Developing Bertrand's Ideas: Orbits of Primes under the Map p → prevprime(2p), Partition and Power-Law Distribution of Lengths

Bertrand's postulate states that for every integer n > 1 there is a prime between n and 2n. This suggests a natural constructive question: for a given prime p, which prime is closest to 2p from below? Denote it by prevprime(2p) — the largest prime not exceeding 2p — and iterate: p → prevprime(2p) → prevprime(2·prevprime(2p)) → … This yields a chain generated by the prime p, developing Bertrand's idea from existence to construction. We study how such chains are arranged for all primes at once, not for a single p: how they intersect, how their lengths are distributed, and how the picture changes as the limit N grows. It turns out that the set of primes up to N is exactly partitioned into disjoint chains (orbits) by this map. The partition depends on the traversal order: a decreasing order yields π(N) singletons (a theorem, not an observation), while an increasing order yields the richest structure and is the one we study. For N = 10^5, 10^6, 10^7, the chain lengths follow a power law with an exponential cutoff, P(L) ∝ L^(−α) exp(−L/L₀), with α growing slowly from 1.42 to 1.65 and L₀ from 3.63 to 4.47. A maximum-likelihood comparison against lognormal, stretched-exponential, geometric, and pure power-law alternatives decisively favours the power-law-with-cutoff form at every scale (ΔAIC ≥ 28, often ≥ 2000). Mergings of chains into earlier chains form a directed forest. Two things must be distinguished. The length of a chain is how many steps it takes; long chains (18, 21, 24) exist, though they are rare. A tree is a group of chains merged into one: the root is the earliest chain, and the others merge into it either directly or through intermediaries, by the same map p → prevprime(2p) (when the result lands on an already-covered member, the chain stops and is counted as merged). A tree has two characteristics: its size — the total number of chains in it — and its depth — the maximum number of steps from the root to the farthest descendant (the root has depth 0, its direct children 1, grandchildren 2, and so on). At N = 10^5, 84.5% of trees consist of a single chain, the largest tree has size 8, and the deepest tree has depth 2. Thus the power-law (heavy-tail) distribution belongs to chain lengths, not to tree sizes: the tree-size distribution declines sharply and is not power-law. A structural reason is that every orphan is a permanent leaf: a segment's starter can never be a merge target. The fitted value α ≈ 1.6 is an effective exponent — the observed slope of an empirical power law over L ∈ [1, 24] — not a universal critical exponent. Our map p → prevprime(2p) is fully deterministic: no randomness, no phase transition, and no claim of belonging to any specific universality class. The heavy-tailed shape is shared with many critical phenomena, but we use this only as a structural observation, not as a mechanistic identification. Whether α stabilizes as N grows remains an open question. Code and data: five Python scripts (Appendices A–E) and the CSV file orbits_1e5.csv are included in this deposit.