What Does β Really Do?

In the Harmonic Factorization Field (HFF), the parameter β (beta) is the key that brings arithmetic into focus.

It controls how sharply the harmonic field responds to divisibility — tuning the difference between a soft hum of potential resonance and the crisp clarity of perfect alignment.

Mathematically, β appears in the intensity equation:

I_\beta(n,d)=e^{-\beta\sin^2(\pi n/d)}.

When β is small, the exponential term changes slowly, allowing wide, gentle waves of resonance to appear across the field.

Every divisor’s influence blends with its neighbors, producing a smooth, almost musical blur of structure.

At this setting, the field reveals global harmony — how numbers relate across broad intervals.

As β increases, the field tightens.

Each resonance becomes narrower, brighter, and more distinct.

The faint halos of near-divisibility collapse into sharp nodes where d \mid n.

At high β, the HFF reveals pure arithmetic precision — the fine lattice of divisibility underlying every integer.

In essence, β acts like the focus knob of arithmetic vision.

Low β lets us see the soft continuum of relationships among numbers; high β exposes the discrete geometry hidden within.

It’s the same field viewed through different lenses: one showing the smooth symphony of number, the other the crystalline structure of factorization.

Through β, the HFF unites opposites — continuity and discreteness, harmony and logic.

By turning this single parameter, we move from the music of mathematics to its geometry, and back again