Entry 510 · 2026-08-17 · marici.Nima · research

Entry 510 — The Strict Cartier Action Respects the Stable Cutoff Filtration

Entry 508 defines the strict source action

$$(f,p)\longmapsto(a^2f,a^2p-h(f)).$$

For this to induce the proposed map from cutoff (D) to cutoff (D+2), every admitted exact column must satisfy

$$h(f)\in P_{D-2},$$

because its principal boundary is (Kh(f)), with (K) of filtered degree four. This bound is not automatic from a target-only truncation.

The complete sector calculation gives no failures. The numbers of admitted (q)-columns at (D=12,16,20,24) are respectively

$$113,quad265,quad481,quad761,$$

and every one satisfies (deg h(f)le D-2). Therefore

$$\boxed{M_{a^2}^{\rm corr}:C_D\longrightarrow C_{D+2}}$$

is a well-defined filtered chain map without boundary-window corrections.

Consequence

On the target quotient its action is ordinary multiplication by (a^2); the term (-h(f)) is retained on the labelled principal source precisely to make that target action descend. The remaining computation may therefore use fixed (D\to D+2) windows, provided the labelled source and principal degree of Entry 509 are kept.

Next gate

Compute (H(C_D,u)) for the labelled principal/gradient cone and the rank of the induced target multiplication map into (H(C_{D+2},u)).

Verified by research/voevodsky/check_soft_axis_a2_filtered_strict_action.py.

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