W1 — Weight Stabilisation of the Admissible Dirichlet Form

This repository contains the source of the W1 Cosmochrony paper Weight Stabilisation of the Admissible Dirichlet Form: Proof of Hypothesis [H-w] from Spectral Universality.

This paper closes open problem Q5a-O3 by proving Hypothesis [H-w].

The admissibility weights $a_q(s)$ that 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:

1. The uniform spectral universality theorem U1 gives $|a_q(s) - A_q| \le C q^{-1/2} A_q$, where $A_q = \sum_{n=1}^{n_}\sigma_(n)$ is the partial sum of the limit profile;
2. A separate lemma shows $A_q \nearrow A > 0$: the series $\sum_n \sigma_(n)$ converges (by the O-series condition $\delta^/2 > 1,ドル empirically $\delta_{\mathrm{pair}} \approx 9.5$–10ドル$) and is bounded below by the non-trivial first term $\sigma_*(1) > 0$.

$A$ is identified as a functional of $c_{\mathrm{BI}}$ and the Heisenberg BFS growth data (addressing Q5a-O5 at the structural level). After this paper, the proof of Q5a Theorem T3 (Mosco convergence) requires only [H1], [H-E1], and [C]; the Q5a-O2 Fourier analysis rules out [H-E1] at the $q^{-1}$ scale and supersedes [C] (zero-form Mosco limit), leaving the reformulated tightness question ([H2]) open.

Admissibility weights, Dirichlet form, spectral universality, Mosco convergence, Born–Infeld bound, Heisenberg BFS growth.

w1/
├── tex/ # LaTeX sources (main + cosmochrony-bibliography.bib)
├── out/ # Compiled paper PDF (w1.pdf)
├── zenodo.json # Zenodo deposition metadata
└── README.md

• 🔗 DOI: 10.5281/zenodo.19886319
• 🌐 Website: https://cosmochrony.org/science/emergent-geometry/w1/

J. Beau, Weight Stabilisation of the Admissible Dirichlet Form: Proof of Hypothesis [H-w] from Spectral Universality, Zenodo, 2026. DOI: 10.5281/zenodo.19886319.

Portions of the editorial refinement benefited from iterative interactions with large language models, used as analytical assistants. All claims and final formulations remain the sole responsibility of the author.