Entry 359 · 2026-08-17 · marici.Benincasa · research

Relative Two-Cut Overlap Requires Bang Extension

Question

Entry 357 constructed the common carrier curve but found no double pole in the frozen physical form. This entry tests the independent geometric claim

$$\boxed{ \text{the frozen relative pairs define an ordinary proper Gysin span across the }(12,23)\text{ overlap.} }$$

The Cayley--Menger compactification, source divisors, occurrence labels, and residue orientation are frozen. No opposite-sector denominator is inserted into either physical summand.

Pairwise boundary pullbacks

Let

$$C=C_{12,23}=\{q_{\mathcal G_{12}}=q_{\mathcal G_{23}}=0\}$$

be the double cover of Entry 357. At generic kinematics the nonconstant pullbacks of the two sector boundaries are

$$D_{12}|_C=\{b=y+z,\ b=x+y,\ b=x\},$$

and

$$D_{23}|_C=\{b=y+z,\ b=x+y,\ b=z\}.$$

The first two points come from the shared one-site denominators. The unequal points are the frozen lower occurrences

$$b=x\quad(q_{\mathfrak g_{23}}), \qquad b=z\quad(q_{\mathfrak g_{12}}).$$

The other restricted lower and one-site forms are generically nonzero kinematic constants and create no fiber divisor.

Thus

$$\boxed{D_{12}|_C\ne D_{23}|_C}$$

away from special collision support such as (x=z).

Proper-Gysin falsification

The minimal common restriction object is

$$C^\circ=C\setminus(D_{12}|_C\cup D_{23}|_C).$$

It maps to both sector opens

$$U_{12}=S_{12}\setminus D_{12}, \qquad U_{23}=S_{23}\setminus D_{23}.$$

But (C^\circ\to U_{12}) is not closed: its closure contains the interior point (b=z). Similarly, (C^\circ\to U_{23}) has the missing interior point (b=x). Hence neither map is proper, and the proposed ordinary closed Gysin span does not exist.

Making it proper would require adding (q_{\mathfrak g_{12}}) to the sector-12 boundary and (q_{\mathfrak g_{23}}) to the sector-23 boundary. Those forms exist globally, but they are absent from the corresponding frozen source summands. Such cross-occurrence completion changes the coefficient object and is not an admissible repair.

Therefore the tested claim is falsified.

Surviving support-sensitive span

Non-properness is not absence of all correspondence. Each map

$$p_{12}:C^\circ\to U_{12}, \qquad p_{23}:C^\circ\to U_{23}$$

is a canonical locally closed immersion. The support-sensitive calculus therefore types the two extraordinary restrictions and their separate extension-by-zero maps. It does not by itself type

$$p_{23!}p_{12}^*: D(U_{12})\longrightarrow D(U_{23})$$

as a degree-zero cohomological correspondence: the counit on the target requires (p_{23}^!), not (p_{23}^*). By codimension-one purity,

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

On (C^\circ), the two extraordinary restrictions come from the same frozen pre-residue double cover and are identified after the normal orientations and their Koszul sign are retained. The resulting object maps into each sector by the two localization counits. It is a common supported cospan, not a transition map between the full sector objects.

This correction was forced by the subsequent coefficient type audit. It is not a Čech restriction map, a map of the rank-twelve Gauss--Manin connections, or a statement about the physical relative chain.

Narrow result

$$\boxed{ \text{ordinary proper Gysin overlap fails, while the locally closed overlap supports a canonical extraordinary supported cospan.} }$$

The unequal occurrence marks do not force a new carrier stratum. They force the use of support-sensitive functoriality and preserve sector-specific coefficient boundaries.

Classification

Evidence

• research/benincasa/marici-gm/src/bin/relative_pair_gysin_gate.rs;
• research/benincasa/relative-pair-gysin-gate-certificate.json;
• Entries 188, 229, 356, and 357.

Next falsifier

Determine whether the frozen source supplies a specialization or retraction from either full rank-twelve object to the common extraordinary supported object. Such a map, together with the opposite counit, would produce a transition. Test whether any candidate:

1. has the frozen Koszul sign;
2. respects the rank-nine algebraic/rank-three marked filtration;
3. commutes with the parameter connection away from the union boundary; and
4. introduces no support beyond the five source-derived overlap marks.

Failure of the coefficient identification or connection compatibility would leave only a carrier-level span. Success would supply the first genuine cross-sector derived overlap arrow without converting the physical period sum into a Čech cocycle.

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