A Signed Walk of Largest Prime Factors over the Primes: Conditional Laws and Numerics

For a prime p, let P (m) denote the largest prime factor of m and set κ (p) = P (p+1) − P (p−1) and S (x) = Σ₏≤ₗ κ (p). The analogous sum over all integers telescopes and is trivial; restricting the index to the primes removes the telescoping and makes S a genuine signed walk, whose growth is the subject of this note. We show that the walk's second moment satisfies D (x) = Σ₏≤ₗ κ (p) ² ≍ x³/ (log x) ², with the upper bound accessible by standard sieve methods (Brun–Titchmarsh) and the matching lower bound conditional on a Hardy–Littlewood prime-pair estimate; numerically, D (x) / (x³/log²x) is flat at ≈ 0. 143 up to 2×10⁶. Granting a decorrelation estimate for the increments together with a routine maximal inequality, the walk obeys the excursion law |S (x) | = x^3/2+o (1) /log x. We identify this decorrelation estimate, in substance, with the Erdős–Pomerance conjecture that the largest prime factors of neighbouring integers are independent — known to follow from the Elliott–Halberstam conjecture for friable integers — and the sign behaviour of κ with the Erdős–Turán conjecture on P (n) versus P (n+1). A surrogate test to 2×10⁶ places the real walk as an unremarkable member of the decorrelated null, with no dominant increment. The note proves no hard theorem unconditionally; its purpose is to define the object precisely, prove what is accessible, isolate where the difficulty lives, and place it correctly among these classical conjectures.