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

Abstract

We close open problem Q5a-O3 of the Cosmochrony spectral admissibility programme by proving Hypothesis [H-w]: the admissibility weights $a_q(s)$, which enter the filtered Dirichlet form ${\mathcal{E}_q}$, converge to a positive constant $A > 0$ uniformly in the generator $s \in S_q$. The proof has two logically distinct steps. First, the uniform spectral universality theorem U1 gives $|a_q(s) - A_q| \leq C q^{-1/2} A_q$ where $A_q = \sum_{n=1}^{{n^{*}(q)}} {\sigma^{*}}(n)$ is the partial sum of the limit profile. Second, a separate lemma shows $A_q \nearrow A > 0$: the series $\sum_n {\sigma^{*}}(n)$ converges (by the O-series condition $\delta^*/2 > 1$, empirically $\delta_{\mathrm{pair}} \approx 9.5$–$10$) and is bounded below by the non-trivial first term ${\sigma^{*}}(1) > 0$. We identify $A$ as a functional of ${c_{\mathrm{BI}}}$ 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 ${\mathcal{E}_q}$) requires only three further hypotheses: [H1], [H-E1], and [C]. The Q5a-O2 Fourier analysis rules out [H-E1] at the $q^{-1}$ scale and supersedes [C] (the published form has zero-form Mosco limit); the reformulated tightness question ([H2]) is open.