Appendix C: Entropic Speed Limit: Detailed Derivation

In Chapter 10 “The Counter-Flow Circle” of the main text, we introduced the Entropic Speed Limit Axiom as the physical guarantee that life can resist instantaneous thermodynamic collapse. This axiom states that a system’s entropy increase rate is not arbitrary; it is strictly constrained by the universe’s total budget $c_{FS}$.

This appendix will provide a rigorous mathematical proof of this axiom based on continuity bounds in quantum information theory. We will show how the flight speed of the thermodynamic arrow of time is locked by the underlying FS geometric structure.

C.1 From Geometric Distance to Statistical Distance

Consider a composite quantum system whose Hilbert space decomposes into “system” and “environment” parts: $\mathcal{H}={\mathcal{H}}_{sys}\otimes {\mathcal{H}}_{env}$. Assume the system has finite dimension $d_{sys}$ (consistent with QCA’s finiteness assumption).

Although the entire universe is in a pure state $|\psi (\tau )⟩$, for observers focusing only on the local part, the system’s state is described by the Reduced Density Matrix:

$${\rho }_{sys}(\tau )={\text{Tr}}_{env}\left(|\psi (\tau )⟩⟨\psi (\tau )|\right)$$

We want to compare the system states at two moments $\tau$ and $\tau +\Delta \tau$.

In global projective space, the FS distance between the pure states at these two moments is directly given by the intrinsic time definition:

$$d_{FS}([\psi (\tau )],[\psi (\tau +\Delta \tau )])\le c_{FS}\Delta \tau$$

In the local system’s state space, the standard measure for distinguishing two density matrices $\rho$ and $\sigma$ is the Trace Distance:

$$T(\rho ,\sigma )=\frac{1}{2}||\rho -\sigma |{|}_1$$

Since partial trace is a contraction mapping (does not increase distance), the trace distance of local states is always less than or equal to the distance of global pure states. For pure states, trace distance and FS distance have the following relationship:

$$T_{sys}\le T_{glob}=\sin (d_{FS})\le d_{FS}$$

Therefore, we obtain the first inequality connecting macroscopic statistical distance with microscopic geometric budget:

$$T_{sys}(\tau ,\tau +\Delta \tau )\le c_{FS}\Delta \tau$$

This means: the rate of change of the system’s statistical state cannot exceed the geometric rotation speed of the universe’s vector.

C.2 Continuity Bounds for Entropy

Next, we need to relate state changes ($T_{sys}$) to entropy changes ($\Delta S_{vN}$).

Von Neumann entropy is defined as $S_{vN}(\rho )=-\text{Tr}(\rho \ln \rho )$. In quantum information theory, the Fannes-Audenaert inequality provides continuity bounds for entropy with respect to state changes.

For two density matrices $\rho$ and $\sigma$ with trace distance $T$ (where $T\le 1/e$), their entropy difference satisfies:

$$|S(\rho )-S(\sigma )|\le T\ln (d_{sys}-1)+h_2(T)$$

where $h_2(T)$ is the binary entropy function, which tends to zero at rate $O(T\ln T)$ as $T\to 0$.

The physical meaning of this inequality is: entropy is a continuous function; as long as state changes ($T$) are sufficiently small, entropy changes are limited by the dimension of the state space ($d_{sys}$).

C.3 The Bandwidth Limit of Destruction

Now, we combine the above two steps.

Consider an infinitesimal time interval $\Delta \tau \to 0$.

1. Substitute the distance bound: $T_{sys}\le c_{FS}\Delta \tau$.

2. Substitute the entropy bound:

   $$|\Delta S_{vN}|\le (c_{FS}\Delta \tau )\ln (d_{sys}-1)+h_2(c_{FS}\Delta \tau )$$

3. Divide both sides by $\Delta \tau$ and take the limit. Since ${\lim }_{x\to 0}{h}_2(x)/x=0$ (the binary entropy term is a higher-order small quantity), this term vanishes in the instantaneous rate.

Finally, we obtain the differential form of the Entropic Speed Limit:

$$|{\dot{S}}_{vN}(\tau )|\le c_{FS}\ln (d_{sys}-1)$$

C.4 Summary of Physical Meaning

This formula is the bridge connecting microscopic geometry with macroscopic thermodynamics in Vector Cosmology.

• The Role of $c_{FS}$: It is not only the source of light speed but also the bandwidth of information processing. The amount of information the universe can “forget” or “randomize” per second cannot exceed this bandwidth.

• The Role of Dimension: $\ln (d_{sys})$ represents the system’s maximum information capacity. The more complex the system, the more potential channels for entropy increase, but under the premise of finite $c_{FS}$, even complete collapse requires time.

This proves: Death is not instantaneous.

No matter how sharp the thermodynamic arrow, its flight speed is limited by the universe’s underlying FS capacity. It is precisely this finite rate that gives life the precious window to construct order, experience time, and write history before heading toward heat death.