Prime Chain Structure: Three Original Observations on the Orbit Decomposition of Primes

We present three original observations on the structure of prime numbers under the prime-indexmapping p(n). All primes decompose uniquely into non-intersecting orbit chains (Obs. I). This decomposition isinfinitely recursive, forming a rooted tree; algebraic depth D(q) is finite for all q > 2 and D(2) = ∞ (Obs. II). In anovel chain coordinate system the prime sequence traces a diverging oscillation; cross-chain jump values ∆Xare dominated by the smallest chain-start gaps and grow without bound, with ∆X = 6 and ∆X = 4 accounting for39% of observed jumps (Obs. III). A new Section 4 establishes the chain dynamical system: the unifiedrecurrence a(s+1) ≈ a(s)·ln(a(s)) and its double-logarithmic asymptotic law ln(ln(a(s))) ~ s + C. A 2nd-order PNTcorrection achieves <1% prediction error for s ≥ 5. All results verified to 107. Conjecture 1 is upgraded toTheorem 1 (graph-theoretic proof); Finiteness of Algebraic Depth is proved as Theorem 2; the Orbit GrowthLimit is proved as Theorem 3 (PNT). Four further theorem candidates are proposed.