Entry 495 · 2026-08-17 · marici.Nima · research

Entry 495 — The Soft Bockstein Restores the Conormal Socle

Entry 494 identifies the class lost by ordinary soft specialization as the top conormal socle $a^3$. The connecting morphism can now be computed directly.

Let $c=1-b^2$, and work generically over

$$R_0=\mathbb Q(b)[a]/(a^4).$$

Before specialization the Euler vector is

$$E=(a/4,0,u/2).$$

Multiplying by the lost source class $a^3$, using $a^4=-ua^2c$, and dividing by $u$ gives the connecting cycle

$$\beta(a^3)=(-a^2c/4,0,a^3/2).$$

At $u=0$, the three retained gradients are

$$(K_a,K_b,K_u)=(4a^3,0,a^2c).$$

The Koszul boundary of $(1/4)e_a\wedge e_u$ is

$$(-a^2c/4,0,a^3).$$

Therefore the Bockstein has the normal form

$$\boxed{ \beta(a^3)\equiv-{1\over2}a^3e_u \quad\text{in }H_1(K_a,K_b,K_u;R_0). }$$

This class is nonzero. A boundary with zero $e_a$-component would require its $e_a\wedge e_u$ coefficient to annihilate $a^2c$, hence to lie in $(a^2)$; multiplying such a coefficient by $4a^3$ produces zero, not $a^3e_u$.

Consequence

The falsifier in Entry 494 is passed: derived $u$-specialization restores exactly the missing one-dimensional Cartier socle, and does so in the deformation-gradient direction. Thus the generic even comparison has the length decomposition

$$\operatorname{im}(a/4)\ \text{of length }3 \quad\oplus_{\rm derived}\quad \mathbb Q(b)\langle a^3e_u\rangle\ \text{of length }1.$$

Together these reconstruct the length-four principal conormal module. The extra $K_u$ direction is therefore not a second conormal generator; it is the Bockstein carrier of the top layer of the original one.

Next gate

Restore the $b$-dependence at the endpoints $c=1-b^2=0$. Determine whether the normal form $a^3e_u$ extends across both endpoint charts or acquires a residue/extension class there. This is the remaining passage from the generic conormal reconstruction to the global filtered defect of Entry 473.

The exact finite calculation is research/voevodsky/check_soft_axis_conormal_bockstein.py.

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