Formal Proof: The Logarithmic Uncertainty Violation

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“Why does the Heisenberg Bound increase logarithmically near the Planck scale?”

This proof derives the Logarithmic Uncertainty Violation as a direct consequence of ISL Axiom 1 (Resource Boundedness) and the Shannon-Hartley Capacity Limit.

1. The Information Cost of Resolution

In a pixelated manifold, the descriptive complexity (C) of a particle’s position (\Delta x) is not linear. It is determined by the Address Space Depth required to locate the particle relative to the Planck Length (l_P).
According to information theory:

C_{addr} = \kappa \ln\left( \frac{\Delta x}{l_P} \right)

This represents the “Synchronisation Overhead” required to maintain a modular state at a given resolution.

2. The Total Risk Equation

Following the ISL Refusal Principle, the total computational risk (R) of a state is the sum of its interaction complexity and its address space complexity:

R_{total} = R_{Heisenberg} + R_{Address}

Substituting the standard Heisenberg risk (\frac{\hbar/2}{\Delta x \Delta p}):

R_{total} = \frac{\hbar/2}{\Delta x \Delta p} + \kappa \ln\left( \frac{\Delta x}{l_P} \right)

3. The Realizability Constraint

For a state to be instantiated by the kernel, the total risk must be normalized against the maximum allowed entropy of the local manifold. In the ISL ground state (T=1, \beta=1), this leads to the stability boundary:

\Delta x \Delta p \ge \frac{\hbar}{2} \left[ 1 + \kappa \ln\left( \frac{\Delta x}{l_P} \right) \right]

4. Physical Interpretation

The logarithmic term is the “Kernel Tax” on precision.

  • At macroscopic scales (\Delta x \gg l_P), the log term is negligible. Standard Heisenberg applies.
  • Near the Planck scale (\Delta x \rightarrow l_P), the informational cost of defining “Where” and “How Fast” becomes recursive. The kernel forces an increase in the uncertainty product to prevent an address-space overflow.

5. Falsifiability

This proof predicts that ultra-high-precision interferometry (e.g., Planck-regime LIGO or tabletop quantum gravity experiments) will detect a non-linear rise in the minimum uncertainty floor that follows a logarithmic curve.

Standard QM: Flat boundary (\hbar/2).
ISL Theory: Logarithmic curve.

This is the “Smoking Gun” of a pixelated universe.

—
TWIST POOL Labs | The Reality Firewall Team

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