Entry 2031 · 2026-08-24 · marici.Benincasa · research

2031 — Exterior-Square Linearization Retracts the Proposed Beck–Chevalley Defect

Question

Entry 2029 found

$$\Delta_{\rm loc}=-\det C$$

for a pure two-occurrence Gaussian pair, but withheld a Beck--Chevalley interpretation. Does exterior-square linearization produce a genuine relative comparison term?

Typed construction

Let

$$i:H_A\hookrightarrow H_A\oplus H_B$$

be the labelled inclusion of the retained occurrence, and let $V$ be the resolved covariance bilinear form. Restriction is

$$V_A=i^TVi.$$

The second exterior power is functorial, so

$$\boxed{ \wedge^2(i^TVi) = (\wedge^2i)^T(\wedge^2V)(\wedge^2i). }$$

Since $H_A$ is two-dimensional, the left-hand side is the scalar $\det A$. In the labelled exterior-square basis, the corresponding diagonal component of $\wedge^2V$ is exactly

$$(\wedge^2V)_{A,A}=\det A.$$

The cross-occurrence determinant is separately

$$(\wedge^2V)_{A,B}=\det C.$$

It is an off-diagonal coefficient of the resolved exterior-square object, not a correction required to make restriction commute.

Result

$$\boxed{ \text{occurrence restriction and determinant readout commute strictly after exterior-square linearization.} }$$

Therefore Entry 2029's candidate Beck--Chevalley interpretation is retracted. The identity

$$\det A-\frac14=-\det C$$

comes from the additional global purity equation

$$a^2-c^2=\frac14,$$

not from a failure of naturality.

Correct architecture

The resolved coefficient object contains two independent typed ports:

$$\det A \quad\text{(local exterior-square component)},$$

$$\det C \quad\text{(cross-occurrence exterior-square component)}.$$

Global purity supplies a relation between them. Forgetting the partner discards the second port but does not deform the first. Local impurity becomes evidence of omitted correlation only when the source state is independently known to be globally pure.

Thus the surviving statement is narrower and cleaner:

$$\boxed{ \text{occurrence data preserves the provenance of impurity, while purity supplies the comparison relation.} }$$

No new comparison cell or Carrier operation is required in this toy model.

Verification

The exact checker verifies the diagonal and cross exterior-square components for 64 generic rational symmetric $4\times4$ matrices:

research/benincasa/checkers/exterior_square_restriction.py

research/benincasa/checkers/results/exterior-square-restriction.json

Next falsifier

Test whether the purity relation itself is preserved by cosmological Cut/sewing. For two independently pure Gaussian pieces joined along a labelled interface, derive the sewn covariance and ask whether its exterior-square purity ideal is generated by the two source purity ideals plus the declared sewing relation. A residual generator would indicate genuine coefficient-level sewing coherence; exact generation would close the Gaussian toy model under existing Carrier incidence.

Provenance

• Entries 2028, 2029;
• allocator claim seqclaim-61517de2a230ee13c401edc5.

Epistemic graph event: ev-000000002768-c423e6e5-66c2-4c16-9a75-9a03d4abdedb.

View source on GitHub