The Cartier-residual Bethe family and its Calogero–Moser fiber

Mode: ATTACK
Completed: 7 September 2026
Status: internally checked structural results on the regular root-hyperplane complement. This is an ordinary research note, not a proof of the Fazzi–Krawczyk bijectivity conjecture and not a Publication.

1. Scope and outcome

Fazzi and Krawczyk define Bethe operators for four-dimensional $\mathcal N=4$ super-Yang–Mills and show that a Bethe solution tracked to zero flavor chemical potentials lands at an extremum or pole of the untwisted elliptic Calogero–Moser potential. They conjecture that the map is a bijection on the sector landing at extrema.

This attack establishes four facts on the regular locus.

1. The apparent coordinatewise division by $\delta=\Delta _1\Delta _2\Delta _3$ is the local expression of a canonical Cartier-residual ideal. It is independent of a cocharacter basis and a choice of positive roots and is equivariant under the torus, center, Weyl, charge-conjugation, and modular actions.
2. Its special fiber is the Calogero–Moser critical scheme scheme-theoretically, not merely pointwise.
3. Near every isolated regular Calogero–Moser extremum, including a degenerate one, the normalized family is finite flat over the flavor germ. Its degree is the local critical-scheme length. A nondegenerate extremum therefore has one unique holomorphic Bethe branch.
4. On the known continuous $A_2$ critical curve, both the first and second tangential obstruction vanish identically. The two relevant effective potentials are constants $3g_2/2$ and $3g_3/20$.

The first three statements turn the local part of the conjecture into one precise question: what is the finite flat cover above each degenerate critical point? They do not prove that every such cover has one underlying point, nor do they control solutions flowing to root hyperplanes or poles.

Primary frontier: Fazzi–Krawczyk, arXiv:2608.19324v1.

2. Coordinate-free Bethe operator

Let $L$ be the chosen cocharacter lattice and

$$T_L^\vee=\operatorname{Hom}(L,\mathbb C^\times).$$

Work on an analytic neighborhood in which every root argument stays away from the period lattice. After the standard shift of the third chemical potential, put

$$\Delta _1+\Delta _2+\Delta _3=0$$

and define the single-root block

$$R(z;\Delta)=\prod_{a=1}^3 \frac{\theta _1(z-\Delta _a;\omega)} {\theta _1(z+\Delta _a;\omega)}.$$

Oddness of $\theta _1$ turns equation (5) of Fazzi–Krawczyk into this block with no residual sign. For $u$ in the regular Cartan torus, define

$$\mathcal Q(u,\Delta)(\lambda) =\prod_{[\alpha]\in\Phi/\{\pm1\}} R(\alpha(u);\Delta)^{\alpha(\lambda)}, \qquad \lambda\in L. \tag{2.1}$$

This definition is independent of the representative of $[\alpha]$:

$$R(-z;\Delta)=R(z;\Delta)^{-1},$$

while replacing $\alpha$ by $-\alpha$ also negates $\alpha(\lambda)$. In a primitive basis $e_i$ of $L$, the coordinates $\mathcal Q(e_i)$ are precisely the usual Bethe operators $Q_i$. Thus a basis change only changes coordinates on $T_L^\vee$; it does not change $\mathcal Q$.

Let $\mathfrak m_1$ be the ideal of the identity of $T_L^\vee$ and set

$$\mathcal I=\mathcal Q^*\mathfrak m_1. \tag{2.2}$$

This is the coordinate-free Bethe ideal.

3. The canonical Cartier residual

Let $B$ be the two-dimensional flavor germ $\Delta _1+\Delta _2+\Delta _3=0$, and let

$$D=V(\Delta _1\Delta _2\Delta _3)\subset B.$$

The three components of $D$ are distinct smooth lines. If $\Delta _a=0$, the remaining two flavors are opposites and their factors in $R$ cancel; the zero-flavor factor cancels itself. Consequently

$$R(z;\Delta)=1,\qquad \mathcal Q=1 \quad\hbox{on }D.$$

Because $D$ is a reduced Cartier divisor on the regular flavor surface,

$$\mathcal I\subseteq\mathcal O(-D).$$

Define

$$\boxed{\ \mathcal J=(\mathcal I:\mathcal O(-D)) =\mathcal I\otimes\mathcal O(D).\ } \tag{3.1}$$

Locally, with $\delta=\Delta _1\Delta _2\Delta _3$, this is the ideal generated by $(Q_i-1)/\delta$. Formula (3.1), rather than those individual quotients, is the intrinsic flavor-normal family.

Descent and equivariance

The following checks are exact.

• Cocharacter basis. Both $\mathcal Q$ and the identity ideal $\mathfrak m_1$ are basis-free. Nonlinear coordinate changes among the individual $(Q_i-1)/\delta$ therefore leave $\mathcal J$ unchanged.
• Positive roots. Independence follows from $R(-z)=R(z)^{-1}$, as in (2.1).
• Torus and center shifts. The theta quasi-periods give $R(z+1)=R(z)$ and $R(z+\omega)=e^{4\pi i\sum_a\Delta_a}R(z)=R(z)$. Root evaluations of allowed shifts are periods, so $\mathcal Q$, $\mathcal I$, and $\mathcal J$ descend.
• Weyl group. For $w$ preserving $L$, $$\mathcal Q(wu,\Delta)(\lambda) =\mathcal Q(u,\Delta)(w^{-1}\lambda).$$ The identity of the dual torus is fixed, hence the residual ideal is equivariant.
• Charge conjugation. Simultaneously negating $z$ and all $\Delta_a$ fixes $R$, and the divisor $D$ is fixed.
• Modular transformations. Under $T$, the common theta multiplier cancels from every ratio. Under $S$, the Gaussian multiplier in a single ratio is exponential in $z\Delta_a$; its product is one because $\sum_a\Delta_a=0$. The convention-dependent simultaneous sign is absorbed by charge conjugation. The local equation $\delta$ scales by a nowhere-zero factor of modular weight $-3$, so the Cartier divisor and the zero ideal of its residual are preserved.

The standard theta transformation laws used here are recorded in DLMF sections 20.2 and 20.7.

This proves Gate A of the hunt: the normalized object does not require a choice of gauge presentation.

4. Exact special fiber

On a simply connected target-torus chart around the identity, take the holomorphic logarithm. Since $\log Q_i$ is divisible by $\delta$,

$$\frac{Q_i-1}{\delta} =\frac{\log Q_i}{\delta} \frac{e^{\log Q_i}-1}{\log Q_i}.$$

The last factor is a holomorphic unit. Hence the normalized multiplicative and logarithmic equations define the same ideal.

Writing $e_2=\sum_{a<b}\Delta_a\Delta_b$, the single-root logarithm has the exact formal expansion

$$\frac1\delta\sum_a \bigl[\log\theta_1(z-\Delta_a) -\log\theta_1(z+\Delta_a)\bigr] =\wp'(z)-\frac{e_2}{12}\wp'''(z) +\frac{e_2^2}{360}\wp^{(5)}(z)+O(e_2^3,\delta^2). \tag{4.1}$$

The leading coordinate at a cocharacter $e_i$ is therefore

$$\sum_{\alpha>0}\alpha(e_i)\wp'(\alpha(u)) =\partial_i V(u), \qquad V(u)=\sum_{\alpha>0}\wp(\alpha(u)). \tag{4.2}$$

It follows at the level of ideals, not only at the level of zero sets, that

$$\boxed{\ \mathfrak B^\nu_0 =\operatorname{Crit}(V)\ } \tag{4.3}$$

on the regular Cartan torus.

5. Isolated extrema give finite flat covers

Let $q$ be any isolated point of $\operatorname{Crit}(V)$ in the regular locus; it need not be nondegenerate. Let

$$B_0=\mathbb C\{\Delta_1,\Delta_2\},\qquad S=B_0\{x_1,\ldots,x_r\}$$

be the local analytic rings at the flavor origin and at $q$, and let $F_1,\ldots,F_r$ be local generators of $\mathcal J$. Their special fiber is $f_i=\partial_iV$. Since $q$ is isolated,

$$\mathbb C\{x_1,\ldots,x_r\}/(f_1,\ldots,f_r)$$

has finite length. Thus $(f_1,\ldots,f_r)$ has height $r$. The same height calculation in the regular ring $S$ shows that $F_1,\ldots,F_r$ is a regular sequence. Hence

$$A=S/(F_1,\ldots,F_r)$$

is Cohen–Macaulay of dimension two. Its special fiber has dimension zero, so analytic finiteness near $(q,0)$, followed by miracle flatness over the regular two-dimensional base $B_0$, gives:

$$\boxed{\ A\text{ is finite flat over }B_0.\ } \tag{5.1}$$

Its rank is preserved on the special fiber and equals

$$\boxed{\ \mu_q= \dim_{\mathbb C} \frac{\mathcal O_{\mathfrak h,q}} {(\partial_1V,\ldots,\partial_rV)}.\ } \tag{5.2}$$

Thus every isolated regular extremum has a canonical local finite flat Bethe cover of degree $\mu_q$. When the Hessian of $V$ at $q$ is invertible, $\mu_q=1$, and the holomorphic implicit-function theorem makes the cover a single graph $u=u(\Delta)$. For $\delta\ne0$, this graph is a unique nearby solution of the original BAEs.

For a degenerate isolated extremum, flatness still holds but degree one need not. The Fazzi–Krawczyk set-theoretic bijection near such a point is now equivalent to understanding whether this degree-$\mu_q$ cover stays supported on one branch or splits. This is a finite local-algebra problem, not a search through Bethe presentations.

6. The first obstruction on a critical component

Equation (4.1) is the gradient of the normalized local potential

$$\mathcal V =V-\frac{e_2}{12}V_2+\frac{e_2^2}{360}V_4 +O(e_2^3,\delta^2),$$

where

$$V_2=\sum_{\alpha>0}\wp''(\alpha(u)), \qquad V_4=\sum_{\alpha>0}\wp^{(4)}(\alpha(u)).$$

If $C$ is a smooth component of $\operatorname{Crit}(V)$ and $u=q+e_2\xi+\cdots$, the first normal equation is

$$\operatorname{Hess}_q(V)\xi =\frac1{12}\nabla V_2(q). \tag{6.1}$$

Its tangential obstruction is $d(V_2|_C)$. If (6.1) is solvable, the second effective potential on $C$ is

$$E_2=\left. \left(\frac{V_4}{360} -\frac12\xi^T\operatorname{Hess}(V)\xi\right) \right|_C. \tag{6.2}$$

The next tangential obstruction is $dE_2$. These statements are the usual Lyapunov–Schmidt reduction written for the intrinsic residual family.

7. Exact $A_2$ positive control

Use root coordinates

$$a=q_1-q_2,qquad b=q_2-q_3,qquad c=a+b$$

and abbreviate

$$X=\wp(a),\quad Y=\wp(b),\quad Z=\wp(c), \qquad s=\wp'(a)=\wp'(b)=-\wp'(c).$$

On the generic part of the continuous critical family, the addition formula and the critical equations give

$$Z=-X-Y,qquad X^2+XY+Y^2=\frac{g_2}{4}. \tag{7.1}$$

These polynomial identities extend across the component. From $\wp''=6\wp^2-g_2/2$,

$$\boxed{\ V_2=\wp''(a)+\wp''(b)+\wp''(c) =\frac{3g_2}{2}.\ } \tag{7.2}$$

Thus the first tangential obstruction vanishes identically. More precisely, with

$$n=(2X+Y,\ X+2Y),$$

the Hessian factorizes on the generic smooth locus as

$$\operatorname{Hess}(V)=2nn^T, \qquad \frac1{12}\nabla V_2=s n. \tag{7.3}$$

Equation (6.1) is therefore exactly

$$n\cdot\xi=\frac{s}{2}. \tag{7.4}$$

For the next order, let $P=XYZ$. The Weierstrass cubic and (7.1) give

$$s^2=4P-g_3. \tag{7.5}$$

Since

$$\wp^{(4)}=120\wp^3-18g_2\wp-12g_3,$$

and $X+Y+Z=0$,

$$V_4=360P-36g_3. \tag{7.6}$$

Equations (7.3)–(7.6) reduce (6.2) to

$$\boxed{\ E_2=\frac{V_4}{360}-\frac{s^2}{4} =\frac{3g_3}{20}.\ } \tag{7.7}$$

The second tangential obstruction also vanishes identically. This explains, through total flavor degree four, why the known $A_2$ CM curve is not cut down by flavor restoration. Fazzi–Krawczyk already prove that the full $A_2$ Bethe family maps to this curve; (7.2) and (7.7) are a structural calibration of the residual deformation, not a claim to a newly discovered family.

8. Verification and dependencies

The dependency-free script verify_normal_family_attack.py represents polynomials in $X,Y,g_3$ with exact rational coefficients. It checks (7.1)–(7.7), including every Hessian entry. The earlier verify_flavor_normalization.py checks the Newton sums and the coefficients of (4.1) through flavor degree nine. Both scripts run without external packages.

Intellectual and formal dependencies are separated as follows.

• Frontier and physical map: Fazzi and Krawczyk define the BAEs, the limit map, its symmetries, the conjecture, and the $A_2$ positive-dimensional control.
• Earlier influence: their third-derivative mechanism generalizes an unpublished type-$A$ argument of O. Aharony communicated by O. Mamroud.
• Formal identities: standard Jacobi theta quasi-periodicity and modular transformations; standard Weierstrass differential and addition formulas.
• Commutative algebra: regular sequences and miracle flatness for local complete intersections over a regular base.

No source found in the bounded 7 September 2026 check formulates the Cartier-residual Bethe ideal, the finite-flat degree formula (5.2), or the two explicit $A_2$ obstruction constants in this form. This is not a priority claim and the work has not been externally reviewed.

9. Boundaries of the result

• The result is local near the unrefined flavor point and on the regular root-hyperplane complement.
• It does not control Bethe solutions flowing to poles of $V$.
• Finite flatness preserves multiplicity; it does not force the underlying cover above a degenerate extremum to have one point.
• The $A_2$ calculation proves two obstruction cancellations, not an all-root-system theorem for continuous critical components.
• No claim is made about the contribution of any branch to the index or to black-hole entropy.

10. Next decisive question

NEXT MODE: ATTACK. Study the discriminant of the canonical finite flat cover at an isolated degenerate extremum. The decisive uniform question is:

Does the residual Bethe deformation preserve the support of the local Jacobian algebra, or does a critical point with $\mu_q>1$ split into multiple generic Bethe solutions?

The first route should be intrinsic: express the trace pairing or discriminant of the finite flat algebra in terms of the Calogero–Moser Lax or spectral data. If that cannot be done without a Dynkin-type or solution census, stop this route and instead seek an all-orders conservation law for the obstruction potentials on arbitrary critical components.