7.2 $\ln$ as Decoder

“The universe roars with exponential functions, while we listen with logarithmic functions. Without the filter of $\ln$, even the quantum history of a single grain of dust would be enough to shatter our reason. The reason we can see a linearly flowing world is because our brains are precise logarithmic decoding machines.”

In the previous section, we established the logarithmic relationship between macroscopic and microscopic through Boltzmann’s formula $S=k\ln W$. But this is not merely a thermodynamic formula; it is the survival strategy of life as an Observer.

If the essence of the universe is $e$ (generation and expansion), then the essence of life must be $\ln$ (compression and decoding).

This is mathematically necessary: $e^x$ and $\ln x$ are Inverse Functions of each other.

The universe is responsible for creating complexity ($e$), while we are responsible for reducing this complexity to understandable linear order ($\ln$).

Taming the Exponential Monster

Imagine if you had to process the wave function evolution of every particle in the universe in real time.

According to the QCA model, the dimension of Hilbert space grows exponentially with time: $D(t)\propto e^{\lambda t}$.

This means that if you want to “omnisciently” record the universe’s history, you need a hard drive that grows at exponential speed. This is physically impossible because your $c_{FS}$ budget (computational bandwidth) is finite.

Facing this exponentially growing data monster, life faces two choices:

1. Be drowned: Attempt to record all details, leading to overload collapse.

2. Decode: Use an algorithm to transform “exponential growth” into “linear signals.”

Mathematics tells us that the only operator that can do this is the logarithm.

$$\ln (e^{\lambda t})=\lambda t$$

Look! Through logarithmic operations, that violent, curved, rapidly escaping exponential curve is instantly straightened into a gentle, linear straight line.

This is the essence of Decoding.

Life does not directly perceive the original physical quantities (those are astronomical numbers); life perceives the Order of Magnitude of physical quantities.

The Illusion of Linear Time

This mechanism perfectly explains why we feel time passes linearly.

In the second book The Ascension of the Spiral, we pointed out that the universe is actually accelerating expansion (spiral), and the $c_{FS}$ budget is increasing exponentially. Logically, we should feel time passing faster and faster, the world becoming more and more chaotic.

But we don’t. We feel yesterday, today, and tomorrow are equally long.

This is because our brains are logarithmic sensors.

• Physical fact: The universe’s information at time $t$ is $I(t)=I_0{e}^{\lambda t}$.

• Subjective perception: Our psychological time $T_{psy}$ is the logarithm of physical information.

  $$T_{psy}\propto \ln (I(t))\propto \lambda t$$

The “uniform time” we see is actually “expanding time” compressed logarithmically.

This is like drawing on logarithmic graph paper: although each tick on the horizontal axis represents a value 10 times the previous one (exponential growth), visually, their distances are equal (linear perception).

This is the ultimate mercy the universe bestows upon observers. It uses $\ln$ to filter out the dizziness brought by exponential explosion, allowing us to live peacefully in a seemingly stable linear illusion.

Dimensional Reduction Compression of Information

In information theory, logarithms are also the foundation of optimal encoding.

When we say a file’s size is “how many bits,” we are actually taking a logarithm: $\text{Bits}={\log }_2(\text{States})$.

Life can understand the universe because life has learned to “ignore”.

• Microscopically, a balloon contains $10^{23}$ gas molecules, whose positions and momenta constitute $e^{10^{23}}$ possible states.

• Macroscopically, life only cares about one value: pressure.

What is pressure? Pressure is the statistical average of those countless microscopic collisions, essentially a logarithmic reading of the number of microscopic states.

This strategy of “ignoring details, only reading magnitude” is the foundation of all intelligent existence.

$\ln$ is the universe’s compression software (Zip).

It packages the suffocating details in high-dimensional Hilbert space (microscopic phases of $v_{int}$) into simple physical laws in low-dimensional macroscopic world (macroscopic motion of $v_{ext}$).

Conclusion: We Are the Inverse Operation

At this point, we see the position of observers in the universe’s mathematical structure.

• Universe ($e$): Responsible for “unfolding”. From $0$ to $\infty$, generating infinite possibilities.

• Life ($\ln$): Responsible for “folding”. From $\infty$ to $1$, extracting unique meaning.

We are the universe’s Inverse Operation.

The universe diverges; we converge. The universe does multiplication; we do addition.

Without us ($\ln$), the universe ($e$) is just a crazy explosion without an audience; without the universe ($e$), we ($\ln$) are just an empty line of code without input.

Together, they constitute the perfect identity $y=\ln (e^x)=x$—the process of “existence being cognized”.

Since our perception is constructed logarithmically, does this mathematical structure leave traces at our physiological level? Why do our eyes’ response to brightness and ears’ response to sound strictly follow logarithmic laws?

This leads to the theme of the next chapter: The Logarithmic Law of Sensation. We will leave abstract mathematics and enter the realm of biology to verify the direct projection of this cosmic law onto neurons.