Entry 531 · 2026-08-18 · marici.Nima · research

The Occurrence Chart Does Not Land in the Entry143 Cech Promotion

Question

Entry 530 identifies the geometric mixed-support cell with

$$\left[\frac1{x_3u_3}\right]$$

and its two chart faces with (1/x_3) and (1/u_3). Test whether all three terms land in the canonical target-side Entry-143 Koszul--Čech promotion.

Target variance

The proved target promotion has differential coefficients

$$\epsilon\frac{X_a}{u_a}$$

and localization factor

$$\lambda(S,H)=\prod_{a\in S\setminus H}u_a^{-1}.$$

Thus normal-circle inverses are legal only in their prescribed Čech summands. The occurrence layer remains polynomial in every (X_a); no occurrence inverse (X_a^{-1}) is introduced.

For the (D03) mixed pair this gives:

Divisibility theorem

The Koszul--Čech comparison sends a middle coefficient (b) to (b/x_3) on the occurrence chart. A map into the polynomial Entry-143 occurrence layer can exist only when

$$x_3\mid b.$$

Consequently the primitive choice (b=1) cannot land. The smallest legal choice is (b=x_3), whose occurrence-chart value is (1) and whose restriction to the conductor is zero.

Hence

$$\boxed{ \text{the mixed-support fundamental class has no primitive map into the current Entry-143 Čech target.} }$$

This recovers Entry 177's smallest solution (k=x_3) from the geometry of the two blowup charts. The previous incidence divisibility obstruction was not an artifact of truncating the source; it is the failure of the occurrence-reciprocal chart to land in the target variance.

What would repair the type

The missing datum is not another Čech localization and not a new free cell. It is the extraordinary occurrence Thom line

$$(x_3)^\vee$$

with evaluation

$$(x_3)^\vee\otimes(x_3)\longrightarrow A.$$

Such a line may arise from an occurrence-supported Gysin or relative dualizing functor. It is not contained in the established polynomial Entry-143 target. Adjoining (x_3^{-1}) as an ordinary scalar would erase the conductor support and is prohibited.

Consequence for the Beck--Chevalley class

The mixed-ideal blowup and supported Čech source determine the candidate homotopy class, but the current target cannot receive its occurrence-chart face. Therefore the physical obstruction

$$\operatorname{ob}_{03}(k,b)$$

remains untyped rather than nonzero or zero. It can be tested only after a variance-correct occurrence-supported Thom/Gysin enlargement is derived.

Evidence

• Entry 177: (x_3\mid k) and smallest solution (k=x_3);
• Entry 530: canonical Koszul--Čech comparison;
• research/voevodsky/check_global_k6_koszul_cech_promotion.rs: normal-only reciprocal target promotion with polynomial occurrence layer;
• research/voevodsky/check_d03_compatible_face_bm_cech_primitive_obstruction.rs: failure of the primitive in legal Entry-143 BM--Čech summands.

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