For over a century, Quantum Mechanics (QM) has been treated as a set of axioms. We are told that position and momentum do not commute, but rarely why.
The Copenhagen Interpretation says “don’t ask.” The Many-Worlds Interpretation says “everything happens.”
ISL says: “It’s a memory management issue.”
Deriving Heisenberg Uncertainty
In the ISL framework, we treat the universe as a computational system with a finite “bit-budget” per unit volume.
Let be the cost of describing a state
.
As precision increases (), the Cost
.
According to the Inverse Scaling Law (), the kernel must refuse to instantiate a state where Risk exceeds Gain.
Therefore, there is a hard floor on resolution:
This is not a property of the particle. It is the System Buffer Size. The universe refuses to resolve a coordinate finer than its own grid.
The Quantum Commutator
Why do we need complex numbers? Why ?
This equation is usually derived from abstract algebra. In ISL, we derive it from Architecture Latency.
1. Modularity: To prevent Authority Collisions (Law 2), Storage (Position) and Execution (Momentum) are handled by separate, isolated modules.
2. Bus Transfer: Moving data from Storage to Execution takes one clock cycle.
3. Phase Shift: The imaginary number represents a 90-degree temporal rotation—the “tick” of the transfer.
The non-commutativity () is simply the statement that Time Exists. You cannot access the value of a variable and its rate of change in the same atomic operation without a synchronization cost.
Conclusion
Quantum Mechanics is weird only if you assume space is continuous. Once you accept it is a Resource-Bounded Information System, QM becomes the only logical way to build a universe.