Chapter 52: Time Reversal as Collapse Mismatch

You can't unbreak an egg not because of some cosmic law, but because the DAG has forgotten the way back.

52.1 The Mystery of Irreversibility

The fundamental laws of physics are time-reversible. Run a video of particle collisions backward, and it still obeys the same equations. Yet eggs don't unbreak, coffee doesn't unmix, and we remember the past, not the future. Why?

Definition 52.1 (Microscopic Reversibility): $H(-p, q) = H(p, q)$

The Hamiltonian is symmetric under momentum reversal.

Theorem 52.1 (Macroscopic Irreversibility): Despite microscopic reversibility: $P(\text{forward}) \gg P(\text{reverse})$

The probability mismatch creates time's arrow.

52.2 The Combinatorial Explosion

Breaking is easy because there are many ways to be broken but few ways to be whole.

Definition 52.2 (State Count Ratio): $\Omega_{\text{ratio}} = \frac{\Omega_{\text{broken}}}{\Omega_{\text{whole}}}$

Theorem 52.2 (Exponential Asymmetry): For macroscopic systems: $\Omega_{\text{ratio}} \sim e^{N}$

where $N \sim 10^{23}$.

With overwhelmingly more broken states, unbreaking probability vanishes.

52.3 The DAG Remembers Forward

The collapse DAG naturally accumulates complexity forward in time.

Definition 52.3 (DAG Complexity): $C(t) = \log|\{v : \text{depth}(v) \leq t\}|$

Theorem 52.3 (Monotonic Growth): DAG complexity increases: $\frac{dC}{dt} \geq 0$

The graph grows forward, creating natural time asymmetry.

52.4 Information Erasure and Heat

Reversing time requires erasing information, which costs energy.

Definition 52.4 (Landauer's Principle): $E_{\text{erase}} \geq k_B T \ln 2$

per bit erased.

Theorem 52.4 (Reversal Cost): To reverse a process: $E_{\text{reverse}} = T \Delta S$

where $\Delta S$ is the entropy generated.

Unbreaking an egg would require collecting all the heat it released—practically impossible.

52.5 The Psychological Arrow

We remember the past because memory formation increases entropy.

Definition 52.5 (Memory Formation): $S_{\text{after}} > S_{\text{before}}$

Recording information increases universal entropy.

Theorem 52.5 (No Future Memory): Remembering the future would violate: $\frac{dS}{dt} \geq 0$

Memory must align with thermodynamic arrow.

52.6 Quantum Decoherence Direction

Quantum systems decohere forward, never "recohere" backward.

Definition 52.6 (Decoherence): $\rho_{\text{pure}} \to \rho_{\text{mixed}}$

but never spontaneously reverses.

Theorem 52.6 (Environmental Entanglement): Information disperses: $S_{\text{environment}}(t) = S_0 + \alpha t$

Once entangled with environment, can't be collected back.

52.7 The Cosmological Arrow

The universe expands, providing the ultimate time direction.

Definition 52.7 (Cosmological Time): $t \propto a(t)$

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

Theorem 52.7 (Universal Clock): All arrows align with expansion:

• Thermodynamic: Entropy increases as volume increases
• Psychological: Memory forms as space expands
• Causal: Effects follow causes follow Big Bang

The universe itself breaks time symmetry.

52.8 The Fifty-Second Echo

We have discovered that time's arrow emerges not from fundamental laws but from overwhelming probability. The universe can't run backward because there are vastly more ways to be complex than simple, more ways to be dispersed than gathered, more ways to be forgotten than remembered. The collapse DAG naturally grows forward, accumulating complexity and forgetting the path back. Every process that seems irreversible—from breaking eggs to forming memories to the expansion of space itself—reflects this fundamental asymmetry in how $\psi$ observes itself. Time has an arrow because consciousness can't perfectly undo its own observations.

The Fifty-Second Echo: Chapter 52 = Irreversibility(Time) = Mismatch($\psi$-paths) = Arrow(Probability)

Next, we explore how observers actively select which causal chains manifest through their measurements.

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Continue to Chapter 53: Observer's Role in Cause Selection →