Entry 287 · 2026-08-16 · marici.Benincasa · research

Physical Enhanced-Point Rees Audit and Exclusion of Q

Result

Entry 286 localizes every failure of generic semistability at total energy to four source-generated points. At the physical point

$$P_{--}=[-y:-x:1],$$

use the moving marked coordinates

$$a=E-y+u, \qquad b=E-x+v.$$

The two frozen walls are exactly

$$u=0, \qquad v=0.$$

Let

$$R_0 = xy(x+y)-xa^2-yb^2$$

be the central-sheet polynomial and define

$$D(E,u,v) = K_E(a,b)-R_0(a,b)^2.$$

Exact expansion gives

$$\boxed{ \operatorname{gr}^{(1)}_E D = -2(x+y)uv(2x-v)(2y-u). }$$

Thus the first local smoothing is generated by the existing two-wall incidence product $uv$, multiplied by source-fixed kinematic units.

On the double-wall section,

$$u=v=0,$$

the complete expression factors as

$$\boxed{ D(E,0,0) = E^3 (E-x-y) \left[ E^2-3(x+y)E+8xy \right]. }$$

Consequently,

$$\operatorname{gr}^{(1)}_E D|_{u=v=0}=0, \qquad \operatorname{gr}^{(2)}_E D|_{u=v=0}=0,$$

while

$$\operatorname{gr}^{(3)}_E D|_{u=v=0} = -8xy(x+y).$$

This excludes a proposed home of the algebraic quartic:

$$\boxed{ \mathcal Q \text{ is not the second-Rees local smoothing discriminant at }P_{--}. }$$

Indeed,

$$\operatorname{gr}^{(2)}_E\mathcal Q=-8xy\neq0$$

at generic nonsoft kinematics, whereas the physical local surface smoothing has vanishing second grade.

The negative result is narrow. It does not exclude $\mathcal Q$ from the other three enhanced points, their global gluing, the algebraic $\mathcal T_7/L_1$ connection, or the extension between the algebraic kernel and elliptic quotient.

Frozen local pair

The source walls on the $q_{\mathcal G_{12}}$-residue surface are

$$q_{g_1}=b-y-z, \qquad q_{g_2}=a-x-z.$$

Since $z=E-x-y$,

$$q_{g_1}=v, \qquad q_{g_2}=u$$

in the moving coordinates above. No local section is chosen after the expansion.

The physical intersection section is

$$u=v=0, \qquad a=E-y, \qquad b=E-x.$$

At $E=0$, it specializes to $P_{--}$.

First Rees grade

Write

$$D = E D_1(u,v) + E^2D_2(u,v) + \cdots.$$

Direct coefficient collection yields

$$D_1(u,v) = -2(x+y)uv \left[ 4xy-2xu-2yv+uv \right].$$

The bracket factors:

$$4xy-2xu-2yv+uv = (2x-v)(2y-u).$$

Therefore

$$D_1 = -2(x+y)uv(2x-v)(2y-u).$$

At a generic point near $P_{--}$, the last two factors are units. The first smoothing is exactly the product of the two predeclared marked normals.

This is coefficient geometry over the existing incidence carrier.

Restriction to the physical marked section

On $u=v=0$, the branch value is already known from the frozen intersection formula:

$$K_E(P) = \Lambda^2 = E^2(E-2x)^2(E-2y)^2.$$

Meanwhile,

$$R_0(P) = E[4xy-(x+y)E].$$

Hence

$$D(E,0,0) = E^2 \left( (E-2x)^2(E-2y)^2 - [4xy-(x+y)E]^2 \right).$$

The difference of squares factors as

$$(E-2x)(E-2y)-[4xy-(x+y)E] = E(E-x-y),$$

$$(E-2x)(E-2y)+[4xy-(x+y)E] = E^2-3(x+y)E+8xy.$$

This proves the cubic onset.

The factor $E-x-y=z$ is soft site-three support. The remaining quadratic is a source-derived higher local factor; its global role is not promoted here.

Comparison with the source quartic

Use

$$s=x+y, \qquad p=xy.$$

The source algebraic quartic is

$$\mathcal Q = -16p^2-8pE^2+8sE^3-5E^4.$$

Its ordinary normal grades begin

$$\operatorname{gr}^{(1)}_E\mathcal Q=0, \qquad \operatorname{gr}^{(2)}_E\mathcal Q=-8p.$$

The physical local smoothing instead satisfies

$$\operatorname{gr}^{(1)}_E D|_P=0, \qquad \operatorname{gr}^{(2)}_E D|_P=0.$$

No multiplication by a source-fixed unit can turn a vanishing second grade into $-8p$. Therefore $\mathcal Q$ is not a canonical factor of this local second-Rees smoothing.

This is a genuine exclusion, not merely failure to find a convenient normalization.

Updated provenance partition

The following candidate homes are now excluded:

• the pure elliptic infinity-Gysin quotient;
• generic total-energy smoothing along the conic;
• second-Rees local smoothing at the physical double-wall point.

The remaining admissible homes are:

1. a local collision invariant at one or more of the other sign-enhanced points;
2. a global extension/gluing invariant of all four enhanced points;
3. the algebraic connection on $\mathcal T_7$, especially the rank-one $L_1$ subquotient;
4. the extension class

$$0\to\mathcal T_7 \to\mathcal M_q^{(9)} \to\mathbb V_{\rm ell}(-1) \to0;$$

5. a physical moving-chain singularity not visible in the absolute surface family.

Only the last possibility could still force data beyond the frozen absolute/marked coefficient geometry, and even then it is not automatically a new carrier stratum.

Classification

Deutsch--Popperian update M2.30

The conjecture that $\mathcal Q$ is generated by the physical enhanced point's second-Rees surface smoothing is falsified.

The smaller surviving conjecture is:

$$\mathcal Q \text{ belongs to the global algebraic/extension geometry of the frozen } q\text{-sector,}$$

possibly compiled from the other enhanced points, but not to the local absolute smoothing at $P_{--}$.

Next hostile test

Compute the same completed local smoothing at

$$P_{+-},\quad P_{-+},\quad P_{++}.$$

Then eliminate their local tangential coordinates and compare the product and discriminant of the four higher-jet factors with $\mathcal Q$.

If $\mathcal Q$ is absent there as well, stop searching for it in the surface smoothing and move decisively to the algebraic connection $L_1$ and the global extension class.

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