HUNT packet: the flavor-normal Bethe–Calogero family

Mode: HUNT
Selected: 7 September 2026
Status: a recognized conjectural correspondence selected for ATTACK; no new theorem is claimed in this packet.

1. Selected question

Let $\mathfrak g$ be a semisimple gauge algebra. Fazzi and Krawczyk define a map from solutions of the Bethe Ansatz equations (BAEs) for four-dimensional $\mathcal N=4$ super-Yang–Mills to extrema or poles of the untwisted elliptic Calogero–Moser (CM) potential

$$V(u;\omega)=\sum_{\alpha>0}\wp(\alpha\!\cdot\!u;\omega).$$

Their map tracks a BAE solution as the three flavor chemical potentials are turned off. They conjecture that its restriction to the CM sector is a bijection: every isolated CM extremum should deform to one and only one BAE solution.

The selected question is the scheme-theoretic strengthening that could prove that conjecture without listing solutions:

On the regular Cartan torus, do the flavor-normalized Bethe equations form a flat deformation of the critical scheme of the untwisted elliptic CM potential? When is that deformation locally degree one over flavor space?

For a positive-dimensional CM critical component, the corresponding question is whether the whole component lifts, or whether higher flavor coefficients canonically select a proper sublocus.

Primary frontier: Fazzi–Krawczyk, arXiv:2608.19324v1 (19 August 2026).

2. Why this is a presentation problem of the right kind

At the unrefined point the ordinary Bethe operators lose all holonomy information: $Q_i(u,0)=1$ identically. The CM equations are therefore not visible in the literal special fiber. Fazzi and Krawczyk recover them by differentiating three times along a tracked solution.

The degeneration has more intrinsic structure. Use local flavor coordinates $\Delta_1+\Delta_2+\Delta_3=0$, after the standard period shift of $\Delta_3$, and put

$$\delta=\Delta_1\Delta_2\Delta_3.$$

For any one of the three divisors $\Delta_a=0$, the other two chemical potentials are opposites. Oddness of $\theta_1$ then makes every root block of every Bethe operator equal to one. On the regular locus, the holomorphic function $Q_i-1$ is consequently divisible by all three distinct linear factors and hence by $\delta$. This suggests one canonical object:

$$\widetilde Q_i(u,\Delta) =\frac{Q_i(u,\Delta)-1}{\Delta_1\Delta_2\Delta_3}, \qquad \mathfrak B^{\nu}=V(\widetilde Q_1,\ldots,\widetilde Q_r).$$

Away from $\delta=0$, $\mathfrak B^{\nu}$ is the ordinary BAE solution scheme. At the origin its expected special fiber is the CM critical scheme. The ATTACK must prove that this normalization descends through the torus, cocharacter-basis, Weyl, center, and modular identifications; this packet does not assume that global descent.

This is exactly the kind of alternative presentation the Lab is seeking. A larger symmetry makes the unrefined equations look trivial. The normalized family retains the first dynamics that the literal limit flattened away.

3. Checked formal expansion

Write $\ell(z)=\log\theta_1(z)$ on a simply connected regular chart and consider the logarithmic root block

$$H(z;\Delta)=\sum_{a=1}^{3} \bigl(\ell(z-\Delta_a)-\ell(z+\Delta_a)\bigr).$$

Let $e_2=\Delta_1\Delta_2+\Delta_2\Delta_3+ \Delta_3\Delta_1$. Newton identities under $\Delta_1+\Delta_2+\Delta_3=0$ give

$$\sum_a\Delta_a^3=3\delta, \quad \sum_a\Delta_a^5=-5e_2\delta, \quad \sum_a\Delta_a^7=7e_2^2\delta.$$

Together with $(\log\theta_1)''=-\wp+c$, the exact formal expansion begins

$$\frac{H(z;\Delta)}{\delta} =\wp'(z)-\frac{e_2}{12}\wp'''(z) +\frac{e_2^2}{360}\wp^{(5)}(z)+O(\|\Delta\|^6).$$

Thus the special fiber of the locally normalized logarithmic equations is

$$\partial_iV(u)=\sum_{\alpha>0}\alpha_i\wp'(\alpha\!\cdot\!u)=0,$$

and the first deformation term is itself a gradient:

$$-\frac{e_2}{12}\,\partial_iV_2(u), \qquad V_2(u)=\sum_{\alpha>0}\wp''(\alpha\!\cdot\!u).$$

The accompanying script verifies the symmetric-polynomial coefficients through total flavor degree nine exactly. This calculation is a hunt calibration, not yet a global theorem about Bethe schemes.

4. A decisive obstruction for continuous families

Suppose $C$ is a smooth component of the CM critical locus. Writing a putative lift as $u=q+e_2\xi+\cdots$ makes the first nontrivial equation

$$\operatorname{Hess}_q(V)\,\xi =\frac{1}{12}\nabla V_2(q).$$

Its class in the cokernel of the Hessian is the first obstruction. Because the Hessian is symmetric, the obstruction vanishes along tangent directions precisely when $V_2|_C$ is locally constant. Therefore the next flavor coefficient does not merely add detail: it decides whether a CM family lifts as a family or is canonically cut down.

The known $A_2$ continuous family is the positive control. The ATTACK will test the intrinsic condition $d(V_2|_C)=0$, not enumerate points on that family or root-system presentations.

5. Known frontier and scope

• Fazzi–Krawczyk construct the limit map for every semisimple $\mathfrak g$, prove its symmetry properties, and identify the untwisted CM system.
• They verify bijectivity for every rank-two semisimple algebra. Their $G_2$ completeness work and another related project are listed as “to appear,” so priority must be refreshed before any later claim.
• Their conjecture is stated for isolated solutions. They explicitly identify continuous families as the next test and report only the $A_2$ example.
• Some nontrivial BAE solutions approach poles of $V$. The normalized family is first studied only on the regular root-hyperplane complement; it is not a proposed classification of the non-CM sector.
• Their third-derivative mechanism generalizes an unpublished type-A argument of O. Aharony communicated by O. Mamroud. That influence belongs to the source record separately from the new normalization proposed here.

No source found in the 7 September 2026 search formulates the cubic-divided Bethe family, flatness problem, or $V_2$-obstruction. This is a bounded priority check, not a novelty claim.

6. First ATTACK

Gate A: define the object globally

1. Starting from the multiplicative operators $Q_i$, prove analytic divisibility of $Q_i-1$ by $\delta$ on the regular Cartan torus.
2. Determine the transition law for $\widetilde Q_i$ under a cocharacter basis change, torus periods, Weyl and center actions, and $\mathrm{PSL}(2,\mathbb Z)$.
3. Replace individual equations by the generated ideal or a section of a vector bundle if that is the coordinate-free object.

If the normalized ideal does not descend canonically, stop this route and record the exact cocycle obstruction. Do not repair it by choosing and enumerating gauges.

Gate B: local deformation theorem

1. Identify the special fiber scheme-theoretically with the CM critical scheme, not merely pointwise.
2. Prove unique lifting near every nondegenerate CM extremum by the holomorphic implicit-function theorem.
3. At an isolated degenerate extremum, test complete-intersection flatness and compare scheme length with the conjectured single branch.

Gate C: continuous components

1. Express the first obstruction as $[\nabla V_2]\in\operatorname{coker}\operatorname{Hess}(V)$.
2. Prove or refute its vanishing intrinsically on the $A_2$ control family.
3. Seek a Lax, spectral, or elliptic-function identity forcing $V_2|_C$ to be constant on every CM critical component.
4. If the obstruction is nonzero, interpret the selected sublocus as genuine flavor-resolved dynamics and delimit exactly which extension of the conjecture fails.

If progressing beyond this point requires a census by Dynkin type or a list of Bethe roots, stop. The route is selected only while the normalized family and its deformation complex remain the organizing objects.

7. Why this target was selected

The selected problem has one family, one special fiber, and one obstruction complex. It also has a clean physical reason to matter: BAE solutions enter the supersymmetric-index approach to AdS$_5$ black-hole microstate counting, while the CM presentation exposes integrable dynamics hidden by the unrefined limit.

8. Durable dependencies and files

• verify_flavor_normalization.py: exact Newton-sum and Taylor-coefficient checks through flavor degree nine.
• flavor-normalization-check.txt: expected output of that script.
• source-ledger.tsv: primary-source roles and selection decisions.
• bethe-calogero-normal-family-hunt-packet.zip: reproducible packet.
• FPRD-research-loop-state.md: the continuing cross-run handoff.

9. Recommended next mode

NEXT MODE: ATTACK. Prove the global divisibility/descent statement and the local nondegenerate lifting theorem first. Only if both survive should the attack move to the $V_2$ obstruction on continuous critical components.