The Balance Factor: A Scale Law for the Anti-Persistence of the Neighbour-Factor Walk, and a Correction to the Program's Own Mechanism

The Iterated-Logarithm Law (DOI 10.5281/zenodo.21198955) closed the Purple Program by discovering that the walk S(x) = Σ_{p≤x} κ(p), κ(p) = P(p+1) − P(p−1), is anti-persistent: it carries less variance than its own quadratic variation, by a factor named the Balance Factor φ. Two questions were left open — does φ converge, and is it ½? — and one mechanism was proposed. This paper answers the questions and corrects the mechanism, in that order of importance. The correction: the attribution of the deficit to residue anti-clustering feeding through the Determination Law is quantitatively insufficient by roughly two orders of magnitude; all short-range channels together (twin sharing at −1.9% of the variance, all lags ≤ 10 at ≈ −9%) account for φ ≈ 0.9, not the measured 0.6–0.7. The program states this plainly, as it once corrected its own conjecture (b). The questions: a block-ladder measurement over all 455 million primes below 10^10 — fourteen cells spanning three decades of block length and four of scale — shows that φ is not a constant at all. It is a two-variable function, and the fourteen measurements collapse onto one line: 1 − φ(b,x) = C·ln b/ln x, with C = 0.493 ± 0.023 and 12.8% rms residual. The scale-dependence explains why every earlier estimate (0.82, 0.70, 0.64, 0.49) differed: they were different points on one law. Evaluated at the walk's own scale (b ~ π(x), where ln b/ln x → 1) the law gives φ_∞ = 1 − C = 0.507, so the answer to the second question is yes, to measurement precision, and only in that limit: C agrees with ½ to 0.3σ. The mechanism: an exact identity, verified symbolically, S(x) = Σ_{m even} P(m)(χ(m−1) − χ(m+1)) with balanced weights Σ c(m) = 0 — the walk is a sum of a heavy-tailed function against zero-sum coefficients, which forces anti-persistence whenever the P-values are positively correlated within the window, as shared giant factors make them; the Dickman-type variable ln b/ln x is exactly the parameter governing that sharing. All measurements are shuffle-calibrated (estimator bias 1 + 0.014/√n, negligible) and the engine reproduces every published checkpoint exactly.