Overview
Q9 gives a sufficient condition for a nonnegative modulation term to leave a kinetic Mosco limit unchanged: kinetic convergence in a specified Hilbert-space comparison [K], and one common kinetic recovery sequence [R] on which the modulation energy vanishes for each finite-energy limit vector.
The resulting form is $A\|f'\|^2$. It is the free kinetic part on every supplied Schrödinger fibre with nonzero central character. It is not the full oscillator form and selects no such character.
What the theorem establishes
- A sufficient Mosco criterion: positivity gives the weak lower bound; the same recovery sequence gives the strong upper bound for both terms. Convergence of their sum does not imply convergence of each summand.
- A scale-aware modulation bound: $\mathcal V_q(f_q)\leq8\pi^2 r_q a_q(X)\|k f_q\|_{\mathcal C_q}^2/q^2$. Suppression requires its right-hand side to vanish on a kinetic recovery sequence.
- An elementwise estimate: for a vector with $\|xf\|^2\leq M\|f\|^2$, one has $A\|f'\|^2\geq A\|f\|_{H^1}^2/(1+4M)$. The moment-bounded class is a cone, not a derived linear form domain.
Geometric scope
Identifying a fibre, an admissible linear domain and a spatial lift requires further constructions. Q5b's [H-L] and [H-lift] remain separate inputs, while [H-hyp] belongs to its signature statement. Horizontal kinetic positivity provides no central coefficient $A_Z$ or bridge non-obstruction.
Q7 version 2.0 distinguishes Q5b's rank-two spatial principal symbol from a proposed positive rank-three target. Q8, Q10 and Q11 still cite results that Q7 and Q9 have retyped; their coefficient and metric readings await dedicated correction.
Open construction tasks
A concrete route must specify the discrete spaces, isometric embeddings, mesh and energy scale; prove [K]; and construct common recovery sequences satisfying [R]. A geometric reading also needs an identification between the supplied carrier and the measured spatial object.
Reference
Jérôme Beau. Toward a Kinetic Limit on a Supplied Schrödinger Fibre: Conditions for Generator Suppression, version 2.0, 2026. doi:10.5281/zenodo.19880574 (version record).