Entry 288 · 2026-08-16 · marici.Benincasa · research

Global Enhanced-Point Discriminant Exclusion of Q

Result

The four enhanced points of the total-energy central fiber do not generate the source quartic through their global collision discriminant on the normalized double conic.

Let

$$C:\qquad xa^2+yb^2=xy(x+y)$$

be the affine double locus at $E=0$. Put

$$A=\frac ay,\qquad B=\frac bx,$$

so that

$$yA^2+xB^2=x+y.$$

The normalization through $(A,B)=(1,1)$ is

$$d=y+xt^2,$$

$$A(t)=\frac{xt^2-2xt-y}{d}, \qquad B(t)=\frac{y-2yt-xt^2}{d}.$$

The four enhanced points occur at

$$t=0,\qquad t=1,\qquad t=\infty,\qquad t=-\frac yx.$$

Restrict the frozen compact branch polynomial $K_E$ to this normalized central conic and clear the fixed denominator:

$$G_{x,y}(t,E) = d^4K_E\bigl(yA(t),xB(t)\bigr).$$

Because $K_0$ vanishes identically on $C$, $E$ divides $G$. Define the source-fixed moving divisor polynomial

$$H_{x,y}(t,E)=\frac{G_{x,y}(t,E)}{E}.$$

It has degree eight in $t$. Its projective collision locus is detected by

$$\Delta_C(x,y,E) = \operatorname{Res}_t \left( H_{x,y}, \partial_tH_{x,y} \right).$$

At the generic nonsoft specialization

$$x=2,\qquad y=3,$$

exact fraction-free elimination gives

$$\deg_E\Delta_C(2,3,E)=51.$$

The source quartic specializes to

$$\mathcal Q_{2,3}(E) = -576-48E^2+40E^3-5E^4.$$

Exact polynomial reduction gives a nonzero remainder. A compact finite certificate is obtained modulo $101$:

$$\boxed{ \Delta_C(2,3,E) \bmod \bigl(\mathcal Q_{2,3},101\bigr) = 78+98E+77E^2+19E^3 \neq0. }$$

Therefore

$$\mathcal Q_{2,3}\nmid\Delta_C(2,3,E).$$

If the symbolic source quartic were a canonical factor of $\Delta_C(x,y,E)$, divisibility would survive every specialization at which the frozen parametrization and source units remain defined. The point $(x,y)=(2,3)$ avoids

$$xy(x+y)(x-y)=0$$

and all four normalized marked parameters are distinct. Hence this one exact generic specialization is a finite falsifier:

$$\boxed{ \mathcal Q \text{ is not the global collision discriminant of the four enhanced} \text{ points on the normalized total-energy double conic.} }$$

Frozen construction

No section or carrier component was introduced after seeing $\mathcal Q$. The calculation uses only:

1. the source branch polynomial $K_E(a,b)$;
2. the central double conic $C$ derived in entry 286;
3. its canonical rational normalization through a source point;
4. the total-energy normal $E$;
5. the ordinary resultant of the resulting divisor with its fiber derivative.

The resultant is computed before comparison with $\mathcal Q$.

The ordinary degree-eight resultant also detects projective loss of degree: a root moving to $t=\infty$ forces degeneration of the leading coefficient and is not omitted by the fixed-degree Sylvester determinant.

Relation to the local audit

Entry 287 excluded $\mathcal Q$ from the second Rees smoothing at the physical point $P_{--}$. The present calculation is stronger in a different direction. It retains the complete degree-eight moving divisor on the normalized central conic and therefore simultaneously tests:

• the other three sign-enhanced points;
• collisions among all four enhanced loci;
• their global projective gluing on $C$;
• the point at infinity of the normalization.

Thus the remaining possibility that the four pointwise jets combine into $\mathcal Q$ through their global conic discriminant is falsified.

This calculation is an independent surface-family audit. Entries 209--212 already compute the generic algebraic connection, its extension, and the physical relative-chain variation. They exclude $\mathcal Q$ from all three. Therefore the present result must not reopen those closed candidate homes.

Provenance partition after the test

Combining the earlier generic coefficient theorem with the present surface-family calculation excludes $\mathcal Q$ from:

1. the pure elliptic infinity-Gysin quotient;
2. the cyclic algebraic line $L_1$;
3. the complete generic rank-seven algebraic kernel $\mathcal T_7$;
4. the generic algebraic-plane extension class;
5. the generic physical relative-chain monodromy;
6. generic total-energy smoothing along the double conic;
7. the physical enhanced point's second Rees smoothing;
8. the combined projective collision discriminant of all four enhanced points.

Thus entry 212's classification is independently strengthened:

$$\boxed{ \mathcal Q \text{ is apparent cyclic/master-presentation alphabet data at generic} \text{ nonsoft homogeneous kinematics.} }$$

The absolute surface-smoothing search is closed unless a later calculation exhibits a source datum absent from the frozen $K_E$-family. Remaining work concerns intersections of true support divisors, integral normalization, and the global marked rank-twelve extension—not another generic home for $\mathcal Q$.

Classification

Deutsch--Popperian update M2.31

The hard-to-vary claim

$$\mathcal Q \text{ is generated by global collision of the four enhanced smoothing} \text{ points}$$

is falsified by an exact generic specialization.

The smaller surviving statement is:

$$\boxed{ \mathcal Q \text{ has no intrinsic generic home in the frozen absolute, marked,} \text{ algebraic, elliptic, extension, or physical-chain geometry tested.} }$$

This strengthens H2 without inventing a new primitive. Every actual surface singularity found remains coefficient geometry over the unchanged energy/Cut carrier, while the printed quartic remains an apparent presentation letter on the generic homogeneous locus.

Correction note

The first committed version of this entry incorrectly listed $\mathcal T_7/L_1$, the algebraic--elliptic extension, and the physical relative chain as surviving candidate homes. Entries 209--212 had already excluded all three generically. The exact resultant calculation was correct; only that historical interpretation and proposed next step were wrong. This revision repairs the frontier without changing the calculation.

Next hostile test

Use the split total-energy model of entry 286 and the finite collision monodromy of entries 282--285 to compute the remaining rank-twelve marked extension

$$0\to H^2(S_E) \to H^2(S_E\setminus W) \to H^1(W)(-1) \to0$$

at $E=0$. In the rational conductor frame

$$(g_{101},g_{110},g_{111}^{\rm inv}),$$

derive the top-column nearby-cycle map and its integral lattice index. Separate:

• the nodal elliptic rank-one nilpotent;
• the four enhanced-point semistable classes;
• the two conductor Kummer characters;
• the known occurrence $2$-torsion gluing.

The finite falsifier is a residual extension class whose support or integral incidence cannot be generated from the frozen conic, marked walls, and signed-energy degeneration. Only that outcome can require a new carrier datum.

View source on GitHub