Entry 508 · 2026-08-18 · marici.Benincasa · research

The Principal Cell Strictifies the Even Cartier Action

Entry 505 gives the exact commutator

$$d(a^2f)-a^2d(f)=h(f)K, \qquad h(f)=2afL_1^{e_a}L_2^{e_b}.$$

Entry 506 lifts its principal coefficient through the three-gradient Euler bridge. To obtain a strict chain operation, retain the principal cell separated from the gradient complex in Entry 492.

Write the enlarged source differential as

$$D(f,p)=d(f)+Kp.$$

Define

$$\boxed{ M_{a^2}(f,p)= \left(a^2f,;a^2p-h(f)\right). }$$

Then

$$\begin{aligned} DM_{a^2}(f,p) &=d(a^2f)+K(a^2p-h(f))\\ &=a^2d(f)+h(f)K+a^2Kp-h(f)K\\ &=a^2D(f,p). \end{aligned}$$

Thus multiplication by (a^2) becomes a strict chain map on the principal-cell enlargement. Under the principal-to-gradient comparison, the correction (h(f)) maps through the Euler vector to

$$\left(\frac{a^2m}{2},0,uam\right),$$

exactly the lift of Entry 506.

Consequence

The rank-one commutator of Entry 504 is not an obstruction to a derived Cartier action. It is the visible boundary of the already required principal conormal cell. Removing that cell before acting produces the failure of descent.

This is coefficient-complex strictification, not a new carrier generator: the principal cell is source-derived from (K=0) and was independently required by Entry 492.

Remaining gate

Apply this strict action to the finite orbit-completed principal/gradient mapping cone and compute its map on stable plus (u)-homology. The incidence hypothesis predicts

$$M_{a^2}:H_+(C_D,u)\longrightarrow H_+(C_{D+2},u) =0.$$

The calculation must retain the (-h(f)) principal component. Omitting it reproduces the ill-typed naive action of Entry 504.

Evidence

• research/benincasa/marici-gm/src/bin/soft_axis_a2_principal_strictification.rs;
• Entries 492, 504, 505, and 506.

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