The normalized Bethe discriminant and the averaged Calogero force

Mode: ATTACK
Completed: 7 September 2026
Status: internally checked structural continuation of the flavor-normal Bethe–Calogero program. This is an ordinary research note. It neither proves nor refutes the Fazzi–Krawczyk bijectivity conjecture, and it has not received external review.

1. Outcome

The previous attack constructed the canonical Cartier-residual Bethe family and proved it finite flat near every isolated regular Calogero–Moser extremum. The open local question was the discriminant of that finite algebra and whether a length- $\mu_q$ fiber at a degenerate extremum remains one branch or splits.

This attack obtains four exact results.

1. The full normalized logarithmic root force, to all flavor orders, is a triple centered average of the Calogero–Moser force: $$\frac{H(z;\Delta)}{\delta} =\mathscr A_\Delta\wp'(z), \qquad \mathscr A_\Delta= \prod_{a=1}^3 \frac{\sinh(\Delta_a\partial_z/2)} {\Delta_a\partial_z/2}. \tag{1.1}$$ The earlier power series is the expansion of one exact operator, not a collection of unrelated correction potentials.
2. The theta root block itself is the Cayley transform of a rational function on the Weierstrass cubic. Its derivative, and therefore the normalized Bethe Jacobian, is algebraic in the elliptic-curve coordinates $(\wp,\wp')$.
3. If $A_q$ is the local finite flat algebra above an isolated extremum $q$, its different is generated by the determinant of the averaged Hessian, and its trace discriminant is the norm of that determinant: $$\boxed{ \operatorname{disc}(A_q/B) \doteq \operatorname{Norm}_{A_q/B} \det\!\left[ \sum_{\alpha>0}\alpha_i\alpha_j (\mathscr A_\Delta\wp'')(\alpha(u)) \right].} \tag{1.2}$$ Here $\doteq$ denotes equality up to the unit introduced by local bases and determinant-line trivializations.
4. A nonzero discriminant answers the branching question: a generic flavor fiber contains exactly $\mu_q$ distinct solutions. An identically zero discriminant does not imply that the cover stays on one support; it may also have several supports with a nonreduced point. The earlier binary formulation omitted this third possibility.

For a corank-one extremum there is a sharp first test. If $v$ spans the Hessian kernel and

$$dV_2(q)(v)\ne0, \qquad V_2=\sum_{\alpha>0}\wp''(\alpha(u)), \tag{1.3}$$

then the local discriminant begins with a nonzero multiple of $e_2^{\mu_q-1}$, and the degenerate point splits into $\mu_q$ simple solutions at generic small flavor. Thus (1.3), rather than a solution census, is the first decisive gate.

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

2. The exact centered-average operator

On a simply connected regular chart, write

$$\ell(z)=\log\theta_1(z),\qquad H(z;\Delta)=\sum_{a=1}^3 \bigl[\ell(z-\Delta_a)-\ell(z+\Delta_a)\bigr], \tag{2.1}$$

with

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

Let $D=\partial_z$. Translation by $t$ is $e^{tD}$, so

$$H=-2\sum_a\sinh(\Delta_aD)\ell. \tag{2.2}$$

For three commuting quantities with sum zero,

$$\sinh x+\sinh y+\sinh z =4\sinh(x/2)\sinh(y/2)\sinh(z/2). \tag{2.3}$$

Applying (2.3) to (2.2) gives

$$\frac{H}{\delta} =-D^3 \prod_{a=1}^3 \frac{\sinh(\Delta_aD/2)}{\Delta_aD/2}\,\ell. \tag{2.4}$$

The standard identity $\ell''=-\wp+c(\omega)$ implies $-\ell'''=\wp'$. Since every factor in (2.4) is a function of $D$, it commutes with $D$, proving (1.1).

This identity is analytic, not merely formal. For any locally holomorphic function $f$,

$$\frac{\sinh(tD/2)}{tD/2}f(z) =\int_{-1/2}^{1/2}f(z+st)\,ds. \tag{2.5}$$

Thus $\mathscr A_\Delta f$ is the threefold centered average of $f$ over the real cube $[-1/2,1/2]^3$, mapped into the complex $z$-line by $(s_1,s_2,s_3)\mapsto\sum s_a\Delta_a$. In particular the exact local potential for the normalized logarithmic equations is

$$\mathcal V_\Delta(u) =\sum_{\alpha>0} (\mathscr A_\Delta\wp)(\alpha(u)), \qquad G_i=\partial_i\mathcal V_\Delta. \tag{2.6}$$

The flavor deformation is therefore an averaged Calogero pair potential. Expanding the even operator recovers and extends the previous calculation:

$$\begin{aligned} \mathscr A_\Delta ={}&1-\frac{e_2}{12}D^2 +\frac{e_2^2}{360}D^4\\ &+\left(-\frac{e_2^3}{20160} +\frac{\delta^2}{60480}\right)D^6 +\frac{e_2^4-e_2\delta^2}{1814400}D^8+\cdots . \end{aligned} \tag{2.7}$$

This also proves directly that the residual family depends symmetrically on the flavors through $e_2$ and $\delta^2$.

3. An algebraic form of the theta block

There is a second exact presentation. Put

$$x=\wp(z),\qquad y=\wp'(z), \qquad y^2=4x^3-g_2x-g_3.$$

For generic flavors let $P_a=(\wp(\Delta_a),\wp'(\Delta_a))$. The group-law condition $\Delta_1+\Delta_2+\Delta_3=0$ says that these three points are collinear. Write their line as

$$y=mx+b. \tag{3.1}$$

The function $y-mx-b$ has zeros at the three $P_a$ and a triple pole at the identity. The sigma–theta relation has no surviving exponential factor because the flavors sum to zero. Comparing divisors and the value at $z=0$ therefore gives the exact identity

$$R(z;\Delta)= \prod_a\frac{\theta_1(z-\Delta_a)}{\theta_1(z+\Delta_a)} =\frac{y-mx-b}{-y-mx-b}. \tag{3.2}$$

Tangent and coincident cases follow by continuity. With

$$a_\Delta=\frac{m}{b},\qquad c_\Delta=\frac1b,$$

equation (3.2) becomes a Cayley transform,

$$R=\frac{1+a_\Delta x-c_\Delta y} {1+a_\Delta x+c_\Delta y} =\frac{1-T}{1+T}, \qquad T=\frac{c_\Delta y}{1+a_\Delta x}. \tag{3.3}$$

Near the flavor origin,

$$a_\Delta=e_2+O(\|\Delta\|^4), \qquad c_\Delta=-\frac{\delta}{2}+O(\|\Delta\|^5), \tag{3.4}$$

consistent with (2.7). Differentiating (3.3) and using the Weierstrass cubic gives a rational formula for the averaged second derivative:

$$(\mathscr A_\Delta\wp'')(z) =-\frac{2c_\Delta}{\delta} \frac{ 6x^2-\frac{g_2}{2} +a_\Delta\left(2x^3+\frac{g_2}{2}x+g_3\right)} {(1+a_\Delta x)^2 -c_\Delta^2(4x^3-g_2x-g_3)}. \tag{3.5}$$

The prefactor in (3.5) tends to one. Consequently the linearization needed for the discriminant is an algebraic function of the elliptic-curve coordinates even though the local potential itself is logarithmic.

4. The different and trace discriminant

Let $B=\mathbb C\{\Delta_1,\Delta_2\}$, and let

$$A_q=B\{x_1,\ldots,x_r\}/(G_1,\ldots,G_r) \tag{4.1}$$

be the finite flat local algebra constructed in the first attack. It is a finite flat local complete intersection, hence a finite syntomic algebra. Its relative different is the zeroth Fitting ideal of $\Omega_{A_q/B}$. The standard complete-intersection presentation makes that ideal principal, generated by the Jacobian determinant

$$j_\Delta=\det\left(\frac{\partial G_i}{\partial x_j}\right). \tag{4.2}$$

This is the analytic local-ring form of the Kähler-different formula. The finite flat complete intersection is Gorenstein over $B$, so $\omega_{A_q/B}=\operatorname{Hom}_B(A_q,B)$ is an invertible $A_q$-module. Choose a generator $\lambda$. Tate’s formula says that the trace functional is

$$\operatorname{Tr}_{A_q/B}(a)=\lambda(j_\Delta a). \tag{4.3}$$

If $e_1,\ldots,e_\mu$ is a $B$-basis of $A_q$, the matrix of the trace pairing is consequently the product of the multiplication matrix of $j_\Delta$ and the matrix $[\lambda(e_ie_j)]$. The second matrix is invertible because $\lambda$ generates the relative dualizing module. Taking determinants gives

$$\det[\operatorname{Tr}(e_ie_j)] =\operatorname{Norm}_{A_q/B}(j_\Delta)\, \det[\lambda(e_ie_j)], \tag{4.4}$$

and the last determinant is a unit. This proves the norm formula in (1.2), including its unit ambiguity. The standard ingredients are recorded in the Stacks Project’s Kähler-different formula, Tate formula, and finite-locally-free discriminant construction.

By (2.6), the matrix in (4.2) is the averaged Hessian

$$M_{ij}(u,\Delta)= \sum_{\alpha>0}\alpha_i\alpha_j (\mathscr A_\Delta\wp'')(\alpha(u)). \tag{4.5}$$

Equations (4.2)–(4.5) prove (1.2). They also connect the discriminant to the Jacobian $H_{\mathrm{BAE}}$ used in the Bethe-sum formula of Fazzi–Krawczyk. On the solution scheme, $Q_i=1$ and $\log Q_i=\delta G_i$, so

$$H_{\mathrm{BAE}} =\det\left[\frac1{2\pi i}\frac{\partial Q_i}{\partial u^j}\right] =\left(\frac{\delta}{2\pi i}\right)^r\det M. \tag{4.6}$$

Thus the intrinsic ramification generator is exactly the BAE Jacobian after removing its universal $\delta^r$ collapse:

$$\boxed{ j_\Delta=\det M =\left(\frac{2\pi i}{\delta}\right)^rH_{\mathrm{BAE}}.} \tag{4.7}$$

The right side extends holomorphically across the flavor divisor by the left side.

5. The right spectral object

No standard Calogero Lax matrix is needed to interpret (4.5). Introduce the Hamiltonian with the averaged pair potential,

$$\mathcal H_\Delta(p,u)=\frac12p\cdot p+\mathcal V_\Delta(u).$$

At a critical point its linearized Hamiltonian vector field is

$$\mathscr S_\Delta= \begin{pmatrix} 0&I\\ -M&0 \end{pmatrix}, \qquad \det(\lambda I-\mathscr S_\Delta) =\det(\lambda^2I+M). \tag{5.1}$$

Therefore $\det M$ is, up to the conventional sign, the product of the squared stability frequencies. The trace discriminant is the norm, over all points of the local cover, of this stability determinant. It vanishes exactly when at least one branch has a zero stability mode.

This is intrinsic spectral data of the deformed family. Spectral data of the undeformed CM equilibrium alone cannot decide how a deformation branches: the same central critical algebra admits both splitting and persistently nonreduced deformations, as the next section shows.

6. What the discriminant decides

Let $\mu=\operatorname{rank}_B A_q=\mu_q$. The trace discriminant has the following exact consequences.

• If it is not the zero germ, its complement is dense. Every complex flavor point in that complement has a finite étale fiber of degree $\mu$, hence exactly $\mu$ distinct nearby Bethe solutions. A degenerate point with $\mu>1$ therefore splits.
• If it is the zero germ, every nearby fiber is nonétale. This is necessary for a single fat branch to persist, but it is not sufficient.

Three one-variable gradient families demonstrate all possibilities while sharing the same kind of central fat point.

$$\begin{array}{c|c|c} \text{gradient equation}&\text{generic support}&\text{discriminant}\\ \hline x^\mu&\text{one length-}\mu\text{ point}&0\\ x^\mu-s&\mu\text{ simple points}& (-1)^{\mu(\mu-1)/2}\mu^\mu(-s)^{\mu-1}\\ x(x-s)^2\ (\mu=3)&\text{one simple and one double point}&0 \end{array} \tag{6.1}$$

The third row corrects the earlier proposed dichotomy: an identically ramified cover can have more than one support.

This matters directly for the conjecture. The Bethe-sum formula weights an isolated solution by $H_{\mathrm{BAE}}^{-1}$, and the companion rank-two analysis treats isolated numerical solutions as nondegenerate; see Fazzi–Krawczyk, arXiv:2608.18209v1. Whenever the generic local BAE fibers are simple in this sense, a length-$\mu_q>1$ CM extremum cannot have a unique preimage: it has $\mu_q$ of them. Hence local set-theoretic bijectivity in the simple-BAE regime requires the isolated CM critical scheme to be reduced.

This is a necessary condition, not a counterexample. No isolated degenerate regular CM extremum at generic modular parameter is produced here.

7. The corank-one splitting gate

Assume the Hessian of $V$ at $q$ has corank one, and let $v$ span its kernel. The holomorphic splitting lemma separates the nondegenerate variables from one null coordinate $x$. Since the local critical algebra has length $\mu$, the reduced special-fiber equation can be normalized to

$$x^\mu=0. \tag{7.1}$$

The first invariant flavor term in (2.7) is $-e_2V_2/12$. After eliminating the nondegenerate variables, the null equation is therefore

$$x^\mu-\frac{e_2}{12}\,dV_2(q)(v) +O(e_2x,e_2^2,\delta^2)=0, \tag{7.2}$$

up to multiplication and rescaling by units. If (1.3) holds, Weierstrass preparation turns (7.2) into a degree-$\mu$ polynomial whose leading model is $x^\mu-ce_2$, $c\ne0$. Its discriminant gives

$$\operatorname{disc}(A_q/B) =u\,e_2^{\mu-1}+\text{terms of higher total degree in }\Delta, \qquad u(0)\ne0. \tag{7.3}$$

Along a generic physical ray $\Delta_a=t d_a$, this vanishes to order $2(\mu-1)$ in $t$, and every sufficiently small point off the discriminant has $\mu$ distinct solutions.

If $dV_2(q)(v)=0$, the first test is silent. The next decision comes from the effective $e_2^2$ term after eliminating the nondegenerate variables, followed by the independent $\delta^2$ term visible in (2.7). Nothing in flatness alone chooses among the three rows of (6.1).

8. Why undeformed Lax data cannot be sufficient

Consider the central potential

$$V_0(x)=\frac{x^{\mu+1}}{\mu+1}.$$

The constant family $V_s=V_0$ and the deformation $\widetilde V_s=V_0-sx$ have exactly the same central potential, central critical algebra, Hessian, and any spectral invariant computed only at $s=0$. The first remains one fat point; the second splits into $\mu$ simple points. Therefore no invariant of the undeformed equilibrium alone can decide the branching problem.

The required information is the deformation class. Formulae (2.6), (3.5), and (4.5) supply it intrinsically for the Bethe family. A standard CM Lax description could still reorganize those formulae, but it cannot replace the flavor-dependent stability determinant.

9. Verification and source boundary

The dependency-free script verify_discriminant_attack.py checks:

• the coefficient identity behind (1.1) through differential order twelve;
• the invariant coefficients displayed in (2.7) through $D^8$;
• the Weierstrass-cubic reduction in (3.5);
• the trace-discriminant formula in the splitting model for $2\leq\mu\leq9$; and
• zero discriminant for the persistent-fat-point and mixed-support models.

The operator factorization itself is proved exactly by (2.2)–(2.4), and the discriminant formula follows from the cited finite-syntomic different theorems; the finite checks are not substitutes for either proof.

A bounded source search found the standard ingredients separately: the theta/Weierstrass relation in Fazzi–Krawczyk, the classical chord-and-tangent law on the Weierstrass cubic, and the complete-intersection different formula in the Stacks Project. It did not find the centered-average factorization, Cayley block, or normed-stability discriminant applied to this Bethe–Calogero family. No priority claim is made.

10. Boundary and next attack

This attack determines the intrinsic discriminant, but it does not prove that the discriminant is nonzero or identically zero at every possible degenerate CM extremum. Such a universal verdict would require new information about the CM critical scheme.

The next structural target is consequently sharper than a Bethe-root census:

Prove that every isolated regular extremum of the equal-coupling untwisted elliptic Calogero–Moser potential is Morse for generic modular parameter, or exhibit a structural mechanism producing a degenerate isolated extremum and evaluate the gate $dV_2(v)$.

If that question cannot be reached through the stability spectrum, integrability, or the local Jacobian algebra without classification by Dynkin type, this route should stop rather than expand into a solution census.