Entry 501 · 2026-08-17 · marici.Nima · research

Entry 501 — The Even Conormal Block Has No Soft Bockstein

Entry 500 retypes the $a^3$ conormal socle and its $u$-Bockstein as anti-invariant. The invariant block of

$$I/I^2\cong R_0=\mathbb Q(b)[a]/(a^4)$$

is instead

$$(I/I^2)_+=\mathbb Q(b)\langle1,a^2\rangle.$$

Under ordinary $u=0$ specialization, the Euler comparison is multiplication by $a/4$ into the $e_a$-component. Hence

$$1\longmapsto {a\over4}e_a, \qquad a^2\longmapsto {a^3\over4}e_a.$$

These two images are independent in $R_0e_a$. Therefore

$$\boxed{ \text{the even conormal block specializes injectively and has no } u\text{-Bockstein.} }$$

The soft Bockstein mechanism of Entries 494--496 is exclusively the odd top-socle phenomenon.

Incidence lift

For the smallest $(1,1)$ $q$-sector, Entry 487 gives at $u=0$

$$m=f(b+1)a, \qquad H_q=-{3\over2}me_a.$$

Lifting the two even Euler images requires

$$f_0=-{1\over6(b+1)}, \qquad f_2=-{a^2\over6(b+1)}.$$

Both are regular away from the incidence endpoint $b=-1$, and both have the same simple incidence denominator. Orbit conjugation does not remove it because $L_1=b+1-u$ is deck invariant.

Consequence

The plus defect is not created by soft $u$-specialization. Its only remaining location in the conormal comparison is incidence descent across $L_1=0$. Although the even block has two vector-space generators, it is one cyclic module over the even Cartier algebra, generated by $1$ with $a^2$ as its second layer. A single module-level descent obstruction can therefore be compatible with Entry 473's constant defect one.

This is not yet the identification: a filtered calculation must determine whether the two apparent poles define one cyclic obstruction or two independent endpoint classes.

Next gate

Compute the incidence local-cohomology boundary of the cyclic pair $(f_0,f_2)$ at $b=-1$, retaining multiplication by $a^2$. Test whether its span over the filtered even Cartier algebra has one generator and whether adding that derived incidence module restores the plus flatness defect.

The specialization and lift audit is research/voevodsky/check_soft_axis_even_conormal_specialization.py.

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