The two deep analytic problems of the Purple Program (DOI 10. 5281/zenodo. 21168196) concerned constants measured but not understood: the determination constant CD ≈ 1. 07 of the Balance Hypothesis's staircase law, and the envelope constant C_κ ≈ 0. 372 of its excursion law. This paper derives closed forms for both from standard heuristics, and verifies each against measurement — including a four-way falsification test that the first formula passes at the sub-percent level. Problem 1: the determination constant is CD (Q) = BQ · ∫₀^∞ ρ (u) ² (1+u) ^ (−1) du, the Dickman ratio density at 1 boosted by an explicit prime-power regression factor BQ depending on the coil set Q; for Q = 3, 5, 7, 11, 13 this gives 1. 0793 against the measured limit 1. 069 ± 0. 010, and the formula's distinctive prediction — different constants for different coil sets (1. 323, 1. 209, 1. 149, 1. 079) — is confirmed over 50 million primes with ratio deviations of 0. 5–0. 9%. Problem 2, part (a): the envelope constant is C_κ = √ (2A/3) with A = 2C₂ · (1/7) · Πₐ>₂ (1 + (q−1) / ( (q−2) (q³−1) ) ), a twin-constant Euler product arising from the Hardy–Littlewood count of the giant-factor events that dominate the walk's variance; numerically C_κ = 0. 37098 against the measured 0. 3723 ± 0. 0010 — agreement to 0. 36%, with the cross-correlation of the two neighbours shown to be one logarithm smaller, explaining the two-term law observed since Version 2. The derivations are heuristic in the standard sense — Dickman equidistribution and Hardy–Littlewood singular series, both unproven at this depth — and the paper says so plainly; parts (b) and (c) of the excursion law (eternal attainment, no escape) remain open. With these resolutions, four of the program's five problems are settled at the level the program set for itself, and every measured constant of Purple Mathematics now has a name. Note: Sixth paper of the Purple Mathematics series; resolves Problems 1 and 2 (a) of the research program stated in Purple Numbers: Foundations, at the program's stated standard: heuristic closed-form identification verified by prediction (four-way coil-set test; second-order structure). Classical inputs (Dickman 1930; Alladi–Erdős 1977; de Koninck–Ivić; Hardy–Littlewood 1923; Wang 2021; Lemke Oliver–Soundararajan 2016) are attributed in the text. The closed forms, the multiplicity-boost factor, the four-way test, the cross-term suppression argument, and the numerical constants are original to this paper. Parts 2 (b) and 2 (c) of the excursion law remain open.
Samir Hanna Safar (Sat,) studied this question.