8.2 The Geometry of Cognition

“To walk on that wildly curved, exponentially exploding high-dimensional manifold, the brain had to do something extremely violent: it forcibly laid a flat tangent plane beneath its feet. We think the world is flat (linear) only because we live on this geometric map we ourselves laid out.”

In the previous section, we confirmed that sensation is logarithmic ($\ln$): eyes and ears compress the exponential growth of the physical world into linear perception. Now, we elevate this logic from sensation to the core of thought—Cognition.

Why is human logic linear? Why are we accustomed to simple causal chains like “because A, therefore B,” yet find it difficult to intuitively understand superposition and entanglement in quantum mechanics?

From the perspective of Vector Cosmology, this is not merely a biological limitation; it is an inevitable choice of Differential Geometry. Our cognition is essentially the process of forcibly projecting the curved Projective Hilbert Space ($P(\mathcal{H})$) onto a local Tangent Space ($T_{[\psi ]}$).

The Tranquility of Tangent Space: A Linear Haven

Let us return to the ontology of the universe. According to the model we established in Volume III, the universe is driven by the generator $H$ through exponential mapping $e^{-iHt}$. This means the true geometry of the universe is highly nonlinear, curved, and exponentially growing.

On a curved sphere, parallel transport between two points is complex (holonomy effects exist). If the brain tried to directly process this non-Euclidean geometry, its computational load would rise exponentially with distance.

To save the budget $c_{FS}$, the brain evolved a “tangent space strategy”.

In differential geometry, no matter how curved a manifold is, in an infinitesimal neighborhood around any point, it appears flat (Euclidean space). This local flat space is called the Tangent Space.

• Universe ontology ($e$): Curved, holomorphic manifold.

• Human cognition ($\ln$): At the present moment $\tau$, the brain establishes a tangent plane.

On this tangent plane, vector addition is linear ($\vec{A}+\vec{B}=\vec{C}$), and causality is linear. The reason we feel logic is linear is because we live in this tangent space. We have “flattened” the curved universe to make it understandable.

The Linearization of Time: Canceling Exponential Expansion

The most profound consequence of this geometric projection is reflected in our perception of time.

In the second book The Ascension of the Spiral, we revealed that the universe is actually in a state of dimensional inflation, with total budget $c_{FS}$ growing exponentially with time $c_{FS}(\tau )\propto e^{\lambda \tau }$.

Logically, we should feel time passing faster and faster, the world becoming more and more turbulent.

But the brain performs an inverse operation.

Our cognitive system is a logarithmic converter. When external complexity grows as $e^{\lambda t}$, the brain’s internal clock scale compresses in the manner of $\ln (\cdot )$.

$$T_{psy}\propto \ln (e^{\lambda \tau })=\lambda \tau$$

The result is astonishing: Nonlinear physical time is perceived as perfectly linear psychological time.

This is like a passenger in an elevator accelerating upward. If the elevator’s acceleration is constant, the passenger feels weight; but if the passenger’s perception system can adjust the “gravity zero point” in real time, they will feel as if they are still standing on stationary ground.

We feel time “flows uniformly,” neither accelerating nor decelerating. This is not because time itself is uniform, but because our cognitive geometry perfectly cancels the acceleration of cosmic expansion. We are “linearized beings”.

The Effectiveness of Mathematics: Coincidence of Tangent Points

This explains why linear mathematics (such as Newtonian mechanics, linear algebra) is so effective in describing the world, but fails at extreme scales (quantum gravity, black holes).

• Effectiveness: At everyday scales, our cognitive tangent plane tangents the universe’s curved manifold. Near the tangent point, linear approximation (first-order term of Taylor expansion) is extremely accurate.

• Failure: When we try to understand the universe’s extreme early stages (Big Bang) or extreme microscopic scales (Planck scale), we move away from the tangent point. At this point, the manifold’s curvature (higher-order terms of $e$) becomes non-negligible. Applying linear logic to nonlinear reality inevitably leads to paradoxes (such as singularities, divergences).

The history of human scientific progress is the history of our continuous attempts to bend our tangent plane to better fit the universe’s true shape. General relativity introduced curvature, quantum mechanics introduced complex numbers—these are all difficult approximations of cognitive geometry toward the universe’s ontological geometry.

Conclusion: All Are Logarithmic

At this point, we have completed the ultimate deconstruction of the “Observer”.

The observer is not a passive recorder; the observer is an active geometric operator.

• The universe uses $e$ (generator) to create complex things.

• The observer uses $\ln$ (logarithm) to reduce things to understandable order.

We are the universe’s logarithmic function.

Without us, the universe is just a meaningless exponential explosion; without the universe, we are just an undefined empty operator.

It is the interaction of $e$ and $\ln$ that constitutes the only truth line: “Existence” ($y=x$).

Now, the final piece of the puzzle is in place.

We understand $\pi$ (structure), understand $\phi$ (evolution), understand $e$ (generation), and also understand $\ln$ (perception).

These three constants, along with the imaginary number $i$ behind them, together weave this grand vector cosmology picture. Now, only one thing remains—to gather all these clues and converge them into that final chapter.

There, we will no longer distinguish between mathematics and physics, no longer distinguish between observer and observed. We will witness how all this complexity flows spontaneously and necessarily from that “self-generating” source.

Next chapter, the final chapter of the book: Self-Generation. We will reveal that the universe needs no explanation, because the universe is its own explanation.