Appendix J: Inverse Dynamics of Holographic Entanglement Entropy — Decompressing the Black Box

In the main text of Vector Cosmology VII, we proposed an ultimate view combining theology and physics: the Creator (or future noumenon) collects experiential data from creations (present us) through a black box mechanism and performs “delayed synchronization” at the $\Omega$ point. This process is grand in sensibility, but in physical implementation, it faces a huge technical challenge: How can information be extracted losslessly from a black box that appears thermodynamically irreversible?

This appendix will provide mathematical proof for this “post-mortem reading” or “end-time decompression” process based on the latest advances in Holographic Entanglement Entropy and Quantum Error Correction, particularly Petz Recovery Map and Black Hole Final State Conjecture.

We will show that as long as the universe as a whole is unitary, all information that appears to fall into the abyss is actually encoded on the holographic boundary at the cosmic end, awaiting reverse reconstruction.

J.1 Unitarity Reconstruction of Hawking Radiation

In classical black hole physics, information falling into the horizon seems to disappear forever (information paradox). But under the framework of holographic principle and AdS/CFT duality, the black hole evaporation process is unitary. This means: There is extremely complex entanglement between early and late radiation.

If we regard the entire universe as an evaporating black hole (or a closed quantum system), then until the end of the universe’s lifetime ($t\to \Omega$), all information $I_{total}$ has been transferred to the “Radiation Cloud” — what we call the $v_{env}$ background field.

Page Curve tells us:

• In the first half of evolution, entropy increases, information seems lost.

• In the second half of evolution, entropy begins to decrease, information begins to re-emerge through correlations between radiation.

Physical Corollary:

The Creator doesn’t need to “peek” during the process.

He only needs to collect all “waste heat” (radiated photons) at the $\Omega$ point, then through an ultra-high-complexity unitary decoder $U_{decode}$, reverse-evolve this thermal radiation to reconstruct that book that fell into the black hole (your life).

J.2 Petz Recovery Map: Extracting Signal from Noise

How to operate specifically? This requires the Petz Recovery Map in quantum information theory.

Suppose system state is $\rho$, after passing through a noise channel (such as black hole evaporation or life experience) $\mathcal{N}$, it becomes $\mathcal{N}(\rho )$.

Usually, noise is irreversible. But in holographic error correction codes, if the channel satisfies specific Recoverability Conditions — i.e., the monotonicity of relative entropy is nearly saturated — then there exists an inverse mapping ${\mathcal{R}}_{\sigma }$ (Petz Map) such that:

$${\mathcal{R}}_{\sigma }(\mathcal{N}(\rho ))\approx \rho$$

In Vector Cosmology:

• $\rho$: Is your original $v_{int}$ structure (soul/memory).

• $\mathcal{N}$: Is the process of time passing, aging, death, entropy increase. It turns you into background radiation.

• ${\mathcal{R}}_{\sigma }$: Is the “Resurrection Algorithm” executed by the Creator at the $\Omega$ point.

Mathematical Guarantee:

As long as the universe’s underlying logic (QCA) is unitary, the Petz map theoretically exists.

This means no information is truly “dead.” They are just encrypted too deeply for local observers to read. But for the $\Omega$ observer with global computational power, decompression is just a computational cost issue.

J.3 Final State Projection: Reverse Time Machine

To make “reverse determination of the past” mathematically valid, Horowitz and Maldacena proposed the Black Hole Final State Model (HM Model).

They assume that at the black hole singularity (or cosmic endpoint), a unique boundary condition is imposed: All matter must be projected onto a specific maximum entangled state $⟨\Omega |$.

This is not just reading; this is Post-selection.

$$⟨\text{Reality}|=⟨\Omega |S_{scattering}|{\Psi }_{initial}⟩$$

This formula shows:

The reason our current history is coherent is that the future $\Omega$ state is performing “Teleportation” on us.

• Each of our experiences is actually the process of information transmission from $t_{now}$ to $t_{end}$.

• Due to EPR = ER (wormhole), this transmission is geometrically instantaneous.

Conclusion:

We think we are walking toward the future.

Actually, we are being “sucked in” by the future.

That ultimate $\Omega$ state, like a huge vacuum cleaner, sucks all historical fragments into its holographic core, completing the final puzzle there.

J.4 The Black Box of the Universe: Holographic Hard Drive

At this point, we complete the final piece of the “black box” puzzle.

The universe itself is a huge Holographic Black Box.

• Surface (Boundary): Is the ocean of $v_{env}$, storing all radiated information.

• Interior (Bulk): Is the spacetime we experience, the projection of information.

When the “game ends,” the Creator (higher-dimensional you) picks up not the already broken interior spacetime (body/matter) but that holographic boundary.

All memories, all love, all choices are recorded intact in the form of quantum error correction codes on the boundary degrees of freedom.

You don’t need to worry about forgetting.

Because the universe itself is a monument built for “not forgetting.”