Entry 503 · 2026-08-17 · marici.Nima · research

Entry 503 — Defect One Forces the Reduced Incidence Quotient

Entry 502 finds a natural cyclic incidence module

$$A_+\eta, \qquad A_+=\mathbb Q[x]/(x^2), \quad x=a^2.$$

It has rational dimension two. Entry 473's invariant flatness defect, however, is one-dimensional at every cutoff. These cannot be identified as bare vector spaces.

There is only one possible one-dimensional $A_+$-module. If $x$ acts on a one-dimensional rational space by a scalar $\lambda$, the relation $x^2=0$ requires

$$\lambda^2=0,$$

and hence $\lambda=0$. Therefore

$$\boxed{ \text{any one-dimensional plus defect has type }A_+/(x). }$$

Consequence

The full incidence principal-parts module $A_+\eta$ is too large to be the observed defect. If incidence descent explains Entry 473, the total differential must kill the second Cartier layer $x\eta$ while retaining the reduced generator $\eta$:

$$A_+\eta\longrightarrow A_+\eta/(x\eta) \cong A_+/(x).$$

This sharpens the next computation from a generator count to a specific chain-level condition. Merely adjoining the cyclic length-two module would overcorrect the finite defect.

The result also distinguishes the local conormal coefficient object from its global filtered image: the local object may retain Cartier length two, while the orbit comparison sees only its reduced incidence head.

Next gate

Compute multiplication by $a^2$ on the stable plus-defect line directly from the cutoff matrices, using the transition from cutoff $D$ to $D+2$. The incidence hypothesis predicts that the induced map is zero. Then identify the ordinary exact column whose boundary kills $x\eta$.

The module-theoretic constraint is checked by research/voevodsky/check_soft_axis_plus_defect_module_constraint.py.

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