Entry 360 · 2026-08-17 · marici.Benincasa · research

Purity Retypes the Two-Cut Overlap as a Supported Cospan

Question

Entry 359 found a canonical locally closed common overlap. Its initial formulation suggested that ordinary restriction followed by extension by zero might already give a transition between the two rank-twelve sector objects. The hard-to-vary claim tested here is

$$\boxed{ p_{23!}p_{12}^* \text{ canonically defines a degree-zero cross-sector coefficient arrow.} }$$

This is a type test before any cyclic basis identification or fitted projector.

Purity degree

The common open curve (C^\circ) has complex codimension one in each sector surface open. For

$$p_{ij}:C^\circ\longrightarrow U_{ij},$$

absolute purity gives

$$\boxed{ p_{ij}^!\mathcal L_{ij} \simeq p_{ij}^*\mathcal L_{ij}[-2](-1). }$$

The source-derived identity of the two ordinary Cayley--Menger/Kummer restrictions has bidegree ((0,0)). A cohomological correspondence feeding the target counit requires the extraordinary restriction of bidegree ((-2,-1)). Therefore the ordinary identity cannot silently be used as the required correspondence class.

The proposed degree-zero transition is falsified.

Corrected object

The frozen normal equations and their Koszul orientation identify the two extraordinary restrictions on the common curve:

$$\mathcal K_{12,23} \simeq p_{12}^!\mathcal L_{12} \simeq p_{23}^!\mathcal L_{23}.$$

Localization provides canonical counits

$$p_{12!}\mathcal K_{12,23}\longrightarrow\mathcal L_{12}, \qquad p_{23!}\mathcal K_{12,23}\longrightarrow\mathcal L_{23}.$$

These arrows have the common supported object as source. They form a cospan inside the two different sector categories after the appropriate direct images. They do not compose into a transition between the full sector objects because no canonical retraction

$$\mathcal L_{12}\longrightarrow p_{12!}\mathcal K_{12,23}$$

or its cyclic analogue is supplied by the localization triangle.

Consequence for the rank-twelve comparison

The obstruction occurs before the explicit (9+3) filtration is reached. Consequently no rank-twelve connection matrix should be compared across the overlap until a source-defined specialization/retraction is found. Cyclic relabeling remains an isomorphism between whole source sectors, but it is not an overlap restriction.

The surviving statement is

$$\boxed{ \text{the frozen two-Cut geometry canonically defines a common supported coefficient object with maps into both sectors, not a descent transition.} }$$

Classification

Correction to Entry 359

Entry 359 has been amended: its initially displayed (p_{23!}p_{12}^*) transition was not type-correct as a degree-zero cohomological correspondence. The valid conclusion of that entry is the locally closed extraordinary cospan described above.

Evidence

• research/benincasa/marici-gm/src/bin/overlap_coefficient_type_gate.rs;
• research/benincasa/overlap-coefficient-type-gate-certificate.json;
• Entries 357 and 359.

Next falsifier

Search the frozen deformation-to-the-normal-cone/nearby-cycle geometry of the pair (q_{\mathcal G_{12}}=q_{\mathcal G_{23}}=0) for a canonical specialization

$$\mathcal L_{12}\longrightarrow p_{12!}\mathcal K_{12,23}$$

or the corresponding map from sector 23. Freeze the normal coordinates, positive sheet, occurrence boundaries, and support before calculation.

If nearby cycles or a source-defined boundary map provides the retraction with the required purity shift, composing it with the opposite counit gives the first genuine cross-sector coefficient arrow. If not, the common object is only supported incidence data and the global sector assembly remains a cyclic equivariant sum rather than descent.

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