A length-nine collision at the square elliptic curve

Mode: ATTACK
Completed: 7 September 2026
Status: internally checked local theorem in the equal-coupling untwisted elliptic Calogero–Moser system. It is an ordinary research note, not a proof or disproof of the Fazzi–Krawczyk bijectivity conjecture, and it has not received external review.

1. Outcome

The previous attack asked whether every isolated regular Calogero–Moser extremum is Morse for generic modular parameter, or whether a degenerate isolated extremum can be exhibited and its flavor splitting measured.

There is a canonical exact example at the square elliptic curve. In the equal-coupling untwisted $B_2$ potential

$$V(q_1,q_2)=\wp(q_1-q_2)+\wp(q_1+q_2)+\wp(q_1)+\wp(q_2), \tag{1.1}$$

take $q_1,q_2$ to be the two half-periods whose Weierstrass values are $a$ and $-a$. Both mixed roots then land at the third half-period, whose value is zero. This is a regular point: none of the four positive-root arguments is the identity.

At $g_3=0$, equivalently $j=1728$, the full Hessian vanishes. The point is nevertheless isolated. In local coordinates $(x,y)$ its first nonconstant jet is

$$V-V(q)=2a^3(x^4-y^4)+O_5(x,y). \tag{1.2}$$

Consequently the initial gradient ideal is $(x^3,y^3)$, and the local critical algebra has length

$$\boxed{\mu_q=9.} \tag{1.3}$$

The collision has a single leading smoothing coordinate. If $r$ is the nearby value of $\wp$ at the mixed half-period and $\sigma_2=\sum_{a<b}\Delta_a\Delta_b$ is the quadratic flavor invariant, then, up to higher weighted order, the two gradient equations are

$$\begin{aligned} F_x&=2a x\bigl(4a^2x^2-\kappa\bigr),\\ F_y&=-2a y\bigl(4a^2y^2-\kappa\bigr),\\ \kappa&=3r+2a^2\sigma_2. \end{aligned} \tag{1.4}$$

For $\kappa\ne0$, each coordinate has three simple choices. Thus the length-nine point splits into nine Morse extrema. The local trace discriminant has leading order

$$\boxed{\operatorname{disc}\doteq \kappa^{18}.} \tag{1.5}$$

This gives a precise answer at a symmetry-enhanced modular parameter. It also shows that neither the first corank-one flavor gate nor a first-order modular transversality test is adequate in higher corank: here $dV_2(q)=0$, while the next jet gives complete splitting.

2. Exact Hessian calculation

Let $p,q,r$ be the three half-period values, so

$$p+q+r=0, \qquad g_2=2(p^2+q^2+r^2). \tag{2.1}$$

Write

$$A(t)=6t^2-\frac{g_2}{2}. \tag{2.2}$$

This is $\wp''$ at a half-period of value $t$. At the fully rational point with coordinate-root values $p,q$ and mixed-root value $r$, the two mixed terms have equal second derivative. Hence the off-diagonal Hessian entries cancel and the eigenvalues are

$$\lambda_x=2A(r)+A(p)=6r(r-p), \qquad \lambda_y=2A(r)+A(q)=6r(r-q). \tag{2.3}$$

On the square curve, $(p,q,r)=(a,-a,0)$ and $g_2=4a^2$, so both eigenvalues vanish. This degeneration occurs on the smooth elliptic curve: the modular discriminant is $g_2^3\ne0$. It is therefore not a collision with the root-hyperplane polar divisor.

3. The local algebra

Every odd derivative of $\wp$ vanishes at a half-period. Differentiating $\wp''=6\wp^2-g_2/2$ twice gives, at such a point,

$$\wp''''=12\wp\,\wp''. \tag{3.1}$$

For the three values $(a,-a,0)$, equation (3.1) yields

$$\wp''''=48a^3,quad -48a^3,quad 0. \tag{3.2}$$

The mixed-root quartic terms vanish; the two coordinate-root terms give exactly (1.2). Its gradient initial forms are

$$8a^3x^3,qquad -8a^3y^3. \tag{3.3}$$

They form a regular sequence with isolated common zero. Standard local intersection theory therefore identifies the critical multiplicity with the product of their degrees, $3\cdot3=9$. Equivalently, the nine monomials $x^iy^j$, $0\le i,j\le2$, form a basis of the initial local algebra.

4. Modular and flavor smoothing

Keep the fully rational half-period branch and move away from $r=0$. Equation (2.3) gives

$$\lambda_x=-6ar+O(r^2), \qquad \lambda_y=6ar+O(r^2). \tag{4.1}$$

The normalized Bethe potential from the preceding attack begins

$$\mathcal V_\Delta =V-\frac{\sigma_2}{12}V_2+O(\sigma_2^2,\delta^2), \qquad V_2=\sum_{\alpha>0}\wp''(\alpha(q)). \tag{4.2}$$

At the square point, $dV_2=0$. But (3.2) gives

$$\operatorname{Hess}(V_2) =\begin{pmatrix}48a^3&0\\0&-48a^3\end{pmatrix}. \tag{4.3}$$

Combining (1.2), (4.1), and (4.2) gives (1.4). The two ostensibly different deformations are therefore aligned to leading order: the local catastrophe sees only $\kappa=3r+2a^2\sigma_2$.

The cubic $4a^2t^3-\kappa t$ has discriminant $16a^2\kappa^3$. The local algebra is the tensor product of its two one-variable rank-three factors at leading order. The discriminant identity

$$\operatorname{disc}(A\otimes B) =\operatorname{disc}(A)^{\operatorname{rk}B} \operatorname{disc}(B)^{\operatorname{rk}A} \tag{4.4}$$

then gives the exponent $3\cdot3+3\cdot3=18$ in (1.5). At every one of the nine solutions with $\kappa\ne0$, the two diagonal second derivatives are nonzero, so the splitting is fully Morse.

5. Scope and stopping boundary

Fazzi and Krawczyk already identify the fully rational $B_2$ extrema. The new content here is the square-lattice local algebra, its length, the aligned modular/flavor smoothing, and the discriminant order. No priority claim is made.

The example proves that the word “generic” in a Morse statement is essential. It does not produce a degenerate isolated extremum at generic modular parameter; instead it supplies an exact local model showing how a special modular collision resolves. A proof of generic reducedness still has to exclude a component that is nonreduced over the modular function field. Generic smoothness disposes of reduced dominant components but does not exclude such persistent nilpotence.

The next viable structural move is therefore not a list of root systems or critical points. It is to prove that a generically nonreduced component of the universal critical scheme must underlie a positive-dimensional critical family. The known $A_2$ family is the control. If that implication cannot be obtained from the gradient algebra or the commuting Hamiltonians, this route stops.

6. Sources and verification

• Fazzi–Krawczyk, Black Holes, the Bethe Ansatz, and Elliptic Calogero–Moser Systems, submitted 19 August 2026: external conjecture, equal-coupling untwisted potential, and the fully rational $B_2$ extrema.
• Fazzi–Eghrari–Krawczyk, rank-two Bethe analysis, submitted 18 August 2026: companion evidence that the nearby generic Bethe solutions are isolated and have nonzero Bethe Jacobian.
• The exact verification script checks (2.3), the square-lattice fourth derivatives, the length-nine initial algebra, the joint smoothing jet, and the discriminant exponent. No floating-point computation is used.
• The downloadable archive contains this note, the dependency-free script, its exact output, and the source ledger. The continuing handoff is kept as a separate stable site artifact.