Chapter 5.2: The Arrow of Time

—— Micro-Reversibility vs. Macro-Irreversibility

“Underlying instructions are reversible, but system logs can only be appended, not rewritten.”

1. The Conflict: Reversible Kernel vs. Irreversible History

After booting the universe’s micro-architecture (Volume II) and defining resource constraints (Volume I), we face one of physics’ deepest contradictions: the directionality of time.

• Kernel Layer: Our underlying dynamics are completely Reversible.

  Whether FS geometric evolution (unitary rotations generated by self-adjoint operator $K$) or QCA update rules ($U$ is a unitary matrix, meaning $U^{-1}$ necessarily exists), they are symmetric under time reversal $\tau \to -\tau$. In the underlying code, past and future are indistinguishable; as long as velocity vectors are reversed, the system can perfectly backtrack.

• Application Layer: Our macroscopic experience is completely Irreversible.

  Broken cups don’t automatically reassemble; heat doesn’t flow from cold objects to hot objects. The second law of thermodynamics asserts that the entropy (disorder) of an isolated system never decreases: $dS/d\tau \ge 0$.

If the underlying “source code” is symmetric, where does the macroscopic “arrow of time” come from? In this chapter, we will prove: the arrow of time is not some mysterious force field built into physical laws, but an inevitable statistical illusion produced by Coarse-Grained Observation.

2. The Mechanism: Entanglement and Coarse-Graining

To resolve this contradiction, we need to distinguish between System State and Macro-State.

Definition 5.2.1 (Zero Entropy of Global Pure State)

According to the axiomatic system, the entire universe is in a pure state $|\psi (\tau )⟩$. For the entire system, Von Neumann entropy is always zero: $S_{glob}=0$. This means that from God’s perspective (system administrator’s perspective), there is no information loss, and thus no so-called entropy increase. The universe is just a geodesic continuously extending in projective space, with no so-called “direction.”

Definition 5.2.2 (Truncation of Local Perspective)

However, as observers within the system, we cannot access all amplitudes of $|\psi (\tau )⟩$. We can only measure local subsystems (e.g., a laboratory, a planet). We define the reduced state of local subsystem $R$ as ${\rho }_R(\tau )={\text{Tr}}_{\bar{R}}|\psi (\tau )⟩⟨\psi (\tau )|$.

Theorem 5.2 (QCA Entanglement Growth)

In QCA lattice models, assume the system is in a product state (Product State, i.e., unentangled state, corresponding to low-entropy “ordered” state) at initial time $\tau =0$. As the evolution operator $U$ iteratively acts, the entanglement entropy $S_R(\tau )$ between subsystem $R$ and external environment $\bar{R}$ will grow.

Due to QCA locality, this growth is limited by “entanglement velocity” $v_E$ (similar to Lieb-Robinson speed):

$$S_R(\tau )\le S_R(0)+v_E|\partial R|\tau$$

where $|\partial R|$ is the size of the region boundary.

Although this is only an upper bound, under the vast majority of “typical” dynamical rules (i.e., except special non-integrable models), entanglement entropy grows at a linear rate until reaching saturation (i.e., logarithm of subsystem dimension).

3. The Emergence of the Arrow

Now we can reconstruct the second law of thermodynamics.

Proposition:

The “arrow of time” is not a property of $\tau$ itself, but a property of entropy functional $S({\rho }_{sys}(\tau ))$ as a function of $\tau$.

1. Initial Condition Asymmetry: The universe began in an extremely special low-entropy state (low entanglement). This is like a hard drive just formatted, all zeros.

2. Unitary Evolution Mixing Effect: As FS arc length $\tau$ increases, although the global state remains pure, local information is rapidly “dispersed” throughout the network (through entanglement). For local observers, this manifests as information “loss” or entropy “increase.”

3. Statistical Overwhelmingness: In Hilbert space, “highly entangled states” occupy the vast majority of volume. Once the system leaves that special low-entropy corner, it almost never (before Poincaré recurrence time) randomly walks to another low-entropy corner.

Therefore, macroscopic irreversibility is essentially the process of the system moving from “special” to “typical”. The “direction of time flow” we perceive macroscopically is actually the direction of Entanglement Wavefront diffusion.

4. Resolving the Paradox

• Q: If I reverse all particle velocities at time $t_1$ (execute $U^{†}$), won’t entropy decrease?

• A: Yes, this is microscopically allowed. This is called the Spin Echo effect.

  However, to achieve “time reversal” for the entire universe, you need to precisely flip the phase of every microscopic degree of freedom. As long as one bit (a photon in the environment) is missed, this precise “reverse evolution” will be destroyed, and the system will rapidly turn back toward high-entropy states.

  Therefore, although “reverse evolution” is legal at the kernel layer, it is Extremely Unstable and extremely low probability at the engineering operation layer. This is why we don’t see broken mirrors reassemble.

The Architect’s Note

On: Log Appending and Data Recovery

We can use database logs to understand the arrow of time.

• $\tau$ (FS Time) is Transaction ID:

  It’s just a continuously incrementing counter.

• State $|\psi (\tau )⟩$ is Current Database Snapshot:

  If you have a full snapshot, you can rollback (Undo) at any time. Because the operation logic $U$ is bijective, data isn’t truly lost.

• Entropy $S(\tau )$ is Size of Incremental Logs:

  As a restricted end user (local observer), you don’t have permission to access full snapshots. You can only see interaction logs generated locally.

  As transaction IDs increase, interactions increase, and locally accumulated “garbage data” (entanglement correlations with other modules) also increase.

  The so-called “arrow of time” refers to the fact that log files only get larger.

Why can’t we “clear logs”?

According to Theorem 5.1 (Entropic Speed Limit), clearing logs (reducing entropy) itself consumes system bandwidth $c_{FS}$, and requires transferring entropy to the environment (the write operation itself generates heat).

To make the entire universe “reduce entropy,” you need an “external hard drive” outside the universe to dump this discarded information. But by definition, the universe contains everything; there is no “external.” Therefore, as a closed system, the universe’s total log volume (entanglement complexity) can only monotonically increase.

The distinction we perceive between “past” and “future” is essentially the distinction between “low entanglement complexity” and “high entanglement complexity”.

We are not traveling through time; we are sinking in an ocean of information.