Persistent nilpotence is not converted into a critical family by integrability

Status

This note is a stopping audit for the flavor-normal Bethe–Calogero research program. It proves a no-go statement about the proposed next argument. It does not refute the Fazzi–Krawczyk conjecture, produce a counterexample in an elliptic Calogero–Moser system, or establish a generic Morse theorem.

The conclusion is precise: complete-intersection gradient structure, finite flatness, generic smoothness of the reduced support, and Liouville integrability do not imply that a generically nonreduced critical component must become positive-dimensional inside a fiber. Any successful continuation therefore needs a genuinely Calogero–Moser-specific versality theorem.

1. The question left by the previous attack

The preceding work constructed a canonical flavor-normal Bethe scheme whose special fiber is the critical scheme of the equal-coupling untwisted elliptic Calogero–Moser potential. Near any isolated regular critical point it is a finite flat complete intersection. Its trace discriminant is, up to a unit, the norm of its averaged-Hessian determinant.

For the universal Calogero critical scheme over the modular curve, generic smoothness controls every reduced component dominating the base. The only remaining local obstruction is a component that is nonreduced over the modular function field while its reduced support remains zero-dimensional in each modular fiber.

The proposed route asked whether complete integrability forces such a nilpotent direction to exponentiate to a positive-dimensional family of critical points. The following exact countermodel shows that it does not.

2. A nonconstant finite-flat countermodel

Let $k$ be a characteristic-zero field, let $B=k[t]$, and fix $r\geq 2$. On affine configuration space with coordinates $x_1,\ldots,x_r$, consider

$$V_t(x)=\frac{x_1^4}{4}+t\frac{x_1^5}{5} +\frac12(x_2-t)^2 +\frac12\sum_{i=3}^r x_i^2.$$

Its relative critical ideal is

$$I=(x_1^3(1+tx_1),\ x_2-t,\ x_3,\ldots,x_r).$$

Consequently the universal critical algebra is

$$\widehat A=B[[x_1,x_2-t,x_3,\ldots,x_r]]/I \cong B[[\varepsilon]]/(\varepsilon^3).$$

This establishes all of the relevant properties at once.

1. Gradient complete intersection. The ideal is generated by the derivatives of one potential and is a regular sequence of length $r$.
2. Finite flatness. The algebra is free of rank three over $B$, with basis $1,\varepsilon,\varepsilon^2$.
3. Nonconstant force and motion over the base. The force in the null direction contains the parameter-dependent term $t x_1^4$, and its reduced support is the moving section $(0,t,0,\ldots,0)$. Locally, $1+tx_1$ is a unit, so neither variation removes the cubic thickening.
4. Persistent nilpotence. Every geometric fiber is $\bar k[\varepsilon]/(\varepsilon^3)$, hence nonreduced of length three.
5. No critical family in a fiber. The reduced critical locus in every fiber is one point. It has dimension zero.

The Hessian determinant in $\widehat A$ is

$$j=3\varepsilon^2.$$

It is nonzero but nilpotent. Multiplication by $j$ has determinant zero, so the different is nontrivial and the trace discriminant vanishes identically over $B$, exactly as it should for a persistently ramified finite-flat cover.

3. Complete integrability does not repair the example

Pass to phase space with canonical coordinates $(x_i,p_i)$. Define

$$H_1=\frac{p_1^2}{2}+\frac{x_1^4}{4}+t\frac{x_1^5}{5},\qquad H_2=\frac{p_2^2}{2}+\frac{(x_2-t)^2}{2},$$

and, for $i\geq3$,

$$H_i=\frac{p_i^2+x_i^2}{2}.$$

The functions $H_1,\ldots,H_r$ Poisson commute because each depends on a disjoint canonical coordinate pair. Their differentials are independent on a dense open set, so they define a Liouville-integrable system. The natural Hamiltonian $H=\sum_iH_i$ has potential $V_t$.

In the completed neighborhood under study, its equilibrium over each value of $t$ is the unique reduced point

$$p_1=\cdots=p_r=0,qquad x_1=0,quad x_2=t,quad x_3=\cdots=x_r=0,$$

but the equilibrium scheme has the cubic nilpotent direction $x_1^3=0$. Thus even a full set of commuting Hamiltonians does not turn the Hessian kernel into a curve of equilibria.

The mechanism is elementary. Poisson commutativity constrains Hamiltonian vector fields on phase space. At an equilibrium all those vector fields may vanish. Their commutativity then supplies no nonzero tangent vector along the critical locus and no versal deformation class in the local Jacobian algebra.

4. What generic smoothness actually gives

Let $S$ be an integral characteristic-zero base and let

$$A=\mathcal O_S[x_1,\ldots,x_r]/ (\partial_1V,\ldots,\partial_rV)$$

be finite flat near a component. Over the function field $K(S)$, this is a zero-dimensional complete intersection. The following conditions are equivalent:

• the generic algebra is reduced;
• it is finite etale over $K(S)$;
• its Jacobian determinant, the Hessian determinant of $V$, generates the unit ideal in every geometric local factor;
• its trace discriminant is nonzero.

Generic smoothness applies after reducedness has been supplied. It cannot prove reducedness of a scheme whose generic algebra is already $K(S)[\varepsilon]/(\varepsilon^3)$. In the countermodel, the reduced support is a smooth section, yet the nilpotent thickening persists over the entire base.

Therefore the desired implication

$$\text{persistent nilpotence} \Longrightarrow \text{positive-dimensional critical fiber}$$

is not a consequence of generic smoothness, finite flatness, gradient form, or integrability, separately or together.

5. The exact missing Calogero–Moser bridge

The argument could still be true for elliptic Calogero–Moser potentials, but only because of additional spectral geometry. One sufficient input would be either of the following.

A. A local versality theorem

At every isolated regular CM critical point, the modular derivative together with the higher commuting Hamiltonians would have to generate every socle direction of the local Jacobian algebra. Equivalently, the relevant Kodaira–Spencer map would have to meet every nilpotent direction. Then an identically vanishing discriminant could not hide on an isolated reduced support.

B. A spectral exponentiation theorem

A kernel vector of the CM stability Hessian would have to determine a nonzero tangent vector to a fixed-spectral-data family of equilibria, and the formal tangent vector would have to algebraize. Ordinary involutivity does not provide either assertion.

Neither theorem is supplied by the standard existence of commuting Calogero–Moser Hamiltonians. The current literature source used for the frontier treats rank-two extrema explicitly and says continuous families are expected generically in higher rank; it does not state either versality principle. Establishing one from Lax or Hitchin geometry would be a new major project, not a missing lemma in the current attack.

6. Stopping verdict

The intrinsic deformation object remains useful:

• the Cartier-residual Bethe family is canonical;
• its special fiber is the CM critical scheme;
• isolated local fibers are finite flat with computable length;
• the trace discriminant is a normed stability determinant;
• the square-lattice $B_2$ collision supplies an exact higher-corank local model.

But these results do not move the global bijectivity conjecture past the persistent-nilpotence obstruction. The next available route would be a Dynkin-type or extremum-by-extremum classification, precisely the residual census this program was designed to avoid. The Bethe–Calogero attack should therefore pause.

The route may be reopened only if a CM-specific versality or spectral exponentiation theorem is found. Otherwise the recommended next mode is HUNT.

Sources and credit

• Marco Fazzi and Kuba Krawczyk, Black Holes, the Bethe Ansatz, and Elliptic Calogero–Moser Systems, submitted 19 August 2026. They define the limiting map, conjecture bijectivity on the CM sector, verify all rank-two cases, and identify continuous families as the next testing ground.
• Pavel Etingof, Giovanni Felder, Xiaoguang Ma, and Alexander Veselov, On elliptic Calogero–Moser systems for complex crystallographic reflection groups. This is context for the existence of commuting elliptic Calogero–Moser Hamiltonians; no generic-reducedness conclusion is imported from it.
• The finite-flat countermodel and the stopping implication are exact internal derivations. They have not been externally reviewed, and no priority claim is made.