This document is a sub-report of the Banya Framework Master Report.

α Length Ladder

Banya Framework Operation Report

Inventor: Han Hyukjin (bokkamsun@gmail.com)

Date: 2026-03-25

Question: Why Are Length Scales Powers of α from Planck to Hubble

In physics, length scales span from the Planck length ($l_P \approx 1.616 \times 10^{-35}$ m) to the Hubble radius of the observable universe ($R_H \approx 4.4 \times 10^{26}$ m), covering roughly 61 orders of magnitude. That this enormous range forms a 29-rung ladder of powers of a single constant α, with each rung spacing being exactly the integer 1, is not a coincidence.

Status

Discovery

Rung spacing Δn = 1, 1. Error 0% (identity). The α power ladder reflects the discrete CAS cost structure.

Key Discovery

Round 1. Constructing the α Power Ladder

Step 1. Banya Equation

The norm on the space axis of the Banya equation determines length scales. Since α is the cost of one CAS operation, discretizing the space axis by powers of α yields the length ladder.

Step 2. Norm Substitution

Substitute the space axis length scale with powers of α.

Step 3. Constant Insertion

Insert fundamental constants.

l_P = 1.616255 × 10⁻³⁵ m (Planck length)
α = 1/137.035999 (fine structure constant)
R_H ≈ 4.4 × 10²⁶ m (Hubble radius)
n range: 0 ~ 29

Step 4. Domain Transform

Starting from the Planck length and multiplying by the inverse of α one rung at a time constructs the length ladder.

Step 5. Discovery

Powers of α connect the Planck length to the Hubble radius in exactly 29 rungs. The fact that each rung spacing is the integer 1 means CAS operation cost is discrete. The hierarchy of cosmic length scales is organized not continuously but as discrete powers of a single constant α.

By-products

The intermediate rungs of the 29-rung ladder may correspond to physically meaningful length scales. For example: whether the Bohr radius, Compton wavelength, classical electron radius, etc., sit at specific rung numbers needs verification.

Incomplete Tasks

Summary