Entry 286 · 2026-08-16 · marici.Benincasa · research

Global Total-Energy Split Model and Four Enhanced Points

Result

At the total-energy boundary

$$E=x+y+z=0,$$

the compactified Cayley--Menger surface is globally reducible. Its branch quartic is the exact square

$$\overline K_0 = R_h^2, \qquad R_h = xy(x+y)s^2-xa^2-yb^2.$$

Therefore

$$\overline S_0 = S_+\cup S_-, \qquad S_\pm:\ W=\pm R_h,$$

and each component is a copy of $\mathbb P^2$. They meet along the smooth conic

$$C:\quad R_h=0.$$

The infinity boundary consists of one line on each sheet. Those lines meet at the two points $C\cap\{s=0\}$, so their dual graph has two vertices and two edges. Its first homology is rank one, giving the expected Tate limit of the elliptic boundary.

The first total-energy smoothing coefficient factors canonically:

$$\boxed{ \left. \frac{\partial\overline K_E}{\partial E} \right|_{E=0} = -2(x+y) (a^2-y^2s^2) (b^2-x^2s^2). }$$

Away from its zeros on $C$, the family has the semistable local form

$$(W-R_h)(W+R_h)=E\cdot\text{unit}.$$

The coefficient vanishes on $C$ at exactly four finite points,

$$[a:b:s] = [\pm y:\pm x:1].$$

Each is a double zero of the restricted first smoothing coefficient. The physical marked collision is

$$[-y:-x:1].$$

Thus the generic $E=0$ degeneration is completely Tate/Kummer. All enhanced total-energy behavior is localized at four points already generated by the frozen branch conic and occurrence/sign data. No new carrier stratum has appeared.

Exact compact central fiber

At $E=0$, write

$$z=-x-y.$$

In the affine chart $s=1$, entry 282 gives

$$K_0 = \left[ xy(x+y)-xa^2-yb^2 \right]^2.$$

Homogenizing on $[a:b:s]\in\mathbb P^2$ gives

$$R_h = xy(x+y)s^2-xa^2-yb^2.$$

The weighted-projective double cover

$$W^2=R_h^2$$

is the union

$$W=R_h \qquad\text{and}\qquad W=-R_h.$$

Both sheets project isomorphically to $\mathbb P^2$. Their double locus is the conic $C$. For generic nonsoft $x,y,x+y$, this conic is smooth.

Affine and boundary double loci

The affine surface removes $s=0$. The conic intersects the infinity line in the two solutions of

$$xa^2+yb^2=0.$$

Hence

$$C_{\rm aff} = C\setminus(C\cap\{s=0\}) \simeq \mathbb G_m.$$

This already shows that the central affine gluing data are Tate/Kummer.

On the two sheets, the anticanonical boundary specializes to lines

$$L_+\subset S_+, \qquad L_-\subset S_-.$$

They meet at the same two points. The normalization-incidence graph is

$$L_+ \mathrel{\substack{\text{---}\\[-1mm]\text{---}}} L_-,$$

with

$$b_1=2-2+1=1.$$

This rank-one graph class is the Tate nearby-cycle remnant of the elliptic boundary. It agrees with the standard nodal Legendre degeneration but is now visible directly in the compact surface geometry.

First total-energy smoothing coefficient

Keep

$$z=E-x-y, \qquad c=-E$$

in the frozen Cayley--Menger polynomial and expand

$$\overline K_E = R_h^2+EJ_h+O(E^2).$$

Exact collection gives

$$J_h = 2(x+y) \left[ x^2a^2s^2+y^2b^2s^2-a^2b^2-x^2y^2s^4 \right].$$

It factors as

$$J_h = -2(x+y) (a^2-y^2s^2) (b^2-x^2s^2).$$

This factorization is source-derived. No normalization has been changed to produce it.

Set

$$U=W-R_h, \qquad V=W+R_h.$$

Then

$$UV = EJ_h+O(E^2).$$

At a point of $C$ where $J_h\neq0$, analytic rescaling gives

$$UV=E.$$

Therefore the total-energy degeneration is already semistable along the generic double curve.

The four enhanced points

Restrict $J_h$ to $C$. Its zeros satisfy

$$a^2=y^2s^2 \qquad\text{or}\qquad b^2=x^2s^2.$$

The conic equation forces both conditions. Thus the zero set is

$$Z = \{ [\epsilon y:\delta x:1] : \epsilon,\delta\in\{\pm1\} \}.$$

Because the restriction of a quartic to a conic has degree eight and there are four distinct generic points, each zero has multiplicity two. Equivalently, the two factors in $J_h$ vanish to first order together along $C$.

The source marked-wall intersection at total energy is

$$a=x+z=-y, \qquad b=y+z=-x,$$

so it selects the occurrence-resolved point

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

The other three points are the remaining sign choices of the same frozen geometry. They are not new cells added after the degeneration was seen.

Nearby-cycle grade away from the enhanced points

On

$$C\setminus Z,$$

the central fiber is a two-component simple normal crossing. Its strata are:

• two rational surfaces $S_\pm$;
• the rational conic $C$;
• after removing infinity, the Tate curve $\mathbb G_m$;
• the two-point incidence of the boundary lines.

Consequently the generic nearby-cycle weight-graded object is assembled from Tate/Kummer pieces. No elliptic coefficient survives as a smooth genus-one object at $E=0$; its information is carried by the logarithmic extension and the rank-one dual-graph cycle.

This is the global geometric version of the nearby-cycle statement in entry 128.

Relation to the second Rees frontier

The generic semistable locus is controlled at first order by $J_h$. At the four points in $Z$, that first-order coefficient vanishes. Therefore higher total-energy jets are genuinely required there.

This is exactly the location where the existing warning becomes active:

$$\text{first normal jet} \not\Rightarrow \text{control of the integrated loop deformation}.$$

The quartic

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

has no linear $E$-term. The next test is not to fit $\mathcal Q$ to the generic double curve. It is to compute the higher local smoothing at the four forced enhancement points and eliminate the local fiber coordinate.

Classification

Deutsch--Popperian conjecture M2.29

The complete total-energy nearby-cycle system is obtained from the two-sheet/conic semistable model plus local corrections at the four source-generated enhanced points. Those corrections are Tate/Kummer and their higher normal discriminant generates every non-elliptic algebraic letter, including $\mathcal Q$, without adding a carrier stratum.

The finite falsifier is now localized. Compute the completed local family at each point of $Z$, including the marked walls, through second Rees order. If $\mathcal Q$ cannot be obtained from these four local relative models or the global extension among them, the conjecture fails.

Next hostile test

At the physical point

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

use moving coordinates

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

Then:

1. expand $UV=\overline K_E-R_h^2$ through the first nonzero local weighted order;
2. include the two marked walls $u=0$, $v=0$;
3. semistabilize the resulting local pair;
4. compute the second Rees specialization;
5. eliminate the local variables and test whether the source quartic $\mathcal Q$ is the canonical collision discriminant;
6. repeat the sign analysis only if the physical point does not already decide the provenance.

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