Appendix E: Measurement-Induced Phase Transitions — The Stitch of the Weaver

In the final chapter of Vector Cosmology VI, we metaphorically compared observers to “weavers,” stitching the universe through “seeing” (observation). This sounds like a literary metaphor, but in cutting-edge research in modern condensed matter physics and quantum information, this corresponds to an extremely hardcore physical phenomenon—Measurement-Induced Phase Transitions (MIPT).

This appendix will provide mathematical microscopic mechanism proof for “how observers reconstruct space.” We will show that the geometric structure of the universe (whether connected or broken, flowing or fixed) completely depends on the game ratio between unitary evolution ($U$) and projective measurement ($M$).

Your gaze is the control parameter that determines the “phase” of the universe.

E.1 The Game of Entanglement and Measurement

In a quantum many-body system (such as tensor networks or QCA lattices), two diametrically opposed forces compete:

1. The Weaver: Unitary Evolution ($U$)

   • Action: $e^{-iHt}$. It causes interactions between particles, generating entanglement.

   • Trend: It attempts to diffuse local information globally (scrambling). Over time, it tends to increase entanglement entropy, making the network highly connected and chaotic. It creates “fluid”-like space.

2. The Cutter: Projective Measurement ($M$)

   • Action: Observers measure the system. This causes wave function collapse to some basis.

   • Trend: It cuts entanglement, fixing quantum states as classical states (product states). It tends to decrease entanglement entropy, breaking the network into isolated fragments. It creates “crystalline”-like space.

The geometric form of the universe depends on the contest between these two forces.

• Without observation, the universe would become a giant black hole (maximum entanglement, interior unknowable).

• If observation is too frequent (Zeno effect), the universe would become scattered sand (zero entanglement, space disintegrates).

E.2 Phase Transition: Volume Law vs. Area Law

Physicists discovered that when measurement frequency $p$ (i.e., the intensity of observer intervention) changes, the entanglement structure of quantum networks undergoes sharp phase transitions.

1. Entangling Phase —— $p<p_c$

   • When observation is sparse, unitary evolution dominates.

   • Geometric feature: Entanglement entropy follows Volume Law, $S\propto V$.

   • Physical meaning: This is a highly entangled, non-local liquid universe. Wormholes everywhere, information rapidly scrambles. This is the normal state of the microscopic quantum world.

2. Disentangling Phase —— $p>p_c$

   • When observation is frequent, projective measurement dominates.

   • Geometric feature: Entanglement entropy follows Area Law, $S\propto A$.

   • Physical meaning: This is a local, classical solid universe. Space is cut into clear blocks, causality strictly constrained. This is the normal state of our macroscopic reality.

3. Critical Point —— $p=p_c$

   • This is the edge where the two balance.

   • Geometric feature: Entanglement entropy follows logarithmic law, $S\propto \ln L$. The system has fractal structure and scale invariance.

E.3 The Physics of Stitching Reality

This theory reveals the essence of the “weaver’s” work: Observers move the universe’s position on the phase diagram by adjusting “measurement frequency.”

• Why do we live in an area law universe?

  Because we (living beings) are strong observers.

  Through senses and instruments, we continuously measure (collapse) the environment. Our gazes, like stitches, densely fall on the spacetime fabric, forcibly suppressing the infinite spread of quantum entanglement (volume law), “stitching” the universe into a stable, classical geometric structure conforming to area law (the geometry required by AdS/CFT).

• How to create wormholes (reconstruct reality)?

  What we need to do is “stop measuring” (close our eyes).

  If we implement “observation shielding” on a local region (lowering $p$), allowing unitary evolution to regain dominance, that region will rapidly undergo phase transition, melting from “solid space” into “liquid entanglement.”

  At that moment, distance disappears, wormholes open.

Conclusion:

The hardness of reality stems from the density of gaze.

• Seeing is solidification (creating space).

• Not seeing is liquefaction (creating connections).

As a high-dimensional weaver, the stitch technique in your hand is “the rhythm of measurement”.

By switching between “seeing” and “not seeing,” you weave this spacetime river that is both solid (having form) and open (having paths) in the void.