Q9 — Proof of the Lifting Hypothesis [H-lift]

This repository contains the source of the Q9 Cosmochrony paper Proof of the Lifting Hypothesis [H-lift].

The companion paper Q5b introduces Hypothesis [H-lift]: the operator $L_\Pi = -A\partial_k^2$ produced by Q5a as the Mosco limit of the admissibility forms $\mathcal{E}q$ is the image, under the Schrödinger representation $\pi_1$ at unit central character, of the kinetic sector of the sub-Laplacian $\Delta{\mathrm{Heis}}$ on $\mathrm{Heis}_3(\mathbb{R})$, restricted to the admissible subspace.

Status revision (version 1.1). Q5a version 3.0 withdraws the convergence framework consumed here (theorems T1–T3 under hypotheses [H1], [H2], [H-w], [H-E1], [C]). That framework is now consumed as a single frozen, unestablished hypothesis [H-F], and the result of this paper is the implication [H-F] ⟹ [H-lift]. In the reformulation of Q5b version 2.0, the residual conditionality is carried by the spatial limit hypothesis [H-L].

Core Result

This paper resolves Q5b Open Problem O1 by establishing [H-lift] under the admissibility-controlled spectral bound (from Q5a hypotheses [H1], [H2], [H-w], [H-E1], [C]). The Born–Infeld admissibility constraint forces the Mosco limit to select precisely the kinetic sector of $d\pi_1(\Delta_{\mathrm{Heis}})$.

The mechanism: the modulation generator $\rho_q(X)$ contributes a position-energy term that vanishes at rate $O(q^{-1})$ (interplay of the $O(q^{-2})$ phase-oscillation factor and the $O(q)$ discrete position spread controlled by the BI bound), while the shift generator $\rho_q(Y)$ contributes the surviving kinetic term $|\partial_k f|^2$.

As a consequence, Q5b Theorems 5.2 and 6.1 (effective metric and Lorentzian signature) become conditional only on [H-F], not on the independent hypothesis [H-lift]; since [H-F] is not established, these results remain conditional (carried by [H-L] in Q5b 2.0).

Keywords

Lifting hypothesis, Mosco limit, Schrödinger representation, sub-Laplacian, Born–Infeld admissibility, kinetic sector, Heisenberg group.

Repository Contents

q9/
├── tex/         # LaTeX sources (main + cosmochrony-bibliography.bib)
├── out/         # Compiled paper PDF (q9.pdf)
├── zenodo.json  # Zenodo deposition metadata
└── README.md

Links

• 🔗 DOI: 10.5281/zenodo.19880574
• 🌐 Website: https://cosmochrony.org/science/emergent-geometry/q9/

Citation

J. Beau, Proof of the Lifting Hypothesis [H-lift], Zenodo, 2026. DOI: 10.5281/zenodo.19880574.

Acknowledgements

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.