The Alpha–Beta–Omega–Sigma Law (ABOS): Exact Constants, Prime Numbers, Prime Spiral, and the 7-Fold Geometry of Asymmetry

This work presents exact, analytical expressions for the Alpha–Beta–Omega–Sigma (ABOS) constants and demonstrates their geometric origin in angular symmetries of the plane: $$\frac{\pi}{3},\ \frac{\pi}{4},\ \frac{\pi}{5},\ \frac{\pi}{8}.$$ From these, we derive the angle of asymmetry $$\theta = \frac{180^\circ}{3 + \frac{1}{1+\alpha}} \approx 52.060718^\circ,$$ which governs a 7-ray quasiperiodic spiral in the distribution of prime numbers via the transform $x_n = p_n^{1/\alpha}$. Key results: - Exact linearity of $(n, p_n^{1/\alpha})$ with slope $\tan\theta = 1.282798712\ldots$, no fitting required (R² = 0.999936 for $n=1\ldots1000$).- 7θ = 360° + Δ, with $\Delta = 4.42502775^\circ$ — universal spatial excess invariant (Δₛ), geometric signature of hyperbolic embedding («chips» curvature, XIII).- α⁹ = δ_Feigenbaum − 1/50 exactly (error < 10⁻⁴⁵), revealing δ as a derived constant: $$ \delta = \alpha^9 + \frac{1}{50},\quad \text{where } 50 = \beta \cdot 5^2 = 2 \cdot 5^2 $$ — the geometric volume of the ABOS cell in the $\{\pi/4, \pi/5, \pi/6\}$ basis (XVII).- Π₀-basis hypothesis: The set $$ \Pi_0 = \{1, 2, 3, 5, 7\} $$ constitutes the complete set of geometrically irreducible symmetries in ABOS: - 1 — origin of causality ($t_0$), - 2 — β-symmetry (cos π → cos π/2 → cos π/3), - 3 — △-triad (minimal causal chain), - 5 — ☆-transition (φ = β·cos π/5), - 7 — Ω-closure (7θ = 360° + Δₛ). All integers $n \notin \Pi_0$ are expressible as finite $\mathbb{Z}_{\ge 0}$-linear combinations of $\Pi_0$; the norm $$ \|n\| = \min\{a + b + c + d + e \mid n = a\cdot1 + b\cdot2 + c\cdot3 + d\cdot5 + e\cdot7\} $$ quantifies dissipation in physical realization (e.g., $p = 11 = 5 + 2\cdot3$, $\|11\| = 3$).- Pentagonal bridge: $$ \alpha = \beta \cdot \cos\frac{\pi}{5} \cdot \omega_m \cdot (1 + \varepsilon),\quad |\varepsilon| < 0.00135, $$ shows α not as isolated, but as the endpoint of a geometric cascade: β (symmetry) → φ (hierarchical packing) → α (localized jump). Thus, Fibonacci growth is not alternative to ABOS — it is its φ-mode intermediate regime (XV). All constants are given in closed analytic form (no decimals in definitions). None require numerical fitting. The ABOS law is not an empirical fit — it is a geometric lattice embedded in time, number, and causality. Primes do not “lie on a line” — they restore the orientation of the Ω-scaffold from noise. This report serves as empirical validation and extension of the core ABOS law: → Alpha–Beta–Omega–Sigma Law (v12) (https://zenodo.org/records/17831253) --- Note on language and authorship: This text was composed in English with the assistance of an AI language model to ensure international accessibility. The author does not claim native English proficiency; all scientific content — including formulas, hypotheses, derivations, and conceptual framework — originates entirely from the author. The AI served solely as a linguistic tool, with no contribution to the theory, mathematics, or interpretation.