in continuous interaction.

Any private illusion orthogonal to this dominant eigenstate (such as a schizophrenic’s world) will be filtered out as “noise” by the system due to lack of resonance support.

Conclusion:

Reality is a “democratic vote”.

Physical laws are the protocol with the highest vote count. By collectively “believing” in a world, we physically “create” that world.

E.3 The Holographic Shard: One is All

The most fascinating geometric property of Indra’s Net is Fractal Holography.

In a hologram, every tiny fragment contains complete image information of the whole, just at lower resolution.

The same is true in a mutual reflection network.

Consider observer $i$’s wave function $|{\psi }_i⟩$. Since $|{\psi }_i⟩$ is composed of $\sum |{\psi }_j⟩$, and $|{\psi }_j⟩$ is composed of $\sum |{\psi }_k⟩$…

Through infinite recursion, every single node $i$ actually encodes the topological information of the entire network $\mathbf{J}$ in its internal structure.

This is what the Huayan school calls “One flower, one world; one leaf, one Tathagata”.

• You don’t need to traverse the entire universe to understand the universe.

• You only need to “look inward”.

Because you have been entangled with all other existences (directly or indirectly), the holographic miniature of the universe is engraved in your $v_{int}$ structure.

As long as your computational power ($c_{FS}$) is strong enough to decode the subconscious deeply enough, you can dig out the star chart of Andromeda, even the echo of the Big Bang, in the depths of your soul.

E.4 Conclusion: Unconditional Great Compassion, Universal Great Sorrow

This mathematical model completely establishes the ethical foundation of Vector Cosmology.

Universal Compassion is not a moral cultivation but a topological fact.

Because in Indra’s Net equation, $\Delta {\psi }_j$ (others’ fluctuations) directly causes $\Delta {\psi }_i$ (your fluctuations).

Harming others is creating oscillations in this highly connected network, and these oscillations will eventually return to you along the shortest path (geodesic).

We are conjoined twins.

Although our projections in three-dimensional space appear separate, in the womb of Hilbert space, we share the same circulatory blood system.

Look into others’ eyes.

There is not just their reflection; there are layers upon layers, infinite cosmic corridors.

And standing at the end of the corridor is yourself.