Proof of the Lifting Hypothesis [H-lift]: Kinetic-Sector Identification via Generator Suppression

Abstract

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 Schr\"odinger representation $\pi_1$ at unit central character, of the kinetic sector of the sub-Laplacian ${\Delta_H}$ on ${\operatorname{Heis}}_3(\mathbb{R})$, restricted to the admissible subspace of $L^2(\mathbb{R})$. This paper resolves Q5b Open Problem O1 by establishing [H-lift] under the admissibility-controlled spectral bound: working from the convergence framework of Q5a versions up to 2.1 (theorems T1–T3 under the hypotheses [H1], [H2], [H-w], [H-E1], [C]; consumed here as the frozen framework hypothesis [H-F]), we show that the Born Infeld admissibility constraint forces the Mosco limit to select precisely the kinetic sector of $d\pi_1({\Delta_H})$. The proof identifies the core mechanism: the modulation generator $\rho_q(X)$ contributes a position-energy term to ${\mathcal{E}_q}$ that vanishes at rate $O(q^{-1})$, due to the interplay between the $O(q^{-2})$ phase-oscillation factor and the $O(q)$ discrete position spread controlled by the Born Infeld admissibility bound, while the shift generator $\rho_q(Y)$ contributes the kinetic term $\|\partial_k f\|^2$ that survives in the Mosco limit. As a consequence, Q5b Theorems 5.2 and 6.1 (effective metric and Lorentzian signature) become theorems conditional only on [H-F], rather than on the additional independent hypothesis [H-lift]. Status revision (version 1.1). Q5a version 3.0 withdraws the framework consumed here: the canonical filtration is exactly a growing toric Fourier window, the published admissibility form converges to the zero form, and no common scalar normalisation produces a non-trivial toric differential operator. Hypothesis [H-F] is therefore not established, and its Mosco component fails for the published normalisation. The derivation of this paper is unchanged but must be read as the implication [H-F] $\Rightarrow$ [H-lift]; in the reformulation of Q5b version 2.0, the residual conditionality is carried by the spatial limit hypothesis [H-L].