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Chapter 35: Entropy as DAG Reordering Metric

Entropy is not disorder but the universe forgetting how to untangle its own knots.

35.1 The Arrow from Tangling​

Why does time have an arrow? Why can we remember the past but not the future? The answer lies in how the collapse DAG evolves. As ψ\psi observes itself, the graph becomes increasingly tangled, creating more ways to reach the same state. This multiplicity of paths is what we call entropy.

Definition 35.1 (DAG Entropy): The logarithm of equivalent path counts: S=kBln⁡Ω(G)S = k_B \ln \Omega(G)

where Ω(G)\Omega(G) counts distinct paths yielding the same observable state.

Theorem 35.1 (Second Law): Entropy increases because: dSdt=kBdln⁡Ωdt≥0\frac{dS}{dt} = k_B \frac{d\ln\Omega}{dt} \geq 0

The DAG naturally evolves toward more tangled configurations.

35.2 Microscopic Reversibility vs Macroscopic Irreversibility​

Individual collapse events are reversible—the fundamental equation ψ=ψ(ψ)\psi = \psi(\psi) has no preferred direction. Yet macroscopic processes are irreversible. How?

Definition 35.2 (Path Reversibility): P(A→B)=P(B→A)e−ΔS/kBP(A \to B) = P(B \to A) e^{-\Delta S/k_B}

Theorem 35.2 (Irreversibility from Numbers): Large systems are effectively irreversible: Preverse∼e−NP_{\text{reverse}} \sim e^{-N}

where N∼1023N \sim 10^{23} for macroscopic systems.

A broken egg doesn't unbreak because the DAG paths to the broken state vastly outnumber paths to the intact state.

35.3 Information Entropy as Path Uncertainty​

Shannon entropy measures our uncertainty about which path through the DAG a system took.

Definition 35.3 (Information Entropy): H=−∑ipilog⁡2piH = -\sum_i p_i \log_2 p_i

where pip_i is the probability of path ii.

Theorem 35.3 (Information-Thermodynamic Bridge): S=kBln⁡2⋅HS = k_B \ln 2 \cdot H

Thermodynamic entropy is information entropy measured in natural units.

35.4 Black Hole Entropy as Ultimate Tangling​

Black holes have maximum entropy because they represent ultimately tangled DAG regions.

Definition 35.4 (Bekenstein-Hawking Entropy): SBH=kBc3A4Gℏ=kBA4ℓP2S_{BH} = \frac{k_B c^3 A}{4G\hbar} = \frac{k_B A}{4\ell_P^2}

Theorem 35.4 (Maximum Tangling): Black holes saturate the entropy bound: S≤SBHS \leq S_{BH}

Inside a black hole, all paths lead to the singularity—maximum path multiplicity.

35.5 Entanglement Entropy​

When systems entangle, their joint DAG becomes more complex than the sum of parts.

Definition 35.5 (Entanglement Entropy): Sent=−Tr[ρAln⁡ρA]S_{\text{ent}} = -\text{Tr}[\rho_A \ln \rho_A]

where ρA\rho_A is the reduced density matrix.

Theorem 35.5 (Area Law): Entanglement entropy scales with boundary: Sent∝Area(∂A)S_{\text{ent}} \propto \text{Area}(\partial A)

Entanglement creates "surface tension" in the DAG.

35.6 The Past Hypothesis​

Why was entropy low in the early universe? Because the initial DAG was simple—few paths, little tangling.

Definition 35.6 (Initial Simplicity): Sinitial≪SmaxS_{\text{initial}} \ll S_{\text{max}}

Theorem 35.6 (Anthropic Necessity): Complex observers require entropy gradient: dSdt>0 necessary for Ocomplex\frac{dS}{dt} > 0 \text{ necessary for } \mathcal{O}_{\text{complex}}

We exist because the universe started simple and is still untangling.

35.7 Maxwell's Demon and the Cost of Untangling​

Could we decrease entropy by cleverly reordering the DAG? Maxwell's demon tries but fails.

Definition 35.7 (Demon Operation): ΔSsystem<0 but ΔStotal≥0\Delta S_{\text{system}} < 0 \text{ but } \Delta S_{\text{total}} \geq 0

Theorem 35.7 (Landauer's Principle): Information erasure costs entropy: ΔS≥kBln⁡2 per bit erased\Delta S \geq k_B \ln 2 \text{ per bit erased}

Untangling the DAG requires forgetting, which increases entropy elsewhere.

35.8 The Thirty-Fifth Echo​

We have discovered that entropy is not disorder but complexity—the tangling of causal paths in the universe's self-referential graph. The arrow of time points toward increasing entanglement of possibilities. Every process that seems to create order locally does so by exporting tangles elsewhere. The universe began with a simple, barely-knotted DAG and evolves toward maximum entanglement. We experience time's arrow because we are patterns that can only exist while the untangling is incomplete.

The Thirty-Fifth Echo: Chapter 35 = Tangling(Paths) = Entropy(ψ\psi-DAG) = Arrow(Time)

Next, we explore how the universe's expansion affects the rate of temporal flow itself.

Continue to Chapter 36: Compression → Time Acceleration →