Entry 507 · 2026-08-17 · marici.Nima · research

Entry 507 — Symbolic Audit of the Three-Gradient Cartier Lift

Benincasa Entry 506 solves the lifting equation for the failure of naive multiplication by (a^2). This entry independently audits that solution in the exact polynomial model used for the deck-completed sector operators.

Entry 505 isolates the commutator as

$$[q,a^2](f)=h(f)K, \qquad h(f)=2afL_1^{e_a}L_2^{e_b},$$

while Entry 504 requires a source-derived homotopy in the retained gradient/Kodaira--Spencer complex. The principal factorization alone is not that homotopy, as emphasized by Entry 492.

Entry 493 supplies the missing typed comparison. For

$$K=a^4+ua^2(1-b^2), \qquad u^2=0,$$

the three-gradient Euler identity is

$$K={a\over4}K_a+{u\over2}K_u.$$

Multiplying this existing chain certificate by (h(f)) gives

$$\boxed{ [q,a^2](f) =K_aH_a(f)+K_uH_u(f), }$$

where

$$H_a(f)={a^2\over2}fL_1^{e_a}L_2^{e_b}, \qquad H_b(f)=0, \qquad H_u(f)=uafL_1^{e_a}L_2^{e_b}.$$

All coefficients are polynomial and the formula holds in every sector and both deck lattices. Thus the rank-one commutator of Entry 504 is genuinely nullhomotopic in the retained three-gradient complex. The deformation direction is essential: deleting (K_u) recovers Entry 493's singular frozen-relative certificate.

Consequence

The complete orbit cokernel still has no naive (A_+)-module structure, but its derived three-gradient enhancement does carry multiplication by (a^2) as a chain operation up to the canonical Euler homotopy above. No new carrier cell and no fitted syzygy have been introduced.

This confirms Entry 506 symbolically rather than by scalar sampling and resolves Entry 504's first gate, but not Entry 503's reduced-incidence prediction. The homotopy component can act nontrivially on the stable defect homology even though its target boundary is exact.

Next gate

Insert ((H_a,0,H_u)) into the actual finite-cutoff mapping cone and compute the induced (a^2)-map on the stable plus defect. The reduced-incidence hypothesis predicts a one-dimensional defect on which this corrected map is zero.

The all-sector calculation is checked by research/voevodsky/check_soft_axis_a2_gradient_lift.py.

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