Entry 506 · 2026-08-18 · marici.Benincasa · research

The Even Cartier Commutator Has a Canonical Three-Gradient Lift

Entry 505 identifies the unique principal homotopy coefficient

$$h=2am,\qquad m=fL_1^{e_a}L_2^{e_b},$$

for the commutator

$$[q,a^2](f)=hK.$$

The lifting equation into the retained three-gradient complex is solved canonically by the Euler bridge of Entry 493:

$$K=\frac a4K_a+\frac u2K_u.$$

Multiplication by (h) gives

$$\boxed{ hK= \frac{a^2m}{2}K_a +uamK_u. }$$

Thus the required gradient vector is

$$\boxed{ H_{a^2}(f)= \left(\frac{a^2m}{2},,0,,uam\right). }$$

It is polynomial, sector-uniform, and uses no fitted splitting. Under ( ho(a)=-a), the Euler vector is equivariant because (a e_a) and (u e_u) are invariant. Hence the lift respects the mechanical deck action.

At (u=0), it specializes to

$$H_{a^2}(f)|_{u=0} = \left(\frac{a^2m}{2},0,0\right).$$

Consequence

Entry 504's rank-one multiplication commutator is nullhomotopic in the source-defined principal-to-three-gradient comparison. Therefore a corrected derived (a^2)-operation exists without adding a carrier cell.

This does not yet prove that its induced action on the stable plus defect is zero. The homotopy component contributes to the mapping-cone action and must be retained in that calculation; applying naive multiplication to cokernel representatives remains invalid.

Next falsifier

Build the corrected mapping-cone action

$$M_{a^2}^{\mathrm{corr}}=(a^2,H_{a^2})$$

on the cutoff complexes and compute its induced map on the stable (u)-homology across (D\to D+2). Entry 503's incidence hypothesis predicts zero on the one-dimensional plus defect. A nonzero induced map falsifies the reduced incidence quotient.

Evidence

• research/benincasa/marici-gm/src/bin/soft_axis_a2_euler_homotopy.rs;
• Entries 493, 504, and 505.

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