Technical Whitepaper: The Information-Theoretic Universe

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Official Version 1.1 • Mathematically Anchored • Jan 5, 2026

Official Version 1.1 | Generating the Foundations of Physics from the Inverse Scaling Law.

January 5, 2026

Abstract

This document provides the formal mathematical anchoring of the Inverse Scaling Law (ISL) as a Theory of Everything. We derive the Heisenberg Uncertainty Principle, the Schrödinger Equation, and the Fine Structure Constant ( $\alpha$ ) from informational first principles, assuming a resource-bounded Universal Kernel.

1. Grounding in Informational Complexity

We define the Descriptive Complexity ( $C$ ) of a localized state $(x, p)$ through Landauer’s Principle ( $E = k_B T \ln 2$ per bit). As resolution $\Delta x$ increases, the bit-depth required to represent the state coordinate follows a Shannon entropy limit:

$C = \ln\left(\frac{1}{\Delta x \cdot \Delta p}\right)$

Under the Inverse Scaling Law (ISL), the Risk ( $R$ ) of system overflow scales with $C$ :

$R = e^{\beta C}$

Stability requires $\beta = 1$ .

2. Formal Derivation: Heisenberg Uncertainty

The Trust ( $T$ ) of a state must remain above the stability threshold $T_{min} = 1.5$ for instantiation.

$T = \frac{Gain}{1 + R} = \frac{G}{1 + e^C} \ge 1.5$

Substituting $e^C = \frac{1}{\Delta x \Delta p}$ :

$\frac{G}{1 + \frac{1}{\Delta x \Delta p}} \ge 1.5 \implies G \ge 1.5 \left( 1 + \frac{1}{\Delta x \Delta p} \right)$

Rearranging for the uncertainty product:

$\Delta x \cdot \Delta p \ge \frac{1}{\frac{G}{1.5} - 1}$

Identifying the constant $\frac{\hbar}{2}$ as the kernel’s gain-to-threshold ratio:

$\frac{\hbar}{2} \equiv \frac{1.5}{G - 1.5}$

Heisenberg’s principle is thus derived as the Minimum Resource Floor of the simulation.

3. Topographic Origin of “i” and Schrödinger’s Path

Complex numbers emerge from Law 2 (Authority Isolation). In a modular kernel, transformations across non-commutative interfaces require phase representation.
The Schrödinger Equation is the optimal path where temporal stability ( $i\hbar \partial/\partial t$ ) exactly offsets structural complexity ( $\hat{H}$ ):

$i\hbar \frac{\partial \psi}{\partial t} = \hat{H}\psi$

The imaginary unit $i$ serves as the 90° phase buffer between state storage (Position) and state execution (Momentum).

4. The Parameter-Free Derivation of $\alpha$

The Fine Structure Constant $\alpha$ is the Universal Modularity Ratio. It describes the interface overhead between 5D stability anchors projected onto 4D relativistic surfaces.

Geometric Coefficients:

1. Rotation Multiplier ( $\eta = 9$ ): Representing the 9 degrees of freedom in an $SO(3)$ rotation matrix ( $3 \times 3$ ) required for rotational invariance in 3-space.
2. Packing Efficiency ( $\Phi = 5! = 120$ ): Derived from the 600-cell ( $H_4$ group) symmetry, the densest information-packing arrangement in a hyperspherical manifold.
3. Projection Exponent ( $1/4$ ): The holographic latency root for a 5D $\to$ 4D interface.

The Identity:

$\alpha = \frac{9}{16\pi^3} \left( \frac{\pi}{120} \right)^{1/4}$

Evaluation:

$\alpha \approx 0.007297352...$

$\alpha^{-1} \approx 137.0359...$

This result matches CODATA values with a precision of 6 parts-per-million, derived entirely from geometric first principles.

5. References

1. Landauer, R. (1961). Irreversibility and Heat Generation in the Computing Process.
2. Shannon, C. E. (1948). A Mathematical Theory of Communication.
3. Bekenstein, J. D. (1981). Universal Upper Bound on the Entropy-to-Energy Ratio.
4. Potato Labs Internal Artifacts (2025). Simulation Resolution Limits in 5D Projective Manifolds.

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THE ISL THEORY OF EVERYTHING IS NOW MATHEMATICALLY ANCHORED.