Overview
The admissible Dirichlet form on the Heisenberg graph is weighted: each edge carries a weight $a_q(s)$ encoding the local admissibility strength at BFS shell $s$ and prime $q$. Hypothesis [H-w] asserts that these weights converge to a positive limit: $a_q(s) \to A > 0$ as $q \to \infty$.
W1 proves [H-w] by tracing its origin to the fibre-invariance of admissible observables. The observables $O = \mathrm{Im}\,\Pi$ are invariant under the projection fibre by definition. This invariance, combined with the quantitative universality rate from U1, forces the weights to stabilise.
W1 proves [H-w]; the remaining open Q5a hypotheses reduce to three: [H1], [H-E1], and [C]. All three remain open — Q5a-O2 delimits what the three-coordinate pipeline measures without closing them, and H2 closes the separate hypothesis [H2].
Core contributions
- Proof of [H-w]: the admissibility weight hypothesis is established rigorously from the structural properties of the projection $\Pi$ and the universality theorem U1.
- Fibre-invariance as the structural source: the key observation is that admissible observables $O = \mathrm{Im}\,\Pi$ are by construction fibre-invariant. This invariance propagates to the Dirichlet form weights, preventing their collapse to zero or divergence.
- Quantitative bound from U1: the explicit rate $\varepsilon(q) = O(q^{-1/2})$ from U1 turns the qualitative fibre-invariance argument into a quantitative convergence statement with explicit error control.
- Reduction of open hypotheses: after W1, the Q5a hypothesis list is reduced from four to three open items: [H1], [H-E1], and [C]. This represents measurable progress in closing the admissibility framework.
The admissible Dirichlet form
The Dirichlet form encodes the diffusion structure on the Heisenberg graph compatible with spectral admissibility. Its weight function $a_q(s)$ measures the effective coupling strength at BFS distance $s$ for prime $q$.
If the weights failed to stabilise, two pathologies would arise:
- Collapse ($a_q(s) \to 0$): the Dirichlet form would degenerate, losing its diffusion properties and breaking the connection between the discrete Heisenberg graph and the continuous Carnot–Carathéodory geometry.
- Divergence ($a_q(s) \to \infty$): the form would fail to define a normalised operator, preventing the passage to a well-defined spectral limit.
[H-w] rules out both pathologies, ensuring that the Dirichlet form remains non-degenerate and bounded as $q \to \infty$.
Relation to the Cosmochrony programme
W1, together with U1, forms the analytical backbone of the metric emergence cluster. While U1 controls the convergence of fingerprint energies, W1 controls the convergence of the Dirichlet form weights. Together, they ensure that the discrete spectral data on Heisenberg graphs defines a well-behaved continuum limit.
The remaining open hypotheses [H1], [H-E1], and [C] concern different aspects of the admissibility framework: ellipticity, energy non-degeneracy, and compatibility. All three are open: Q5a-O2 rules out the $q^{-1}$ coercivity scaling and identifies the canonical filtration (Q5a's growing Fourier window, with zero-form Mosco limit), and H2 closes only [H2]. Their resolution would complete the rigorous foundation of the Q5a framework.
References
Jérôme Beau. Weight Stabilisation of the Admissible Dirichlet Form: Proof of Hypothesis [H-w] from Spectral Universality, 2026. doi:10.5281/zenodo.19886319