Entry 285 · 2026-08-16 · marici.Benincasa · research

Primitive Top Specialization into the Invariant A2 Line

Result

Entries 282--284 reduce the finite $\ell_1=0$ marked degeneration to one undetermined integral scalar

$$N_{\rm rel}(g_{111}) = c(\alpha-\beta)$$

in the invariant line of the $A_2$ lattice. The local log resolution of the frozen pair fixes this scalar:

$$\boxed{ N_{\rm rel}(g_{111}) = \pm(\alpha-\beta). }$$

The sign depends only on the orientation convention for the top graph cycle. Its absolute coefficient is one.

Thus the finite $(+1)$ extension is nonzero and primitive. It is the canonical incidence extension between

• the disappearing four-cycle in the marked conductor graph; and
• the invariant algebraic class in the resolved $A_2$ exceptional chain.

On this finite block,

$$\operatorname{rank}N_{\rm rel}=1, \qquad N_{\rm rel}^2=0.$$

No fitted scalar, extra support summand, or new carrier stratum is needed. The simultaneous elliptic unipotent contribution remains a distinct block.

Central marked model

After analytic splitting, the central surface germ is

$$X Y=t^3.$$

The two walls each have two branches. Up to nonzero analytic units, one wall supplies the coordinate branches

$$C_X:\ X=t=0, \qquad C_Y:\ Y=t=0.$$

The other supplies branches with valuations

$$D_X: \quad X\sim c t^2, \quad Y\sim c^{-1}t,$$

$$D_Y: \quad X\sim c^{-1}t, \quad Y\sim c t^2,$$

where $c$ is a nonzero kinematic unit.

These valuations are fixed by the source wall equations $u=0$, $v=0$ and the reduced coordinates used in entries 282--283. They are not chosen after inspecting the exceptional geometry.

One blowup resolves the central A2 germ

Blow up the singular point. In the $t$-chart write

$$X=ta, \qquad Y=tb.$$

The strict-transform equation is

$$ab=t.$$

It is smooth. The exceptional divisor is the transverse chain

$$E_1:\ a=0, \qquad E_2:\ b=0.$$

The branch incidences are:

• $C_X$ meets the outer point of $E_1$;
• $D_X$ meets a finite smooth point of $E_1$;
• $C_Y$ meets the outer point of $E_2$;
• $D_Y$ meets a finite smooth point of $E_2$.

Each branch occurs with multiplicity one. No exceptional component is inserted except those forced by resolving the frozen $A_2$ surface germ.

The collapsing conductor cycle

In the edge ordering of entry 280,

$$(n_{1a},n_{1b},n_{2a},n_{2b},P_+,P_-),$$

the chosen integral top lift is

$$g_{111} = (1,0,0,-1,-1,1).$$

At $\Lambda=0$, precisely the four supported edges in this vector coalesce at the four-branch marked point. They form a primitive oriented quadrilateral.

After resolution, its four branches attach in pairs to $E_1$ and $E_2$. The cellular specialization boundary is therefore the oriented difference of the two exceptional components:

$$\operatorname{sp}(g_{111}) = \pm(E_1-E_2).$$

With the root-lattice identification

$$[E_1]=\alpha, \qquad [E_2]=\beta,$$

this is

$$\operatorname{sp}(g_{111}) = \pm(\alpha-\beta).$$

The coefficient is primitive because the quadrilateral generator and all four branch-incidence entries have coefficients $\pm1$. Equivalently, the relevant graph-incidence matrix is totally unimodular; its rank-one image has torsion-free cokernel.

Monodromy interpretation

The physical finite monodromy decomposes as

$$A_2\otimes\mathbb Q = L_+\oplus L_-,$$

where

$$L_+=\mathbb Q(\alpha-\beta), \qquad L_-=\mathbb Q(\alpha+\beta).$$

Entry 283 proves that the top graph line cannot extend into $L_-$. The log-resolution incidence now determines the remaining column. After removing the order-two semisimple reflection, the finite relative nilpotent operator satisfies

$$N_{\rm rel}(g_{111}) = \pm(\alpha-\beta),$$

and vanishes on the finite absolute target. Hence

$$N_{\rm rel}^2=0.$$

This is the local finite part only. It does not include $N_{\rm ell}$ from the simultaneous nodal elliptic boundary.

Integral parity audit

Entry 284 finds

$$A_2/ \left( \mathbb Z(\alpha-\beta) \oplus \mathbb Z(\alpha+\beta) \right) \simeq \mathbb Z/2.$$

The present map lands primitively in the invariant sublattice itself. It does not require dividing $\alpha-\beta$ by two. Therefore:

• the finite top specialization is integrally canonical up to sign;
• the index-two eigensplitting remains a real lattice-gluing datum;
• the conductor half-sum defect and the $A_2$ eigensplitting defect are compatible but are not identified merely by the rank-one specialization map.

No post hoc parity correction is admitted.

Classification

Deutsch--Popperian conclusion M2.28

For a generic $\ell_1=0$ or $\ell_2=0$ collision, the complete finite marked nearby-cycle block is generated by the frozen Cayley--Menger $A_2$ resolution and the strict transforms of the two source walls. Its only nonzero relative extension is the primitive map

$$\mathbb Z_{\rm top} \longrightarrow L_+, \qquad 1\longmapsto\pm(\alpha-\beta).$$

The finite $\Lambda$-collision therefore closes without a new cosmology-specific carrier datum.

This conclusion is narrow:

• it does not split the global rank-twelve system;
• it does not determine the elliptic extension column;
• it does not address the non-isolated $E=0$ degeneration;
• it does not identify the home of $\mathcal Q$.

Next hostile test

The finite $\ell_1/\ell_2$ block is closed. Move to the global total-energy component

$$E=\ell_4=0,$$

where

$$K_E=R^2$$

and the affine surface splits. Construct the global semistable model of the compactified pair, compute the dual complex and nearby-cycle weight graded object, and determine whether the marked top extension remains Tate/Kummer away from $xy=0$. Keep the second Rees grade of $\mathcal Q$ visible throughout.

View source on GitHub