Why Three?: On Budget Exhaustion and the Visibility Threshold

We conjecture that the number three—appearing as both spatial dimensions and particle generations—derives from a universal 45° angular budget divided exponentially. The fourth structure exists but decouples below a visibility threshold. We call this the Caprazli Budget Conjecture (CBC). To the author’s knowledge, this is the first proposal to unify the generation and dimension counts under a single geometric mechanism.

Conjecture 1 (Caprazli Budget Conjecture — CBC): The 45° internal budget of complementary pairs, divided by the ratio λ ≈ 0.22, yields exactly three visible structures before crossing the visibility threshold θ_min. This mechanism governs both internal degrees of freedom (particle generations) and external degrees of freedom (spatial dimensions).

1. Fourth-generation fermions exist with mixing angles |θ_i4| < 0.5°

2. Fourth spatial dimension exists with effective radius R₄/R₃ < λ ≈ 0.22

3. The division ratio λ is identical across flavour and geometric sectors

4. Sum of mixing angles satisfies Σθₙ ≤ 45°