This note studies the arithmetic walk built from κ (n) = P (n + 1) − P (n − 1), where P is the largest prime factor, and S (x) = Σ ≤ ₗ κ (p) over primes. It asks whether the empirically observed law |S (x) | ~ x^ (3/2) /ln x can be made a theorem, and separates the question into two parts. The scale — the second-moment law Σ ≤ ₗ κ (p) ² ≍ x³/ (ln x) ² — appears reducible to the established machinery for the largest prime factor of shifted primes (Hooley–Motohashi; Bombieri–Vinogradov; Brun–Titchmarsh) ; the reduction is given and confirmed numerically to 2·10⁷, with the constant c ≈ 0. 143 reported as measured, not derived. The excursion — that the partial sums attain this scale — reduces to a single decorrelation estimate plus a maximal inequality, which remains open. A status table separates what is elementary, what appears reducible to known methods, and what is open. No theorem is claimed; the contribution is to define the object precisely and isolate exactly what remains to be proved. Part II (added in v2) is a separate, descriptive companion on a different question — how well a prime is identified by its two neighbours. Read together, the positional Deviation ID and the factor Dig ID give two independent sign-bits per prime and a unique pair for about 74–82% of primes (slowly fading with scale) ; completed with their two magnitude companions (the gap-sum and factor-sum, i. e. the full local data of both gaps and both neighbours' largest prime factors), the reading is unique for ~99. 8% of primes and does not fade through 10⁷, the remaining fraction being "environment twins. " Part II is descriptive, established by exact counting, and does not bear on the open problem of Part I. Part III (added in v3) returns to that open problem and narrows it. The off-diagonal correlation is split into a structured part (shifts shared between nearby primes, chiefly twins) and a generic part. The structured part is argued negligible — O (D/ln x) = o (D), by standard prime-pair counting, confirmed numerically (the lag-1 correlation shrinks like 1/ln x). The generic part is shown to be invisible by a surrogate (independent-increment) test: the real walk's excursion sits at the 42nd percentile of the decorrelated null. What remains is stated as a precise conditional theorem — given (a) a generic-decorrelation estimate and (b) a maximal inequality, the excursion law follows. Part III proves no theorem; it converts a vague open problem into a specific one, isolating exactly where the difficulty lives.
Samir Hanna Safar (Wed,) studied this question.