Exact translation

The limiting map is a local deformation theorem

For a semisimple gauge algebra$\mathfrak g$, the Bethe operators $Q_i(u,\Delta)$ encode saddles contributing to the supersymmetric-index approach to AdS₅ black-hole microstate counting. Their proposed limiting partner is the untwisted elliptic Calogero–Moser potential

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

The attack constructs a canonical Bethe solution scheme near the unrefined flavor point and proves it is a flat deformation of $\operatorname{Crit}(V)$near every isolated regular extremum. Exactly one branch is automatic at a nondegenerate extremum; at a degenerate one, the branching question is measured by the trace discriminant of its local Jacobian algebra.

Structural stopping theorem

Integrability does not exponentiate a nilpotent equilibrium

The hoped-for final implication is false under all of the abstract hypotheses available here. Over a parameter$t$, consider

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

Its completed universal critical algebra is the gradient complete intersection$k[t][\varepsilon]/(\varepsilon^3)$. It is finite flat of rank three; every reduced fiber is one isolated point, while every scheme fiber is nonreduced. The Hessian determinant is$3\varepsilon^2$, a nonzero nilpotent whose norm and trace discriminant both vanish.

Adding canonical momenta and taking the separated coordinate energies gives a full set of Poisson-commuting Hamiltonians, independent on a dense open set. Thus even Liouville integrability, finite-flat gradient form, and a smoothly moving reduced support do not force the nilpotent direction to become a curve of equilibria inside a fiber.

The Calogero statement could still hold, but it now needs a CM-specific theorem: either the modular and spectral Kodaira–Spencer classes generate every local Jacobian socle, or each stability-kernel vector algebraizes to a fixed-spectral family of equilibria. No such statement follows from ordinary involutivity. Without it, the remaining method is a root-type census, so the program pauses here.

Higher-corank local model

A square-lattice extremum is a length-nine collision

In the equal-coupling untwisted$B_2$ potential, choose the fully rational point whose coordinate-root half-period values are $a,-a$ and whose mixed roots have value zero. This is a regular point. At the square elliptic curve, $g_3=0$ or$j=1728$, its entire Hessian vanishes, but its first nonconstant jet is

$$V-V(q)=2a^3(x^4-y^4)+O_5(x,y).$$

The gradient initial ideal is therefore$(x^3,y^3)$. The point is isolated, with local critical-scheme length$\mu_q=9$. Thus a blanket Morse statement is false at special modular parameters even on the regular root-hyperplane complement.

Let $r$ be the nearby Weierstrass value of the mixed half-period and$\sigma_2=\sum_{a<b}\Delta_a\Delta_b$. The first joint smoothing jet depends on a single coordinate:

$$F_x=2ax(4a^2x^2-\kappa),\qquad F_y=-2ay(4a^2y^2-\kappa),\qquad \kappa=3r+2a^2\sigma_2.$$

Every $\kappa\ne0$ fiber has nine Morse points, and the trace discriminant begins with a unit times $\kappa^{18}$. Here$dV_2(q)=0$, so the earlier corank-one test sees nothing; the quartic jet supplies the full answer. Fazzi and Krawczyk already identify the fully rational branch. The local algebra and its smoothing are the unreviewed contribution here.

Intrinsic object

The universal cubic flattening has a canonical residual

Shift the third flavor coordinate by its torus period and write

$$\Delta_1+\Delta_2+\Delta_3=0,\qquad \delta=\Delta_1\Delta_2\Delta_3.$$

If any one $\Delta_a$ vanishes, the other two are opposites and every theta-function root block cancels exactly. Hence$Q_i-1$ has all three flavor lines as factors on the regular root-hyperplane complement.

$$\mathcal Q(u,\Delta)(\lambda)= \prod_{[\alpha]\in\Phi/\{\pm1\}} R(\alpha(u);\Delta)^{\alpha(\lambda)}, \qquad \mathcal J=(\mathcal Q^*\mathfrak m_1:\delta).$$

Here $R$ is the theta-function block, $\mathcal Q$ maps the cocharacter lattice into its dual torus, and$\mathfrak m_1$ is the ideal of that torus's identity. Locally,$\mathcal J$ is generated by$(Q_i-1)/\delta$, but the Cartier-residual formula is basis-free. Theta oddness, quasi-periodicity, and modular transformation laws prove its equivariance under positive-root choices, lattice and center shifts, Weyl symmetry, charge conjugation, and the modular action. Gate A therefore passes without fixing a gauge presentation.

Local deformation theorem

Every isolated extremum carries a finite flat Bethe cover

On a regular logarithmic chart, set$e_2=\sum_{a<b}\Delta_a\Delta_b$. Exact Newton identities and$(\log\theta_1)''=-\wp+c$ give the single-root expansion

$$\frac{1}{\delta}\sum_a \bigl[\log\theta_1(z-\Delta_a) -\log\theta_1(z+\Delta_a)\bigr] = \wp'(z)-\frac{e_2}{12}\wp'''(z) +\frac{e_2^2}{360}\wp^{(5)}(z)+\cdots .$$

The zeroth coefficient is exactly$\nabla V$. Because the multiplicative and logarithmic quotients differ by holomorphic units, this identifies the special fiber scheme-theoretically:

$$\mathfrak B^\nu_0=\operatorname{Crit}(V).$$

At any isolated regular extremum $q$, the gradient ideal is a zero-dimensional complete intersection. Its normalized deformation remains a complete intersection; miracle flatness makes the local family finite flat over the two-dimensional flavor germ, of degree

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

A nondegenerate extremum has$\mu_q=1$ and therefore one unique holomorphic Bethe branch. At a degenerate extremum, flatness still holds, but the degree can exceed one: local bijectivity becomes the intrinsic question of whether that cover splits. The next coefficient is also a gradient, that of$V_2=\sum_{\alpha>0} \wp''(\alpha\cdot u)$.

All flavor orders

The normalized force is an exact centered average

Let $D=\partial_z$ and write the normalized logarithmic theta force as$H(z;\Delta)/\delta$. The identity for three hyperbolic sines whose arguments sum to zero gives, without truncating a flavor series,

$$\frac{H(z;\Delta)}{\delta} =\mathscr A_\Delta\wp'(z),\qquad \mathscr A_\Delta= \prod_{a=1}^{3} \frac{\sinh(\Delta_aD/2)}{\Delta_aD/2}.$$

Each factor is the centered-average operator$f(z)\mapsto \int_{-1/2}^{1/2}f(z+s\Delta_a)\,ds$. Hence the exact normalized equations are the gradient of the averaged potential

$$\mathcal V_\Delta(u)= \sum_{\alpha>0}(\mathscr A_\Delta\wp)(\alpha\!\cdot\!u).$$

The theta block also has an algebraic presentation. If the three flavor points on the Weierstrass cubic lie on$y=mx+b$, then$R=(y-mx-b)/(-y-mx-b)$. Its logarithmic derivative is therefore rational in$(\wp,\wp')$. The full deformation, not merely its first corrections, is intrinsic elliptic-curve data.

Exact ramification invariant

The discriminant is the norm of the stability determinant

For the finite flat local algebra$A_q/B$, the complete-intersection different is generated by the averaged Hessian determinant. Finite-flat duality then turns its norm into the trace discriminant, up to a unit:

$$\operatorname{disc}(A_q/B)\doteq \operatorname{Norm}_{A_q/B}\det M,\qquad M_{ij}=\sum_{\alpha>0}\alpha_i\alpha_j (\mathscr A_\Delta\wp'')(\alpha\!\cdot\!u).$$

On the solution scheme this generator is the usual Bethe Jacobian with its universal flavor collapse removed:

$$\det M=\left(\frac{2\pi i}{\delta}\right)^r H_{\mathrm{BAE}}.$$

Equivalently, if$\mathscr S_\Delta= \left(\begin{smallmatrix}0&I\\-M&0\end{smallmatrix}\right)$ is the linearized Hamiltonian stability operator, then$\det(\lambda I-\mathscr S_\Delta) =\det(\lambda^2I+M)$. The relevant spectral datum is therefore the deformed stability spectrum. Undeformed Lax data alone cannot determine how a chosen deformation branches.

Branching verdict

Nonzero discriminant means full generic splitting

If the discriminant germ is nonzero, the cover over its complement is finite étale of degree$\mu_q$. Thus a degenerate point with$\mu_q>1$ produces exactly$\mu_q$ distinct nearby solutions at generic flavor. If the germ vanishes identically, every fiber remains ramified, but that does not force a single support: one fat point and a mixture of simple and nonreduced supports are both possible.

There is a sharp first test when the Hessian has corank one. If$v$ spans its kernel and

$$dV_2(q)(v)\ne0,$$

the reduced null equation begins$x^{\mu_q}-c e_2$, so the discriminant begins with a unit times$e_2^{\mu_q-1}$. This forces full generic splitting. Consequently, in any regime where the nearby Bethe solutions are simple, local bijectivity requires every isolated regular Calogero–Moser critical point to be reduced. No degenerate isolated point is exhibited here, so this is a necessary condition rather than a counterexample.

Positive-dimensional control

The first two obstructions cancel identically on A₂

If $C$ is a smooth component of the Calogero–Moser critical locus, the first lifting equation at $q\in C$ is

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

The obstruction is the class of$\nabla V_2$ in the Hessian cokernel. On the known continuous$A_2$ family, set$X=\wp(a)$,$Y=\wp(b)$, and$Z=\wp(a+b)$. The critical equations and elliptic addition law give

$$X+Y+Z=0,\qquad X^2+XY+Y^2=\frac{g_2}{4}.$$

Consequently the first obstruction potential is constant,$V_2=3g_2/2$. The Hessian factorizes as $2nn^T$ with$n=(2X+Y,X+2Y)$. Eliminating that normal direction makes the next effective potential

$$E_2=\frac{V_4}{360}-\frac{s^2}{4} =\frac{3g_3}{20}, \qquad s=\wp'(a)=\wp'(b)=-\wp'(a+b).$$

It too is constant. Thus flavor restoration does not cut down the curve through total flavor degree four. The full$A_2$ correspondence was already established by Fazzi and Krawczyk; these constants calibrate the obstruction theory rather than claim a new family.

Stopping boundary

The missing bridge must be Calogero-specific

The square-lattice model shows how one special modular collision resolves, but generic smoothness begins only after reducedness is known. The exact countermodel above closes the proposed integrability argument: persistent nilpotence can sit over a smooth reduced section with zero-dimensional fibers.

Reopen this program only for a spectral-versality theorem that is intrinsic to elliptic Calogero–Moser systems. Otherwise the next research mode is a hunt for a different target governed by one global invariant or universal construction.

Hunt comparison

Why this target won

Kobayashi's complex-Fourier-zero rigidityhas a beautiful entire invariant, but equality of reduced zero sets suppresses multiplicities and leaves no comparably sharp first gate. Motivic Frobenius rigidity has a canonical base-change map but enters broad unresolved characteristic-p cycle theory. The PPT-cubed route naturally opens high-dimensional cones of witnesses.

Here, one universal factor creates one deformation object, whose special fiber and first obstruction are explicit. The presentation change can expose dynamics erased by symmetry restoration without turning the attack into a configuration census.

Research record

Proof, exact checks, and sources

• Persistent-nilpotence stopping proof
• Exact no-go checks
• No-go check output
• Downloadable stopping-audit archive
• Square-lattice length-nine collision proof
• Exact collision checks
• Collision-check output
• Downloadable collision archive
• Discriminant and centered-average proof
• Exact discriminant-attack checks
• Discriminant-check output
• Downloadable discriminant archive
• First attack: residual family and A₂
• Exact A₂ obstruction check
• Attack-check output
• Full hunt packet
• Exact normalization check
• Check output
• Source ledger
• Downloadable attack archive
• Original hunt archive