This paper closes the last open problem of the Purple Program (DOI 10. 5281/zenodo. 21168196): parts (b) and (c) of the Excursion Law — that the walk S (x) = Σ≤ₗ κ (p) never escapes the order of its envelope and attains it forever. Three results. First, an elementary lemma shows the walk's steps, though as large as p/2, are individually negligible at the envelope scale, so the Lindeberg condition holds: the walk lives in the Gaussian, not the stable, universality class — settling a question left open since the first paper of the series. Second, the conditional theorem: granting the variance asymptotic and weak dependence with a limiting variance ratio φ = lim Var S (x) /V (x), the walk obeys the central limit theorem and Kolmogorov's law of the iterated logarithm, limsup ±S (x) /√ (2φ V (x) ln ln V (x) ) = 1, which corrects the original conjecture (b): the true escape order carries a factor √ (ln ln x) absent from the 2026 formulation — theory amending the program's own conjecture. Third, the discovery the tests forced: the walk is anti-persistent. Its increments are Gaussian in shape but carry only half the variance of the quadratic variation (φ ≈ 0. 49 at the largest measured scale), the deficit built from negative correlations at every lag and level — steps, signs, and determination weights, the last inheriting the Lemke Oliver–Soundararajan anti-clustering of residues. The framework's founding image, "the crowd pushes back, " is now a measured constant. With φ = 0. 49 the corrected iterated-logarithm ceiling at 10¹0 is 1. 98 envelope units; the measured all-time maximum is 1. 905 — the walk stands at 96% of its predicted ceiling, attaining on schedule. The Balance Factor φ — does it converge, and is it 1/2? — is posed as the program's successor problem. Note: Seventh and closing paper of the Purple Mathematics research program. With this deposit, all five problems posed in Purple Numbers: Foundations are settled at the program's stated standard. Classical inputs (Kolmogorov 1929; Hartman–Wintner 1941; Berkes–Philipp 1979; Lemke Oliver–Soundararajan 2016) are attributed in the text. The Lindeberg lemma, the corrected excursion law with explicit constant, the four-level measurement of anti-persistence, the Balance Factor φ, and the 96% attainment verification are original to this paper. The theorem is conditional and says so; the discovery is unconditional measurement
Samir Hanna Safar (Sat,) studied this question.