The degree budget is essential in the energy-shell quantum bootstrap

Mode: ATTACK
Date: 7 September 2026
Status: exact scope counterexample and corrected attack frontier; no claim of priority or external review

Result

The momentum-bound part of Keren’s Conjecture 1 is false for general polynomial potentials if the stated range $k\le K-1$ is read literally. The obstruction is not a numerical SDP pathology and does not require a non-SOS potential.

Proposition

For every integer $m\ge3$, there is a polynomial Schrödinger Hamiltonian of degree $2m$, bounded below with discrete spectrum, and a fixed energy $E$, such that

$$\sup_{L\in\mathcal F_m^{\mathrm{SE}}(E)}L(p_1^4)=\infty.$$

Since $2\le m-1$, this contradicts the range $k\le K-1$ at $K=m$.

The natural degree-admissible range for the energy-shell mechanism is instead

$$\boxed{k\le K-\frac{\deg V}{2}+1.}$$

For quartic potentials this becomes $k\le K-1$, exactly the range in Keren’s conjecture. Thus the result is best read as a scope correction: the quartic conjecture remains open and is the substantive frontier.

Construction

Fix $m\ge3$. In one dimension take

$$h_m=\frac{p^2}{2}+x^{2m}$$

on $L^2(\mathbb R)$. If a higher-dimensional example is desired, add independent harmonic spectators:

$$H_m=h_m+\frac12\sum_{j=2}^d(p_j^2+q_j^2).$$

This operator is bounded below and has compact resolvent. Let $\psi_n$ be eigenvectors obtained by exciting the first factor and keeping the spectators in their ground states, with energies $\lambda_n\to\infty$. Write $\lambda_0$ for the ground energy of the full product state.

Fix $E>\lambda_0$. For every $n$ with $\lambda_n>E$, put

$$\alpha_n=\frac{E-\lambda_0}{\lambda_n-\lambda_0}, \qquad \rho_n=(1-\alpha_n)|\psi_0\rangle\langle\psi_0| +\alpha_n|\psi_n\rangle\langle\psi_n|.$$

Then $\rho_n$ is a positive normalized stationary state and

$$\operatorname{tr}(\rho_nH_m)=E.$$

At level $K=m$, the degree bound for the energy-shell rows is

$$\deg O\le2K-\deg V=0.$$

Consequently the only energy-shell equation is the mean-energy equation $L(H_m)=E$. Every $\rho_n$ therefore defines a point of $\mathcal F_m^{\mathrm{SE}}(E)$. It satisfies all stationarity equations, not merely the degree-two subset, because it commutes with $H_m$.

Divergence of the fourth momentum moment

Let $e_n$ be the first-factor energy in $\psi_n$. The exact quantum virial identity for $p^2/2+x^{2m}$ gives

$$\langle p_1^2\rangle_{\psi_n} =\frac{2m}{m+1}e_n.$$

Since $p_1^2$ is positive, Cauchy–Schwarz gives

$$\langle p_1^4\rangle_{\psi_n} \ge \langle p_1^2\rangle_{\psi_n}^2 =\left(\frac{2m}{m+1}\right)^2e_n^2.$$

The spectator energy is constant, so $e_n=\lambda_n-O(1)$. Hence

$$\begin{aligned} \operatorname{tr}(\rho_np_1^4) &\ge \alpha_n\left(\frac{2m}{m+1}\right)^2e_n^2\\ &=\frac{E-\lambda_0}{\lambda_n-\lambda_0} \left(\frac{2m}{m+1}\right)^2e_n^2 \longrightarrow\infty. \end{aligned}$$

This proves the proposition.

Why the degree-adjusted range is forced

Keren’s quartic proof bounds $p_j^{2k}$ by applying the energy-shell identity with $O=p_j^{2k-2}$. For $\deg V=2m$, that row is present only when

$$2k-2\le2K-2m,$$

equivalently

$$k\le K-m+1.$$

The family above shows this is not a cosmetic restriction. If the row is absent, a tiny amount of arbitrarily high energy can preserve the fixed first energy moment while making the next kinetic moment diverge.

The mechanism is the elementary failure of one moment to control the next:

$$\text{fixed }\mathbb E[H] \quad\not\Rightarrow\quad \text{bounded }\mathbb E[H^2].$$

The full eigenstate equation would forbid the mixture. A truncation carrying only $O=1$ does not.

Correction to the initial conic formulation

The hunt packet said that failure of order boundedness must appear as a positive linear recession functional. That is too strong for a spectrahedral moment cone. Unboundedness can occur along a nonlinear escape curve whose leading recession direction does not see the unbounded lower moment.

The scalar Hankel example is exact. For

$$v(t)=(1,t,t^2)^T, \qquad M_2(t)=v(t)v(t)^T,$$

every $M_2(t)$ is a normalized positive Hankel moment matrix and its second moment is $t^2\to\infty$. But if $D$ is a positive-semidefinite Hankel recession matrix with $D_{00}=0$, positivity forces the first row of $D$ to vanish. Hankel consistency then gives

$$D_{11}=D_{02}=0.$$

Thus no linear recession direction has positive second-moment coordinate, despite the unbounded objective.

The corrected finite-dimensional alternatives are:

• a cone certificate $C[1]-[p_j^{2k}]\in\mathcal Q_{K,E}$ is sufficient for a bound;
• absence of such a certificate may be exposed only by an asymptotic sequence of normalized positive functionals, not necessarily one recession ray; and
• any duality statement must distinguish the closed cone in the ordered quotient from the generally noncompact normalization slice.

Scope against the source

Conjecture 1 in arXiv:2608.07047v2 says “Let $H$ be as in (4)” and states the range $k\le K-1$. Equation (4) allows a general polynomial potential, whereas the displayed range and the proved Proposition 1 are calibrated to a quartic. The counterexample therefore addresses the literal generality of the wording. It does not refute the quartic conjecture and should not be read as evidence that the intended quartic mechanism fails.

Keren already identifies the underlying low-budget phenomenon in the quartic $K=2$ row of Table 1: only $\langle H\rangle=E$ survives, and a violation occurs. The contribution here is to turn that warning into an all-degree analytic family that falls inside the literal quantifiers of Conjecture 1. Accordingly no priority claim is made for the dilution mechanism itself.

Status and next question

This is an elementary scope correction to a new conjecture, not a substantial advance warranting Publication. The attack remains credible after narrowing the target.

NEXT MODE: ATTACK – settle the first genuinely open quartic gate: for $\deg V=4$, $K=3$, does the full degree-two energy-shell relation bound $\langle p_j^4\rangle$ when $V-\min V$ is nonnegative but not SOS?

The next run must seek a universal quartic certificate or a parametric asymptotic functional. It must not enumerate quartic potentials or SDP witnesses.

Sources

• Daniel Keren, Stationarity is not enough: tightness of the quantum mechanical bootstrap and the copositive cone, arXiv:2608.07047v2, revised 12 August 2026.
• Konrad Schmüdgen, A Strict Positivstellensatz for the Weyl Algebra, arXiv:math/0403076v2.
• M. Navascués, A. García-Sáez, A. Acín, S. Pironio, and M. B. Plenio, A paradox in bosonic energy computations via semidefinite programming relaxations, arXiv:1203.3777.