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

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.

Before W1, the Q5a framework had several open hypotheses. After W1 proves [H-w], the remaining open hypotheses reduce to three: [H1], [H-E1], and [C]. The latter two are subsequently addressed by Q5a-O2 and H2.

Status. Proved.

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 observables. The stabilisation of weights at $A > 0$ means that the effective Dirichlet form on the emergent geometry has a well-defined, non-trivial diffusion coefficient — a prerequisite for the continuous spectral theory to make contact with the geometric quantities derived in Q10 and Q11.

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. Their resolution (Q5a-O2, H2) completes 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