Weight Stabilisation of the Admissible Dirichlet Form: Proof of Hypothesis [H-w] from Spectral Universality

We close open problem Q5a-O3 of the Cosmochrony spectral admissibility programme by proving Hypothesis [H-w]: the admissibility weights aq(s), which enter the filtered Dirichlet form Eq, converge to a positive constant $A > 0$ uniformly in the generator s ∈ Sq. The proof has two logically distinct steps. First, the uniform spectral universality theorem U1 gives |aq(s) - Aq| ≤ C q-1/2 Aq where Aq = ∑ₙ₌₁^n*(q) σ*(n) is the partial sum of the limit profile. Second, a separate lemma shows Aq A > 0: the series ∑ₙ σ*(n) converges (by the O-series condition δ^*/2 > 1, empirically δₚₐᵢᵣ ≈ 9.5–$10$) and is bounded below by the non-trivial first term σ*(1) > 0. We identify A as a functional of cBI and the Heisenberg BFS growth data, addressing open problem Q5a-O5 at the level of a structural argument. After this paper, the proof of Q5a Theorem T3 (Mosco convergence of Eq) requires only three further hypotheses: [H1] (moderate), [H-E1] (moderate), and [C] (hard).