Editing Openai/693e3ce6-229c-8008-97dc-ab720cb1f95a (section)

==== Two different “limits” get conflated: ====

===== - You run Nε<∞N_\varepsilon<\inftyNε<∞ iterations. =====
* Work is at most nNε<∞nN_\varepsilon<\inftynNε<∞.
* Nothing is infinite.

===== - Nε→∞N_\varepsilon\to\inftyNε→∞ for any method. =====
* If γ>0\gamma>0γ>0, we proved that the spurious support contribution stays bounded even as ε→0\varepsilon\to 0ε→0, because identification happens in finite time.
* If γ=0\gamma=0γ=0, you can have no finite identification time, so spurious activations can persist for arbitrarily many iterations; the spurious contribution can grow with NεN_\varepsilonNε. In that sense, a bound that tries to be uniform “independent of ε\varepsilonε” will necessarily break.

That is exactly what the 1γ2α\frac{1}{\gamma^2\sqrt{\alpha}}γ2α1 term is signaling.