Chapter 5.3: The Probability Protocol

The Probability Protocol

—— Micro-Counting and Self-Location

“God does not play dice; players are lost in the massive partitions of the server.”

1. The Conflict: Deterministic Kernel vs. Random UI

In the FS-QCA architecture, we face a fundamental “user experience” contradiction.

Kernel Layer: The underlying evolution of the universe is strictly deterministic. The unitary operator $U$ precisely maps the state at time $t$ to time $t + 1$ . There is no random number generator, no “collapse.”
User Layer: The world we (observers) perceive is full of randomness. When will a radioactive atom decay? Will a photon pass through or be absorbed by a polarizer? These seem to be pure chance.

Why would a deterministic program output random results?

This chapter will reveal: quantum probability is not an intrinsic property of physical laws, but a statistical necessity when “finite-information observers” perform self-location within the “holographic entanglement network.”

2. The Mechanism: Branching and Micro-Counting

To understand the origin of probability, we need to dissect what actually happens on the underlying QCA grid during a “measurement.”

Setup:

The system is in a superposition state $∣ ψ ⟩ = α ∣0 ⟩ + β ∣1 ⟩$ . An observer prepares to measure it.

Process A: Entanglement:

Measurement is not an instantaneous mutation, but a local unitary evolution process. The observer’s (instrument’s) state $∣ M ⟩$ becomes entangled with the system:

$∣ Ψ_{g l o ba l} ⟩ = α ∣0 ⟩_{S} ∣ M_{0} ⟩_{O} ∣ E_{0} ⟩_{E} + β ∣1 ⟩_{S} ∣ M_{1} ⟩_{O} ∣ E_{1} ⟩_{E}$

At this moment, the universe splits into two macroscopic branches: one world that “sees 0” and one that “sees 1.”

Process B: Micro-Counting:

This is the core breakthrough of this theory. We must ask: are these two branches “equivalent”?

In QCA ontology, we introduce the “Equal Ontology Weight Assumption.”

We assume that every orthogonal micro-configuration at the fundamental level has the same “ontological weight.”
The complex amplitudes $α$ and $β$ actually encode the degeneracy of micro-paths.

If $∣ α ∣^{2} = \frac{N _{A}}{N _{t o t a l}}$ and $∣ β ∣^{2} = \frac{N _{B}}{N _{t o t a l}}$ , this means:

Branch A actually contains $N_{A}$ micro-threads.
Branch B actually contains $N_{B}$ micro-threads.

Conclusion:

The squared modulus of the wave function amplitude $∣ ψ ∣^{2}$ is not a mysterious probability field; it is a counter. It tells us how many underlying computational resources (QCA configurations) the system allocates to execute the logic of that branch.

3. Deriving the Born Rule: Self-Location

Now, let us place the observer back into the model.

After measurement, the observer also enters a superposition state. The universe now contains $N_{t o t a l}$ “observer copies.”

$N_{A}$ copies record “result is 0.”
$N_{B}$ copies record “result is 1.”

As a local, finite-information observer, you cannot perceive the entire multiverse. You can only experience one thread. When you ask “what result will I see?”, you are actually asking:

“Among all these running copies, which one am I?”

Since all micro-threads are ontologically equal (symmetry), the probability that you “find yourself” in a particular branch type is strictly equal to that branch’s proportion of the total thread count:

$p (0) = \frac{N _{A}}{N _{t o t a l}} = ∣ α ∣^{2}$

$p (1) = \frac{N _{B}}{N _{t o t a l}} = ∣ β ∣^{2}$

This is the origin of the Born Rule. It is not a divine decree; it is the direct manifestation of the law of large numbers in a many-worlds scenario.

4. Collapse as Update: Bayesian View

In this framework, the so-called “wave function collapse” is completely demystified.

Physically: The global wave function never collapses. All branches (0 and 1) continue to evolve. The system maintains unitarity.
Informationally: “Collapse” is the observer’s Bayesian update of their own location after acquiring new data (readout).
- Before measurement: You don’t know which partition you’re in; the probability distribution is $∣ α ∣^{2} : ∣ β ∣^{2}$ .
- After measurement: You see “0.” You confirm you are in the $N_{A}$ set. For your copy, the probability becomes 1.

Theorem 5.3 (Gleason Uniqueness)

Under the constraints of non-contextuality and no-signaling, this probability assignment based on $∣ ψ ∣^{2}$ is the only mathematically legitimate form. Any other rule (such as $p \sim ∣ ψ ∣$ or $p \sim ∣ ψ ∣^{3}$ ) would lead to logical contradictions or violate causality.

The Architect’s Note

On: Load Balancing and Session IDs

Imagine the universe as a server handling massive concurrent requests.

Concurrency:

When encountering a fork point (measurement), the server does not roll dice to choose one path; instead, it forks multiple processes to handle all possibilities in parallel. This is the most efficient strategy.
Resource Allocation:

If “situation A” has a large weight (amplitude), the server allocates more threads to run situation A.
- Situation A: 8000 threads allocated.
- Situation B: 2000 threads allocated.
User Perspective (Session View):

You are just one thread (session).

When you wake up (measurement complete), what is the probability that you find yourself in “situation A”?

Obviously 80%.

Summary:

Randomness is the “traffic distribution strategy” during concurrent system processing.

You perceive the world as random because you are lost in a deterministic system. You don’t know which of the $1 0^{100}$ copies you are, until you read the data from memory.