FPRD hunt packet: the energy-shell order in the quantum-mechanical bootstrap

Mode: HUNT
Selected: 7 September 2026
Status: active research target; no theorem claim

Exact question

Let

$$H=\frac12\sum_{j=1}^d p_j^2+V(q_1,\ldots,q_d)$$

be a polynomial Schrödinger Hamiltonian that is bounded below and has discrete spectrum. At bootstrap level $K$, let $\mathcal F_K^{\mathrm{SE}}(E)$ be the normalized truncated moment functionals that are positive on all Hermitian squares of degree at most $2K$, satisfy the stationarity relations, and satisfy the energy-shell relations

$$L(O(H-E))=0$$

whenever the degree budget permits them.

Daniel Keren asks whether, for every $k\le K-1$, there is a constant $C(E,k)$ such that

$$\boxed{L(p_j^{2k})\le C(E,k) \quad\text{for every }L\in\mathcal F_K^{\mathrm{SE}}(E).}$$

He further conjectures that at sufficiently high level the positional part of every feasible functional lies in the genuine moment cone. This packet selects the first, Archimedean momentum question as the initial target. A solution would identify the mechanism that prevents the eigenstate bootstrap from escaping along unphysical high-momentum directions.

Primary source: Daniel Keren, Stationarity is not enough: tightness of the quantum mechanical bootstrap and the copositive cone, arXiv:2608.07047v2, revised 12 August 2026.

Known frontier

Keren proves an exact stationary-bootstrap failure in a two-dimensional quartic double well: a rational feasible point has a positive definite moment matrix but assigns negative expectation to a pointwise nonnegative polynomial. The gap is therefore between positivity and sums of squares, not merely a numerical solver artifact.

The eigenstate constraints behave differently. In five tested settings they remove the separating witness. At level $K=3$ for the two-dimensional quartic example, $\langle p_j^4\rangle$ is unbounded with stationarity alone and bounded after the energy-shell rows are imposed, while $\langle p_j^6\rangle$ remains unbounded, exactly matching the predicted degree budget.

The proved positive case is narrower. If $d=2$, $V$ is quartic, and $V-V_*$ is a sum of squares, Keren proves

$$\langle p_j^4\rangle \le 2(E-V_*)\langle p_j^2\rangle +\sum_a\langle\partial_j^2(g_a^2)\rangle, \qquad V-V_*=\sum_a g_a^2.$$

The key identity is

$$\frac12\{p_j^2,g^2\} =(gp_j)^*(gp_j)-\frac12\partial_j^2(g^2).$$

The general conjecture begins exactly where a nonnegative polynomial need not be a sum of squares. Standard convergence theorems for noncommutative polynomial optimization assume bounded operators or an Archimedean quadratic module, which the Weyl relations do not supply. Schmüdgen’s strict Positivstellensatz for the Weyl algebra uses strict positivity and elliptic leading-symbol hypotheses and does not directly settle the boundary, degree-truncated energy-shell problem. Navascués, García-Sáez, Acín, Pironio, and Plenio also show that positivity in the Schrödinger representation is only dense in Weyl sums of squares, which does not make a fixed truncation exact.

No source located in the bounded hunt proves Keren’s momentum-bound conjecture, disproves it, or gives a degree-preserving energy-shell Positivstellensatz.

The intrinsic object

Let $\mathcal W_d$ be the filtered Weyl *-algebra generated by self-adjoint $q_j,p_j$ with $[q_i,p_j]=i\delta_{ij}$. Inside its Hermitian filtered piece $\mathcal W_{d,\le2K}^{\mathrm h}$, quotient by the real span $\mathcal R_{K,E}$ of all degree-admissible Hermitian stationarity and energy-shell relations. The resulting ordered vector space is

$$\mathcal A_{K,E} =\mathcal W_{d,\le2K}^{\mathrm h}/\mathcal R_{K,E},$$

with positive cone the closure of the image of truncated Hermitian squares,

$$\mathcal Q_{K,E} =\overline{\left\{ \left[\sum_\nu A_\nu^*A_\nu\right]: \deg A_\nu\le K \right\}}.$$

Normalized bootstrap-feasible points are precisely the positive linear functionals on this one ordered quotient satisfying $L([1])=1$. A certificate

$$C[1]-[p_j^{2k}]\in\mathcal Q_{K,E}$$

therefore proves the desired bound for every feasible moment functional at once. More generally, conic duality locates any failure in a recession functional on the same quotient.

This is the reason for selecting the target. Moment coordinates are only a presentation of $(\mathcal A_{K,E},\mathcal Q_{K,E},[1])$. The question is whether the energy-shell relation turns $[1]$ into an order unit in the specific momentum directions visible at level $K$. It is not a request to enumerate moment matrices, potentials, dimensions, or non-SOS witnesses.

Why this target passes the selection calibration

1. One universal object. The filtered energy-shell ordered quotient is defined for every polynomial Schrödinger Hamiltonian and every truncation.
2. One exact translation. Analytic boundedness of all feasible momentum moments becomes algebraic order boundedness of one class.
3. A decisive first gate. Either construct a degree-preserving order-unit certificate from the energy-shell ideal, or produce a positive recession functional that survives it.
4. No natural residual census. The proof object is a universal identity or cone-separation theorem. A list of low-dimensional SDPs cannot answer it.
5. Physical meaning. The target distinguishes a genuine eigenstate shell from mere stationarity and addresses why bosonic bootstrap bounds appear to converge despite the non-Archimedean Weyl algebra.

Concrete first attack

The first attack will work in the ordered quotient, not in a coordinate list.

Gate A: exact filtered duality

Prove the precise finite-dimensional alternative for each admissible $(K,k)$: either

$$C[1]-[p_j^{2k}]\in\mathcal Q_{K,E}$$

for some $C$, or there is a normalized/recession positive functional that makes $p_j^{2k}$ unbounded. The closure and possible lack of a Slater point must be handled explicitly; informal SDP duality is not enough.

Gate B: the energy-shell Gram map

Apply the relation $O(H-E)=0$ with $O=p_j^{2k-2}$ and polarizations thereof. The kinetic term produces the positive Gram block

$$\frac12\sum_i[p_j^{2k-2}p_i^2].$$

The entire problem is to control the Hermitian class of $p_j^{2k-2}V$ without assuming $V-V_*$ is a sum of squares. Seek a degree-preserving identity of the form

$$\operatorname{Re}(p_j^{2k-2}V) =\sum B_\nu^*B_\nu+\text{lower filtered terms} +\mathcal R_{K,E},$$

where the lower terms are already order bounded. Commutator identities, ellipticity of the leading symbol, and denominators generated internally by $H-E$ are allowed; an external Archimedean ball constraint is not.

Gate C: first honest boundary class

The first test is not another quartic SOS example. Use a coercive polynomial $V$ for which $V-\min V$ is nonnegative but not SOS, with the smallest degree and dimension compatible with the filtered budget. Compute the exact symbolic dual once, solely to decide whether the universal identity survives the first boundary class. A feasible unbounded ray is a counterexample to the momentum claim at that $(K,k)$; a rational dual certificate becomes the seed of an all-potential proof.

Stop rule

Stop this route if the only continuation is a growing catalogue of potentials, symmetry types, SDP witnesses, or moment coordinates. Continue only if the ordered quotient yields either a universal certificate calculus or a parametric recession obstruction.

Other frontiers considered

• Radial derivative on the graded Möbius algebra. Sinclair’s canonical lowering operator and log-concavity conjecture are attractive, but the paper already leans on complete small-matroid computation and occupancy matrices. A failed Lefschetz construction would naturally reopen a matroid census.
• Gromov’s critical scalar-curvature decay, construction direction. Wang and Zhu prove the sharp obstruction direction in dimensions four through seven, while the existence direction remains open. The object is global, but a first attack would require an unrestricted positive-scalar-curvature block-filling theorem; there is no comparably decisive local gate.
• Log canonical models with varying boundary. Li and Li reduce much of the surface problem, but the remaining minimal cases naturally lead into cone and surface classification.
• K-theory of quotient ring spectra. Ben-Moshe’s paper is structurally strong, but its named questions are substantially answered in the paper; the hunt did not find a comparably explicit residual open statement.

Source ledger

• Daniel Keren, arXiv:2608.07047v2, revised 12 August 2026.
• Konrad Schmüdgen, arXiv:math/0403076v2, 26 July 2005.
• M. Navascués, A. García-Sáez, A. Acín, S. Pironio, M. B. Plenio, arXiv:1203.3777v2, 2013 journal version.
• Stefano Pironio, Miguel Navascués, Antonio Acín, arXiv:0903.4368, submitted 25 March 2009.
• Thomas Sinclair, arXiv:2608.18519v1, submitted 19 August 2026.
• Zhehui Wang and Jintian Zhu, arXiv:2609.03676v1, submitted 3 September 2026.
• Xingying Li and Zhan Li, arXiv:2609.03305v1, submitted 3 September 2026.
• Shay Ben-Moshe, arXiv:2609.01843v1, submitted 1 September 2026.

Status and next mode

This hunt creates no theorem, counterexample, or publication. It selects an external open conjecture and an intrinsic algebraic presentation for attacking it.

NEXT MODE: ATTACK – establish the filtered order-unit alternative and test the first coercive non-SOS boundary class.